text:
\chapter{Sets of finite perimeter in some infinite-dimensional examples }

In this chapter, we study some infinite-dimensional examples of Dirichlet spaces $X$, associated sets of finite perimeter and discuss the connection with $\mathbf{B}^{1,1/2}(X)$. Our goal is not to study the general theory of Besov spaces related to infinite-dimensional Dirichlet spaces, but rather to look at some specific examples.

\

 In many infinite-dimensional examples, coming back to the original ideas by De Giorgi \cite{DG} it seems interesting to take advantage of the existence of nice integration by parts formulas to construct BV functions and sets of finite perimeter. To provide the reader with a motivation, we briefly present below those ideas in a general setting, avoiding at first the exact assumptions and technicalities associated with a possible infinite-dimensional setting.

\

Let $(X,\mu,\mathcal{E},\mathcal{F}=\mathbf{dom}(\mathcal{E}))$ be a quasi regular local symmetric Dirichlet space. We assume that $\mathcal{E}$ can be written as
\[
\mathcal{E}(f,f)=\| Df \|_\mathcal{H}^2 ,
\]
where $D$ is a closed unbounded operator from $L^2(X,\mu)$ to some (separable) Hilbert space $\mathcal{H}$. In many situations (see for instance \cite{Ebe99}, {Theorem 3.9} and the examples below), we are able to think of $\mathcal{H}$ as a $L^2$ space of sections of a vector bundle over $X$: more precisely, $\mathcal{H}$ is isometrically isomorphic to some $ \int_{X}^\oplus \mathcal{H}_x \ d\mu(x)$ and for any $\eta_1,\eta_2\in \mathcal{H}$,
\[
\langle\eta_1,\eta_2\rangle_\mathcal{H} = \int_{X} \langle \eta_1,\eta_2\rangle_{\mathcal{H}_x} \ d\mu(x).
\]
In this situation, following De Giorgi in \cite{DG}, for $f \in L^1(X,\mu)$ it is then natural to define:
\[
\mathbf{Var} f = \sup\left\{ \int_X f D^* \eta d\mu
\,
:
\,
 \eta \in \mathbf{dom} (D^*), \| \eta \|_{\mathcal{H}_x} \leq 1, \mu \text{-almost everywhere} \right\},
\]
where $D^*$ is the adjoint operator of $D$ and one defines then:
\[
BV(X) := \left\{f\in L^1(X,\mu)\, :\,
\mathbf{Var} f < +\infty \right\}.
\]
This allows one to define the perimeter of a measurable set $E\subset X$ with $\mathbf 1_E\in BV(X)$ by
\[
P(E,X) = \mathbf{Var} \mathbf{1}_E.
\]
When $X$ is a complete Riemannian manifold and $D=\nabla$ is the Riemannian gradient, this construction yields the usual $BV$ space and sets with finite perimeters are Caccioppoli sets.

In the following examples, the strong Bakry-\'Emery condition is satisfied
\[
\| DP_t f \|_{\mathcal{H}_x} \le (P_t \| Df \|_{\mathcal{H}.})(x).
\]
and one shall prove:

\begin{theorem}
Let $X$ be one of the Dirichlet spaces studied in Sections \ref{Wiener} or \ref{gibbs}. Let $E$ be a measurable set with finite measure. The following are equivalent:

\begin{enumerate}
\item $P(E,X) <+\infty$;
\item The limit $\lim_{t \to 0^+} \int_X \| DP_t \mathbf{1}_E \|_{\mathcal{H}_x} d\mu(x)$ exists.
\end{enumerate}
Moreover, if $E$ is a set satisfying one of the above conditions, then $\mathbf{1}_E \in \mathbf{B}^{1,1/2}(X)$ and
\[
\| \mathbf{1}_E \|_{1,1/2}\le 2 \sqrt{2} P(E,X) =2 \sqrt{2} \lim_{t \to 0^+} \int_X \| DP_t \mathbf{1}_E \|_{\mathcal{H}_x} d\mu(x).
\]
\end{theorem}

The theorem is stated in a class of specific examples, but the method which is developed is actually relatively general.

\section{Wiener space}\label{Wiener}

In this section, we discuss sets of finite perimeters in abstract Wiener space, The theory of BV functions is already very well established and understood in that case (see Fukushima-Hino \cite{FuHi}, and also the survey \cite{MNP} and the references therein).

Let $(X,\mathcal{H},\mu)$ be an abstract Wiener space in the sense of L. Gross \cite{Gross}, so $X$ is a separable real Banach space, $\mathcal{H}$ is a separable real Hilbert space continuously and densely embedded into $X$ and $\mu$ is a Gaussian measure satisfying for $l \in X^* \subset \mathcal{H^*} \simeq \mathcal{H}$,
\[
\int_X \exp \left( i \langle l , x \rangle \right) \mu(dx)=\exp \left(-\frac{1}{2} \| l \|^2_{\mathcal{H}} \right).
\]
We define the Ornstein-Uhlenbeck semigroup on $L^2(X,\mu)$ by the formula:
\begin{align}\label{OU semigroup}
P_t f(x) =\int_X f \left( e^{-t} x +\sqrt{1-e^{-2t} }y \right) d\mu(y), \quad f \in L^2(X,\mu), x \in X.
\end{align}
The associated Dirichlet form may be written in terms of the Malliavin derivative. We now describe this Dirichlet form. We say that a functional $F: X \to \mathbb{R}$ is a smooth cylindric functional if, for some $n$, it can be written in the form
\[
F(x)=f\left( \langle x, l_1 \rangle, \cdots, \langle x ,l_n \rangle \right),
\]
where $f$ is smooth and bounded, and $l_1,\cdots, l_n \in \mathcal{H}^*$ . Here $\langle x, l\rangle$ stands for the Wiener integral. The set of smooth cylindric functionals will be denoted by $\mathcal{C}$ and will be used as nice core algebra. Note that $\mathcal{C}$ is dense in $L^2(X,\mu)$. If $F \in \mathcal{C}$, one defines its Malliavin derivative by the formula,
\begin{align}\label{Malliavin derivative}
DF(x)=\sum_{i=1}^n \frac{\partial f}{\partial x_i} \left( \langle x, l_1 \rangle, \cdots, \langle x ,l_n \rangle \right) l_i \in \mathcal{H}.
\end{align}

We can then consider the pre-Dirichlet form $(\mathcal{E}, \mathcal{C}, \mu)$ given for $F,G \in \mathcal{C}$ by
\[
\mathcal{E} (F,G)=\int_X \langle DF (x), DG(x) \rangle_\mathcal{H} d\mu(x).
\]
The closure of this pre-Dirichlet form yields a quasi-regular local symmetric Dirichlet space $(X,\mathcal{F}, \mu)$ whose semigroup is precisely the Ornstein-Uhlenbeck semigroup (see \cite{Kusuoka}).

Note that the Dirichlet form $\mathcal{E}$ admits a carr\'e du champ and that one has for $f \in \mathcal{F}$, and $x \in X$,
\[
\Gamma(f)(x)= \| Df (x) \|_\mathcal{H}^2.
\]
where $D$ stands for the closed extension of the formula \eqref{Malliavin derivative}. One can check directly from the formula \eqref{OU semigroup} that for $f \in \mathcal{C}$,
\[
DP_t f (x) = \vec{P}_t D f (x)
\]
where $\vec{P}_t $ is a semigroup on $L^2 (X , \mathcal{H} ,\mu)$ that satisfies for every $\eta \in L^2 (X , \mathcal{H} ,\mu)$,
\begin{align}\label{BE vector}
\| \vec{P}_t \eta (x) \|_{\mathcal{H}} \le e^{-t} (P_t \| \eta \|_\mathcal{H}) (x)
\end{align}
In particular, one has the strong Bakry-\'Emery curvature condition:
\[
\| DP_t f (x) \|_\mathcal{H} \le (P_t \| Df \|_{\mathcal{H}})(x), \quad f \in \mathcal{F}.
\]
We are now ready to define sets of finite perimeter, following Fukushima-Hino \cite{FuHi}.

One can think of $D$ as a closed unbounded operator $L^2(X,\mu) \to L^2 (X , \mathcal{H} ,\mu)$. Denote its adjoint by $D^*$ (this is the so-called Skorohod integral).
If $E \subset X$ is a measurable set,
\[
P(E,X) = \sup\left\{ \int_E D^* \eta d\mu\, :\,
\eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1, \mu \text{-almost everywhere} \right\},
\]
 and we say that $E$ has a finite perimeter if $P(E,X) <+\infty$.
As everywhere in the work, for $f \in L^p(X,\mu)$, we define:
\[
\| f \|_{p,\alpha}= \sup_{t >0} t^{-\alpha} \left( \int_X P_t (|f-f(y)|^p)(y) d\mu(y) \right)^{1/p}.
\]
and
\[
\mathbf{B}^{p,\alpha}(X)=\{ f \in L^p(X,\mu)
\,
:
\,
 \| f \|_{p,\alpha} <+\infty \}.
\]

The theorem is the following:

\begin{theorem}\label{Set Wiener}
Let $E$ be a measurable set. The following are equivalent:

\begin{enumerate}
\item $P(E,X) <+\infty$;
\item The limit $\lim_{t \to 0^+} \int_X \| DP_t \mathbf{1}_E (x) \|_{\mathcal{H}} d\mu(x)$ exists.
\end{enumerate}
Moreover, if $E$ is a set satisfying one of the above conditions, then $\mathbf{1}_E \in \mathbf{B}^{1,1/2}(X)$ and
\[
\| \mathbf{1}_E \|_{1,1/2}\le 2 \sqrt{2} P(E,X) =2 \sqrt{2} \lim_{t \to 0^+} \int_X \| DP_t \mathbf{1}_E \|_{\mathcal{H}_x} d\mu(x).
\]
\end{theorem}

The fact that (1) is equivalent to (2) is actually already essentially known (see \cite{MNP}), but we give here a proof that immediately adapts to the case considered in the following section. The proof will be divided in several lemmas.

\begin{lemma}
Let $E$ be a measurable set. If the limit $$\lim_{t \to 0^+} \int_X \sqrt{\Gamma(P_t \mathbf{1}_E) } d\mu(x)$$ exists then $\mathbf{1}_E \in \mathbf{B}^{1,1/2}(X)$ and
\[
\| \mathbf{1}_E \|_{1,1/2} \le 2 \sqrt{2} \lim_{t \to 0^+} \int_X \sqrt{\Gamma(P_t \mathbf{1}_E) } d\mu(x).
\]
\end{lemma}

\begin{proof}
The strong Bakry-\'Emery estimate implies that for every $f \in \mathcal{F} $ and $t \ge 0$, one has $\mu$-almost everywhere
\[
\| P_t f -f \|_1 \le \sqrt{2t} \int_X \sqrt{\Gamma(f)} d\mu.
\]
Let $E$ be a measurable set such that $\lim_{t \to 0^+} \int_X \sqrt{\Gamma(P_t \mathbf{1}_E) } d\mu(x)$ exists. One has for every $s \ge 0$,
\[
\| P_{t+s} \mathbf{1}_E -P_s\mathbf{1}_E \|_1 \le \sqrt{2t} \int_X \sqrt{\Gamma(P_s \mathbf{1}_E)} d\mu.
\]
Taking the limit when $s \to 0^+$ yields,
\[\
\| P_{t} \mathbf{1}_E -\mathbf{1}_E \|_1 \le \sqrt{2t} \lim_{s\to 0^+} \int_X \sqrt{\Gamma(P_s \mathbf{1}_E)} d\mu
\]
and so $\mathbf{1}_E \in \mathbf{B}^{1,1/2}(X)$ with $$ \| \mathbf{1}_E \|_{1,1/2} \le 2 \| P_{t} \mathbf{1}_E -\mathbf{1}_E \|_1 \le 2 \sqrt{2} \lim_{t \to 0^+} \int_X \sqrt{\Gamma(P_t \mathbf{1}_E) } d\mu(x).$$
\end{proof}

\begin{lemma}
For every $\eta \in L^2 (X , \mathcal{H} ,\mu)$, and $t>0$, $\vec{P}_t \eta \in \mathbf{dom} (D^*)$.
\end{lemma}

\begin{proof}
For $f \in \mathcal{F}$ and $\eta \in L^2 (X , \mathcal{H} ,\mu)$, one has
\[
\int_X \langle D f (x) , \vec{P}_t \eta (x)\rangle_\mathcal{H} d\mu(x)= \int_X \langle \vec{P}_t D f (x) , \eta (x)\rangle_\mathcal{H} d\mu (x)=\int_X \langle DP_t f (x) , \eta (x)\rangle_\mathcal{H} d\mu (x).
\]
Since the operator $DP_t$ is bounded in $L^2$, the conclusion follows.
\end{proof}

\begin{lemma}
If $f \in \mathcal{F}$, then
\[
 \sup\left\{ \int_X f D^* \eta d\mu\, :\,
  \eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1, \mu \text{-a.e.} \right\}=\int_X \| Df (x) \|_\mathcal{H} d\mu(x).
\]
\end{lemma}

\begin{proof}
Let $f \in \mathcal{F}$. It is clear that for $\eta \in \mathbf{dom} (D^*)$ such that $ \| \eta (x) \|_{\mathcal{H}} \leq 1$, one has
\[
\int_X f D^* \eta d\mu =\int_X \langle Df , \eta \rangle_\mathcal{H} d\mu \le \int_X \| Df \|_\mathcal{H} d\mu
\]
We now prove the converse inequality. Let $f \in \mathcal{F}$. Let $s, \ve >0$ and consider
\[
\eta= \vec{P}_s \left( \frac{Df}{\| Df \|_\mathcal{H} +\ve} \right).
\]
From the previous lemma $ \eta \in \mathbf{dom} (D^*) $ and from \eqref{BE vector}, one has
\[
\| \eta (x) \|_\mathcal{H} \le 1, \quad \mu \text{-a.e.}
\]
Moreover,
\[
\int_X f D^* \eta d\mu=\int_X \left\langle Df, \vec{P}_s \left( \frac{Df}{\| Df \|_\mathcal{H} +\ve} \right)\right\rangle d\mu.
\]
Letting $s \to 0^+$ and $\ve \to 0$ yields
\[
\int_X \| Df \|_\mathcal{H} d\mu \le \sup\left\{ \int_X f D^* \eta d\mu
\,
:
\,
\eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1, \mu \text{-a.e.} \right\}.
\]
\end{proof}

We are now ready to prove Theorem \ref{Set Wiener}.

\begin{proof}
It just remains to prove that 1 is equivalent to 2.

Let $E$ be a measurable set and $\eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \text{ a.e.}$. We have
\[
\int_X (P_t \mathbf{1}_E) D^* \eta d\mu=\int_X \langle DP_t \mathbf{1}_E, \eta \rangle_\mathcal{H}d\mu \le \int_X \| DP_t \mathbf{1}_E \|_\mathcal{H}d\mu.
\]
Since we have in $L^2$, $\lim_{t \to 0} P_t \mathbf{1}_E=\mathbf{1}_E$, one deduces
\[
\int_E D^* \eta d\mu \le \liminf_{t \to 0^+}\int_X \| DP_t \mathbf{1}_E \|_\mathcal{H}d\mu.
\]
Thus,
\[
\sup\left\{ \int_E D^* \eta d\mu\, :\,
 \eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \right\} \le \liminf_{t \to 0^+}\int_X \| DP_t \mathbf{1}_E \|_\mathcal{H}d\mu.
\]
On the other hand, again if $E$ be a measurable set and $\eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \text{ a.e.}$, then we have:
\[
\int_X (P_t \mathbf{1}_E) D^* \eta d\mu=\int_X \mathbf{1}_E P_t D^* \eta d\mu=\int_X \mathbf{1}_E D^* \vec{P}_t \eta d\mu \le \sup\left\{ \int_E D^* \eta d\mu   \, :\,
 \eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \right\},
\]
where in the last inequality we used the fact that $ \| \vec{P}_t \eta \|_\mathcal{H} \le 1$. Since from the previous lemma we know that
\[
\sup\left\{\int_X (P_t \mathbf{1}_E) D^* \eta d\mu
\,
:
\,
\eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \right\} =\int_X \| DP_t \mathbf{1}_E (x) \|_\mathcal{H} d\mu(x),
\]
and so,
\[
\limsup_{t \to 0^+}\int_X \| DP_t \mathbf{1}_E (x) \|_\mathcal{H} d\mu(x) \le \sup\left\{ \int_E D^* \eta d\mu\,
:
\,
 \eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1 \right\}.
\]
The proof is complete.
\end{proof}

\section{Path spaces with Gibbs measure}\label{gibbs}

In this section, we present new examples of infinite-dimensional Dirichlet spaces for which one can construct sets of finite perimeter as in the Wiener space.
The examples are Dirichlet spaces built on the space of continuous functions $C(\mathbb{R},\mathbb{R}^d)$, where the reference measure is the Gibbs measure associated with the formal Hamiltonian:

\[
\mathcal{H}(w) =\frac{1}{2} \int_\mathbb{R} \| w'(x) \|^2 dx+ \int_{\mathbb{R}} U(w(x)) dx, \quad w \in C(\mathbb{R},\mathbb{R}^d),
\]
where $U:\mathbb{R}^d \to \mathbb{R}$ is an interaction potential.
The corresponding diffusion processes are defined through the Ginzburg-Landau type stochastic partial differential equations:
\begin{align}\label{spde}
dX_t(x)=\left( \frac{1}{2}\frac{d^2}{dx^2} X_t(x) -\nabla U(X_t(x)) \right) dt+dW_t(x), \quad t>0, x \in \mathbb{R},
\end{align}
where $(W_t)$ is a white noise process.

In the sequel we will assume that the interaction potential $U$ satisfies the following conditions:

\begin{enumerate}
\item $U$ is two times continuously differentiable and $\nabla^2 U \ge 0$;
\item There exists $A>0$ and $p>0$ such that for every $x \in \mathbb{R}^d$:
\[
\| \nabla U (x) \| + \| \nabla^2 U(x) \| \le A (1+\| x \|^p);
\]
\item $\lim_{ \| x \| \to +\infty} U(x) =+\infty$.
\end{enumerate}

We now briefly sketch, omiiting some details, the construction of the relevant Dirichlet space. For the details, we refer to Kawabi \cite{Kawabi} or Kawabi-R\"ockner \cite{KR}.

We denote $X=C(\mathbb{R},\mathbb{R}^d)$\footnote{Strictly speaking, the construction needs to be performed on a subspace $X$ of $C(\mathbb{R},\mathbb{R}^d)$.} . We first define a set of smooth cylindric functionals. We say that a functional $F: X\to \mathbb{R}$ is a smooth cylindric functional if, for some $n$, it can be written in the form
\[
F(w)=f\left( \langle w, \phi_1 \rangle, \cdots, \langle w ,\phi_n \rangle \right),
\]
where $f$ is smooth and bounded, and $\phi_1,\cdots,\phi_n$ are smooth and compactly supported. Here, we use the notation $\langle w ,\phi_i \rangle=\int_{\mathbb{R}} \langle w (x) ,\phi_i (x) \rangle dx$. The set of smooth cylindric functionals will be denoted by $\mathcal{C}$ and will be used as nice core, as in the Wiener space example.

We denote by $\mathcal{H}$ the space of square integrable functions $\mathbb{R} \to \mathbb{R}^d$. If $F \in \mathcal{C}$, one defines its derivative by the formula,
\[
DF=\sum_{i=1}^n \frac{\partial f}{\partial x_i}\left( \langle w, \phi_1 \rangle, \cdots, \langle w ,\phi_n \rangle \right) \phi_i \in \mathcal{H}.
\]

Let now $\mu$ be the $U$-Gibbs measure defined by (2.1) in \cite{Kawabi}. Note that $\mathcal{C}$ is dense in $L^2(X,\mu)$. We can then consider the pre-Dirichlet form $(\mathcal{E}, \mathcal{C}, \mu)$ given for $F,G \in \mathcal{C}$ by
\[
\mathcal{E} (F,G)=\int_X \langle DF (w), DG(w) \rangle_\mathcal{H} d\mu(w).
\]
The closure of this pre-Dirichlet form yields a quasi-regular local symmetric Dirichlet space $(X,\mathcal{F}, \mu)$. We shall denote by $\{ P_t,t\ge 0\}$ the semigroup of this Dirichlet form. It is worth noting that $\{ P_t,t\ge 0\}$ is conservative and is indeed the semigroup associated to the spde \eqref{spde} (see \cite{KR}).

Note that the Dirichlet form $\mathcal{E}$ admits a carr\'e du champ and that one has for $f \in \mathcal{F}$,
\[
\Gamma(f)(w)= \| Df (w) \|_\mathcal{H}^2.
\]
In this framework, one has the strong Bakry-\'Emery estimate (see Proposition 2.1 in \cite{Kawabi06}):
\[
\| DP_t f (w) \|_\mathcal{H} \le (P_t \| Df \|_{\mathcal{H}})(w), \quad f \in \mathcal{F},
\]
Actually, according to Lemma 3.1. in \cite{Kawabi}, as in the Wiener space example, for $f \in \mathcal{F}$,
\[
DP_t f (w) = \vec{P}_t D f (w)
\]
where $\vec{P}_t $ is a semigroup on $L^2 (X , \mathcal{H} ,\mu)$ that satisfies for every $\eta \in L^2 (X , \mathcal{H} ,\mu)$,
\begin{align}\label{BE vector2}
\| \vec{P}_t \eta (w) \|_{\mathcal{H}} \le (P_t \| \eta \|_\mathcal{H}) (w).
\end{align}

From there, the situation is exactly similar to the Wiener space case that was studied in the previous section and, if $E \subset X$ is a measurable set, we set
\[
P(E,X) = \sup\left\{ \int_E D^* \eta d\mu
\,
:
\,
 \eta \in \mathbf{dom} (D^*), \| \eta (x) \|_{\mathcal{H}} \leq 1, \mu \text{-almost everywhere} \right\},
\]
 and we say that $E$ has a finite perimeter if $P(E,X) <+\infty$. The same proof that we gave in the Wiener space example works and Theorem \ref{Set Wiener} therefore holds.

config: arxiv


TITLE: Norm of an operator with $L^2$
QUESTION [1 upvotes]: Let $x(t)$ be continuous on [-1,1] in a space with the $L^2$ norm (i.e. $\lVert x \rVert_2=\sqrt{\int_{-1}^{1} \lvert x(t)\rvert^2dt}$). I have to find the norm of the operator $f(x)$, where $f(x)=\int_{-1}^0x(t)dt -\int_{0}^1x(t)dt$.
I suspect $|f|=\sqrt2$, as
$$|f(x)|=\lvert\int_{-1}^0x(t)dt -\int_{0}^1x(t)dt \rvert$$ $$\le \int_{-1}^0 \lvert x(t) \rvert dt +\int_{0}^1 \lvert x(t) \rvert dt $$ $$= \int_{-1}^1 \lvert x(t) \rvert dt$$ $$\le \sqrt{\left( \int_{-1}^1 \lvert 1 \rvert^2 dt\right)}\sqrt{\left( \int_{-1}^1 \lvert x(t) \rvert^2 dt\right)}$$ by the Cauchy-Schwarz inequality for integrals.
$$\implies|f(x)| \le \sqrt2 \lVert x \rVert_2$$
However, I have no idea how to prove that $\sqrt 2$ is the norm by showing it's the infimum. Any suggestions?

REPLY [2 votes]: You can make all your inequalities into equalities if you take $x=\frac1{\sqrt2}\,(1_{[-1,0]}-1_{[0,1]})$. As $\|x\|=1$ and $|f(x)|=\sqrt2$, you get that $\|f\|\geq\sqrt2$. And you have already proven the reverse inequality.
As this $x$ is not continuous, we need to approximate. Namely, we need
$$
x_n(t)=\begin{cases}
1,&\ -1\leq t\leq -1/n\\ \ \\
-nt,&\ -1/n\leq t\leq 1/n\\ \ \\
-1,&\ 1/n\leq t\leq 1
\end{cases}
$$


text:
\section{Characteristics of Floor and Ceiling Function}
Tags: Floor Function, Ceiling Function

\begin{theorem}
Let $f: \R \to \Z$ be an [[Definition:Integer-Valued Function|integer-valued function]] which satisfies both of the following:
:$(1): \quad \map f {x + 1} = \map f x + 1$
:$(2): \quad \forall n \in \Z_{> 0}: \map f x = \map f {\dfrac {\map f {n x} } n}$
Then either:
:$\forall x \in \Q: \map f x = \floor x$
or:
:$\forall x \in \Q: \map f x = \ceiling x$
\end{theorem}

\begin{proof}
From $(1)$, by [[Principle of Mathematical Induction|induction]] we have:
:$\forall n \in \N: \map f {x + n} = \map f x + n$
and
:$\forall n \in\N: \map f {x - n} = \map f x - n$
and therefore, in particular:
:$(3): \quad \forall n \in \Z: \map f n = \map f 0 + n$
From $(2)$, we get
{{begin-eqn}}
{{eqn | l = \map f 0
      | r = \map f {\map f 0}
}}
{{eqn | r = \map f 0 + \map f 0
      | c = using $(3)$ and $f$ being [[Definition:Integer-Valued Function|integer valued]]
}}
{{end-eqn}}
Hence
:$\map f 0 = 0$
Thus from $(3)$ it follows that:
:$\forall n \in \Z: \map f n = n$
Suppose that $\map f {\dfrac 1 2} = k \le 0$.
Then:
{{begin-eqn}}
{{eqn | l = k
      | r = \map f {\dfrac 1 {1 - 2 k} \map f {\dfrac 1 2 - k} }
      | c =
}}
{{eqn | r = \map f {\dfrac 1 {1 - 2 k} \paren {\map f {\dfrac 1 2} - k} }
      | c =
}}
{{eqn | r = \map f 0
      | c =
}}
{{eqn | r = 0
      | c =
}}
{{end-eqn}}
We show that in this case, by [[Principle of Mathematical Induction|induction]]:
:$\map f {\dfrac 1 n} = 0$ for all $n \in \N$
Induction hypothesis:
:$\map f {\dfrac 1 {n - 1} } = 0$
Then from $(1)$:
:$\map f {\dfrac n {n - 1} } = \map f {\dfrac 1 {n - 1} + 1} = 0 + 1 = 1$
so:
{{begin-eqn}}
{{eqn | l = \map f {\dfrac 1 n}
      | r = \map f {\dfrac 1 n \map f {\dfrac n {n - 1} } }
      | c =
}}
{{eqn | r = \map f {\dfrac 1 {n - 1} }
      | c = using $(2)$
}}
{{eqn | r = 0
      | c =
}}
{{end-eqn}}
Finally, we show by induction on $m$ that even:
:$\map f {\dfrac m n} = 0$
for $m \in \set {1, \ldots, n - 1}$.
Above we have shown this for $m = 1$.
Let $1 \le m < n$.
If $m \divides n$, then
:$\map f {\dfrac m n} = \map f {\dfrac 1 {n / m} } = 0$
Otherwise, write:
:$n = \paren {k - 1} m + r$
Then:
:$k > 1$
and:
:$1 \le r \le m - 1$
We therefore have:
:$k m = n + m - r$
so by the induction hypothesis:
:$\map f {\dfrac {k m} n} = \map f {1 + \dfrac {m - r} n} = 1 + \map f {\dfrac {m - r} n} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \map f {\dfrac m n}
      | r = \map f {\dfrac 1 m \map f {\dfrac {m k} n} }
      | c =
}}
{{eqn | r = \map f {\dfrac 1 m}
      | c =
}}
{{eqn | r = 0
      | c =
}}
{{end-eqn}}
By $(1)$, this shows that
:$\map f {\dfrac 1 2} \le 0 \implies \map f x = \floor x$
for all [[Definition:Rational Number|rational]] $x$.
Suppose that $\map f {\dfrac 1 2} = k > 0$.
Then the [[Definition:Integer-Valued Function|integer-valued function]] $g: \R \to \Z$ satisfies:
:$\map g x = -\map f {-x}$
satisfies $(1)$ and $(2)$, and also:
{{begin-eqn}}
{{eqn | l = \map g {\dfrac 1 2}
      | r = 1 - \map f {\dfrac 1 2}
      | c =
}}
{{eqn | o = \ge
      | r = 0
      | c =
}}
{{eqn | ll= \leadsto
      | l = \map f x
      | r = -\map g {-x}
      | c =
}}
{{eqn | r = -\floor {-x}
      | c =
}}
{{eqn | r = \ceiling x
      | c = [[Floor of Negative equals Negative of Ceiling]]
}}
{{end-eqn}}
Thus:
:$\map f {\dfrac 1 2} > 0 \implies \map f x = \ceiling x$
for all [[Definition:Rational Number|rational]] $x$.
{{qed}}
\end{proof}

config: wiki


TITLE: Is a harmonic function in $L^p(\mathbb{R}^2)$ equal to zero?
QUESTION [7 upvotes]: Let $u :\mathbb{R}^2 \to \mathbb{R}$ be a smooth harmonic function ($\Delta u = 0$). Furthermore, suppose that $u \in L^p(\mathbb{R}^2)$, with $1 < p < \infty$. Does it follow that $u = 0$ everywhere?
By the theorem of Liouville, this follows immediately if $u \in L^\infty(\mathbb{R}^2)$. For $u \in L^p(\mathbb{R}^2)$, this would also follow if we knew that $u$, for example, is uniformly continuous ($u$ is necessarily unbounded, find a sequence of points $x_k$ such that $u(x_k) \to \infty$, and apply the definition of uniform continuity in a neighbourhood of those points to get a contradiction with $u \in L^p(\mathbb{R}^2)$).
The enemy seems therefore higher and narrower ``spikes'' going off to infinity. Any help would be much appreciated!

REPLY [5 votes]: This may settle it:
By the mean value property, we have, leting $r >0$,
$$
|u(x)| \leq \frac 1 {\pi r^2} \int_{B_r(x)} |u(y)| \ d y.
$$
Using H\"older on the RHS, we have
$$
\frac 1 {\pi r^2} \int_{B_r(x)} |u(y)| \ d y \leq \frac 1 {{\pi}^{1-1/q}} r^{-2 + 2/q} \|u \|_{L^p(\mathbb{R}^2)}.
$$ Where $1/q + 1/p =1$. Hence $-2+2/q = -2/p$. Then send $r \to \infty$.