text:
\begin{document}
\maketitle
\begin{abstract}
We prove bilinear inequalities for differential operators in~$\mathbb{R}^2$.  Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However, here we study the phenomenon in itself. We consider elliptic case, where our analysis is complete, and non-elliptic, where it is not. The latter case is related to Strichartz estimates in a very easy case of two dimensions.
\end{abstract}

\tableofcontents

\section{Introduction}\label{SIntro}
The aim of this paper is to provide some generalizations  of the famous Gagliardo--Nirenberg inequality in its easiest form:
\begin{equation}\label{GagliardoNirenberg}
\big|\scalprod{f}{g}_{L_2(\mathbb{R}^2)}\big| \lesssim \|\partial_1f\|_{L_1(\mathbb{R}^2)}\|\partial_2g\|_{L_1(\mathbb{R}^2)}.
\end{equation}
Here and in what follows we  write ``$a \lesssim b$''  instead of ``$a \leq Cb$ for some uniform constant~$C$'' for brevity; we also write~$a \asymp b$ when~$a \lesssim b$ and~$b \lesssim a$. The symbol~$\partial_j$,~$j=1,2$, denotes the differentiation with respect to the~$j$th variable.

To be more precise, we study estimates on a scalar product of two functions
in some Hilbert space (in this paper, some Sobolev space of fractional order) in terms of a product of~$L_1$-norms of some differential polynomials applied to these functions.  For the author, the interest in inequalities of
such type originated from the work on non-isomorphism problems for Banach spaces of smooth functions and embedding theorems used there, see the short report~\cite{KMS} and the preprint~\cite{KMSpreprint}.

We are going to use some formalism to make our statements shorter. Suppose~$k$ and~$l$ to be natural numbers,~$\alpha$ and~$\beta$ to be real non-negative numbers, and~$\sigma$ and~$\tau$ to be complex non-zero numbers. By the symbol~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ we mean the statement that the inequality
\begin{equation}\label{BilinearEmbeddings}
\left|\langle f,g\rangle_{W_2^{\alpha,\beta}(\mathbb{R}^2)}\right| \lesssim \left\|(\partial_1^k -\tau\partial_2^l)f\right\|_{L_1(\mathbb{R}^2)}\left\|(\partial_1^k -\sigma\partial_2^l)g\right\|_{L_1(\mathbb{R}^2)}
\end{equation}
holds true for any Schwartz functions~$f$ and~$g$. The scalar product on the left is taken in the fractional Sobolev space,
\begin{equation}\label{norma}
\langle f,g\rangle_{W_2^{\alpha,\beta}(\mathbb{R}^2)}=
\int_{\mathbb{R}^2}\hat{f}(\xi,\eta)\overline{\hat{g}(\xi,\eta)}|\xi|^{2\alpha}|\eta|^{2\beta}\,d \xi d \eta.
\end{equation}
Here and in what follows, by~$\hat \varphi$ we denote the Fourier transform of a function~$\varphi$. When~$\alpha$ and~$\beta$ are integers,~$W^{\alpha,\beta}_2$ reduces to the classical anisotropic homogeneous Sobolev space defined by the semi-norm~$\|\partial_1^{\alpha}\partial_2^{\beta}f\|_{L_2(\mathbb{R}^2)}$. Moreover, if~$p$ and~$q$ are integers, then
\begin{equation}\label{DerivationOfScalarProduct}
\scalprod{\partial_1^p\partial_2^q f}{\partial_1^p\partial_2^q g}_{W^{\alpha,\beta}_2} = (2\pi) ^{2p + 2q} \scalprod{f}{g}_{W_2^{\alpha + p,\beta + q}}.
\end{equation}

Inequalities~\eqref{BilinearEmbeddings} are homogeneous not in a usual sense, they are \emph{anisotropic} homogeneous. The transformations that preserve the right-hand side of~\eqref{BilinearEmbeddings}
 (and thus have to preserve the left-hand side) are not of the form~$(x,y) \to (\lambda x, \lambda y)$, but of the form~$(x,y)\to(\lambda^l x,\lambda^k y)$. The basic theory of such type embedding theorems can be found, e.g. in~\cite{BIN} (see also~\cite{Kol} and~\cite{S}, where the case of the limiting order summation exponent is considered). Using this rescaling, we see that~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ may hold true only when
\begin{equation}\label{line}
\frac{\alpha + \frac{1}{2}}{k} + \frac{\beta + \frac{1}{2}}{l} = 1.
\end{equation}

It may seem that the validity of~$\BE$ should not depend on~$\sigma$ and~$\tau$ too much. However, at least one point is obvious from the very beginning:  \emph{elliptic} cases should be distinguished from non-elliptic ones.
\begin{Def}\label{Ellipticity}
We say that the set of numbers~$(k,l,\sigma,\tau)$ is elliptic if both polynomials~$\xi^k - \sigma_1 \eta^l$ and~$\xi^k - \tau_1 \eta^l$\textup, $\tau_1=(2\pi i)^{l-k}\tau$\textup,~$\sigma_1=(-1)^{l-k}(2\pi i)^{l-k}\overline{\sigma}$\textup, do not have real roots except~$0$.
\end{Def}
Here and in what follows,~$\tau_1$ always denotes~$(2\pi i)^{l-k}\tau$,~$\sigma_1$ always denotes~$(-1)^{l-k}(2\pi i)^{l-k}\overline{\sigma}$.
In the case where one of the numbers~$k$ and~$l$ is odd, ellipticity means that the numbers~$\tau_1$ and~$\sigma_1$ are not real. When both~$k$ and~$l$ are even, the numbers~$\sigma_1$ and~$\tau_1$ may be real negative. We begin with the elliptic case.
\begin{Th}\label{T3}
Suppose that~$k$ and~$l$ are natural and~$(k,l,\sigma,\tau)$ is elliptic. The statement~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ holds true if and only if one of the numbers~$k$ and~$l$ is odd\textup,~$\alpha = \frac{k-1}{2}$\textup,~$\beta=\frac{l-1}{2}$\textup, and the numbers~$\sigma_1$ and~$\tau_1$ have non-zero imaginary parts of the same sign.
\end{Th}
The ``if'' part of this theorem was proved in the preprint~\cite{KMS}. However, we repeat it here both for the sake of completeness and because it serves as a good introduction to a similar theorem for the non-elliptic case. Theorem~\ref{T3} has a special particular case~$\sigma = \tau$.
\begin{Cor}\label{sigmaEqualstau}
The statement~$\BE(k,l,\alpha,\beta,\sigma,\sigma)$ does not hold if~$(k,l,\sigma,\sigma)$ is elliptic.
\end{Cor}
Moreover, it will follow from the proof that the inequality
\begin{equation*}
\|f\|_{W^{\alpha,\beta}_2} \lesssim \big\|(\partial_1^k - \sigma\partial_2^l)\big\|_{L_1}
\end{equation*}
does not hold when~$(k,l,\sigma,\sigma)$ is elliptic.
Theorem~\ref{T3} is, in some sense, a surprise. Usually, if one has an embedding into~$W^{\frac{k-1}{2},\frac{l-1}{2}}_2$, then it is natural to expect that it is also valid for all other pairs~$(\alpha,\beta)$ that satisfy equation~\eqref{line} (for example, all the embedding theorems from~\cite{Kol} and~\cite{S} are of this type). However, here this principle does not work, mainly due to the fact that we are working with bilinear embeddings.

The limiting case~$\sigma = \infty,\tau = 0$ was considered in~\cite{PS}, where the inequality
\begin{equation*}
\left|\langle f,g\rangle_{W_2^{\frac{k-1}{2},\frac{l-1}{2}}(\mathbb{R}^2)}\right| \lesssim \left\|\partial_1^k f\right\|_{L_1(\mathbb{R}^2)}\left\|\partial_2^l g\right\|_{L_1(\mathbb{R}^2)}
\end{equation*}
was proved. As a consequence, this inequality leads to the inequality
\begin{equation*}
\|f\|_{W^{\frac{k-1}{2},\frac{l-1}{2}}_2}^2 \lesssim \|\partial_1^k f\|_{L_1}\|\partial_2^l f\|_{L_1},
\end{equation*}
which is a very particular case of Solonnikov's embedding theorem, see~\cite{S} (see also~\cite{Kol} for even stronger results and~\cite{KS} for a detailed explanation why this inequality follows from much more general theorems in~\cite{S}). By rescaling, the latter inequality is equivalent to
\begin{equation*}
\|f\|_{W^{\frac{k-1}{2},\frac{l-1}{2}}_2} \lesssim \|\partial_1^k f\|_{L_1} + \|\partial_2^l f\|_{L_1} \asymp \left\|(\partial_1^k -\tau\partial_2^l)f\right\|_{L_1} + \left\|(\partial_1^k -\sigma\partial_2^l)g\right\|_{L_1}
\end{equation*}
provided~$\sigma \ne \tau$. We signalize that~$\BE(k,l,\frac{k-1}{2},\frac{l-1}{2},\sigma,\tau)$ does not follow from this inequality (because by Theorem~\ref{T3} the former statement may not hold);  the multiplicative and the additive forms are not equivalent here.

When we consider non-elliptic cases and~$k \ne l$ (the case~$k=l$ is special), we are working with something like the Strichartz estimates (see, e.g.~\cite{T}). This hints us that the treatment of non-elliptic cases will lead us to some oscillatory integrals (fortunately, one-dimensional) that are unavoidable when dealing with the Strichartz estimates. However, the main difficulty is still the~$L_1$-space on the right-hand side of~\eqref{BilinearEmbeddings}.
\begin{Th}\label{TVanishing}
Suppose that~$k$ and~$l$ are natural\textup, one of them is odd\textup, and~$k\ne l$. Suppose that~$\sigma$ and~$\tau$ are complex numbers such that both numbers~$\tau_1=(2\pi i)^{l-k}\tau$ and~$\sigma_1=(-1)^{l-k}(2\pi i)^{l-k}\overline{\sigma}$ are real. In this case\textup,~$\BE(k,l,\frac{k-1}{2},\frac{l-1}{2},\sigma,\tau)$ holds true.
\end{Th}
A simple algebraic trick leads to some positive results for the case where both~$k$ and~$l$ are even.
\begin{Cor}\label{EvenParametersVanishingOp}
Suppose that both numbers~$k$ and~$l$ are even\textup, but one of them contains~$2$ only in the first power. In this case\textup, the statements~$\BE(k,l,\frac{3}{4} k -\frac12,\frac14 l - \frac12,\sigma,\tau)$ and~$\BE(k,l,\frac14 k -\frac12,\frac34 l - \frac12,\sigma,\tau)$ hold true whenever~$\sigma_1$ and~$\tau_1$ are distinct positive real numbers.
\end{Cor}

We also have some negative results for non-elliptic operators.
\begin{Le}\label{KnappExample}
The statement~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ does not hold when~$\sigma = \tau$ and~$k\ne l$.
\end{Le}
This lemma will be an easy consequence of a rather standard trick called the Knapp example~(see~\cite{T}). We did not find an exact reference for our case, so we will repeat this standard stuff. We signalize that the effect that breaks~$\BE$ in this case is totally different from the one we had in Theorem~\ref{T3}. In Lemma~\ref{KnappExample} it is of oscillatory integral nature, whereas in the said theorem it is a singular integral that ``diverges''. We specialize the case~$k\ne l$, because it is more difficult and interesting. Surely, the conclusions of~Theorem~\ref{TVanishing}, Corollary~\ref{EvenParametersVanishingOp}, and~Lemma~\ref{KnappExample} hold true (with a bit different but easier proofs) when~$k=l$. We leave this cases.

The text is organized as follows: in Section~\ref{SElliptic} we deal with the elliptic case (prove Theorem~\ref{T3}); in Section~\ref{SVanish} we work on non-ellipticity cases; finally, in Section~\ref{SFurther}, we give an explanation why our results for non-elliptic cases are not complete and give some related results and conjectures for linear and quadratic inequalities. In this last section we omit detailed proofs, but provide sketches (the results presented there are incomplete, they need further study; moreover, this is a related, but different story).

This paper is of technical character: we believe that neither the statements nor the proofs presented here are general enough (the worst thing is that we are working not with many variables, but with only two of them). However, the effect itself (especially for non-elliptic cases) seems to be new in its nature. Both these facts urge us to write down too many details in Sections~\ref{SElliptic} and~\ref{SVanish}, we apologize for that.

I am grateful to my scientific adviser S. V.~Kislyakov for statement of the problem and attention to my work and to A. I. Nazarov for useful comments.

\section{Elliptic case}\label{SElliptic}
We are going to prove Theorem~\ref{T3} in the following order: first, we prove that~$\BE$ holds if the parameters satisfy the assumptions; second, we construct counterexamples that disprove~$\BE$ in all the remaining cases. Unfortunately, we did not manage to formalize the latter part in a good way: though the ideology of the counterexample is clear, the technical treatment varies for different cases of parameters.

\subsection{Proof of~$\BE$ in the case of an odd number~$k$}\label{SsOddk}
At least one of the numbers~$k$ and~$l$ is odd; by symmetry, we may suppose that~$k$ is odd.  In what follows, we assume that~$\sigma_1$ and~$\tau_1$ are distinct (the case where they are equal is a bit different, we treat it afterwards).
The functions~$f$ and~$g$ belong to the Schwartz class, so, their Fourier transform is infinitely differentiable and decays rapidly at infinity. Thus, we can represent the scalar product on left-hand side of~\eqref{BilinearEmbeddings}
as a limit,
\begin{equation*}
\int\limits_{\mathbb{R}^2}\hat{f}(\xi,\eta)\overline{\hat{g}(\xi,\eta)}\xi^{k-1}|\eta|^{l-1}\,d \xi d \eta =
\lim_{\substack{\varepsilon\rightarrow 0 \\ R \rightarrow \infty}}\,\int\limits_{\Omega_{\varepsilon,R}} \hat{f}(\xi,\eta)\overline{\hat{g}(\xi,\eta)}\xi^{k-1}|\eta|^{l-1}\,d \xi d \eta,
\end{equation*}
where~$\Omega_{\varepsilon,R}=\{(\xi,\eta)\in\mathbb{R}^2\mid \varepsilon\leqslant |\eta|\leqslant R\}$.
Next, we replace the Fourier transforms in the integrand by their expressions
in terms of~$f_1$ and~$g_1$ found from the formulas
\begin{equation*}
\big((2\pi i\xi)^k - \tau (2\pi i\eta)^l\big)\hat{f}(\xi,\eta) =
\hat{f}_1(\xi,\eta); \quad \big((2\pi i\xi)^k - \sigma (2\pi i\eta)^l\big)\hat{g}(\xi,\eta) = \hat{g}_1(\xi,\eta),
\end{equation*}
so, the expression~$\|f_1\|_{L_1}\|g_1\|_{L_1}$ coincides with the right-hand side of~\eqref{BilinearEmbeddings}. We must estimate the quantity
\begin{equation*}
\lim_{\substack{\varepsilon\rightarrow 0 \\ R \rightarrow \infty}}\,
\int\limits_{\Omega_{\varepsilon,R}}\frac{|\eta|^{l - 1}\xi^{k - 1}\hat{f}_1(\xi,\eta)\overline{\hat{g}_1(\xi,\eta)} \,d\xi d\eta}{\big((2\pi i\xi)^k -
\tau (2\pi i\eta)^l\big)\big((-2\pi i\xi)^k - \overline{\sigma} (-2\pi i\eta)^l\big)}.
\end{equation*}
The denominator of the integrand does not vanish on~$\mathbb{R}^2$ except at zero, due to the assumptions of ellipticity.
Then (for~$\varepsilon$ and~$R$ fixed) we replace~$\hat{f}_1$ and~$\hat{g}_1$ in the last
formula by their definitions in terms of~$f_1$ and~$g_1$, and change the order of integration:
\begin{equation}\label{quantity}
\lim_{\substack{\varepsilon \rightarrow 0 \\ R \rightarrow \infty}}
\iint F(\varepsilon,R,x_1,x_2,y_1,y_2)
f_1(x_1,x_2)\overline{g_1(y_1,y_2)} \,dy_1 dy_2 dx_1 dx_2,
\end{equation}
where
\begin{equation}\label{integrand}
F(\varepsilon,R,x_1,x_2,y_1,y_2)=\int\limits_{\Omega_{\varepsilon,R}}\frac{|\eta|^{l-1}\xi^{k-1}
e^{2\pi i ((x_1 - y_1)\xi + (x_2 - y_2)\eta)}}{((2\pi i\xi)^k - \tau (2\pi i\eta)^l)((-2\pi i\xi)^k -
\overline{\sigma} (-2\pi i\eta)^l)} d\xi d\eta.
\end{equation}

To prove inequality~\eqref{BilinearEmbeddings}, we must show that the modulus of the quantity
\eqref{quantity} does not exceed the value~$C\|f_1\|_1\|g_1\|_1$. For this, we prove that the function
\eqref{integrand} is bounded uniformly in all~$\eps$ and~$R$, $0<\varepsilon\leqslant R<\infty$, and almost all quads of reals~$x_1,\,x_2,\,y_1$, $y_2$. The things become slightly more transparent if we integrate in~\eqref{integrand} first with respect to~$\xi$ and then with respect to~$\eta$, and in the inner integral introduce
a new variable~$\rho$ by setting~$\xi=\rho |\eta|^{l/k}$. This yields the formula
\begin{equation*}
F(\varepsilon,R,x_1,x_2,y_1,y_2)=\frac{1}{(2\pi)^{2k}}\!\!\!\!\int\limits_{\varepsilon\leqslant |\eta|\leqslant R}
\!\!\!\!\!|\eta|^{-1}\!\!\!\int\limits_{-\infty}^{+\infty}\frac{\rho^{k-1}e^{2\pi i(a\rho|\eta|^{l/k}+b\eta)}}{\big(\rho^k -
\tau_1(\sign\eta)^l\big)\big(\rho^k -\sigma_1(\sign\eta)^l\big)}\,d\rho d\eta,
\end{equation*}
where we have put~$a=x_1 - y_1$,~$b=x_2-y_2$, and~$\tau_1$ and~$\sigma_1$ were introduced in Definition~\ref{Ellipticity}.

Here integration in~$\eta$ is over the union~$[-R,-\varepsilon]\cup [\varepsilon,R]$, and we will prove
the boundedness of the integral over each of these two intervals separately. Because of symmetry, we
consider only the interval~$[\varepsilon,R]$ (then~$\sign\eta$ in the denominator disappears). For definiteness,
we assume that $a>0$ (the opposite case reduces to this one if we change $\rho$ by $-\rho$; note that we may drop the case of $a=0$ because
it corresponds to a set of measure $0$). After that, we take
$(2\pi a)^{k/l}\eta$ for a new variable in the outer integral; this will modify the parameters $b$, $\varepsilon$, and $R$,
but will allow us to assume that $2\pi a=1$. So, finally, we must show that the following integral is bounded uniformly
in~$\eps$,~$R$,~$0\leqslant\varepsilon <R$, and~$b$:
\begin{equation}\label{ff}
\int\limits_{\varepsilon}^R \eta^{-1}\left(\int\limits_{-\infty}^{\infty}
\frac{\rho^{k-1}e^{i(\rho|\eta|^{l/k}+b\eta)}}{(\rho^k -\tau_1)(\rho^k -\sigma_1)}\,d\rho\right)\,d\eta.
\end{equation}
We will calculate the
integral with respect to~$\rho$ with the
help of the residue formula, perceiving~$\rho$ as a complex variable. Since the integrand
decays rapidly at infinity in the upper half-plane, integration over the contour that consists of the
interval~$[-r, r]$ and the upper part of the circle of radius~$r$ and centered at zero shows,
after the limit passage as~$r\to\infty$, that the integral in question is~$2\pi i$ times the
sum of the residues at the poles of the integrand in the upper half-plane. All these poles are simple and are~$k$th
roots of~$\tau_1$ or~$\sigma_1$ (we have assumed that~$\sigma_1 \ne \tau_1$). Let~$u^k=\tau_1$ (and~$\Re u > 0$). Perceiving the integrand in question as~$\frac{\varphi(\rho)}{\psi(\rho)}$ with~$\psi(\rho)=\rho^k -\tau_1$, we use the fact that~$\varphi$ is regular at~$u$ to conclude that the residue at~$u$ is
\begin{equation*}
\frac{\varphi(u)}{\psi^{\prime}(u)}=\frac{u^{k-1}e^{i(u|\eta|^{l/k}+b\eta)}}{ku^{k-1}(\tau_1 -\sigma_1)}
=\frac{e^{i(u|\eta|^{l/k}+b\eta)}}{k(\tau_1 -\sigma_1)}.
\end{equation*}
Similarly, if~$v^k=\sigma_1$ (and~$\Re v>0$), the residue at~$v$ is equal to
\begin{equation*}
\frac{e^{i(v|\eta|^{l/k}+b\eta)}}{k(\sigma_1 -\tau_1)}.
\end{equation*}
Under our assumptions on~$\sigma$ and~$\tau$, the equations~$\rho^k=\tau_1$ and~$\rho^k=\sigma_1$ have one and the same number of
roots in the upper half plane. This shows that, up to a constant factor, the integral~\eqref{ff} is equal
to a sum of~$(k\pm 1)/2$ expressions of the form
\begin{equation}\label{SimpleIntegral}
\frac{1}{k(\tau_1-\sigma_1)}\int\limits_{\varepsilon}^R e^{ib\eta}\frac{e^{iu|\eta|^{l/k}}-e^{iv|\eta|^{l/k}}}{\eta}d\eta,
\end{equation}
where~$u$ and~$v$ are some~$k$th roots of, respectively,~$\tau_1$ and~$\sigma_1$ in the upper half-plane
(note that, happily, the denominators in the above two formulas for residues are opposite to each other).

We recall that~$b$ is real, so we estimate the absolute value of the integrand by~$C|u-v|\eta^{\frac lk -1}$ when~$\eta \leq 1$ and by~$e^{-\Im u |\eta|^{l/k}}+e^{-\Im v |\eta|^{l/k}}$ when~$\eta > 1$. Both integrals~$\int_0^1 \eta^{\frac lk - 1}\,d\eta$ and~$\int_1^{\infty}(e^{-\Im u |\eta|^{l/k}}+e^{-\Im v |\eta|^{l/k}}) \,d\eta$ converge. This finishes the proof
of~$\BE(k,l,\frac{k-1}{2},\frac{l-1}{2},\sigma,\tau)$ when~$k$ is odd and~$\sigma_1$ and~$\tau_1$ are distinct numbers with imaginary parts of the same sign.

The case where~$\sigma_1 = \tau_1$ can be treated in a similar way. In this case,  all the poles of the function~$\rho \mapsto\frac{\rho^{k-1}e^{i(\rho|\eta|^{l/k}+b\eta)}}{(\rho^k -\tau_1)(\rho^k -\sigma_1)}$ are of the second order. After calculations, the two-dimensional integral~\eqref{ff} appears to be a linear combination of one-dimensional integrals of the form
\begin{equation*}
\int\limits_{\varepsilon}^R e^{ib\eta}|\eta|^{\frac{l}{k} - 1}e^{iu|\eta|^{l/k}}d\eta.
\end{equation*}
Alternatively, this can be justified by passing to the limit as~$\tau_1 \to \sigma_1$ in formula~\eqref{SimpleIntegral}. This integral is uniformly bounded, again because~$\Im u > 0$.
\begin{Rem}
In fact\textup, we have proved that the distribution~$M_{k,l,\sigma,\tau}$ defined by formula
\begin{equation}\label{Distribution}
\scalprod{M_{k,l,\sigma,\tau}}{\phi} = \lim\limits_{\substack{R \to \infty\\ \eps \to 0}}\,
\int\limits_{\Omega_{\varepsilon,R}}\frac{|\eta|^{l - 1}\xi^{k - 1}\phi(\xi,\eta) \,d\xi d\eta}{\big((2\pi i\xi)^k -
\tau (2\pi i\eta)^l\big)\big((-2\pi i\xi)^k - \overline{\sigma} (-2\pi i\eta)^l\big)},\quad \phi \in \mathcal{S}(\mathbb{R}^2),
\end{equation}
has bounded Fourier transform. We did not formulate this as a statement\textup, because it is not clear a priori why does formula~\textup{\eqref{Distribution}} define a distribution. By the classical Malgrange--Ehrenpreis theorem
\begin{equation*}
\frac{1}{\big((2\pi i\xi)^k -
\tau (2\pi i\eta)^l\big)\big((-2\pi i\xi)^k - \overline{\sigma} (-2\pi i\eta)^l\big)}
\end{equation*}
 is a distribution belonging to~$\mathfrak{D}'(\mathbb{R}^2)$ \textup(and even to~$\mathcal{S}'(\mathbb{R}^2)$\textup, see~\textup{\cite{H}}\textup)\textup, however\textup, it is not clear whether it can be multiplied by a non-smooth function~$|\eta|^{l-1}$.
\end{Rem}

\subsection{Counterexamples}\label{SsCounterexamples}
Assume that~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ holds. In this case, we know that~$\alpha$ and~$\beta$ must satisfy equation~\eqref{line}. In this subsection, we construct a pair of functions~$f$ and~$g$ (depending on the parameters) that will give further restrictions on the numbers~$k,l,\alpha,\beta,\sigma,\tau$. We also suppose that if at least one of the numbers~$k$ and~$l$ is even, then~$l$ is even.

Let~$\varphi$,~$0 \leq \varphi \leq 1$, be an infinitely differentiable compactly supported function on~$\mathbb{R}$ equal to~$1$ near zero.
We put~$\psi(\xi,\eta) = \varphi\big(\sqrt{\xi^{2k}+\eta^{2l}}\big)$ and~$\psi_t(\xi,\eta) = \psi(\frac{\xi}{t^l},\frac{\eta}{t^k})$, where~$t > 0$
should be thought of as a large number. Next, let~$V = \psi_t - \psi$, and let~$h = \check{V} \in \mathcal{S}(\mathbb{R}^2)$.
Then the~$L_1(\mathbb{R}^2)$-norm of~$h$ is dominated by a constant independent of~$t$. At the same time,~$\hat{h} = V$ is equal
to~$0$ near the origin and is equal to~$1$ on a large set if~$t$ is large. Note also that~$V(\xi,\eta) = v(\sqrt{\xi^{2k} + \eta^{2l}})$ for
some~$v \in \mathfrak{D}(\mathbb{R})$.

Now, we find two functions~$F,G \in \mathcal{S}(\mathbb{R}^2)$ from the equations
\begin{equation*}
(\partial_1^k - \tau\partial_2^l)F = (\partial_1^k - \sigma\partial_2^l)G = h.
\end{equation*}
These equations are easily solvable after passage to Fourier transforms:
\begin{equation*}
\hat{F}(\xi,\eta) = \frac{V(\xi,\eta)}{(2\pi i \xi)^k - \tau(2\pi i \eta)^l},\quad \hat{G}(\xi,\eta) = \frac{V(\xi,\eta)}{(2\pi i \xi)^k - \sigma(2\pi i \eta)^l},
\end{equation*}
and it is clear that the solutions are in the Schwartz class indeed (recall that the polynomials in the denominators do not have zeros except~$0$).

Now, inequality~\eqref{BilinearEmbeddings} with these~$F$ and~$G$ yields
\begin{equation*}
\bigg|\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\frac{|V(\xi,\eta)|^2|\xi|^{2\alpha}|\eta|^{2\beta}\,d\xi d\eta}{\big((2\pi i \xi)^k - \tau(2\pi i \eta)^l\big)\big((2\pi i \xi)^k - \sigma(2\pi i \eta)^l\big)}\bigg| \lesssim 1
\end{equation*}
independently of~$t$. We transform the integral as we did it before,
including the change of variables~$\xi = \rho|\eta|^{\frac{l}{k}}$, to obtain
\begin{equation*}
\bigg|\int\limits_{0}^{+\infty}|\eta|^{-1}\int\limits_{-\infty}^{+\infty} \frac{V\big(\rho|\eta|^{\frac{l}{k}},\eta\big)^2|\rho|^{2\alpha}}{\big(\rho^k - \tau_1\big)\big(\rho^k - \sigma_1\big)} \, d\rho d\eta\bigg| \lesssim 1.
\end{equation*}
The power~$|\eta|^{-1}$ in the outer integral arises after a short calculation involving equation~\eqref{line}, and the signs of~$\eta$ in the denominator disappear, if~$l$ is even (if both~$k$ and~$l$ are odd, then one has twice the same integral after splitting the integral with respect to~$\eta$ into integrals over~$(-\infty,0]$ and~$[0,\infty)$ and making the change of variable~$\rho \to -\rho$ in the first one). Changing the order of integration, we arrive at
\begin{equation*}
\int\limits_{-\infty}^{+\infty}\bigg[\frac{|\rho|^{2\alpha}}{\big(\rho^k - \tau_1\big)\big(\rho^k - \sigma_1\big)}\int\limits_{0}^{+\infty}\frac{v^2\big(\sqrt{\rho^{2k}\eta^{2l} +\eta^{2l}}\big)}{\eta}\,d\eta\bigg]d\rho.
\end{equation*}

An obvious change of variable shows that the integral with respect to~$\eta$ is equal to~$\int_{0}^{\infty}\frac{v^2(u^l)}{u}\,du$, which does not depend
on~$\rho$ and can be made arbitrarily large if~$t$ is large. So,~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ can only be fulfilled if
\begin{equation}\label{a3}
\int\limits_{-\infty}^{\infty}\frac{|\rho|^{2\alpha}\,d\rho}{\big(\rho^k - \tau_1\big)\big(\rho^k - \sigma_1\big)} = 0.
\end{equation}

\subsubsection{Cases where~$k$ is even (both numbers~$k$ and~$l$ are even)}
In this case the integral~\eqref{a3} can be rewritten as
\begin{equation*}
2\int\limits_{0}^{\infty}\frac{|\rho|^{2\alpha}\,d\rho}{\big(\rho^k - \tau_1\big)\big(\rho^k - \sigma_1\big)} = \frac{2}{k}\int\limits_{0}^{\infty}\frac{|\rho|^{\frac{2\alpha-k+1}{k}}\,d\rho}{\big(\rho - \tau_1\big)\big(\rho - \sigma_1\big)}.
\end{equation*}
Both numbers~$k$ and~$l$ are even, so, the situation is symmetric and we may assume that~$\alpha \leq \frac{k-1}{2}$. The case~$\alpha = \frac{k-1}{2}$ differs a bit from the other cases.

\paragraph{Case~$\alpha = \frac{k-1}{2}$ and~$\sigma_1 \ne \tau_1$.} In this case, the integral in question can be computed directly:
\begin{equation*}
\int\limits_{0}^{\infty}\frac{\,d\rho}{\big(\rho - \tau_1\big)\big(\rho - \sigma_1\big)}= \frac{\log (-\sigma_1) - \log(-\tau_1)}{\sigma_1 - \tau_1} \ne 0,
\end{equation*}
provided~$\sigma_1 \ne \tau_1$. Here~$\log$ is the branch of logarithm defined for~$\arg z \in [0,2\pi)$ such that it is real when~$\arg z = 0$.
\paragraph{Case~$\alpha = \frac{k-1}{2}$ and~$\sigma_1 = \tau_1$.} If~$\sigma_1 = \tau_1$, then the integral~\eqref{a3} equals~$\tau_1^{-1}$, which is non-zero too.
\paragraph{Case~$\alpha < \frac{k-1}{2}$,~$\sigma_1 \ne \tau_1$.} In this case, the integral can be rewritten as
\begin{equation*}
\int\limits_{0}^{\infty}\frac{|\rho|^{\frac{2\alpha-k+1}{k}}\,d\rho}{\big(\rho - \tau_1\big)\big(\rho - \sigma_1\big)} = \frac{1}{\sigma_1 - \tau_1}\bigg(\int\limits_{0}^{\infty}\frac{|\rho|^{\frac{2\alpha-k+1}{k}}\,d\rho}{\rho - \sigma_1} - \int\limits_{0}^{\infty}\frac{|\rho|^{\frac{2\alpha-k+1}{k}}\,d\rho}{\rho - \tau_1}\bigg).
\end{equation*}
These integrals converge absolutely, because we have assumed that~$\frac{2\alpha -k + 1}{k} < 0$.
Thus, we have to prove that the function~$\Phi$ given by the formula
\begin{equation*}
\Phi(\zeta) = \int\limits_{0}^{\infty}\frac{|\rho|^{\frac{2\alpha-k+1}{k}}\,d\rho}{\rho - \zeta}
\end{equation*}
is an injective function on~$\mathbb{C}\setminus\mathbb{R}_+$. We list the properties of the function~$\Phi$.
\begin{enumerate}
\item The function~$\Phi$ is holomorphic as a function on~$\mathbb{C}\setminus\mathbb{R}_+$.
\item The function~$\Phi$ is non-zero, e.g. it is positive on~$\mathbb{R}_-$.
\item For any~$\lambda \in \mathbb{R}_+$ one has~$\Phi(\lambda\zeta) = \lambda^{\frac{2\alpha - k+ 1}{k}}\Phi(\zeta)$.
\end{enumerate}
The third property follows by a change of variable in the integral that defines~$\Phi$. Let~$\Delta(z)$ be the branch of~$z \mapsto z^{\frac{2\alpha -k+1}{k}}$ defined on~$\mathbb{C}\setminus\mathbb{R}_+$ that is real positive on the negative real  half-line. By the third property of the function~$\Phi$,~$\Phi(\zeta) = \Phi(-1)\Delta(\zeta)$ when~$\zeta$ is a negative real. By the uniqueness theorem for analytic functions,~$\Phi(\zeta) = \Phi(-1)\Delta(\zeta)$ for~$\zeta \in \mathbb{C} \setminus\mathbb{R}_+$. But the function~$\Delta$ is injective, because~$\frac{2\alpha -k + 1}{k} > -1$ due to equation~\eqref{line}.
\paragraph{Case~$\alpha < \frac{k-1}{2}$,~$\sigma_1 = \tau_1$.} To treat the remaining case, we note that the integral~\eqref{a3} in question in this case equals~$\Phi'(\sigma_1)$ (where~$\Phi$ is the holomorphic function introduced above). This value is non-zero, because a holomorphic injective function cannot have vanishing derivative.
\subsubsection{Cases where~$k$ is odd}
In these cases we have~$|\rho|^{k-1} = \rho^{k-1}$,
and~\eqref{a3} can be rewritten as follows ($s = \rho^k$):
\begin{equation*}
\int\limits_{-\infty}^{\infty} \frac{|\rho|^{2\alpha}}{\big(\rho^k - \tau_1\big)\big(\rho^k - \sigma_1\big)}\,d\rho=
\frac{1}{k}\int\limits_{-\infty}^{\infty} \frac{|s|^{\frac{2\alpha - k + 1}{k}}}{(s-\tau_1)(s - \sigma_1)}\,ds.
\end{equation*}
\paragraph{Case~$\alpha = \frac{k-1}{2}$,~$\sigma_1$ and $\tau_1$ have imaginary parts of different signs.}
In this case the modulus disappears, we are integrating an analytic function. Integrating over the same contour as we did in Subsection~\ref{SsOddk}, we see that the integral equals zero if~$\sigma_1$ and~$\tau_1$ have imaginary parts of the same sign, and does not equal zero if they do not.
\paragraph{Case~$\alpha \ne \frac{k-1}{2}$,~$\sigma_1 \ne \tau_1$.}
We do the same trick as before and rewrite the integral as
\begin{equation*}
\frac{1}{\tau_1 - \sigma_1}\lim\limits_{R \to +\infty}\int\limits_{-R}^R\Big(\frac{|s|^{\frac{2\alpha-k+1}{k}}}{s-\tau_1} - \frac{|s|^{\frac{2\alpha-k+1}{k}}}{s-\sigma_1}\Big)\, ds
\end{equation*}
and introduce the function~$\Phi$ given by formula
\begin{equation*}
\Phi(\zeta) = \lim\limits_{R \to +\infty}\int\limits_{-R}^R\frac{|s|^{\frac{2\alpha-k+1}{k}}}{s - \zeta}\,ds.
\end{equation*}
Here are the properties of the functions~$\Phi$ (note that we still have to prove that the limit in~$R$ exists).
\begin{enumerate}
\item The function~$\Phi$ is analytic in~$\mathbb{C} \setminus \mathbb{R}$ (in particular, the limit in~$R$ exists).
\item The function~$\Phi$ is non-zero, e.g.  it is non-zero on the imaginary axis (it is pure imaginary there).
\item For all admissible~$\zeta$,~$\Phi(-\zeta) = -\Phi(\zeta)$.
\item For any~$\lambda \in \mathbb{R}_+$ one has~$\Phi(\lambda\zeta) = \lambda^{\frac{2\alpha - k+ 1}{\alpha}}\Phi(\zeta)$.
\end{enumerate}
Only the first property needs a proof (together with the fact that the function~$\Phi$ is well-defined). Writing
\begin{equation*}
\lim\limits_{R \to +\infty}\int\limits_{-R}^R\frac{|s|^{\frac{2\alpha-k+1}{k}}}{s - \zeta}\,ds = \int\limits_{0}^R s^{\frac{2\alpha - k + 1}{k}}\Big(\frac{1}{s - \zeta} - \frac{1}{s+ \zeta}\Big)\,ds = \int\limits_{0}^R \frac{2\zeta s^{\frac{2\alpha - k + 1}{k}}}{(s - \zeta)(s+ \zeta)}\,ds,
\end{equation*}
we again obtain an absolutely convergent integral, because~$\frac{2\alpha - k + 1}{k} < 1$ by virtue of equality~\eqref{line}.

We must prove that~$\Phi$ is injective on~$\mathbb{C} \setminus\mathbb{R}$. We denote by~$\Delta$
the branch of the power~$z^{\frac{2\alpha - k + 1}{k}}$ that arises if we allow the argument to vary within~$(-\pi,\pi]$ and maps~$\mathbb{R}_+
$ to itself.
By the fact that~$|\frac{2\alpha - k + 1}{k}| < 1$,~$\Delta$ is injective. Moreover, since~$\Delta$ takes the upper half-plane either into
the upper or into the lower half-plane (depending on the sign of~$2\alpha -k +1$), we see that
\begin{equation}\label{a6}
\Delta(\zeta_1) + \Delta(\zeta_2) \ne 0
\end{equation}
whenever~$\zeta_1,\zeta_2 \in \mathbb{C}_{+}$.

Now, consider a point~$\zeta_0$ that lies inside the first quarter. If~$|\arg \lambda| < \delta$, where~$\delta$ is
sufficiently small, then~$\lambda\zeta_0 \in \mathbb{C}_+$, and the function~$\lambda \mapsto \Phi(\lambda \zeta_0)$
is analytic. But for~$\lambda > 0$ we have
\begin{equation*}
\Phi(\lambda\zeta_0) = \Delta(\lambda)\Phi(\zeta_0)
\end{equation*}
by the fourth property of~$\Phi$,
whence~$\Phi(\lambda \zeta_0) = C\Delta(\lambda \zeta_0)$ for~$|\arg \lambda| < \delta$.
By the uniqueness theorem,~$\Phi(\zeta) = C\Delta(\zeta)$ for all~$\zeta \in \mathbb{C}_{+}$. Clearly,~$C \ne 0$ because~$\Phi$ is nonzero on
the imaginary axis.

Now, we see that the restrictions~$\Phi\mid_{\mathbb{C}_+}$ and~$\Phi\mid_{\mathbb{C}_-}$ are injective.
If~$\zeta_1 \in \mathbb{C}_+$ and~$\zeta_2 \in \mathbb{C}_-$, then~$\Phi(\zeta_1) \ne \Phi(\zeta_2)$
by~\eqref{a6} and the fact that~$\Phi$ is odd.

\paragraph{Case~$\alpha \ne \frac{k-1}{2}$,~$\sigma_1 = \tau_1$.} As in the case of even parameters, the integral in question equals~$\Phi'(\sigma_1)$. This value is non-zero, because it is a derivative of an injective analytic mapping.

\section{Non-elliptic case}\label{SVanish}
\subsection{Proof of Theorem~\ref{TVanishing}}
It is more convenient to work with compactly supported~$f$ and~$g$. The following claim is a straightforward consequence of the fact that~$\mathfrak{D}$ is dense in~$\mathcal{S}$.
\begin{Fact}\label{Compact support}
Suppose that inequality~\textup{\eqref{BilinearEmbeddings}} holds for some~$k,l,\alpha,\beta,\sigma,\tau$ with any~$f$ and~$g$ that belong to~$\mathfrak{D}(\mathbb{R}^2)$. Then~$\BE(k,l,\alpha,\beta,\sigma,\tau)$ also holds.
\end{Fact}
So, when proving Theorem~\ref{TVanishing}, we may assume that~$f$ and~$g$ are compactly supported. Moreover, using rescaling, we may make their support lie inside a unit disc centered at zero.

By symmetry, we may assume that~$k$ is odd. We remind the reader that~$\BE(k,l,\frac{k-1}{2},\frac{l-1}{2},\sigma,\tau)$ follows from the uniform boundedness of the kernel given by formula~\eqref{integrand}. However, in the case of real~$\sigma_1$ and~$\tau_1$, the situation is more complicated and the set~$\Omega_{\eps,R}$ is defined as
\begin{equation*}
\Omega_{\eps,R} = \Big\{(\xi,\eta)\in\mathbb{R}^2\big|\,\,\eps < |\eta| < R,\,\,|\xi^k - \sigma_1\eta^l| > \eps g(|\eta|),\,\,|\xi^k - \tau_1\eta^l| > \eps g(|\eta|)\Big\},
\end{equation*}
where~$g$ is some positive function decaying rapidly at zero and infinity. We note that by our assumption about the supports of~$f$ and~$g$ we may assume that~$|a| < 2$ and~$|b| < 2$, where~$a = x_1 - x_2$ and~$b = y_1 - y_2$. We may assume that~$a > 0$ and~$\eps < 1$.

We make our traditional change of variable~$\xi = \rho |\eta|^{\frac lk}$:
\begin{equation*}
\int\limits_{\Omega_{\eps,R}}\frac{\xi^{k-1}
e^{2\pi i (a\xi + b\eta)}\, d\xi}{(\xi^k - \tau_1 \eta^l)(\xi^k -
\sigma_1 \eta^l)}  = \frac{1}{(2\pi)^{2k}}\int\limits_{\varepsilon\leqslant |\eta|\leqslant R}
|\eta|^{-1}\int\limits_{B_{\eps}(\eta)}\frac{\rho^{k-1}e^{2\pi i(a\rho|\eta|^{l/k}+b\eta)}}{\big(\rho^k -
\tau_1(\sign\eta)^l\big)\big(\rho^k -\sigma_1(\sign\eta)^l\big)}\,d\rho d\eta.
\end{equation*}
Here~$B_{\eps}(\eta)$ is given by
\begin{equation*}
B_{\eps}(\eta) = \big\{\rho \in \mathbb{R}\,\big|\,\,\dist(\rho^k,\{\sigma_1(\sign \eta)^l,\tau_1(\sign \eta)^l\}) >  \eps |\eta|^{-l}g(|\eta|)\big\}.
\end{equation*}
We take~$(2\pi a)^{\frac{k}{l}}\eta$ to be a new variable, redefine~$R$ and~$b$ (but we still carry~$\eps$ and~$a$ and do not change them; it will be convenient to use the restrictions~$\eps$ < 1 and~$a < 2$) and rewrite the integral as
\begin{equation}\label{2Int}
\int\limits_{(2\pi a)^{-\frac{k}{l}}\varepsilon\leqslant |\eta|\leqslant R}
|\eta|^{-1}\int\limits_{B_{\eps,a}(\eta)}\frac{\rho^{k-1}e^{ i(\rho|\eta|^{l/k}+b\eta)}}{\big(\rho^k -
\tau_1(\sign\eta)^l\big)\big(\rho^k -\sigma_1(\sign\eta)^l\big)}\,d\rho d\eta,
\end{equation}
where
\begin{equation*}
B_{\eps,a}(\eta) = \big\{\rho \in \mathbb{R}\,\big|\,\,\dist(\rho^k,\{\sigma_1(\sign \eta)^l,\tau_1(\sign \eta)^l\}) >  \eps |(2\pi a)^{-\frac{k}{l}}\eta|^{-l}g(|(2\pi a)^{-\frac{k}{l}}\eta|)\big\}.
\end{equation*}
The integral with respect to~$\rho$ can be represented as a contour integral (after rewriting~$\int_{\mathbb{R}} = \lim_{r \to \infty}\int_{-r}^r$), see Figure~\ref{fig:con}.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 1 \linewidth]{Contour.pdf}
\caption{Contour of integration.}
\label{fig:con}
\end{center}
\end{figure}
We denote a unique real root of the equation~$z^k = s$,~$s \in \mathbb{R}$, by~$z_0(s)$. It is not hard to see that this equation has exactly~$\frac{k-1}{2}$ roots in the upper half-plane~$\Im z > 0$. So, if~$r$ is sufficiently large, there are exactly~$k-1$ poles of the integrand inside the contour ($\frac{k-1}{2}$ matching~$s = \sigma_1(\sign \eta)^l$ and~$\frac{k-1}{2}$ matching~$s = \tau_1 (\sign \eta)^l$). We also take the function~$g(\eta)$ to be so small that the small semicircles do not intersect ($g(\eta) \lesssim \eta^l$ will do because~$\eps < 1$). We need a simple lemma on estimating the difference between the integral over a small semicircle and the ``semi-residue''; this is a quantitative version of the Sokhotski--Plemelj formula.
\begin{Le}\label{error}
If a meromorphic function~$\psi$ has a simple pole at a point~$x_0$ on the real line\textup, then
\begin{equation*}
\Bigg|\int\limits_{\substack{|z-x_0|=r \\ \Im z >0}}\psi(z)\,dz +\pi i\Res_{x_0}\psi\Bigg|\,
\leqslant\pi r\max_{\substack{|z - x_0|\leqslant r,\\\Im z >0}}|h'|,
\end{equation*}
where~$h(z)=(z-x_0)\psi(z)$ \textup{(}the semicircle of integration is oriented clockwise\textup{)}.
\end{Le}

\begin{proof}
Since~$\psi$ has simple pole at~$x_0$, it can be written in the form~$\psi(z)=\frac{c}{z - x_0}+\psi_1(z)$, where $\psi_1$ is regular near $x_0$. Accordingly, we write
\begin{equation*}
\int\limits_{\substack{|z-x_0|=r \\ \Im z >0}}\!\!\!\!\!\!\!\psi(z)\,dz=\int\limits_{\substack{|z-x_0|=r \\ \Im z >0}}\!\!\!\!\!\frac{c\,dz}{z-x_0}+
\int\limits_{\substack{|z-x_0|=r \\ \Im z >0}}\!\!\!\!\!\psi_1(z)\,dz.
\end{equation*}
Since $\psi_1(z)=\frac{h(z)-h(x_0)}{z-x_0}$, the second integral is estimated as follows:
\begin{equation*}
\Bigg|\int\limits_{\substack{|z-x_0|=r \\ \Im z>0}}\psi_1(z)\,dz\Bigg|
\leqslant\pi r\max_{\substack{|z - x_0|\leqslant r,\\ \Im z >0}}|\psi_1|
\leqslant\pi r\max_{\substack{|z - x_0|\leqslant r,\\ \Im z >0}}|h'|.
\end{equation*}
The first integral can be calculated with ease:
\begin{equation*}
\int\limits_{\substack{|z - x_0| = r \\ \Im z > 0}} \frac{c\,dz}{z - x_0} =
\int\limits_{\substack{|z| = r,\\ \Im z > 0}} \frac{c\,dz}{z} =
\int\limits_{\frac{1}{2}}^0 \frac{c\,d(re^{2\pi i \theta})}{re^{2\pi i \theta}} =
-\pi i c.
\end{equation*}
\end{proof}
Therefore, the sum of the integrals over the semicircles is
\begin{equation*}
-\pi i e^{ib\eta}\frac{e^{ i z_0(\sigma_1 (\sign\eta)^{l})|\eta|^{\frac lk}} - e^{ i z_0(\tau_1 (\sign\eta)^{l})|\eta|^{\frac lk}}}{k(\sigma_1 - \tau_1)} + \eps O\bigg( \max(1,|\eta|^{\frac{l}{k}})|(2\pi a)^{-\frac{k}{l}}\eta|^{-l}g(|(2\pi a)^{-\frac{k}{l}}\eta|)\bigg).
\end{equation*}
The sum of residues result in a sum of integrals of the type~\eqref{SimpleIntegral}, as we have seen, such integrals are uniformly bounded. However, the integral that comes from semi-residues is subtler. We begin with estimating the error (i.e. the part of the integral~\eqref{2Int} that comes from the~$O$):
\begin{equation*}
\begin{aligned}
\eps\int\limits_{-\infty}^{\infty}\max(1,|\eta|^{\frac{l}{k}})|(2\pi a)^{-\frac{k}{l}}\eta|^{-l}g\big(|(2\pi a)^{-\frac{k}{l}}\eta|\big) \,\frac{d\eta}{|\eta|} =
\eps\int\limits_{-\infty}^{\infty}\max\big(1,2\pi a|\eta|^{\frac{l}{k}}\big)|\eta|^{-l}g(|\eta|) \,\frac{d\eta}{|\eta|} \stackrel{\scriptscriptstyle a < 2}{\leq}\\
\eps \int\limits_{-\infty}^{\infty}\big(1 + |\eta|^{\frac{l}{k}}\big)|\eta|^{-l}g(|\eta|) \,\frac{d\eta}{|\eta|} \leq \eps
\end{aligned}
\end{equation*}
provided~$g(z) = |z|^{2l + 1} e^{-|z|}$ (we also multiply this function by a small constant to fulfill~$g(|\eta|)\lesssim\eta^l$). So, the contribution of the error to~\eqref{2Int} is uniformly bounded (and even small if~$\eps$ is small). The integral coming from the semi-residues looks like this (we have redenoted~$\eps$):
\begin{equation*}
\int\limits_{\eps}^{R} e^{i b \eta}\frac{\big(e^{i c_1|\eta|^{\frac lk}} - e^{i c_2|\eta|^{\frac lk}}\big)\,d\eta}{\eta},
\end{equation*}
here~$c_1$ and~$c_2$ are some real constants (equal to~$z_0(\sigma_1 (\sign\eta)^{l})$ and~$z_0(\tau_1 (\sign\eta)^{l})$ correspondingly).
The part of the integral over the interval~$[0,1]$ is bounded, so we can drop it. On the ray~$(1,\infty)$, we use triangle inequality
\begin{equation*}
\bigg|\int\limits_{1}^{R} e^{i b \eta}\frac{\big(e^{i c_1|\eta|^{\frac lk}} - e^{i c_2|\eta|^{\frac lk}}\big)\,d\eta}{\eta}\bigg| \leq \bigg|\int\limits_1^R e^{i (b\eta +c_1|\eta|^{\frac lk})}\,\frac{d\eta}{\eta}\bigg| + \bigg|\int\limits_1^R e^{i (b\eta + c_2|\eta|^{\frac lk})}\,\frac{d\eta}{\eta}\bigg|
\end{equation*}
and make a change of variable~$|c_{1,2}|^{\frac kl}\eta = e^x$, redefine~$b$, and arrive at
\begin{equation*}
\Big|\int\limits_0^{e^R} e^{i (b e^x \pm e^{\alpha x})} \,dx\Big|,
\end{equation*}
where~$\alpha = \frac{l}{k} \ne 1$.
It remains to prove a simple lemma on oscillatory integrals.
\begin{Le}\label{OscillatoryIntegral}
Let~$\alpha$\textup,~$\alpha \ne 1$\textup, be a positive fixed parameter. Then
\begin{equation*}
\Big|\int\limits_0^{R} e^{i (b e^x \pm e^{\alpha x})}\,dx\Big| \lesssim 1.
\end{equation*}
\end{Le}
This lemma will be deduced from the second Van-der-Corput lemma (for example, see~\cite{HE},~$\S 2.5.2$) cited below.
\begin{Le}\label{VdC}
Suppose that the function~$F$ is twice differentiable on~$(a,b)$ and~$F''$ does not have roots on this interval. In this case\textup,
\begin{equation*}
\Big|\int\limits_a^b e^{iF(x)}\,dx \Big| \leq \frac{8 \sqrt{\pi}}{\sqrt{\min\limits_{\xi \in (a,b)} |F''(\xi)|}}.
\end{equation*}
\end{Le}

\paragraph{Proof of Lemma~\ref{OscillatoryIntegral}.}
Consider the case where~$\pm$  is~$-$, the remaining case is similar.
For brevity, we introduce the functions~$h_b$ given by~$h_b(x) = be^x - e^{\alpha x}$. Surely,~$h''_b(x) = be^x - \alpha^2e^{\alpha x}$, which can be zero (and thus does not allow to apply Lemma~\ref{VdC} directly). However,~$h''_b(x)$ can be small only on a set of small measure. It is enough to prove that
\begin{equation*}
\Big|\int\limits_C^{R} e^{i (b e^x - e^{\alpha x})}\,dx\Big| \lesssim 1,
\end{equation*}
where~$C$ is some numeric constant that does not depend on~$b$ (but will depend on~$\alpha$). The function~$h''_b$ changes monotonicity on~$[C,\infty)$ at most once. For a monotone function, the set where its modulus does not exceed one, is an interval (or ray). Therefore, the set~$\{x\in[C,\infty) \mid |h''_b(x)| < 1\}$ is a union of at most two intervals (this set is finite, because~$|h''_b|(x) \to \infty$ as~$x\to \infty$). We are going to prove that if~$|h''_b(z)| < 1$, then~$|h''_b(z\pm 1)| > 1$ for~$z \in [C,\infty)$. If this assertion is proved, it is not hard to see that the intervals constituting the set~$\{x\in[C,\infty)\mid |h''_b(x)| < 1\}$ have common length at most~$2$. The complement of this set in~$[C,\infty)$ is a union of at most three intervals (one of them is a ray). On each of them, the integral can be estimated by~$20$ by Lemma~\ref{VdC}, on the complement it is estimated by~$2$ and the lemma is proved.

To prove the assertion, we denote~$b e^z$ by $p$, and~$\alpha^2e^{\alpha z}$ by $q$. If both~$|h''_b(z)| < 1$ and~$|h''_b(z + 1)| < 1$ (we consider this case, the case~$|h''_b(z-1)| < 1$ is similar), then
\begin{equation*}
q-1 < p < q+1\quad \hbox{and}\quad e^{\alpha} q -1< ep < e^{\alpha} q + 1.
\end{equation*}
In this case,~$e^{\alpha} q - 1 < e(q+1)$ and~$e(q-1) < e^{\alpha} q + 1$, which is~$q< \frac{e + 1}{|e^{\alpha} - e|}$. Taking~$C >10 +  \alpha^{-1}\ln\frac{e + 1}{\alpha^2|e^{\alpha} - e|}$, we get a contradiction. The assertion is proved.
\qed

\subsection{Proof of Corollary~\ref{EvenParametersVanishingOp}}\label{SsEvenParam}
Corollary~\ref{EvenParametersVanishingOp} will be obtained by a simple algebraic trick.
\begin{Le}\label{Implication}
The following implication holds provided~$\sigma\tau \ne 0$\textup:
\begin{equation*}
\begin{aligned}
\BE(k,l,\alpha,\beta,\sigma,\tau)\,\, \&\,\,\BE(k,l,\alpha,\beta,-\sigma,-\tau) \,\,\&\,\,\BE(k,l,\alpha,\beta,\sigma,-\tau)\,\, \&\,\, \BE(k,l,\alpha,\beta,-\sigma,\tau) \Rightarrow\\ \BE(2k,2l,\alpha + 2k,\beta,\sigma^2,\tau^2)\,\,\&\,\,\BE(2k,2l,\alpha,\beta + 2l,\sigma^2,\tau^2).
\end{aligned}
\end{equation*}
\end{Le}
\begin{proof}
By applying the statement~$\BE(k,l,\alpha,\beta,\pm\sigma,\pm\tau)$ to the pair of functions~$(\partial_1^k \pm\sigma\partial_2^l)f$ and~$(\partial_1^k \pm \tau\partial_2^l)g$, we get estimates on four scalar products:
\begin{equation*}
\Big|\scalprod{(\partial_1^k \pm \tau\partial_2^l)f}{(\partial_1^k \pm \sigma \partial_2^l )g}_{W^{\alpha,\beta}_2}\Big| \lesssim \big\|(\partial_1^{2k} - \tau^2\partial_2^{2l})f\big\|_{L_1}\big\|(\partial_1^{2k} - \sigma^2\partial_2^{2l})g\big\|_{L_1}.
\end{equation*}
It is not hard to see that one can express both~$\scalprod{\partial_1^k f}{\partial_1^k g}_{W^{\alpha,\beta}_2}$ and~$\scalprod{\partial_2^l f}{\partial_2^l g}_{W^{\alpha,\beta}_2}$
as a linear combination of the four scalar products on the left-hand side, and thus estimate them by the expression on the right-hand side. We only have to use equation~\eqref{DerivationOfScalarProduct} to obtain~$\BE(2k,2l,\alpha + 2k,\beta,\sigma^2,\tau^2)$ and~$\BE(2k,2l,\alpha,\beta + 2l,\sigma^2,\tau^2)$.
\end{proof}
\paragraph{Proof of Corollary~\ref{EvenParametersVanishingOp}.} By Theorem~\ref{TVanishing}, the statments~$\BE(\frac{k}{2},\frac{l}{2},\frac{k}{4} - \frac12,\frac{l}{4}-\frac12,\pm\sigma',\pm\tau')$, where~$\sigma'_1 = \sqrt{\sigma_1}$ and~$\tau_1'=\sqrt{\tau_1}$, hold true (because one of the numbers~$\frac{k}{2}$ and~$\frac{l}{2}$ is odd and the numbers~$\pm\sigma_1'$ and~$\pm\tau_1'$ are real distinct). Lemma~\ref{Implication} finishes the proof.
\subsection{Proof of Lemma~\ref{KnappExample}}
We are going to disprove even a weaker statement, namely, we disprove the inequality
\begin{equation}\label{LinearCoercive}
\|f\|_{W_2^{\alpha,\beta}} \lesssim \|(\partial_1^k - \sigma\partial_2^l)f\|_{L_1}.
\end{equation}
The case where~$(k,l,\sigma,\sigma)$ is elliptic, has been already considered. We assume that~$(k,l,\sigma,\sigma)$ is not elliptic.

Let~$\zeta = (\zeta_1,\zeta_2)$ be some non-zero point in~$\mathbb{R}^2$ such that~$\zeta_1^k - \sigma_1\zeta_2^l = 0$ and~$|\zeta_1| > 2$,~$|\zeta_2| > 2$. We will consider only the functions~$f$ whose spectrum lies in a~$\frac12$-neighbourhood of~$\zeta$; for such functions~$f$
\begin{equation*}
\|f\|_{W_2^{\alpha,\beta}} \asymp \|f\|_{L_2}
\end{equation*}
because the weight~$|\xi|^{2\alpha}|\eta|^{2\beta}$ in formula~\eqref{norma} is bounded from below and above on the support of~$\hat{f}$.
Let~$\phi \in \mathfrak{D}(\mathbb{R})$ be a real-valued function supported on~$[-\frac12,\frac12]$. By~$\phi_{\lambda}$ we denote the function~$x \to \phi(\lambda x)$. The sequence~$\{f_n\}_n$ that disproves~\eqref{LinearCoercive} is given by formula
\begin{equation*}
\hat{f}_n(\xi,\eta) = \phi_n(\eta - \zeta_2)\phi_{n^2}(\xi - (\sigma_1\eta^l)^{\frac{1}{k}}), \quad |\xi - \zeta_1| \leq 1, |\eta - \zeta_2| \leq 1,
\end{equation*}
and zero when~$\xi$ or~$\eta$ in all other cases. We see that the function~$\hat{f}_n$ is supported on a small~$n^{-1}\times n^{-2}$-rectangle~$R_n$ that covers the~$n^{-2}$-neighbourhood of the curve~$\gamma = \{(\xi,\eta)\in\mathbb{R}^2\mid \xi^k = \sigma_1\eta^l\}$ near the point~$\zeta$ (namely, the bigger side of~$R_n$ is parallel to the tangent to~$\gamma$ at the point~$\zeta$). The~$L_2$-norm on the left-hand side of~\eqref{LinearCoercive} can be computed with ease,
\begin{equation}\label{L2estimate}
\|f_n\|_{L_2}^2 = \|\hat{f}_n\|_{L_2}^2 = \!\!\!\!\int\limits_{|\eta - \zeta_2| < \frac1n}\!\!\!\!\!\!\!\!\phi_n^2(\eta - \zeta_2)\!\!\!\!\!\!\!\!\int\limits_{|\xi - \zeta_1| < \frac12}\!\!\!\!\!\!\!\!\phi_{n^2}^2(\xi - (\sigma_1\eta^l)^{\frac{1}{k}})\,d\xi d\eta= \frac{1}{n^2}\|\phi\|_{L_2}^2\int\limits_{\mathbb{R}}\phi_n^2(\eta - \zeta_2) \,d\eta = \frac{1}{n^3}\|\phi\|_{L_2}^4.
\end{equation}
If~$n$ is big enough, then
\begin{equation*}
\big|(\xi^k - \sigma_1\eta^l)\hat{f}_n(\xi,\eta)\big| \lesssim n^{-2},
\end{equation*}
because on~$R_n$, where~$\hat{f}_n$ is supported, the polynomial~$(\xi^k - \sigma_1\eta^l)$ does not exceed~$n^{-2}$. Consequently,
\begin{equation*}
\big\|(\partial_1^k - \sigma\partial_2^l)f_n\big\|_{L_{\infty}} \lesssim \big\|(\xi^k - \sigma_1\eta^l)\hat{f}_n\big\|_{L_1} \lesssim n^{-5},
\end{equation*}
because the latter function does not exceed~$n^{-2}$ and is supported on~$R_n$ whose measure is~$n^{-3}$. Consider the polar set~$R^{\circ}_n$ of~$R_n$. This is a rectangle~$n\times n^2$ whose ``longer axis'' is perpendicular to the ``longer axis'' of~$R_n$ (it is parallel to the normal at~$\zeta$ to the curve~$\{\xi^k = \sigma_1\eta^l\}$).
\begin{Fact}
For any~$\eps > 0$ and any~$r > 0$ fixed we have an estimate
\begin{equation*}
\big|(\partial_1^k - \sigma\partial_2^l)f_n(x,y)\big| \lesssim n^{-3}(x^2 + y^2)^{-r},\quad~\dist\big((x,y),R_n^{\circ}\big) \geq n^{\eps}.
\end{equation*}
\end{Fact}
The proof of this fact is rather standard, but takes some place if written in detail. It is the same as the proof of the fact ``the Fourier transform of an~$r$ times continuously differentiable compactly supported function decays at infinity as~$|\xi|^{-r}$'' (one has to do lots of integrations by parts). With this fact at hand ($B_r(z)$ stands for the ball of radius~$r$ centred at~$z$),
\begin{equation*}
\begin{aligned}
\iint\limits_{\mathbb{R}^2}\big|(\partial_1^k - \sigma\partial_2^l)f_n\big| = \int\limits_{R_n^{\circ} + B_{n^{\eps}}(0)}\!\!\!\!\!\big|(\partial_1^k - \sigma\partial_2^l)f_n\big| +\!\!\!\!\! \int\limits_{\mathbb{R}^2 \setminus (R_n^{\circ} + B_{n^{\eps}})}\!\!\!\!\!\big|(\partial_1^k - \sigma\partial_2^l)f_n\big| \lesssim\\
n^{-5}n^{3 + 2\eps} + \!\!\!\!\!\int\limits_{\mathbb{R}^2 \setminus (R_n^{\circ} + B_{n^{\eps}})}\!\!\!\!\! n^{-3}(x^2 + y^2)^{-r}\,dx\,dy \lesssim n^{-2 + 2\eps}.
\end{aligned}
\end{equation*}
This inequality, together with~\eqref{L2estimate}, contradicts~\eqref{LinearCoercive} when~$n$ is large enough (we have~$n^{-\frac32}$ on the left-hand side and~$n^{-2 + 2\eps}$ on the right).

\section{Further development}\label{SFurther}
Though we were studying bilinear inequalities, we got some additional information about quadratic inequalities of the form
\begin{equation}\label{QuadraticEmbeddings}
\|f\|_{W_2^{\alpha,\beta}(\mathbb{R}^2)}^2\lesssim\|(\partial_1^k - \sigma\partial_2^l)f\|_{L_1(\mathbb{R}^2)} \|(\partial_1^k - \tau\partial_2^l)f\|_{L_1(\mathbb{R}^2)}.
\end{equation}
The statement that inequality~\eqref{QuadraticEmbeddings} holds true will be called~$\QE(k,l,\alpha,\beta,\sigma,\tau)$. Surely,~$\BE$ implies~$\QE$ with the same parameters. So, our positive results for~$\BE$ transfer to~$\QE$. Moreover, by an application of inequality between arithmetic and geometric means to formula~\eqref{norma} we see that
\begin{equation*}
\|f\|_{W_2^{\alpha,\beta}} \lesssim \|f\|_{W_2^{\alpha^-,\beta^-}} + \|f\|_{W_2^{\alpha^+,\beta^+}},\quad (\alpha_-,\beta_-)\,\,\hbox{and}\,\,(\alpha_+,\beta_+)\,\,\hbox{satisfy~\eqref{line}}\,\, \hbox{and}\,\,\alpha \in (\alpha_-,\alpha_+).
\end{equation*}
This, together with Corollary~\ref{EvenParametersVanishingOp} shows that~$\QE$ holds true when~$k$ and~$l$ are even, but one of~$\frac{k}{2}$ and~$\frac{l}{1}$ is odd,~$\sigma_1$ and~$\tau_1$ are real positive,~$(\alpha,\beta)$ satisfies~\eqref{line} and lies between~$(\frac34 k - \frac12,\frac14 l -\frac12)$ and~$(\frac14 k -\frac12,\frac34 l -\frac12)$.

The negative results for~$\QE$ are poorer. We surely know that~$\QE$ does not hold when~$\sigma = \tau$ (this follows from the corresponding part of the proof of Theorem~\ref{T3} and Lemma~\ref{KnappExample}). It seems that the quadratic inequalities resemble standard embedding theorems more than the bilinear ones.
\begin{Conj}
The statement~$\QE$ holds whenever~$\sigma \ne \tau$.
\end{Conj}

Our positive results for~$\BE$ are similar in elliptic and non-elliptic cases. The situation for negative results is far from being the same. The main difficulty in constructing counter-examples in the style of Subsection~\ref{SsCounterexamples} for the non-elliptic case is explained by the following fact.
\begin{Fact}\label{Spaces}
Consider the closure in~$L_p(\mathbb{R}^2)$\textup,~$1 \leq p < \frac43$\textup, of the set
\begin{equation*}
\Big\{(\partial_1^k - \sigma\partial_2^l)f \,\Big|\,\, \hat{f}\in\mathfrak{D}(\mathbb{R}^2)\Big\}
\end{equation*}
and denote this linear space by~$G_p^{k,l,\sigma}$. Suppose~$k\ne l$. If the polynomial~$\xi^ k -\sigma_1\eta^l$ is elliptic\textup, this space coincides with the space~$L_p(\mathbb{R}^2)$ provided~$p > 1$ and with the space~$L_1^0(\mathbb{R}^2)$ that consists of function having zero integral provided~$p=1$. When~$\xi^ k -\sigma_1\eta^l$ is not elliptic\textup, the space~$G_p^{k,l,\sigma}$ is not complemented in~$L_p$.
\end{Fact}
We will not prove this fact. The first part is standard. We give an explanation about the nature of the effect hidden in the second part. Consider the set~$\gamma = \{(\xi,\eta)\in\mathbb{R}^2\mid \xi^k = \sigma_1\eta^l\} \subset \mathbb{R}^2$. Out of zero, this is a smooth~$C^{\infty}$ curve (a submanifold of~$\mathbb{R}^2$), moreover, this curve is convex. If~$p < \frac43$ and~$K$ is some compact subset of~$\gamma$ (we suppose that~$0 \notin K$), then the operator defined by formula
\begin{equation*}
R_{K}[f] = \hat{f}\big|_{K},\quad f \in \mathcal{S}(\mathbb{R}^2),
\end{equation*}
is continuous from~$L_p(\mathbb{R}^2)$ to~$L_1(K)$ (we equip~$K$ with the Lebesgue measure on~$\gamma$), see e.g.~\cite{T}. Therefore, the kernel of this operator is closed in~$L_p$, denote it by~$L_p^K$. The intersection of all such spaces is again closed in~$L_p$, call it~$L_p^{\gamma}$. Surely,~$G_p^{k,l,\sigma} \subset L_p^{\gamma}$. It is a small surprise, but~$G_p^{k,l,\sigma} = L_p^{\gamma}$. This is a consequence of the fact that if~$\varphi \in \mathfrak{D}(\mathbb{R}^2)$ and~$\varphi
\equiv  0$ on~$\gamma$, then~$(\xi^k - \sigma_1\eta^l)^{-1}\varphi$ is also in~$\mathfrak{D}(\mathbb{R}^2)$, at least when~$0 \notin \supp \varphi$ (some problems with zero appear, but they are negligible for the spaces of functions). The fact that~$L_p^{\gamma}$ is not complemented follows from a well-known principle (see, e.g.~\cite{R}, Lemma~$3.1$ for~$p > 1$ and Theorem~$1.4$ for the case~$p=1$) that if a translation-invariant subspace of~$L_p$ is complemented, then there is a projector onto it that is also translation-invariant (i.e. a multiplier).

We state, again without proof, that when~$\frac43 \leq p < \infty$, the spaces~$L_p$ and~$L^{\gamma}_p = G_p^{k,l,\sigma}$ coincide.

Though Fact~\ref{Spaces} forbids counter-examples in the style of Subsection~\ref{SsCounterexamples}, it may give some additional hope for embedding theorems to hold. We end the story with a small discussion of a related conjecture.
\begin{Conj}
Let the set~$(k,l,\sigma,\sigma)$ be non-elliptic. The inequality
\begin{equation*}
\|f\|_{W_q^{\alpha,\beta}} \lesssim \|(\partial_1^k - \sigma \partial_2^l)f\|_{L_p}
\end{equation*}
holds true whenever~$\frac{1}{p} - \frac{1}{q} \geq \frac23$\textup,~$1 < p < \frac43$\textup,~$q < \infty$ and
\begin{equation}\label{HomogeneityCondition}
\frac{\alpha}{k} + \frac{\beta}{l} = 1 - \Big(\frac{1}{p} - \frac{1}{q}\Big)\Big(\frac{1}{k} + \frac{1}{l}\Big).
\end{equation}
\end{Conj}
The space on the left-hand side of the inequality is defined as the closure of~$\mathcal{S}(\mathbb{R}^2)$ with respect to the norm
\begin{equation*}
\|f\|_{W_q^{\alpha,\beta}} = \Big\|\mathcal{F}^{-1}\big[\mathcal{F}[f](\xi,\eta)|\xi|^{\alpha}|\eta|^{\beta}\big] \Big\|_{L_q},
\end{equation*}
where~$\mathcal{F}$ stands for the Fourier transform. When the set~$(k,l,\sigma,\sigma)$ is elliptic, the statement holds true whenever~$p > 1$ and follows from classical embedding theorems and the fact that pure derivatives are expressed in terms of~$(\partial_1^k - \sigma \partial_2^l)$ by a singular integral operator.

We are able to prove the conjecture with additional assumption~$q > 4$. This is a combination of known facts. First, the problem can be localized in spectrum, i.e. reduced to the case where~$\hat{f}$ is supported near some point on the curve~$\gamma$. This is done by decomposing~$f = \sum P_jf$, where~$P_j$ is the Littlewood--Paley smoothed spectral projector, application of Littlewood--Paley inequality~$\|f\|_{L_p} \asymp \|(\sum |P_j f|^2)^{\frac12}\|_{L_p}$, H\"older inequality (for the last step, the inequalities~$p \leq 2$ and~$q \geq 2$ are of crucial importance), and anisotropic dilatation invariance of the problem given by equation~\eqref{HomogeneityCondition}. After the problem has been localized, the space on the left-hand side turns into the standard Lebesgue space~$L_q$. Thus, we are studying the continuity of the operator with the symbol~$\frac{1}{\xi^k - \sigma_1\eta^l}$ as an operator from the localized space~$L_p^{\gamma}$ to~$L_q$. This is, modulo convolutions with Schwartz functions, the same as to study the operator with the symbol~$\vp(\dist(\cdot,\gamma))^{-1}$. The continuity of a similar operator, but between standard Lebesgue spaces, had been completely studied in~\cite{B}. Namely, in this paper, the negative powers of the Bochner-Riesz operator were studied (i.e. the same-type operators, but where the degree~$-1$ can vary from~$-\frac32$ to~$0$ and even be complex, but~$\gamma$ is a circumference). However, the method can be applied verbatim to the case of arbitrary smooth convex curve in place of the unit circumference. It finishes the proof of the conjecture for the cases where~$q > 4$.

However, it is not hard to see that when~$q < 4$, the corresponding operator is not continuous as an operator between the Lebesgue spaces. So, the interesting cases of the conjecture are where~$q  <4$. We believe that this is the place where the space~$L^{\gamma}_p$ can play a definite role.


TITLE: Evaluate Surface Integral
QUESTION [0 upvotes]: Evaluate the $$\int_S { \vec{D} \cdot} d\vec{s}$$ over the surface $\vec{D} = 2y\vec{a_x} +  xz\vec{a_y} +z\vec{a_z}$. The cube's coordinates are from $x,y,z\in{(0,2)}$. I am struggling to find it none zero, because I do $$\int_{top} {\vec{D} \cdot} d\vec{s} - \int_{bot} {\vec{D} \cdot} d\vec{s} + \int_{right} {\vec{D} \cdot} d\vec{s} - \int_{left} {\vec{D} \cdot} d\vec{s} + \int_{front} {\vec{D} \cdot} d\vec{s} - \int_{back} {\vec{D} \cdot} d\vec{s} = 0$$ But then I check using the divergence theorem and get 8. I am not sure what I am doing wrong to get 0 on the surface integral since it seems to me it all cancels out. Any help?

REPLY [0 votes]: The integral on the top face is
$$\int_{top}\vec D\cdot d\vec s = \int_{(z=2)\cap \partial S} 0\cdot 2y+0\cdot xz+1\cdot z\ dxdy=\int_{x=0}^2\int_{y=0}^2 2\,dxdy = 8.$$
While the same for the bottom face (the normal vector now looks in the opposite direction: $\pmatrix{0\\0\\-1}$ -- that corresponds to the negative sign in your formulation) gives $\ 0\ $ as $z=0$ on the bottom face.


text:
\section*{Introduction}

The following theorem is the main result of this paper.

\begin{thm}\label{mainthm}
Let $\phi$ be an automorphism of a finitely generated free group $F_n$.
\begin{itemize}
\item There exists an explicit algorithm that, given two elements $u,
v\in F_n$, decides whether there exists some exponent $N$ such that
$u\phi^N=v$.
\item There exists an explicit algorithm that, given two elements $u,
v\in F_n$, decides whether there exists some exponent $N$ such that
$u\phi^N$ is conjugate to $v$.
\end{itemize}
If such an exponent $N$ exists, then the algorithms will compute $N$ as
well.  The words $u, v$ are specified as words in the generators of
$F_n$, and $\phi$ is specified in terms of the images of generators.
\end{thm}

The results in this paper was motivated by work that first appeared in
\cite{rect}.  \thmref{mainthm} plays a role in the computation of fixed
subgroups of free group automorphisms \cite{fixedsubgroup}, and it
constitutes one part of the recent solution of the conjugacy problem in
free-by-cyclic groups due to Bogopolski, Maslakova, Martino, and Ventura
\cite{conjprob}.

Our main technical tool is an algorithmic extension of the theory of
relative train track maps \cite{hb1, hb2}.  Specifically, we present
algorithmic (and possibly even practical) ways of finding {\em
efficient} relative train track maps that share many the properties of
{\em improved} relative train track maps as introduced (in a
nonconstructive fashion) in \cite{tits1}.

One intriguing aspect of our argument is that it suggests that the
detection of orbits in free groups and the computation of efficient maps
are closely related problems.  Orbit detection and computation of
efficient maps leapfrog each other, with orbit detection providing a
crucial step in the computation of efficient maps, and efficient maps
enabling the detection of orbits.

In \secref{bccsec}, we review well-known results on homotopy
equivalences of finite graphs, with an emphasis on computational aspects
of the constants involved.  \secref{trainsec} contains a brief review of
the theory of relative train track maps, including first steps towards
improvements.  \secref{improvesec} contains the first part of our
construction of efficient train track maps.  \secref{pathsec} presents
an algorithm that detects orbits of paths, and \secref{fixedsec} builds
upon the results of \secref{improvesec} and \secref{pathsec} to provide
the last, and most difficult, step in our construction of efficient maps,
the detection of fixed points of certain lifts of homotopy equivalences
of finite graphs.  Finally, in \secref{mainsec}, we translate our
results from the realm of homotopy equivalences of graphs to the realm
of automorphisms of free groups.

I would like to express my gratitude to Oleg Bogopolski and Armando
Martino for their hospitality, encouragement, and many helpful
discussions.

config: arxiv


text:
\begin{document}

\title[Configurations and compactness]{Configurations of points on degenerate varieties and properness of moduli spaces}

\author[D. Abramovich]{Dan Abramovich}
\thanks{Research of D.A. partially supported by NSF grant DMS-0335501 and DMS-0603284}
\address{Department of Mathematics, Box 1917, Brown University,
Providence, RI, 02912, U.S.A}
\email{abrmovic@math.brown.edu}

\author[B. Fantechi ({\tiny\today})]{Barbara Fantechi}
\thanks{Research activities of B.F. partially supported by Italian research grant PRIN 2006 ``Geometria delle variet\`a proiettive", ESF Research Network MISGAM, Marie Curie RTN ENIGMA, and GNSAGA}
\address{SISSA,
Via Bonomea 265,
34136 Trieste, Italy}
\email{fantechi@sissa.it}

\date{\today}

\maketitle
\setcounter{tocdepth}{1}

\tableofcontents

 \section*{Introduction}

Consider a smooth variety $X$ and a smooth divisor $D\subset X$.  Kim and Sato in \cite[1.1, esp. Theorem 1]{Kim-Sato} define

a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a

moduli space of stable configurations of $n$ points lying on expansions of $(X,D)$ in the sense of \cite{Li1}.

The purpose of this note is to generalize \cite[Theorem 1]{Kim-Sato}  to the case where $X$ is an algebraic stack; and to construct  an analogous projective moduli space $W_\pi^{[n]}$ for a degeneration $\pi:W \to B$. We construct $X^n_D$ and $W_\pi^{[n]}$  and prove their properness using a universal construction introduced in \cite{ACFW}.
We then use these spaces for a concrete application, as explained in the next paragraph.

In \cite{AF}, a degeneration formula for Gromov--Witten invariants of schemes and stacks is developed, generalizing the approach of Jun Li \cite{Li1,Li2}. This in particular requires proving properness of Li's stack of pre-deformable stable maps  in the case where the target $(X,D)$ or $W\to B$ is a  Deligne--Mumford stack. One could simply adapt Li's proof, or follow the age-old tradition of imposing such endeavor as an exercise on ``the interested reader".

Instead, we prefer to provide a different proof here, which  uses the properness of $X_D^{[n]}$ and $W_\pi^{[n]}$. Similar ideas are used in \cite{KKO} to prove the properness of their space of ramified maps.

\begin{conv*}  To keep ideas simple we work over an algebraically closed base field $K$; a scheme will be a scheme of finite type over $K$; a curve will be a purely one-dimensional scheme; we will use the word algebraic stack in the sense of Artin, and assume locally finite type over $K$; we will write DM for Deligne-Mumford stacks. A point in an algebraic stack, and in particular in a scheme, will be a $K$-valued point.
\end{conv*}

\paragraph{\bf Achnowledgements.} Thanks to Bumsig Kim, Andrew Kresch, and Jonathan Wise, whose shared ideas and comments are manifest in this note.

\section{Stable expanded configurations}

Suppose $X$ is a smooth algebraic stack. Then $X^n$ is a parameter space for $n$ ordered points in $X$. If $D\subset X$ is a smooth divisor, we may wish to consider a space of $n$ ordered points on $X$ where points are not allowed to land in $D$, but rather $X$ is replaced by an expansion. We construct in Section \ref{confpairs} such a compactification, denoted $X^{[n]}_D$: it is an immediate generalization of the first construction in \cite{Kim-Sato} to algebraic stacks, though our method is different.

Similarly, suppose {$\pi:W\to B$ is a flat morphism with $W$ a smooth algebraic stack, $B$ a smooth curve and all fibers $W_b$ smooth except for $W_0:=W_{b_0}$ which is the union of two smooth stacks $X_1$ and $X_2$ intersecting transversely along a divisor $D$. }

Then the self fibered product $W^n_B$ can be viewed as a parameter space of $n$ ordered points on fibers of $\pi$.

{We} construct in Section \ref{confdeg} a space of $n$-tuples of points on the fibers which are not allowed to land in the singular locus of $\pi$ but rather $W$ is replaced by an expansion.

\subsection{Notation for degenerations}\label{notdeg}

\begin{conv}\label{degeneration_conv}
In this section we fix $\pi: W \to B$, a flat morphism such that $B$ is a smooth curve, $W$ is a smooth algebraic stack, and $b_0\in B$ is the unique critical value of $\pi$; we set $W_0:=\pi^{-1}(b_0)$ and assume $W_0= X_1\sqcup_D X_2$  is the union of two smooth closed substacks $X_1$ and $X_2$ intersecting transversally along $D$, a smooth divisor in each $X_i$. This implies that $W_0$ is  first-order smoothable along its singular locus $D$, i.e., $N_{D/X_1}$ is dual to $N_{D/X_2}$.
\end{conv}

We use the notation of \cite[Section 2.3]{ACFW}. The {\em expansion of length $\ell\ge0$ of $W_0$} is \begin{equation*}
  W(\ell) := X_1 \hskip -1ex \mathop{\sqcup}\limits_{\hphantom{_-}D=D_1^-}  P_1 \mathop{\sqcup}\limits_{D_1^+=D_2^-}  \cdots   \mathop{\sqcup}\limits_{D_{\ell-1}^+=D_\ell^-}  P_\ell\mathop{\sqcup}\limits_{D_\ell^+=D}\hphantom{_-}
  X_2
\end{equation*}
where the {\em exceptional components} $P_j$ are all isomorphic to the $\PP^1$-bundle $\PP(N_{D/X_1}\oplus \cO_D)=\PP(\cO_D\oplus N_{D/X_2})$ and $D_j^{-}$, $D_j^{+}$ are
{the} zero and infinity sections {in $P_j$} (see \cite[Definition 2.3.1]{ACFW} for details). {The automorphism group of $W(\ell)$ is defined to be $\GG_m^\ell$ where the $j$-th factor acts on the fibers of $P_j$ fixing $D_j^-$ and $D_j^+$.}

Let $\cA$ be the stack quotient $ [\AA^1/\GG_m]$, so that morphisms $S\to \cA$ are{ pairs $(L,s)$ consisting of a line bundle and section} on $S$, see e.g. \cite[Lemma 2.1.1]{cadman}. We denote by $\fT$ the universal stack of expansions of the degeneration $\varpi:\cA^2 \to \cA$ induced by $t=xy$, with universal expansion $(\cA^2)'\to \cA^2$.

The {\em moduli stack of expansions} of $W\to B$ is $\fT_B = \fT\times_\cA B$ where $B \to \cA$ is the morphism associated to the Cartier divisor $\{b_0\} \subset B$; the universal expansion is $W' = (\cA^2)' \times_{\cA^2} W\to W$, where $W\to\cA^2$ is the smooth morphism induced by the divisors $X_1$ and $X_2$: see \cite[Definition 2.3.6]{ACFW} for details.

\subsection{Configurations on degenerations}\label{confdeg}

\begin{definition}
A {\em stable expanded configuration} $(\cW, \sigma_i)$ of degree $n$ on $W\to  B$ consists of
\begin{enumerate}
\item
a point $\cW$ of $\fT_B$, hence {either a smooth fiber $W_b$ or an expansion of $W_0$}, and
\item an ordered  collection of  $n$ smooth points $\sigma_i\in\cW^\smooth$,
\end{enumerate}
such that, in case $\cW$ is an expansion, the following stability condition holds:
\begin{itemize}
\item each exceptional component $P_j$ contains at least one $\sigma_{i}$.
\end{itemize}

An {\em isomorphism} $\rho: (\cW,\sigma_i) \to (\cW',\sigma_i')$ is an isomorphism of expanded degenerations $\rho:\cW \to \cW'$ such that $\rho\circ \sigma_i = \sigma_i'$.{ Note that the stability condition is equivalent to requiring that the only automorphism of $\cW$ fixing all the $\sigma_i$'s is the identity automorphism.}

\end{definition}

\begin{defn}
A {\em family of stable expanded configurations} of degree $n$ over a $B$-scheme $S$ is given by
\begin{enumerate}
\item a family $\cW_S\to W\times_B S$ of expanded degenerations over $S$, i.e. an object of $\fT_B(S)$;
\item $n$ sections $\sigma_i:S\to \cW_S$;
\end{enumerate}
such that for every $s\in S$ the fiber $(\cW_s,\sigma_{i,s})$ over $s$ is a stable expanded configuration of degree $n$.
\end{defn}

Morphisms of families of stable expanded configurations are defined in the obvious way; the resulting fibered category $W_{\pi}^{[n]}$ is clearly a stack, and indeed a sheaf if $W$ is a scheme or algebraic space. Composing $\sigma_i$ with $\cW \to W$, we obtain an object of the fibered power $W^n_\pi$ of $W$ over $B$, giving a morphism $W_{\pi}^{[n]} \to W^n_\pi$.

The notation should not be confused with the notation for the degree-$n$ Hilbert scheme of a surface.

The following lemma is an immediate consequence of the definition:
\begin{lemma}\label{Lem:config-open} Assume $V\subset W$ is open and $\varpi$ is the restriction of $\pi$. Then $V^{[n]}_\varpi = V^n_\varpi \times_{W^n_\pi} W^{[n]}_\pi$.

\end{lemma}

We construct $W_{\pi}^{[n]}$ starting with a special case, described in the following lemma:

\begin{lemma}\label{Lem:basic-deg} Assume $B= \AA^1$ and $W=W_{\PP}\subset \PP^2 \times \AA^1$ is the pencil $tZ^2 = XY$. Then for every $n\ge 1$ the  stack $W_{\pi}^{[n]}$ is naturally isomorphic to the moduli space of stable weighted $n$-pointed genus 0 maps $\ocM_{0, (\epsilon,\ldots,\epsilon)}(W,\beta_F)$ in the sense of Hassett, where $\beta_F$ is the class of the fiber and $\epsilon$ is a real number in $(0,1/n)$.

In particular in this case $W_{\pi}^{[n]}$ is a smooth variety, projective over $B$ and hence over $W^n_\pi$.
\end{lemma}
\begin{proof}Fix an integer $n>0$, an $\epsilon$ as in the statement, and write $M:=\ocM_{0, (\epsilon,\ldots,\epsilon)}(W,\beta_F)$ for brevity.

Let $S$ be a scheme, and $(\cW,\sigma_i)$ a family of stable expanded configurations over $S$. Then every fiber $\cW_s$ of $\cW$ is a nodal curve of arithmetic genus zero; moreover, if a component $\bar C$ of $\cW_s$ is contracted by the natural morphism $f:\cW\to W$, then $\bar C$ is rational and contains at least two nodes and a marked point, hence is stable since $1+1+\epsilon>2$.
This defines a natural map $W_\pi^{[n]}\to M$ as sheaves.

To define the inverse, let $S$ be a scheme and $(C_S,\sigma_{i,S},f_S)\in M(S)$; we want to show that it is a stable expanded configuration. Let $(C,x_i,f)$ be a point in $M$; since the global genus is zero, every irreducible component of $C$ is smooth and rational. Moreover, no contracted component can be a tail, i.e., intersect the rest of the curve in only one point, since if such a component contains $r$ marked points we have $1+r\eps\le 1+n\eps<2$, contradicting stability.

Let $t\in B$ such that $f(C)=W_t$. If $t\ne 0$, $W_t$ is integral, hence there is a unique component $C_0$ of $C$ mapping isomorphically to $W_t$. If $C$ had other components, than at least one of them must be a tail since every tree with at least two vertices has at least two end vertices; since all other components are contracted, this yields a contradiction. Therefore $C\to W_t$ must be an isomorphism.

If $t=0$, then $W_t$ is the union of two irreducible components $X_1$ and $X_2$ meeting in one point $p$, and $C$ has two non-contracted components $C_1$ and $C_2$ such that $f$ maps $C_i$ to $X_i$ isomorphically. Any other irreducible component must be contracted, hence $C_1$ and $C_2$ are the only tails of $C$, which implies that $C$ is a chain of rational curves, with $C_1$ and $C_2$ as extremes. Moreover, the stability condition implies that each irreducible component must contain at least one marked point $x_i$. This concludes the proof, since other conditions (flatness of $C_S\to S$, properness of $W_S\to W\times_TS$) are part of the definition of stable maps.

{It follows that} $W_{\pi}^{[n]}$ is a smooth variety, projective over $B$ and hence over $W_{\pi}^n$ (see \cite[Theorem 1.9]{AG}, \cite[Theorem 1.1.4]{BayerManin}, \cite[Theorem 1.7]{MustataMustata}, see also \cite[Theorem 1]{Hassett}).
\end{proof}
We now return to the general case.

\begin{proposition}\label{Prop:conf-deg}
The stack $W_{\pi}^{[n]}$ is smooth and algebraic, and the morphism $W_{\pi}^{[n]} \to W^n_\pi$ is projective. In particular, if $W$ is a scheme, or a Deligne--Mumford stack, so is $W_{\pi}^{[n]}$.
\end{proposition}

\begin{proof}
{\sc Case 1:} $B= \AA^1$ and $W= W_\AA =\AA^2$ mapping via $t=xy$.  This case is the restriction of the case treated in  Lemma \ref{Lem:basic-deg} to an open set  $W_\AA \subset W_{\PP}$, so it follows by Lemma \ref{Lem:config-open}.

Note that in this case there is an action of $\GG_m$ on $B$ and a compatible action of $\GG_m^2$ on $W$; if we write $(s_1,s_2)$ for a point on $\GG_m^2$, it acts on $W$ via $(x,y) \mapsto (s_1x,s_2y)$, and $s=s_1s_2\in \GG_m$ acts on $B$ via $t\mapsto st$. This induces an action of the $n$-th fiber product $\GG_m^{n+1}$ of  $\GG_m^2$ over $\GG_m$
 on $W_{\pi}^{[n]} \to W^n_\pi$ equivariant with respect to  the $\GG_m$ action on $B$.

{\sc Case 2:}  $B= [\AA^1/\GG_m]=\cA$ and $W= W_\cA= [\AA^2/\GG_m^2]=\cA^2$.  This is the quotient of the previous case by the action of $\GG_m^{n+1}$.
Projectivity is preserved, since the relatively ample line bundle admits an equivariant structure; indeed, every line bundle on a toric variety admits such structure, see Section 3.4 of \cite{fulton1993introduction}.
For reference below we denote the  morphism described in this case $\varpi:W_\cA \to \cA$.

{\sc General case:} We use the notation in Section  \ref{notdeg}.

Consider the smooth morphisms $B\to\cA$ given by the divisor $b_0$ and $W\to W_\cA=\cA^2$ given by the divisors $X_1$ and $X_2$:
Composing any family of stable expanded configurations of degree $n$ on $W$ with these morphisms   gives a family of stable configurations in the fibers of $W_\cA$, hence we get a natural morphism   $$W_{\pi}^{[n]} \to W_{\pi}^n \times_{(W_\cA)_{\varpi}^n} (W_\cA)_{\varpi}^{[n]}.$$

We construct an inverse of this morphism as follows. Let $S$ be a $B$--scheme, and $S\to (W_\cA)_{\varpi}^{[n]}$ a morphism over $\cA$ given by a family of stable expanded configurations  $\tilde \sigma_i:S\to (\cW_\cA)'$; then to every $B$--morphism $g:=(g_1,\ldots,g_n):S\to W_\pi^n$ we can associate a family of stable expanded configurations  on $W$ by letting $\sigma_i:S\to \cW$ be the morphisms induced by $\tilde\sigma_i$ and $g_i:S\to W$.

It is easy to see that these two constructions are inverse of each other.\end{proof}

\subsection{Other degenerate cases}\label{Sec:other} The construction given in the general case of Proposition \ref{Prop:conf-deg} apply in all cases introduced in \cite[Section 2.2]{ACFW}.
In particular one can consider the situation of having a degenerate fiber with no chosen smoothing as introduced in  \cite[2.2.2]{ACFW}: here
$B$ is a point {$B=\{b_0\}$} and $W$ is first order smoothable, i.e. the union of two smooth components $X_1$ and $X_2$ meeting transversally along a smooth divisor $D$, with $N_{D/X_1}$ dual to $N_{D/X_2}$.

The result is the following:

\begin{proposition}\label{Prop:conf-deg-fiber}
The stack $W_{\pi}^{[n]}$ is algebraic, and the morphism $W_{\pi}^{[n]} \to W^n_\pi$ is projective. In particular, if $W$ is a scheme, or a Deligne--Mumford stack, so is $W_{\pi}^{[n]}$.
\end{proposition}
\begin{proof} We can modify the previous proof as follows. First, consider {$B\to \cA$} induced by the unique morphism $B\to 0\subset \AA^1$; the fiber product {$W_B:=W_\cA\times_\cA B$} is the transversal union of two smooth irreducible components $W_{B,i}$, each isomorphic to {$\cA$}, meeting transversally along $\GG_m$. The pairs $(X_i,D)$ define smooth morphisms $X_i\to W_{B,i}$ and the condition on the normal bundles ensures that they glue to define a smooth morphism $W\to W_B$. The rest of the proof is the same as the general case above.
\end{proof}

Note that in this case $W_{\pi}^{[n]}$ is not smooth. Our construction gives a smooth map to  $\{b_0\} \times_\cA ({W_\cA})_{\varpi}^{[n]}$; its singularities are therefore modeled on the boundary of  $\ocM_{0, (\epsilon,\ldots,\epsilon)}(W,\beta_F)$, which is a normal crossings divisor.

\subsection{Notation for pairs}
\begin{conv}\label{pairs_conv} In this section we fix a pair $(X,D)$, where $X$ is a smooth algebraic stack and $D$ a smooth divisor in $X$.
\end{conv}

The {\em expansion of length $\ell\ge0$} of $(X,D)$ is the pair $(X(\ell),D(\ell))$ with \begin{equation*}
X(\ell) := X_1 \hskip -1ex \mathop{\sqcup}\limits_{\hphantom{_-}D=D_1^-}  P_1 \mathop{\sqcup}\limits_{D_1^+=D_2^-}  \cdots   \mathop{\sqcup}\limits_{D_{\ell-1}^+=D_\ell^-}  P_\ell
\end{equation*}
and $D(\ell):=D_\ell^-\subset P_\ell$. As in \ref{degeneration_conv} the {\em exceptional components} $P_j$ are all isomorphic to $\PP(N_{D/X_1}\oplus \cO_D)$ and $D_j^{-}$, $D_j^{+}$ are their zero and infinity sections (see \cite[Definition 2.3.1]{ACFW}). Its automorphism group is again defined to be $\GG_m^\ell$ acting componentwise on the $P_j$'s.

Following the notation of \cite[Section 2.1]{ACFW} we {let again $\cA:=[\AA^1/\GG_m]$ and} denote by $\cD\subset \cA$ the smooth divisor $[0/\GG_m]$. We denote by $\cT$ the universal stack of expansions of pairs: it parametrizes expansions of $(\cA,\cD)$, with universal expansion denoted by $(\cA',\cD')$. The stack $\cT$ also parametrizes expansions of any pair $(X,D)$, with universal family $(X',D')$ where $X' = \cA'\times_\cA X$ and  $D' = \cD' \times_\cD D$, see \cite[Definition 2.1.6]{ACFW}.

\subsection{Configurations on pairs}\label{confpairs}

\begin{definition}
A {\em stable expanded configuration} $(X',D', \sigma_i)$ of degree $n$ on $(X,D)$ consists of
\begin{enumerate}
\item
a point $(X',D')$ of $\cT$, {that is an expansion of the pair $(X,D)$} and
\item an ordered  collection of  $n$ smooth points $\sigma_i\in(X')^\smooth\setminus D'$,
\end{enumerate}
such that the following stability condition holds:
\begin{itemize}
\item each exceptional component $\PP_j$ contains at least one $\sigma_{i}$.
\end{itemize}

An {\em isomorphism} $\rho: (X',D',\sigma_i) \to (\bar X',\bar D',\bar\sigma_i)$ is an isomorphism of expanded pairs  $\rho:(X',D')\to(\bar X',\bar D') $ such that $\rho\circ \sigma_i = \bar\sigma_i$.
\end{definition}

\begin{defn}
A {\em family of stable expanded configurations} of degree $n$ over a scheme $S$ is given by
\begin{enumerate}
\item {a family of expansions $(X'_S, D'_S) \to (X,D)\times S$ of the pair $(X,D)$ parametrized by $S$, that is} an object of $\cT(S)$;
\item $n$ sections $\sigma_i:S\to X'_S$;
\end{enumerate}
such that for every $s\in S$ the fiber over $s$ is a stable expanded configuration of degree $n$.
\end{defn}

Morphisms of families of stable expanded configurations are defined as before; we denote the resulting category $X_{D}^{[n]}$, and again it is a stack with a natural map to $X^n$.

\begin{lemma}\label{Lem:basic-pairs}
Assume $X= \PP^1$ and $D=\{0\}$. Consider the moduli stack of weighted $(n+1)$-pointed stable maps $\ocM:=\ocM_{0; (\epsilon,\ldots,\epsilon,1)}(\PP^1,1)$ of degree 1, with its last evaluation map $ev:=ev_{n+1}: \ocM \to \PP^1$. Then $X_{D}^{[n]} = ev^{-1} (\{0\})$ {and hence} is a smooth projective variety, {therefore the morphism $X_D^{[n]}\to X^n$ is projective}.
\end{lemma}

\begin{proof}
We briefly review the argument which is  very similar to that of Lemma \ref{Lem:basic-deg}. The morphism $X_{D}^{[n]}\to \ocM$ is defined by using $D$ as $(n+1)$-st section; it clearly maps to $ev^{-1}(0)$. To define the opposite morphism, note that if $(C,x_i,f)$ is a point in $\ocM$, the curve $C$ can have only two tails, one mapping isomorphically to $\PP^1$ and the other containing the $(n+1)$-st section, so it is a chain of rational curves.

As in Lemma \ref{Lem:basic-deg}, smoothness and projectivity follow from \cite[Theorem 1.9]{AG}, \cite[Theorem 1.1.4]{BayerManin}, \cite[Theorem 1.7]{MustataMustata}, \cite[Theorem 1]{Hassett}).

\end{proof}

\begin{proposition}\label{Prop:conf-pairs}
The stack $X_{D}^{[n]}$ is smooth and algebraic, and the morphism $X_{D}^{[n]} \to X^n$ is projective. In particular, if $X$ is a scheme, or a Deligne--Mumford stack, so is $X_{D}^{[n]}$.
\end{proposition}

\begin{proof}

{\sc Case 1:} $X= \AA^1$ and  $D= \{0\}$. This case is the restriction of Lemma \ref{Lem:basic-pairs} to the open subscheme $(\AA^1)^n\subset (\PP^1)^n$, and the analog of Lemma \ref{Lem:config-open} holds.

Note that in this case there is an action of $\GG_m$ on $\AA^1$ fixing $\{0\}$. This induces an action of
$\GG_m^{n}$ on $X_{D}^{[n]} \to X^n$.

{\sc Case 2:}  $X= [\AA^1/\GG_m]=\cA$ and $D= [\{0\}/\GG_m]$.  As in the proof of Proposition \ref{Prop:conf-deg}, this is the quotient of the previous case by the action of $\GG_m^{n}$.
Projectivity is preserved, since again the relatively ample line bundle admits an  equivariant structure.

This is a universal case, the resulting  configuration stacks are denoted by $\cA_{\cD}^{[n]}\to \cA^n$.

{\sc General case:} The pair $(X,D)$ defines a smooth morphism $X \to \cA$. Composing any family of stable expanded configurations of degree $n$ on $(X,D)$ with this structure morphism gives a family of stable configurations of $(\cA,\cD)$, hence we get a natural morphism   $$X_{D}^{[n]} \to X^n \times_{\cA^n} \cA_{\cD}^{[n]}.$$ This is an isomorphism exactly as in Proposition \ref{Prop:conf-deg}.
\end{proof}

\section{Properness of the stack of predeformable maps}\label{Sec:proper}

Defining numerical invariants via virtual classes requires constructing a suitable moduli stack of DM type, proving its properness and constructing a perfect obstruction theory.

For instance if $\pi:W\to B$ is a smooth projective morphism, with $B$ an affine scheme, then the connected components of the stack $K(W)$ of stable maps to the fibers of $\pi$ are proper over $B$ and carry a relative perfect obstruction theory; this allows to define GW invariants and prove their invariance under deformations. Properness of $K(W)\to B$ holds under the weaker assumption that $W\to B$ be flat and projective; however, in this case the natural obstruction theory is not perfect.

Both the properness results and the construction of perfect obstruction theories are the key technical core of Jun Li's statement and proof of the degeneration formula for GW invariants in the language of algebraic geometry. In this section we want to give an alternative proof of the properness part, extending it to the case of DM stacks; we describe the set-up following the notation of \cite[Definitions 3.1.4, 3.2.9, C.1.6]{AF}, where we combine the cases of pairs and degenerations.

\subsection{Notation and background}

We let $W\to B$ be either one of the cases of interest: a degeneration in the sense of Convention \ref{degeneration_conv}, or a first-order smoothable variety $W\to B=\{b_0\}$ as in Section \ref{Sec:other}, or the trivial map from $X$ to a point in the context of Convention \ref{pairs_conv}; the notation $K(W)$ denotes the algebraic stack of stable maps to the fibers of $W\to B$.  {The {\em special locus} $\cW^{\spe}$ of $\cW$ is either  $\cW^{\spe} = \cW^\sing$ the singular locus for a degeneration, or $\cW^{\spe} = \cW^\sing\cup D$ in the case of a pair.}

We use $T$ to denote the corresponding stack of expansions:  either the stack of expanded degenerations $\fT_B$ in the first two cases,  or the stack of expanded pairs $\cT$; we have a natural morphism $T\to B$ and a universal expansion $W' \to W\times_BT$. If $P_j$ is an exceptional components in an expansion, we call its {\em interior} the complement $P_j^o$ of {$D_j^+\cup D_j^-$}.

We denote by $\fK$ the stack of twisted stable maps to the fibers of $W' \to T$;
while our target is allowed to be a Deligne Mumford stack, we keep the terminology lighter by always using the shorter term {\em stable maps} to refer to either stable maps in case $W$ is a scheme, or {\em twisted stable maps} as in \cite{AV} in case it is a stack.

A point of $\fK$ is a tuple $(C,\Sigma, \cW, f)$ where $\cW\to W$ is a point in $T$, and $(C,\Sigma, f)$ is a stable map to $\cW$, where $\Sigma$ is the ordered set of marked points. Such a stable map is called {\em degenerate} if there is an irreducible component $X$ of $C$
mapping to $\cW^\spe$; otherwise it is called {\em non-degenerate}.  A non-degenerate map is called {\em predeformable} if {all points on $C$ mapping to the divisor $D\subset \cW$ are marked, and if } near each point mapping to $\cW^\sing$ the map is given on strict henselizations  by $$(k[x,y,z_1,\ldots,z_d]/xy)^\sh \to (k[u,v]/uv)^\sh\qquad x\mapsto u^r, y\mapsto v^r$$ for some $r>0$. Here and later  {\em sh} means strict henselization and may be thought of as a choice of local analytic coordinates.

We denote by $\fK_{\nd}$ the open substack of $\fK$ whose points are non-degenerate maps, and by $\fK_\pd\subset \fK_{\nd}$ the locally closed substack of predeformable maps, with its reduced substack structure; while in \cite[Pages 541-547]{Li1} the locus $\fK_\pd$ is endowed with a  delicate schematic structure, we do not need it since all we are interested in is properness. Finally, we denote by $K\subset \fK$ the maximal open DM substack, and we let $K_{\nd}:=K\cap \fK_{\nd}$ and $K_\pd:=K\cap \fK_\pd$.

Let $(C,\Sigma, \cW, f)$ be a point of $\fK$; we call a smooth rational irreducible component $X$ of $C$  {\em semistable}, if it maps to a fiber of an exceptional component  $P_j$ of $\cW$, with exactly one point of $X$ meeting each of $D_j^+$ and $D_j^-$, and no other marked or singular points of $C$ on $X$.

 (The terminology in \cite[Proof of Lemma 3.2]{Li1} is {\em trivial component}.)

Semistable components can be used to characterize points of $\fK$ belonging to the Deligne--Mumford locus $K$, see \cite[Lemma 3.2]{Li1}, \cite[Definition 3.1.5]{AF}.

\begin{lemma}\label{pointsincK} A point $(C,\Sigma,\cW,f)$ of $\fK$ is not in the Deligne--Mumford locus $K$ if and only if there is an exceptional component $P_j$  such that every component of $C$ whose image meets  the interior $P_j^o=P_j\setminus (D_j^+\cup D_j^-)$  is a semistable component.
\end{lemma}
\begin{proof}
The following argument can be found in the proof of \cite[Lemma 3.2]{Li1}. A point is in $K$ if and only if no positive dimensional subgroup of the group of automorphisms of the expansion $\cW$ lifts to an automorphism of $C$. The Deligne--Mumford condition is equivalent to ensuring that for each $P_j$ there is at least one component $X$ mapping to $P_j$ to which no covering of the $\GG_m$ action lifts. It is easy to see that the only components whose image meets $P_j^o$ to which the action lifts are exactly the semistable components.
\end{proof}

\begin{notation}\label{notval}
 When we use the valuative criterion, we always take $\Delta=\Spec R$ for $R$ a discrete valuation ring, with generic point $\eta$ and closed point $s$.

 Given a commutative diagram $$
\xymatrix{
\eta\ar[r]^{f_\eta}\ar[d] &X\ar[d]^p\\
\Delta\ar[r]_{g} & Y}
$$
we will view liftings of $g$ to $X$ as a groupoid whose objects are the two-commutative diagrams
$$
\xymatrix{
\eta\ar[r]^{f_\eta}\ar[d] &X\ar[d]^p\\
\Delta\ar[ru]^{f}\ar[r]_{g} & Y}
$$
and whose morphisms $f\Rightarrow f'$ are those sending the two-commutative diagram of $f$ to that of $f'$. We will sometimes write ``there is a unique lifting with certain properties'' to mean ``there exists a lifting with certain properties, and it is unique up to unique isomorphism''.
\end{notation}

\subsection{Statement and proof}

The aim of this section is to prove the following theorem:

\begin{theorem}\label{propermappd} The  morphism $K_{pd}\to K(W)$ induced by $K_{pd} \subset \fK$ is proper.
\end{theorem}

\begin{lemma}\label{predefisclosed} The  substack  of predeformable maps

 is closed in $\fK^u_{\nd}\subset \fK^u$, the open locus of nondegenerate maps.
\end{lemma}
\begin{proof} This is a local statement in the \'etale topology; hence we can use the proof given by Jun Li, see \cite[Lemma 2.7]{Li1}.
\end{proof}

\begin{lemma}\label{Lem:chain} Let $(C,\Sigma, f:C\to\cW)\in \fK$ be a point in the closure of $\fK_\pd$; denote $(\bar C,\bar \Sigma,\bar f)$ its image in  $K(W/T)$, i.e. the stabilisation of the composition of $f$ with the structure morphism $\cW\to W$. Then any connected component $Z$ of the locus in $C$ contracted in $\bar C$ is a chain of rational curves which maps to either a marked or singular point of $\bar C$.
\end{lemma}
\begin{proof} Each contracted component must be rational and map to a fiber of $\cW$ to $W$, hence the second part of the statement is obvious. For the first we must prove that $Z$ has no an end component $X$, i.e. one having only one node and no marked point.

We argue by contradiction: such an end component cannot be contracted in $\cW$ since $f$ is stable, so it  must map nontrivially to some fiber in an exceptional component of $\cW$; since each fiber intersects {$\cW^{\spe}$} in two points, there is a {non-marked} point $x$ on $X$ which is smooth on $C$ but maps to $\cW^\spe$.

We use again Jun Li's argument, since this is an {\'etale local computation. In case $x\in D$, then  in a neighborhood of $x$ the divisor $D$ intersects every nearby fiber. So $x$ is in the closure of the locus of marked points and hence it is marked, contradicting the assumption.  Now consider the case when $x$ maps to $\cW^\sing$.} By assumption there exist  a family $$(f_\Delta:C_\Delta\to\cW_\Delta , \Sigma_{\Delta})\in\fK(\Delta)$$ such that $(f_\eta:C_\eta\to\cW_\eta , \Sigma_{\eta})\in \fK_\pd$ and $$(f_s:C_s\to\cW_s , \Sigma_{s}) = (f:C\to\cW , \Sigma).$$ The problem is local at  $z$ so we may assume $$\cW_\Delta^\sh= \left(\Spec R[u,v,w_1,\ldots,w_m]/(uv-a)\right)^\sh$$ for some $a\in R$,
$$C_\Delta^\sh = \left(\Spec R[x]\right)^\sh,$$ and on the central fiber
$$f^*u=0, \quad f^*v = x^r.$$

Consider the homomorphism $f_\Delta^*:R[u,v,w_i]/(uv-a)\to R[x]$ and let $u_\Delta,v_\Delta$ be the images of $u$ and $v$; they must satisfy $u_\Delta v_\Delta=a$, and be equal to $(0,x^r)$ modulo $m_s$. In particular $v_\Delta$, viewed as a polynomial in $x$, has positive degree; therefore the only possibility that its product with $u_\Delta$ has degree zero is that $u_\Delta=a=0$, which means that $f_\eta$ is not predeformable.
\end{proof}

\begin{lemma}\label{propfp}
The commutative diagram $$\xymatrix{
\fK \ar[r]\ar[d] & T\ar[d]\\
K(W) \ar[r] & B
}$$
induces a proper morphism  $\fK \to K(W)\times_B   T$.
\end{lemma}
\begin{proof}
The fibered product is $\fK(W_T/T)$ where $W_T:=W\times_B T$. The structure map $\cW\to W_T$ induces the morphism from $\fK$ to the fiber product, and it is proper by  \cite[Corollary 9.1.3]{AV}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{propermappd}]
To prove properness of the morphism $K_{pd}\to K(W)$ we use the valuative criterion.  We will fix from now on a commutative diagram
\begin{equation}\label{Eq:set-up}\xymatrix{
\eta \ar[r]\ar[d] & K_{pd}\ar[d]\ar[r] & T\ar[d]\\
\Delta\ar[r] &K(W)\ar[r] & B,}
\end{equation}
and our aim will be showing that, after a base change $\tDelta\to\Delta$, there is a unique lifting of $\Delta\to K(W)$ to $\tDelta\to K_{pd}$.
We will denote by $(\bar C, \bar \Sigma, \bar f)$ the family of stable maps over $\Delta$ corresponding to $\Delta\to K(W)$ and by $(C_\eta,\Sigma_\eta,f_\eta:C_\eta\to \cW)$ the family corresponding to $\eta\to K_{pd}$.

From Lemma \ref{propfp} it follows that
 there exists an equivalence, compatible with base change,  between liftings of $\Delta\to B$ to $T$ and liftings, up to base change $\tDelta\to\Delta$, of $\Delta\to K(W)$ to $\fK$. Given a lifting $a:\tDelta \to T$, we will denote by $(C^a,\Sigma^a, f^a:C^a\to\cW)$ the family of stable maps corresponding to the lifting $\tDelta\to \fK$ induced by $a$.

The key to producing a lifting   of $\Delta\to B$ to $T$ is to use an auxiliary choice of a stable configuration of points as a guide.

\begin{proposition}\label{Prop:sections}
 After a base change  $\tDelta\to\Delta$, there exist a positive integer $N$ and  closed subschemes $(\bar p_1,\ldots,\bar p_N)=:\bar P\subset \bar C$ such that\begin{enumerate}
 \item the induced morphisms $\bar p_i\to \tDelta$ are isomorphisms;
\item each $\bar p_i$ is contained in the smooth, {unmarked} locus of  $\bar C$;
\item $\bar P_\eta$ lifts uniquely to $P_\eta\subset C_\eta$;
\item every irreducible component of $\bar C_s$ intersects $\bar P_s$.
\end{enumerate}
\end{proposition}
\begin{proof} {Write $\bar C^*$ for the open dense substack obtained by deleting the nodal and marked locus of $\bar C$. The inverse image of $\bar C^*$ in $C$ is isomorphic to $\bar C^*$, so once we construct $P$ inside $\bar C^*$,  condition (3) becomes a consequence of (2). Now $\bar C^*$ is a smooth representable curve over $\Delta$.  For each irreducible component  $X_{i}\subset \bar C^*_s$ let $\bar p_{i,s}$ be a closed point and  {$U_{i}\subset \bar C^*$ an affine neighborhood such that $U_i \cap \bar C^*_s\subset X_i$. There exists an element  $h_{s} \in \Gamma(\cO_{U_{i}\cap \bar C^*_s})$} vanishing to order 1 on  $\bar p_{i,s}$. Let $h$ be a lift of $h_s$ in $\Gamma(\cO_{U_{i}})$, and $H$ its zero locus.  Then $H \to \Delta$ is quasi-finite, and replacing $\Delta$ by the localization of $H$ at $\bar p_{i,s}$ we get a section of $U_i \to \Delta$ meeting $X_{i}$. Denoting its image in $\bar C$ by $\bar p_i$ and repeating this for all components we obtain the required set of sections $P$.
}

\end{proof}

We now replace $\Delta$ by $\tDelta$, so we may assume sections as in the proposition exist over $\Delta$.

\begin{proposition}\label{uniqstbl}
\begin{enumerate}
\item The morphism $\eta\to \cW^N$ induced by $f_\eta$ and $P_\eta\subset C_\eta$ defines a stable configuration, and hence a morphism $\eta\to W^{[N]}_\pi$ which lifts the morphism $\Delta\to W^N$ given by $\bar f$ and $\bar P$;
\item There exists a unique lifting of $\Delta\to B$ to $a_0:\Delta\to T$ such that the induced map $s\to \cW^{[N]}$  is a stable configuration.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) If $X\subset C_\eta$ is an irreducible component then $X$ contracts to a point in $\bar C_\eta$ if and only if it contains no $p_i$, if and only if it is semistable. The result then follows from Lemma \ref{pointsincK}.

(2) This follows from the fact that $W^{[N]}_\pi\to W^N_B$ is projective by Proposition \ref{Prop:conf-deg}, hence proper and representable.
\end{proof}

\begin{proposition}\label{ndifstable}
The lifting  $(C^{a_0}_s,\Sigma^{a_0}_s,f^{a_0}_s)\in \fK$ corresponding to $a_0$ lies in $K_\pd$.
\end{proposition}
\begin{proof}
Since $K_\pd$ is closed in $K_{\nd}$ it suffices to show that  $(C^{a_0}_s,\Sigma^{a_0}_s,f^{a_0}_s)\in K_{\nd}$, and since  $K_{\nd} =  K\cap \fK_{\nd}$ it suffices to show that is in both $K$ and $\fK_{\nd}$.

{\sc Step 1}: $(C^{a_0}_s,\Sigma^{a_0}_s,f^{a_0}_s)\in \fK_{\nd}.$
 We need to show that no component of $C^{a_0}$ maps to {$\cW^\spe$}. These components come in two types: those contracted in $\bar C_s$ and those that are not.

{\sc Step 1a}:  components contracted in $C^{a_0}_s\to \bar C_s$. By Lemma \ref{Lem:chain},
any connected component of the locus in $C^{a_0}_s$ contracted in $\bar C_s$ is a chain of rational curves which maps to either a marked or singular point of $\bar C_s$. But in a chain of rational curves over a point in $\bar C$ every component maps to a fiber of $\cW \to W$, so a component mapping to $\cW^\spe$  maps to a point in $\cW$, contradicting stability.

{\sc Step 1b}:   consider a component $X_i\subset \bar C^*_s$. Then $X_i$ is canonically a component of $(C^{a_0})^*_s$  using notation as in the proof of Proposition \ref{Prop:sections}. Since $\bar p_i$ meets $X_i$ and $\bar p_i$ does not land in $\cW^\spe$, it follows that neither does $X_i$.

{\sc Step 2}: It remains to prove that $(C^{a_0}_s,\Sigma^{a_0}_s,f^{a_0}_s)\in K$. Since the map  $s\to \cW^{[N]}$  induced by $a_0$ is stable, the configuration $(p_{i,s})$ in $\cW$ is stable. So every exceptional component of $\cW$ has at least one $f^{a_0}_s(p_i)$ in its interior some component $X_i \subset \bar C^*_s$. Since these are not semistable components, we have that $(C^{a_0}_s,\Sigma^{a_0}_s,f^{a_0}_s)\in K$, as required.

\end{proof}

\begin{proposition}\label{stblifnd}
Let $a:\Delta\to T$ be any lifting of $\Delta\to B$. If  $(C^{a}_s,\Sigma^{a}_s,f^{a}_s)\in K_\pd$ then $a$ is isomorphic to $a_0$.
\end{proposition}

\begin{proof}
By properness of $W^{[n]}_\pi$, it suffices to show that $f^{a}_s(P)$ is a stable configuration. This means that every exceptional component of $\cW$ contains a point of $f^a_s(P)$ in its interior, and that $f_s^a(P)$ is disjoint from $\cW^\spe$.

Since the inverse image of $\cW^\spe$ is contained in its nodes and marked points, and since $P$ is chosen disjoint from those, we have that $f_s^a(P)$ is disjoint from $\cW^\spe$. On the other hand, by Lemma \ref{pointsincK}, every exceptional component of $\cW$ contains in its interior the image of some component $X_i\subset \bar C^*_s$. Since $X_i$ contains a point of $P$, every exceptional component of $\cW$ contains in its interior the image of a point of $P$, as required.

\end{proof}

To conclude the proof of  Theorem \ref{propermappd}
we use the valuative criterion of properness.

It is enough to show that there exists a unique lifting $a$  of $\Delta$ to $T$ such that the induced point $(C^{a},\Sigma^{a}, f^{a})$ is contained  in $K_\pd$.
 Hence it is enough to prove that a lifting $a:\Delta \to T$ induces a map to $K_\pd$ if and only if it is isomorphic to $a_0$, where $a_0$ is defined in Part (2) of Proposition \ref{uniqstbl}.  This is given by Propositions \ref{stblifnd} and  \ref{ndifstable}.
\end{proof}

\begin{corollary}\label{C:pd-proper} Under the assumptions for this section, assume moreover that $W\to B$ is proper and has projective coarse moduli space. Then $K_{pd}$ is proper over $B$.
\end{corollary}
\begin{proof}
This follows since by \cite[Theorem 1.4.1]{AV} the stack $K(W)$ is proper.\end{proof}