text:
\begin{document}
\title{Lorentz--Karamata spaces}
\author{Dalimil Pe{\v s}a}

\address{Dalimil Pe{\v s}a, Department of Mathematical Analysis, Faculty of Mathematics and
Physics, Charles University, Sokolovsk\'a~83,
186~75 Praha~8, Czech Republic}
\email{pesa@karlin.mff.cuni.cz}
\urladdr{0000-0001-6638-0913}

\subjclass[2010]{46E30}
\keywords{Lorentz--Karamata spaces, slowly varying functions, embedding theorems, associate spaces, rearrangement-invariant Banach function norms, quasinormed spaces}

\thanks{This research was supported by the grant no. P201-18-00580S of the Czech Science Foundation and the grant SFG205 of Faculty of Mathematics and Physics, Charles University, Prague.}

\begin{abstract}
In this paper, we consider Lorentz--Karamata spaces with slowly varying fuctions and study their properties.

We consider Lorentz--Karamata functionals over an arbitrary sigma-finite measure space equipped with a non-atomic measure and corresponding Lorentz--Karamata spaces. We characterise non-triviality of said spaces, then study when they are equivalent to a Banach function space and obtain a complete characterisation. We further provide a complete characterisation of embeddings between Lorentz--Karamata spaces. At last, we describe the associate spaces of Lorentz--Karamata spaces in some of the limiting and sublimiting cases.
\end{abstract}

\date{\today}

\maketitle

\makeatletter
   \providecommand\@dotsep{2}
\makeatother

\section{Introduction}

In this paper we focus on one particular scale of function spaces, called Lorentz--Karamata spaces. These function spaces were introduced in 2000 by Edmunds, Kerman and Pick in~\cite{EdmundsKerman00}, and their name reflects the fact that their construction encapsules both the Lorentz-type structure of fine tuning of function spaces and the concept of the so-called slowly-varying functions that had been studied by Karamata. The original motivation for the introduction of these spaces was connected with the investigation of the very important problem of nailing down optimal partner function spaces in Sobolev embeddings on regular domains in the Euclidean space. During the last two decades the Lorentz--Karamata spaces have extended their field of applications for example to Gauss-Sobolev embeddings  in \cite{CianchiPick09}, boundedness of operators on probability spaces in \cite{CianchiPick15} and traces of Sobolev functions in \cite{CianchiPick16}. They were also studied in connection with Bessel potential type spaces in \cite{GogatishviliOpic04} and \cite{Neves02} and their interpolation properties were the focus of \cite{Bathory18}. Their upmost importance seems to consist of the fact that they provide a class of very good and useful examples for various tasks in functional analysis and its applications, which is on one hand quite versatile (note that Lorentz--Karamata spaces contain Lebesgue spaces,  Lorentz spaces, Zygmund classes, Lorentz--Zygmund spaces, a good deal of Orlicz spaces and Marcinkiewicz endpoint spaces, the space of Br\'ezis and Wainger and many more), on the other hand they are relatively easily manageable, which makes them extremely handy.

During the last 20 years, Lorentz--Karamata spaces have been several times put under a detailed scrutiny. For instance, an~alternative characterization by ``norms within norms'' of them is given in~\cite{EdmundsOpic08}. Further characterisation by the means of alternative norms was obtained in \cite{FernandezSignes14}. They are briefly mentioned also in~\cite{FucikKufner13}. A detailed study focused on some of their basic functional properties can be found in \cite{EdmundsEvans04} and \cite{Neves02}. But none of these works provided a completely satisfactory result as they usually contained various restrictions. One of our principal goals is to fill in the gap and provide a comprehensive study of these spaces.

Second of our main goals is to make Lorentz--Karamata spaces less technical. In most instances, see for example \cite{EdmundsEvans04}, \cite{Ho20} and \cite{Neves02}, the definition of Lorentz--Karamata spaces is unnecessarily complicated and furthermore formulated in such a way that the class loses some of its generality, because the definition does not allow for the behaviour of the slowly-varying function around zero being different from that around infinity. Notable exemptions of this rule are the papers \cite{Bathory18} and \cite{GogatishviliOpic04} where the approach si more similar to ours, but \cite{GogatishviliOpic04} does not concern with the properties of the spaces themselves at all while \cite{Bathory18} does so only very briefly --- merely non-triviality and embeddings are treated there. This suboptimal approach to Lorentz--Karamata spaces probably stems from the way they were originally introduced in \cite{EdmundsKerman00}, where this kind of issues did not arise since the spaces were considered only in the case when the underlying measure space was of finite measure. We get rid of this issue by providing a definition which is both simpler and more general. In the light of this less restrictive approach, we elected to provide proofs for all of our results in order not to force on the reader the duty of going through the existing proofs and checking their validity in this different setting.

We would like to point out at this point that even though the Lorentz--Karamata spaces are a fairly general class of function spaces, they are further generalised by the classical Lorentz spaces, which have been studied quite extensively since their introduction by Lorentz in \cite{Lorentz51}. In some cases we use in our proofs this more abstract theory of classical Lorentz spaces, namely the results contained in \cite{CarroPick00}, \cite{CarroSoria96}, \cite{CarroRaposo07}, \cite{CarroSoria93}, \cite{GogatishviliSoudsky14}, \cite{Sawyer90} and \cite{Stepanov93}, while in other cases we opted for a more elementary approach. As a further reading on the topic we would recommend \cite{ArinoMuckenhoupt90}, \cite{CarroGogatishvili08}, \cite{CarroSoria97}, \cite{GogatishviliKrepela17}, \cite{GogatischviliPick03}, \cite{GogatishviliPick06}, \cite{Lorentz61} and \cite{Sinnamon02}.

The paper is structured as follows. We first provide the necessary theoretical background in Section~\ref{SP}. We then provide the comprehensive treatment of Lorentz--Karamata spaces in Section~\ref{SMR}.

To be more specific, we first provide in Section~\ref{SSSV} a definition of slowly varying functions and state some of their important properties. We then define the Lorentz--Karamata spaces in Section~\ref{SSLK} and prove some of their basic properties. In Section~\ref{SSE} the embeddings between Lorentz--Karamata spaces are completely characterised. The equivalence of Lorentz--Karamata spaces defined using the non-increasing rearrangement and those defined using the maximal function  is studied in Section~\ref{SSQN} where we obtain a complete characterisation. In Section~\ref{SSLKAS} we describe the associate spaces of Lorentz--Karamata spaces for most choices of parameters with a notable exemption of one sublimiting case. Finally, in Section~\ref{SSBFS} we provide a complete characterisation of the cases when a Lorentz--Karamata space is a Banach function space.

We would like to emphasize that we make no assumptions on the finiteness of the underlying measure space.

\section{Preliminaries}\label{SP}

The aim of this section is to establish the basic framework for later work. We strive to keep the definitions and notation as standard as possible.

From now on, we will denote by $(R, \mu)$, and sometimes $(S, \nu)$, some arbitrary sigma-finite measure space. When $E \subseteq R$, we will denote its characteristic function by $\chi_E$. The set of all extended complex-valued  $\mu$-measurable functions defined on $R$ will be denoted by $M(R, \mu)$, its subsets of all non-negative functions and of all functions finite $\mu$-almost everywhere on $R$ will be denoted by $M_+(R, \mu)$ and $M_0(R, \mu)$ respectively. As usual, we identify functions that are equal $\mu$-almost everywhere. We will usually abbreviate $\mu$-almost everywhere to $\mu$-a.e.\ and simply write $M$, $M_+$ and $M_0$, instead of $M(R, \mu)$, $M_+(R, \mu)$ and $M_0(R, \mu)$ respectively, whenever there is no risk of confusion. In some special cases when $R = \mathbb{R}^n$ we will denote the $n$-dimensional Lebesgue measure by $\lambda^n$ (or just $\lambda$ when $n=1$).

When $X$ is a set and $f, g: X \to \mathbb{C}$ are two maps satisfying that there is some positive and finite constant $C$, depending only on $f$ and $g$, such that $\lvert f(x) \rvert \leq C \lvert g(x) \rvert$ for all $x \in X$, we will denote this by $f \lesssim g$. We will also write $f \approx g$, or sometimes say that $f$ and $g$ are equivalent, whenever both $f \lesssim g$ and $g \lesssim f$ are true at the same time. We choose this general definition because we will use the symbols ``$\lesssim$'' and ``$\approx$'' with both functions and functionals.

\subsection{Non-increasing rearrangement} \label{SSNR}

In this section, we define the non-increasing rearrangement of a function and some related terms. We proceed in accordance with \cite[Chapter~2]{BennettSharpley88}.

The non-increasing rearrangement of a function $f \in M$, traditionally denoted $f^*$, is defined as the generalised inverse of the distribution function, that is for any $t \in [0, \infty)$
\begin{equation*}
	f^*(t) = \inf \{ s \in [0, \infty); \; \mu_f(s) \leq t \},
\end{equation*}
where the distribution function $\mu_f$ of a function $f \in M$ is defined for $s \in [0, \infty)$ by
\begin{equation*}
	\mu_f(s) = \mu(\{ t \in R; \; \lvert f(t) \rvert > s \}).
\end{equation*}

Some basic properties of the distribution function and the non-increasing rearrangement, with proofs, can be found in \cite[Chapter~2, Proposition~1.3]{BennettSharpley88} and \cite[Chapter~2, Proposition~1.7]{BennettSharpley88}.

A very important classical result is the Hardy-Littlewood inequality which asserts that it holds for all $f, g \in M$ that
\begin{equation}
	\int_R \lvert f g \rvert \: d\mu \leq \int_0^{\infty} f^* g^* \: d\lambda. \label{THLI}
\end{equation}
For details, see for example \cite[Chapter~2, Theorem~2.2]{BennettSharpley88}. As an immediate consequence, we get that, for all $f, g \in M$,
\begin{equation}
\sup_{\substack{\tilde{g} \in M \\ \tilde{g}^* = g^*}} \int_R \lvert f \tilde{g} \rvert \: d\mu \leq \int_0^{\infty} f^* g^* \: d\lambda. \label{DR}
\end{equation}
This leads to the definition of resonant measure spaces, as those spaces where we have equality in \eqref{DR}. It has been proven that in order for a measure space to be resonant, it suffices for its measure to be non-atomic. For details, see \cite[Chapter~2, Theorem~2.7]{BennettSharpley88}.

We will also need in the paper the so called maximal function of $f^*$ (where $f \in M$), which is traditionally denoted $f^{**}$ and is defined for $t \in [0, \infty)$ by
\begin{equation*}
	f^{**}(t) = \frac{1}{t} \int_0^{t} f^*(s) \: ds.
\end{equation*}
Some properties of the maximal function can be found in \cite[Chapter~2, Proposition~3.2]{BennettSharpley88} and  \cite[Chapter~2, Theorem~3.4]{BennettSharpley88}.

\subsection{Function norms and quasinorms} \label{SSFN}

The following two definitions are adapted from \cite[Chapter~1, Definition~1.1]{BennettSharpley88} and \cite[Chapter~2, Definition~4.1]{BennettSharpley88} respectively.

\begin{definition}
	Let $\lVert \cdot \rVert : M \to [0, \infty]$ be some non-negative functional such that it holds for all $f \in M$ that $\lVert f \rVert = \lVert \; \lvert f \rvert \; \rVert$. We then say that $\lVert \cdot \rVert$ is a Banach function norm if its restriction on $M_+$ satisfies the following conditions:

	\begin{enumerate}[label=\textup{(P\arabic*)}, series=P]
		\item \label{P1} $\lVert \cdot \rVert$ is a norm, i.e.
		\begin{enumerate}
			\item it is positively homogeneous, i.e.\ $\forall a \in \mathbb{C} \forall f \in M_+ \: : \: \lVert a \cdot f \rVert = \lvert a \rvert \lVert f \rVert$,
			\item it satisfies $\lVert f \rVert = 0 \Leftrightarrow f = 0$  $\mu$-a.e.,
			\item it is subadditive, i.e.\ $\forall f,g \in M_+ \: : \: \lVert f+g \rVert \leq \lVert f \rVert + \lVert g \rVert$.
		\end{enumerate}
		\item \label{P2} $\lVert \cdot \rVert$ has the lattice property, i.e.\ if some $f, g \in M_+$ satisfy $f \leq g$ $\mu$-a.e., then also $\lVert f \rVert \leq \lVert g \rVert$.
		\item \label{P3} $\lVert \cdot \rVert$ has the Fatou property, i.e.\ if  some $f_n, f \in M_+$ satisfy $f_n \uparrow f$ $\mu$-a.e., then also $\lVert f_n \rVert \uparrow \lVert f \rVert $.
		\item \label{P4} $\lVert \chi_E \rVert < \infty$ for all $E \subseteq R$ satisfying $\mu(E) < \infty$.
		\item \label{P5} For every $E \subseteq R$ satisfying $\mu(E) < \infty$ there exists some finite constant $C_E$, dependent only on $E$, such that for all $f \in M_+$ the inequality $ \int_E f \: d\mu \leq C_E \lVert f \rVert $ holds.
	\end{enumerate}
\end{definition}

We will be interested in particular in the class of rearrangement invariant Banach function norms.

\begin{definition}
	We say that a Banach function norm $\lVert \cdot \rVert$ is rearrangement invariant, abbreviated r.i., if  it satisfies the following additional condition:
	\begin{enumerate}[resume*=P]
		\item \label{P6} If two functions $f,g \in M_+$ satisfy $f^* = g^*$ then $\lVert f \rVert = \lVert g \rVert$.
	\end{enumerate}
\end{definition}

For the properties of Banach function norms and r.i.~Banach function norms see \cite[Chapter~1]{BennettSharpley88} and \cite[Chapter~2]{BennettSharpley88}, respectively.

It will be also useful to define a somewhat weaker version of r.i.\ Banach function norms, namely the rearrangement-invariant quasi-Banach function norms.

\begin{definition} \label{Q}
	Let $\lVert \cdot \rVert : M \to [0, \infty]$ be some non-negative functional such that it holds for all $f \in M$ that $\lVert f \rVert = \lVert \; \lvert f \rvert \; \rVert$. We then say that $\lVert \cdot \rVert$ is an quasi-Banach function norm, if it satisfies the axioms \ref{P2}, \ref{P3}, and \ref{P4} from the definition of r.i.\ Banach function norm and also a weaker version of \ref{P1}, namely
	\begin{enumerate}[label=\textup{(Q\arabic*)}]
		\item \label{Q1} $\lVert \cdot \rVert$ is a quasinorm, i.e.
		\begin{enumerate}
			\item it is positively homogeneous, i.e.\ $\forall a \in \mathbb{R} \forall f \in M_+ \: : \: \lVert a \cdot f \rVert = \lvert a \rvert \lVert f \rVert$,
			\item it satisfies  $\lVert f \rVert = 0 \Leftrightarrow f = 0$ $\mu$-a.e.,
			\item it is subadditive up to a constant, i.e.\ there is some finite constant $C$ such that $\forall f,g \in M_+ \: : \: \lVert f+g \rVert \leq C(\lVert f \rVert + \lVert g \rVert)$.
		\end{enumerate}
	\end{enumerate}
\end{definition}

\begin{definition}
	We say that a quasi-Banach function norm $\lVert \cdot \rVert$ is rearrangement-invariant, abbreviated r.i., if it satisfies \ref{P6}.
\end{definition}

Our definition is standard, the same as for example in \cite{CaetanoGogatishvili16}, \cite{NekvindaPesa20}, and \cite{Lopez08} (note that the requirement of completeness in \cite{Lopez08} is redundant as follows from \cite[Lemma~3.6]{CaetanoGogatishvili16}). The paper \cite{NekvindaPesa20} treats the properties of (r.i.) quasi-Banach function norms in some detail.

\begin{definition}
	Let $\lVert \cdot \rVert$ be a Banach function norm on $M$. Then the set
	\begin{equation*}
	X = \{ f \in M; \; \lVert f \rVert < \infty \}
	\end{equation*}
	equipped with the norm $\lVert \cdot \rVert$ will be called a Banach function space. Further, if $\lVert \cdot \rVert$ is rearrangement invariant, we shall say that $X$ is a rearrangement invariant Banach function space.

	If $\lVert \cdot \rVert$ is an (r.i.)\ quasi-Banach function norm, we define an (r.i.)\ quasi-Banach function space in exactly the same manner.
\end{definition}

To provide an example, let us note that the classical Lebesgue functional  $\lVert \cdot \rVert_q$ is an r.i.\ Banach function norm on $M$ for $q \in [1, \infty]$ and an r.i.\ quasi-Banach function norm on $M$ for $q \in (0,\infty]$.

\subsection{Associate spaces} \label{SSAS}
In this section we define the associate functionals and consequently the associate spaces. We intentionally make the definitions very broad and general. For details on associate spaces of Banach function spaces, see \cite[Chapter~1, Section~2, 3 and 4]{BennettSharpley88}. We will approach the matter in a more general way.

When talking about associate spaces, it is useful to use the notation $q'$ for the number defined by
\begin{equation*}
q' =
\begin{cases}
\frac{q}{q-1} & \text{for } q \in (1, \infty), \\
1  & \text{for } q = \infty, \\
\infty  & \text{for } q = 1,
\end{cases}
\end{equation*}
which we will do from now on.

\begin{definition} \label{DAS}
	Let $\lVert \cdot \rVert: M \to [0, \infty]$ be some non-negative functional and put
	\begin{equation*}
	X = \{ f \in M; \; \lVert f \rVert < \infty \}.
	\end{equation*}
	Then the set
	\begin{equation*}
	X' = \left \{ f \in M; \; \sup_{\substack{g \in M \\ \lVert g \rVert \leq 1}} \int_R \lvert f g \rvert \: d\mu < \infty \right \}
	\end{equation*}
	will be called the associate space of $X$ and the functional $\lVert \cdot \rVert_{X'}$ defined for $f \in M$ by
	\begin{equation*}
	\lVert f \rVert_{X'} = \sup_{\substack{g \in M \\ \lVert g \rVert \leq 1}} \int_R \lvert f g \rvert \: d\mu
	\end{equation*}
	will be called the associate functional of $\lVert \cdot \rVert$.
\end{definition}

As the notation suggests, these terms are interesting mainly when the functional $\lVert \cdot \rVert$ is at least a quasinorm, but we wanted to emphasize that the definition itself lays no requirements on it.

The point of the associate space is to provide a general form of H{\"o}lder inequality, namely that it holds for all $f, g \in M$ that
\begin{equation}
	\int_R \lvert f  g \rvert \: d\mu \leq \lVert g \rVert \lVert f \rVert_{X'}, \label{THAS}
\end{equation}
where $\lVert \cdot \rVert: M \to [0, \infty]$ is assumed to be some positively homogeneous functional and in the case when the right-hand side is of the form $0 \cdot \infty$ it is to be interpreted as $\infty$.

To provide an example, if we take $\lVert \cdot \rVert_q$, $q \in [1, \infty]$ as our $\lVert \cdot \rVert$, we obtain that $\lVert \cdot \rVert_{X'} = \lVert \cdot \rVert_{q'}$. On the other hand, if we take $\lVert \cdot \rVert_q$, $q \in (0,1)$, we get $\lVert f \rVert_{X'} = \infty$ for all $f$ other than zero (and of course $\lVert 0 \rVert_{X'} = 0$) and thus $X' = \{0\}$.

It is quite obvious from Definition~\ref{DAS} that if two positively homogeneous functionals $\lVert \cdot \rVert_X$ and $\lVert \cdot \rVert_Y$ satisfy $\lVert \cdot \rVert_X \lesssim \lVert \cdot \rVert_Y$ then also $\lVert \cdot \rVert_{Y'} \lesssim \lVert \cdot \rVert_{X'}$. Equivalently, we may say that if $X \hookrightarrow Y$ then $Y' \hookrightarrow X'$, where the continuity  of the embedding is understood relatively to the defining functionals, i.e.\ it means that the identity operator is bounded.

It has been proven, see for example \cite[Chapter~1, Theorem~2.2]{BennettSharpley88} and \cite[Chapter~1, Theorem~2.7]{BennettSharpley88}, that the associate functional of a Banach function norm $\lVert \cdot \rVert$ is itself a Banach function norm, and that its associate functional is $\lVert \cdot \rVert$. This result has been improved recently by Gogatishvili and Soudsk{\'y} in \cite{GogatishviliSoudsky14}. Since we will use this result later in the paper, we present it below. The symbol $\lVert \cdot \rVert_{X''}$ denotes the second associate functional, that is an associate functional of $\lVert \cdot \rVert_{X'}$.

\begin{theorem} \label{TFA}
	Let $\lVert \cdot \rVert : M \to [0, \infty]$ be a functional that satisfies \ref{P4} and \ref{P5} and which also satisfies for all $f \in M$ that $\lVert f \rVert$ = $\lVert \: \lvert f \rvert \: \rVert$. Then the functional $\lVert \cdot \rVert_{X'}$ is a Banach function norm. In addition, $\lVert \cdot \rVert$ is equivalent to a Banach function norm if and only if $\lVert \cdot \rVert \approx \lVert \cdot \rVert_{X''}$

\end{theorem}

\subsection{Fundamental function} \label{SSFF}

In this section, we define the fundamental function of a r.i.\ quasi-Banach function space and state some of its properties, and then do the same with the endpoint spaces. We proceed in accordance with \cite[Section~7.9]{FucikKufner13} and \cite[Section~7.10]{FucikKufner13}. We note that this topic has been also covered in \cite[Chapter~2, Section~5]{BennettSharpley88}.

In this section, we will always assume that the space $(R, \mu)$ is completely non-atomic.

Given a r.i.\ quasi-Banach function space $(X, \lVert \cdot \rVert)$, the fundamental function $\varphi_X$ of $X$ is defined for all $t$ in the range of $\mu$ by
\begin{equation*}
	\varphi_X(t) = \lVert \chi_E \rVert,
\end{equation*}
where $E$ is some subset of $R$ of measure $\mu(E) = t$. Note that the set $E$ in the definition always exists by the assumption that $t$ is in range of $\mu$, and that the definition does not depend on the choice of $E$ since the norm satisfies \ref{P6}.

One important result is the relation of the respective fundamental functions of $X$ and of its associate space $X'$. If $X$ is a r.i.\ quasi-Banach function space and $X'$ its associate space, then the corresponding fundamental functions satisfy the estimate
\begin{equation}
\varphi_{X'}(t) \geq \frac{t}{\varphi_X(t)} \label{TFFAS1}
\end{equation}
for all non-zero $t$ in the range of $\mu$. Furthermore, if $X$ is a Banach function space, then we have equality in \eqref{TFFAS1} for all relevant $t$. The weaker result for r.i.\ quasi-Banach function spaces follows directly from the H{\" o}lder inequality \eqref{THAS}. Proof of the stronger result for Banach function spaces can be found for example in \cite[Theorem~7.9.6]{FucikKufner13}.

It has been shown, that fundamental function of any Banach function space over a resonant measure space is quasiconcave, in the sense that is is non-decreasing, it attains zero only at the point zero and that the function $\frac{t}{\varphi(t)}$ is non-decreasing on $(0, \mu(R))$. For details, see for example \cite[Remark~7.9.7]{FucikKufner13}.

We are now equipped to define the Marcinkiewicz endpoint space.

\begin{definition}\label{DMES}
	Let $\varphi$ be a quasiconcave function on $[0, \mu(R))$. Then the set $M_{\varphi}$, defined as
	\begin{equation*}
	M_{\varphi} = \left \{ f \in M(R, \mu); \; \sup_{t \in (0, \mu(R))} \varphi(t) f^{**}(t) < \infty \right \},
	\end{equation*}
	is called the Marcinkiewicz endpoint space.
\end{definition}

It follows, see for example \cite[Proposition~7.10.2]{FucikKufner13}, that the Marcinkiewicz endpoint space equipped with the naturally chosen functional
\begin{align*}
\lVert f \rVert  &= \sup_{t \in (0, \infty)} \varphi(t) f^{**}(t) & \text{for } f \in M(R, \mu)
\end{align*}
is an r.i.\ Banach function space and its fundamental function coincides with $\varphi$ on the range of $\mu$. In fact, it is the largest of such spaces, in the sense that if $X$ is an r.i.\ Banach function space the fundamental function of which coincides with $\varphi$ on the range of $\mu$ then $X \hookrightarrow M_{\varphi}$. Proof can be found for example in \cite[Proposition~7.10.6]{FucikKufner13}.

To find the smallest r.i.\ Banach funtion space with given fundamental function, one must first observe that any quasiconcave function is equivalent to a concave function. This result can be found for example in \cite[Proposition~7.10.10]{FucikKufner13}.  It can be then shown, as in, for example, \cite[Theorem~7.10.12]{FucikKufner13}, that any r.i.\ Banach function space can be equivalently renormed in such way that its fundamental function is then concave (and obviously non-decreasing). For concave $\varphi$, we may then define the Lorentz endpoint space as follows.

\begin{definition} \label{DLES}
	Let $\varphi$ be a non-decreasing concave function on $[0, \mu(R))$. Then the set $\Lambda_{\varphi}$, defined as
	\begin{equation*}
	\Lambda_{\varphi} = \left \{ f \in M(R, \mu); \; \int_0^{\mu(R)}  f^* \: d\varphi < \infty \right \},
	\end{equation*}
	is called the Lorentz  endpoint space.
\end{definition}

We note that the Lebesgue-Stieltjes integral in question is well-defined since $\varphi$ is non-decreasing. We further get, see for example \cite[Proposition~7.10.16]{FucikKufner13}, that the Lorentz endpoint space equipped with the naturally chosen functional
\begin{align*}
\lVert f \rVert  &= \int_0^{\mu(R)}  f^* \: d\varphi & \text{for } f \in M(R, \mu),
\end{align*}
is an r.i.\ Banach function space fundamental function of which coincides with $\varphi$ on the range of $\mu$. It is the smallest space with those properties, in the sense that if $X$ is an r.i.\ Banach function space such that its fundamental function coincides with $\varphi$ on the range of $\mu$, then $\Lambda_{\varphi} \hookrightarrow X$. For proof, see \cite[Proposition~7.10.15]{FucikKufner13}.

The Lorentz and Marcinkiewicz endpoint spaces are mutually associate, in the sense that if $\varphi$ is a concave function, then $\Lambda_{\varphi}' = M_{\bar{\varphi}}$ and $M_{\varphi}' = \Lambda_{\bar{\varphi}}$, where $\bar{\varphi}$ is defined on $(0, \mu(R))$ by
\begin{equation*}
	\bar{\varphi}(t) = \frac{t}{\varphi(t)}.
\end{equation*}
and $\bar{\varphi}(0) = 0$. This follows directly from the discussion above, since if $X$ is an r.i.~Banach function space whose fundamental function is $\bar{\varphi}$, then the fundamental function of $X'$ is $\varphi$ and thus $\Lambda_{\varphi} \hookrightarrow X'$. Hence, $X \hookrightarrow \Lambda_{\varphi}'$, and it follows that $\Lambda_{\varphi}'$ is the largest r.i.~Banach function space whose fundamental function is $\bar{\varphi}$, i.e.~it must coincide with $M_{\bar{\varphi}}$.

In conclusion, we note that the endpoint spaces do not change, up to equivalence of the defining functionals, if $\varphi$ is replaced with an equivalent function.

\section{Lorentz--Karamata spaces} \label{SMR}

From now on we suppose that our measure space $(R, \mu)$ has a non-atomic measure. As noted in Section~\ref{SSNR}, this means that it is always resonant. We will also, unless stated otherwise, assume that $\mu(R) = \infty$.

\subsection{Slowly varying functions} \label{SSSV}

\begin{definition} \label{DSV}
	Let $b:(0,\infty) \rightarrow (0, \infty)$ be a measurable function. Then $b$ is said to be slowly varying, if for every $\varepsilon > 0$ there exists a non-decreasing function $b_{\varepsilon}$ and non-increasing function $b_{-\varepsilon}$ such that $t^{\varepsilon}b(t) \approx b_{\varepsilon}(t)$ on $(0, \infty)$ and $t^{-\varepsilon}b(t) \approx b_{-\varepsilon}(t)$ on $(0, \infty)$.
\end{definition}

We will, for the sake of brevity, usually abbreviate  slowly varying as s.v.

The class of s.v.\ functions includes, for example, constant positive functions and, to provide something at least slightly less trivial, the functions $t \mapsto 1+\lvert \log(t) \rvert$ and $t \mapsto 1+\log(1+\lvert \log(t) \rvert)$ occurring in the definition of generalized Lorentz-Zygmund spaces introduced by Bennett and Rudnick in \cite{BennettRudnick80} and later treated in great detail by Opic and Pick in \cite{OpicPick99}. As a further example of an s.v.~function, this time non-logarithmic, we present a function $b$ defined on $(0, \infty)$ by

\begin{equation*}
b(t) = \begin{cases}
e^{\sqrt{\log t}} & \text{for } t \in [1, \infty), \\
e^{\sqrt{\log t^{-1}}} & \text{for } t \in (0, 1).
\end{cases}
\end{equation*}
Restriction of this function to the interval $(0, 1)$ was used in \cite{CianchiPick09} to characterise self-optimal spaces for Gauss-Sobolev embeddings.

\begin{lemma} \label{LSV}
	Let $b, b_2$ be s.v.\ functions. Then the following holds:
	\begin{enumerate}[label=\textup{(SV\arabic*)}]
		\item \label{SV1} The function $b^r$ is slowly varying for every $r \in \mathbb{R}$.
		\item \label{SV2} The functions $b+b_2$, $bb_2$, $\frac{b}{b_2}$ and $t \mapsto b(\frac{1}{t})$ are	are s.v.
		\item \label{SVC} The function $b$ is bounded by positive constants on compact subsets of $(0, \infty)$.
		\item \label{CSV} Let $c > 0$, then $b(ct) \approx b(t)$ on $(0, \infty)$.
		\item \label{LEFF} \label{LEFFH} Let $\alpha \in (0, \infty)$. Then it holds on $(0, \infty)$ that
		\begin{align}
			\int_0^{t} s^{\alpha-1} b(s) \: ds &\approx t^{\alpha} b(t), & \esssup_{s \in (0,t)} s^{\alpha} b(s) &\approx t^{\alpha} b(t), \label{LEFF1}\\
			\int_t^{\infty} s^{-\alpha-1} b(s) \: ds &\approx t^{-\alpha} b(t), & \esssup_{s \in (t,\infty)} s^{-\alpha} b(s) &\approx t^{-\alpha} b(t).  \label{LEFFH1}
		\end{align}
		\item \label{LTb} \label{LHb} Let us define, for $t \in (0, \infty)$,
		\begin{align*}
			\tilde{b}(t) &= \int_0^{t} s^{-1} b(s) \: ds, & \tilde{b}_{\infty}(t) &= \esssup_{s \in (0, t)} b(s), \\
			\hat{b}(t) &= \int_t^{\infty} s^{-1} b(s) \: ds, & \hat{b}_{\infty}(t) &= \esssup_{s \in (t, \infty)} b(s).
		\end{align*}
		Then $b \lesssim \tilde{b}$, $b \lesssim \tilde{b}_{\infty}$, $b \lesssim \hat{b}$, and $b \lesssim \hat{b}_{\infty}$ on $(0, \infty)$ and the functions $\tilde{b}$, $\tilde{b}_{\infty}$, $\hat{b}$, and  $\hat{b}_{\infty}$ are s.v.~if and only if they are finite $\lambda$-a.e.~on $(0, \infty)$. Furthermore, it holds that
		\begin{align} \label{LTb1}
			\limsup_{t \to 0^+} \frac{\tilde{b}(t)}{b(t)} = \limsup_{t \to \infty} \frac{\tilde{b}(t)}{b(t)} &= \infty, &
			\limsup_{t \to 0^+} \frac{\hat{b}(t)}{b(t)} = \limsup_{t \to \infty} \frac{\hat{b}(t)}{b(t)} &= \infty.
		\end{align}
		\item \label{SV3} It holds for every $ \alpha \neq 0$ that
		\begin{align*}
			\lim_{t \to 0^+} t^{\alpha}b(t) = \lim_{t \to 0^+}t^{\alpha}, \\
			\lim_{t \to \infty} t^{\alpha}b(t) = \lim_{t \to \infty}t^{\alpha}.
		\end{align*}
		\item \label{SV4} Let $\alpha \neq -1$. Then
		\begin{equation*}
			\int_0^1 t^{\alpha}b(t) \: dt < \infty \iff \int_0^1 t^{\alpha} \: dt < \infty
		\end{equation*}
		and
		\begin{equation*}
			\int_1^{\infty} t^{\alpha}b(t) \: dt < \infty \iff \int_1^{\infty} t^{\alpha} \: dt < \infty.
		\end{equation*}
		Consequently,
		\begin{equation*}
			\int_0^{\infty} t^{\alpha} b(t) \: dt = \infty.
		\end{equation*}
	\end{enumerate}
\end{lemma}

\begin{proof}
	For \ref{SV1}, \ref{SV2},  \ref{CSV}, and \ref{LEFF} see \cite[Proposition~2.2]{GogatishviliOpic05}. For \ref{SVC}
	see \cite[Proposition~2.1]{Bathory18} and for the first part of \ref{LTb} see \cite[Lemma~2.2]{Bathory18} and \cite[Lemma~2.1]{GurkaOpic07}. The remaining part of \ref{LTb}, i.e.~\eqref{LTb1}, has been partially proved in \cite[Lemma~2.2]{CaetanoGogatishvili11} and \cite[(2.1) and (2.2)]{NevesOpic20} and the method used in the former paper can be easily adapted to cover all the four cases mentioned in \eqref{LTb1}.

	The properties \ref{SV3} and \ref{SV4} follow directly from the definition. We leave the details to the reader.
\end{proof}

\begin{remark}
	The restrictions $ \alpha \neq 0$ and $ \alpha \neq -1$ in \ref{SV3} and \ref{SV4}, respectively, are necessary, as the property that $b$ is s.v.~is simply too weak to determine the behaviour of  $t^{\alpha}b(t)$ in these limiting cases. The counterexamples are easy to construct.
\end{remark}

\begin{remark}
	There exist three different definitions of slowly varying functions that can be found in the literature. The three defining conditions (for $b \in M((0,\infty), \lambda)$) are:
	\begin{enumerate}
		\item \label{DSVi}
			For every $\varepsilon > 0$ the function $t^{\varepsilon}b(t)$ is non-decreasing on some neighbourhoods of zero and infinity while the function $t^{-\varepsilon}b(t)$ is non-increasing on some neighbourhoods of zero and infinity.
		\item \label{DSVii}
			For every $\varepsilon > 0$ there exists a non-decreasing function $b_{\varepsilon}$ and non-increasing function $b_{-\varepsilon}$ such that
			\begin{equation*}
				\lim_{t \to 0} \frac{t^{\varepsilon}b(t)}{b_{\varepsilon}(t)} = \lim_{t \to \infty} \frac{t^{\varepsilon}b(t)}{b_{\varepsilon}(t)} = \lim_{t \to 0} \frac{t^{-\varepsilon}b(t)}{b_{-\varepsilon}(t)} =\lim_{t \to \infty} \frac{t^{-\varepsilon}b(t)}{b_{-\varepsilon}(t)} = 1.
			\end{equation*}
		\item \label{DSViii}
			$b$ satisfies Definition~\ref{DSV}.
	\end{enumerate}
	The condition \ref{DSVi} was used in the original definition of Lorentz--Karamata spaces in \cite{EdmundsKerman00} and the functions satisfying it are said to belong to the Zygmund class $\mathcal{Z}$ (see \cite{Zygmund35_3} for further details). The condition \ref{DSVii} is the closest to the original definition of slowly varying functions as given by Karamata in \cite{Karamata30} and \cite{Karamata33} --- while it is not the original definition, it is equivalent to it. Functions satisfying this condition are treated thoroughly in \cite{BinghamGoldie87}. The condition \ref{DSViii} is the one currently used in most papers concerning Lorentz--Karamata spaces and it first appeared in either \cite{GogatishviliOpic04} or \cite{GogatishviliOpic05} (the idea to use only the equivalence with monotone functions appeared originally in the paper \cite{Neves02}, but in this case the function was also required to behave the same way near zero as it does near infinity, so the definition was significantly less general).

	Provided that the function $b$ is assumed to be bounded on compact sets (which is very reasonable for our purposes), then clearly the validity of \ref{DSVi} implies that of \ref{DSVii} and the validity of \ref{DSVii} implies that of \ref{DSViii}. On the other hand, for any function $b$ satisfying \ref{DSVii} there is a function $b_0$ satisfying \ref{DSVi} such that
	\begin{equation*}
		\lim_{t \to 0} \frac{b(t)}{b_{0}(t)} = \lim_{t \to \infty} \frac{b(t)}{b_{0}(t)} = 1.
	\end{equation*}

	Some of the properties shown in Lemma~\ref{LSV} can be strengthened if we assume that $b$ satisfies \ref{DSVii}. Specifically, in \ref{CSV} and \ref{LEFF} we can replace the relation ``$\approx$" in all its occurrences with ``the ratio of the left-hand side and the right-hand side converges to $1$ at both $0$ and $\infty$" while	in \eqref{LTb1} we get that
	\begin{align*}
		\lim_{t \to 0^+} \frac{\tilde{b}(t)}{b(t)} = \lim_{t \to \infty} \frac{\tilde{b}(t)}{b(t)} &= \infty, &
		\lim_{t \to 0^+} \frac{\hat{b}(t)}{b(t)} = \lim_{t \to \infty} \frac{\hat{b}(t)}{b(t)} &= \infty.
	\end{align*}
	For details, see \cite[Chapter~1]{BinghamGoldie87}.
\end{remark}

\subsection{Lorentz--Karamata functionals} \label{SSLK}

\begin{definition} \label{DLK}
	Let $p \in (0, \infty], q \in (0, \infty]$ and let $b$ be an s.v.\ function. We then define the Lorentz--Karamata functionals $ \lVert \cdot \rVert_{p, q, b}$ and $ \lVert \cdot \rVert_{(p, q, b)}$,  for $f \in M$, as follows:
	\begin{align*}
	\lVert f \rVert_{p, q, b} = \lVert t^{\frac{1}{p}-\frac{1}{q}}b(t)f^*(t) \rVert_q, \\
	\lVert f \rVert_{(p, q, b)} = \lVert t^{\frac{1}{p}-\frac{1}{q}}b(t)f^{**}(t) \rVert_q,
	\end{align*}
	where $\lVert \cdot \rVert_q$ is the classical Lebesgue functional on $(0, \infty)$.

	We further define the corresponding Lorentz--Karamata spaces $L^{p, q, b}$ and $L^{(p, q, b)}$ as
	\begin{align*}
	L^{p, q, b} &= \{ f \in M; \; \lVert f \rVert_{p, q, b} < \infty \}, \\
	L^{(p, q, b)} &= \{ f \in M; \; \lVert f \rVert_{(p, q, b)} < \infty \}.
	\end{align*}
\end{definition}

Note that the Lorentz functionals $\lVert \cdot \rVert_{p, q}$ and $ \lVert \cdot \rVert_{(p, q)}$, defined for example in \cite[Chapter~4, Definiton~4.1]{BennettSharpley88}, are special cases of Lorentz--Karamata functionals with $b(t) = 1$. If further $p=q$ then we have that $\lVert \cdot \rVert_{p, q, b}$ coincides with the classical Lebesgue functional $\lVert \cdot \rVert_q$, as follows from \cite[Chapter~2, Proposition~1.8]{BennettSharpley88}. Another examples of Lorentz--Karamata functionals are the generalised Lorentz-Zygmung functionals studied in \cite{OpicPick99}.

We begin with the useful observation that in the case when $p=q=\infty$ we may assume $b$ to be non-decreasing. This result first appeared in \cite{Bathory18}, where one can also find its proof.

\begin{proposition} \label{PbND}
	Let $b$ be an s.v.~function. Then $L^{\infty, \infty, b} = L^{\infty, \infty, \tilde{b}_{\infty}}$, up to equivalence of the defining functionals, where $\tilde{b}_{\infty}$ is the function defined in \ref{LTb}, that is
	\begin{equation*}
		\tilde{b}_{\infty}(t) = \esssup_{s \in (0, t)} b(s).
	\end{equation*}
\end{proposition}

\begin{proposition} \label{LQ}
	Let $p, q, b$ be as above. Then the Lorentz--Karamata functional $\lVert \cdot \rVert_{p, q, b}$ is an r.i.~quasi-Banach function norm, and consequently $(L^{p, q, b},\lVert \cdot \rVert_{p, q, b})$ is an r.i.\ quasi-Banach function space, if and only if $p < \infty$, or $p = \infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q < \infty$. Furthermore, if those conditions are not satisfied, then $L^{p, q, b}=\{ 0 \}$.
\end{proposition}

\begin{proof}
	Throughout this proof, the letters $f, g$ will denote some arbitrary functions belonging to $M$. We begin by proving the sufficiency.

	All the properties, except the the subadditivity up to a constant and \ref{P4} are immediate consequences of the definition, properties of the non-increasing rearrangement, and properties of the Lebesgue functionals.

	For the subadditivity up to a constant, if we remember that the non-increasing rearrangement satisfies $(f+g)^*(t_1+t_2) \leq f^*(t_1) + g^*(t_2)$, we may use \ref{CSV} to get the following estimate:
	\begin{equation*}
	\begin{split}
		\lVert f+g \rVert_{p, q, b} &= \lVert t^{\frac{1}{p}-\frac{1}{q}}b(t)(f+g)^*(t) \rVert_q \\
		&\leq \left \lVert t^{\frac{1}{p}-\frac{1}{q}}b(t) \left ( f^* \left (\frac{t}{2} \right )+g^* \left (\frac{t}{2} \right )  \right ) \right  \rVert_q \\
		&\lesssim \left  \lVert \left ( \frac{t}{2} \right )^{\frac{1}{p}-\frac{1}{q}}b \left (\frac{t}{2} \right ) \left (  f^* \left (\frac{t}{2} \right )+g^* \left (\frac{t}{2} \right ) \right ) \right  \rVert_q.
	\end{split}
	\end{equation*}
	Now, it remains only to use the simple change of variables $s = \frac{t}{2}$ and use the subadditivity (up to a constant) of $\lVert \cdot \rVert_q$ to get the desired result. We have thus verified that $ \lVert \cdot \rVert_{p, q, b}$ satisfies \ref{Q1}.

	It remains to verify \ref{P4}. Since it holds for $E \subseteq R$, $\mu(E) < \infty$ that $\chi_E^* = \chi_{(0, \mu (E))}$, it follows that \ref{P4} is equivalent to the question whether $\lVert t^{\frac{1}{p} - \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q < \infty$. If $p \neq \infty$, this is satisfied either by \ref{SV4}  or by \ref{SV3}, while in the remaining case it is assumed.

	The preceding argument also shows that $\lVert \cdot \rVert_{p, q, b}$ does not have the property \ref{P4} when $p=\infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q = \infty$, which establishes the necessity. It also follows immediately that, in this case, $L^{p, q, b}=\{ 0 \}$ which concludes the proof.
\end{proof}

It is worth mentioning that for some special sets of parameters $p, q, b$, $ \lVert \cdot \rVert_{p, q, b}$ may qualify to be a r.i.\ Banach function norm on $M$. Let us name at least the cases $p = q \in [1, \infty]$ while $b(t) = 1$ for all $t \in (0, \infty)$ when $ \lVert \cdot \rVert_{p, q, b}$ coincides with the classical Lebesgue functional $\lVert \cdot \rVert_q$.

In the following lemma we examine for which choices of $p, q, b$ the functional $\lVert \cdot \rVert_{p, q, b}$ satisfies \ref{P5} and provide some sufficient and necessary conditions.

\begin{proposition}\label{LP5}
	Let $b$ be an s.v.\ function.
	\begin{enumerate}
		\item \label{LP5i}
		Suppose that one of the following condition holds:
		\begin{enumerate}[label=\textup{\alph*)}]
			\item \label{LP5ia}
			$1 \leq q < p$,
			\item \label{LP5ib}
			$1 \leq q = p$ and there are some constants $K, t_0 > 0$ such that $b(t) \geq K$ for all $t \in (0, t_0)$,
		\end{enumerate}
		then $\lVert \cdot \rVert_{p, q, b}$ satisfies \ref{P5}.
		\item \label{LP5ii}
		Suppose that one of the following condition holds:
		\begin{enumerate}[label=\textup{\alph*)}]
			\item \label{LP5iia}
			$p<q \leq 1$,
			\item \label{LP5iib}
			$p = q \leq 1$ and $\lim_{t \to 0^+} b(t) = 0$,
		\end{enumerate}
		then $\lVert \cdot \rVert_{p, q, b}$ does not satisfy \ref{P5}.
	\end{enumerate}
\end{proposition}

\begin{proof}
	Suppose first that the assumption of \ref{LP5i} holds. To prove that $\lVert \cdot \rVert_{1, 1, b}$ satisfies \ref{P5}, we fix some $E \subseteq R$ of measure $\mu(E) = r < \infty$ and observe that, thanks to \ref{SVC} and either \ref{SV3} or our assumption, there is a constant $K > 0$ such that $t^{\frac{1}{p}-\frac{1}{q}}b(t) \geq K$ for all $t \in (0, r)$. We now employ the Hardy-Littlewood inequality \eqref{THLI} and the classical Jensen inequality, which can be found for example in \cite[Theorem~4.2.11]{FucikKufner13} and which we may use because we assume $q \geq 1$, to obtain that
	\begin{equation*}
	\begin{split}
		\int_E \lvert f \rvert \: d\mu &\leq K^{-1} r \left ( \left ( \frac{1}{r} \int_0^r t^{\frac{1}{p}-\frac{1}{q}} b(t) f^*(t) \: dt \right)^q \right)^{\frac{1}{q}} \leq K^{-1} r^{1-\frac{1}{q}} \left ( \int_0^r t^{\frac{q}{p}-1} b^q(t) (f^*(t))^q \: dt \right)^{\frac{1}{q}} \\
		&\leq K^{-1} r^{1-\frac{1}{q}} \lVert f \rVert_{p,q,b}.
	\end{split}
	\end{equation*}

	Suppose now that the assumption of \ref{LP5ii} holds. We then have, either from \ref{SV1} and \ref{SV3} or directly from our assumption, that $\lim_{t \to 0^+} t^{\frac{q}{p}-1}b^q(t) = 0$. We fix some $R_0 \subseteq R$ of finite non-zero measure and then find, for arbitrary $n \in \mathbb{N}$, some $r>0$ such that $t^{\frac{q}{p}-1}b^q(t) < \frac{1}{n}$ for all $t \in (0, r)$ and some set $E \subseteq R_0$ of measure $0 < \mu(E) < r$. We then easily obtain the estimate
	\begin{equation*}
	\int_{R_0} \chi_E \: d\mu = \mu(E) = \int_0^{\mu(E)} 1 \: dt > n \int_0^{\mu(E)} t^{\frac{q}{p}-1}b^q(t) \: dt.
	\end{equation*}
	From this, we may obtain the desired estimate by using again the classical Jensen inequality (this time applicable by our assumption that $q \leq 1$), since it yields
	\begin{equation*}
	\begin{split}
	\int_{R_0} \chi_E \: d\mu &\geq \mu(R_0)^{1-\frac{1}{q}} \bigg ( \int_{R_0} \chi_E \: d\mu \bigg )^{\frac{1}{q}} > n^{\frac{1}{q}}\mu(R_0)^{1-\frac{1}{q}} \bigg ( \int_0^{\mu(E)} t^{\frac{q}{p}-1}b^q(t) \: dt \bigg )^{\frac{1}{q}} \\
	&= n^{\frac{1}{q}}\mu(R_0)^{1-\frac{1}{q}} \lVert \chi_E \rVert_{p,q,b}.
	\end{split}
	\end{equation*}
	Since $n \in \mathbb{N}$ was arbitrary, we have shown that $R_0$ is the set testifying that \ref{P5} does not hold.
\end{proof}

Let us point out here that by using much less direct methods one can weaken the condition \ref{LP5ia} in part \ref{LP5i} to just $1 < p$ (see Corollary~\ref{CP5}), while the condition \ref{LP5ib} in part \ref{LP5i} can also be extended to cover the cases when $q <1$ (see Corollary~\ref{CP5b}). Furthermore, the condition \ref{LP5iia} in part \ref{LP5ii} can be weakened to just $p < 1$ (as follows from Proposition~\ref{PAS}) and the condition \ref{LP5iib} in part \ref{LP5ii} can be extended to cover the cases when $q \geq 1$ (see again Corollary~\ref{CP5b}).

We now turn our attention to $\lVert \cdot \rVert_{(p, q, b)}$. Because different properties of $\lVert \cdot \rVert_{(p, q, b)}$ have different requirements, we will treat them separately. We start by examining the non-triviality.

\begin{proposition} \label{PP4}
	Let $p, q, b$ be as in Definition~\ref{DLK}. Then $\lVert \cdot \rVert_{(p, q, b)}$ satisfies \ref{P4} if and only if $p$ and $b$ satisfy one of the following conditions:
	\begin{enumerate}
		\item \label{Casei} $p \in (1, \infty)$,
		\item \label{Caseii}$p = 1$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(1, \infty)}(t) \rVert_q < \infty$,
		\item \label{Caseiii}$p= \infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q < \infty$.
	\end{enumerate}
	Furthermore,  if none of those conditions is satisfied, then $L^{(p, q, b)}=\{ 0 \}$.
\end{proposition}

\begin{proof}
	The situation is more complicated than when we were examining \ref{P4} in Proposition~\ref{LQ}, because, if we fix some $E \subseteq R$ with measure $\mu(E) < \infty$, then $(\chi_E)^{**}$ is not equal to $\chi_{(0, \mu(E))}$ as before but rather to function $\varrho$ defined as follows
	\begin{equation*}
	\varrho(t) =
	\begin{cases}
	1 & \text{for } t \in (0, \mu(E)), \\
	\frac{\mu(E)}{t} & \text{for } t \in [ \mu(E), \infty).
	\end{cases}
	\end{equation*}
	Thus, in order to verify \ref{P4}, it is necessary to concern yourself not only with integrability/boundedness  on deleted neighbourhoods of zero, but also on deleted neighbourhoods of infinity. To be precise, if $q \in (0, \infty)$, then \ref{P4} is equivalent to whether both of the following conditions are satisfied:
	\begin{enumerate}[label=(\alph*)]
		\item \label{LNa} $t^{\frac{q}{p}-1}b^q(t)$ is integrable on some deleted neighbourhood of zero.
		\item \label{LNb} $t^{\frac{q}{p} -q -1}b^q(t)$ is integrable on some deleted neighbourhood of infinity.
	\end{enumerate}
	Just as in Proposition~\ref{LQ}, condition \ref{LNa} is satisfied for $p < \infty$ by \ref{SV4}, and assumed in the remaining case by \ref{Caseiii}. Similarly, \ref{LNb} is satisfied for $p > 1$ by \ref{SV4} and assumed in the remaining case by \ref{Caseii}. The method for $q = \infty$ remains unchanged, only instead of \ref{SV4}, \ref{SV3} is used.

	Those arguments also show that the presented conditions are necessary for \ref{P4}. The remaining claim that $L^{(p, q, b)}=\{ 0 \}$ when they are not satisfies is a direct consequence.
\end{proof}

We now show some properties that are common for all parameters $p, q, b$ allowed in Definition~\ref{DLK}. We note that these results are not interesting in the cases when $L^{(p, q, b)}=\{ 0 \}$, but we want to emphasise that it is only the property \ref{P4} that gets violated in those situations.

\begin{proposition}  \label{PP5}
	Let $p, q, b$ be as in Definition \ref{DLK}. Then $\lVert \cdot \rVert_{(p, q, b)}$ satisfies \ref{Q1}, \ref{P2}, \ref{P3}, \ref{P5} and \ref{P6}.
\end{proposition}

\begin{proof}
	All the properties except \ref{P5} are direct consequences of the definition, the properties of the maximal function and the properties of the Lebesgue functionals. We thus focus on \ref{P5}.

	Fix some $E \subseteq R$ with measure $\mu(E) < \infty$ and arbitrary $f \in M$. We may assume that $f \neq 0$ (in $M$). Thanks to the Hardy-Littlewood inequality \eqref{THLI} it suffices to prove that there is some finite constant $C_E$ satisfying
	\begin{equation}
	\int_0^{\mu(E)} f^*(t) \: dt \leq  C_E \lVert f \rVert_{(p, q, b)} \label{PP51}
	\end{equation}
	for all $f \in M$. It holds that
	\begin{equation*}
	\begin{split}
		\lVert f \rVert_{(p, q, b)} &\geq \left \lVert \chi_{(\mu(E), \infty)}(t) t^{\frac{1}{p}-\frac{1}{q} -1}b(t) \int_0^t f^{*}(s) \: ds \right \rVert_q \\
		&\geq C \int_0^{\mu(E)} f^{*}(s) \: ds,
	\end{split}
	\end{equation*}
	where
	\begin{equation*}
	C = \lVert \chi_{(\mu(E), \infty)}(t) t^{\frac{1}{p}-\frac{1}{q} -1}b(t) \rVert_q > 0,
	\end{equation*}
	since both $t^{\frac{1}{p}-\frac{1}{q} -1}$ and $b$ are non-zero on $(0, \infty)$.

	Now, if $C < \infty$ then we have just proved \eqref{PP51} with $C_E = C^{-1}$, while in the other case we get from Proposition~\ref{PP4} that $\lVert f \rVert_{(p, q, b)} = \infty$ and \eqref{PP51} thus holds trivially.
\end{proof}

Now, it follows that if we take $q \in [1, \infty]$ we will have that both $f \mapsto f^{**}$ and $\lVert \cdot \rVert_q$ are subadditive, which gives us the last property of r.i.\ Banach function norms. Hence, we have the following proposition.

\begin{proposition} \label{LN}
	Let $p, q, b$ be as in Definition~\ref{DLK} and suppose  $q \in [1, \infty]$. Then $\lVert \cdot \rVert_{(p, q, b)}$ is rearrangement invariant Banach function norm on $M$, and consequently $L^{(p, q, b)}$ is r.i.\ Banach function space, if and only if $p$ and $b$ satisfy one of the following conditions:
	\begin{enumerate}
		\item $p \in (1, \infty)$,
		\item $p = 1$ and $b$ is $\lVert t^{- \frac{1}{q}} b(t) \chi_{(1, \infty)}(t) \rVert_q < \infty$,
		\item $p= \infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q < \infty$.
	\end{enumerate}
	Furthermore,  if none of those conditions is satisfied, then $L^{(p, q, b)}=\{ 0 \}$.
\end{proposition}

\subsection{Embeddings between Lorentz--Karamata spaces} \label{SSE}

The first proposition is a generalisation of a result about embeddings between Lorentz spaces $L^{p,q}$, which can be found for example in \cite[Chapter~4, Proposition~4.2]{BennettSharpley88}.

\begin{proposition} \label{PELK}
	Let $b_1, b_2$ be two s.v.~functions such that $b_2 \lesssim b_1$, let $p \in (0, \infty]$ and suppose  $0 < q < r \leq \infty$. Then
	\begin{equation}
	\lVert \cdot \rVert_{p, r, b_2} \lesssim \Vert \cdot \rVert_{p, q, b_1} \label{PELK1}
	\end{equation}
	or equivalently,
	\begin{equation*}
	L^{p, q, b_1} \hookrightarrow L^{p, r, b_2},
	\end{equation*}
	and the embedding is proper, in the sense that there is a function $f \in M$ such that $f \in L^{p, r, b_2}$ but $f \notin L^{p, q, b_1}$.
\end{proposition}

\begin{proof}
	Fix arbitrary $f \in M$ and $q \in (0, \infty)$ and assume first that $r = \infty$. Then we may use either \ref{LEFF} or \ref{LTb}, in dependence on whether $p < \infty$ of $p = \infty$, and the fact that $f^*$ is non-increasing to compute
	\begin{equation} \label{PELK4}
	\begin{split}
	\lVert f \rVert_{p, \infty, b_2} &\lesssim \sup_{t \in (0, \infty)} t^{\frac{1}{p}} b_1(t) f^*(t) \\
	&\lesssim \sup_{t \in (0, \infty)} \left ( \int_0^{t} s^{\frac{q}{p} -1} b_1^q(s) \: ds \right )^{\frac{1}{q}} f^*(t)  \\
	&= \lVert f \rVert_{p, q, b_1}
	\end{split}
	\end{equation}
	which proves \eqref{PELK1} for $r = \infty$. Note that the finiteness of the expressions in question is of no concern even for $p = \infty$ since in the case when $b_1$ satisfies $\lVert t^{- \frac{1}{q}} b_1(t) \chi_{(0, 1)}(t) \rVert_q < \infty$ then it is also bounded on $(0, 1)$, which implies that $b_2$ is bounded on $(0, 1)$ too, while in the remaining case there is nothing to prove.

	We now use \eqref{PELK4} to compute for $r \in (q, \infty)$
	\begin{equation*}
	\begin{split}
	\lVert f \rVert_{p, r, b_2} &\lesssim \left ( \int_0^{\infty} s^{\frac{q}{p} -1} b_1^q(s) (f^*(s))^q \left [s^{\frac{1}{p}} b_2(s) (f^*(s)) \right ]^{r-q} \: ds \right )^{\frac{1}{r}} \\
	&\leq \lVert f \rVert_{p, \infty, b_2}^{\frac{r-q}{r}} \cdot \lVert f \rVert_{p, q, b_1}^{\frac{q}{r}} \\
	&\lesssim  \lVert f \rVert_{p, q, b_1}
	\end{split}
	\end{equation*}
	which together with \eqref{PELK4} proves \eqref{PELK1} for all the desired parameters $p, q, r, b_1, b_2$. Again, the finiteness of the expression is of no concern thanks to a similar argument as the one presented above.

	As for the second part of the statement, is suffices to consider some $f \in M(R, \mu)$ that satisfies $f^*(t) \approx t^{-\frac{1}{p}} \lvert \log(t) \rvert^{-\frac{1}{q}} b_1^{-1}(t) \chi_{(0, \min\{\mu(R), 1\})}(t)$.
\end{proof}

In the next theorem we present a full characterisation of embeddings between Lorentz--Karamata spaces. This result first appeared in \cite{Bathory18}, but we chose to present it here because the original formulation does not present the different conditions in their explicit form. We also chose to present its full proof since the original one fails to consider some of the cases and is at times quite vague. In our proof, we use the abstract theory of classical Lorentz spaces and the previous Proposition to cover most of the cases, while in some of the remaining ones we employ ideas inspired by the work in \cite{Bathory18} and \cite{Neves02}.
\begin{theorem} \label{TELK}
	Let $b_1, b_2$ be two s.v.\ functions and let $p_1, p_2 \in (0, \infty]$ and $q_1, q_2 \in (0, \infty]$. If $p_i$ is infinite, for $i=1,2$, then we further suppose that the corresponding function $b_i$ satisfies $\lVert t^{- \frac{1}{q}} b_i(t) \chi_{(0, 1)}(t) \rVert_{q_i} < \infty$. We then have the following characterisation:
	\begin{enumerate}
		\item
		If $p_1 > p_2$ then the embedding $L^{p_1, q_1, b_1} \hookrightarrow L^{p_2, q_2, b_2}$ holds if and only if $\mu(R) < \infty$.
		\item
		If $p_1 = p_2$ and $q_1 \leq q_2$ then the embedding $L^{p_1, q_1, b_1} \hookrightarrow L^{p_2, q_2, b_2}$ holds if and only if one of the following condition holds:
		\begin{enumerate}[label=(\alph*)]
			\item
			$p_1=p_2<\infty$ and $\frac{b_2}{b_1}$ is bounded on $(0, \mu(R))$,
			\item
			$p_1 = p_2 = \infty$, $q_1, q_2 < \infty$, and the functions $b_1, b_2$ satisfy
			\begin{equation}
			\sup_{r \in (0, \mu(R))} \frac{\left ( \int_0^{r} t^{-1}b_2^{q_2}(t) \: dt \right )^{\frac{1}{q_2}}}{\left ( \int_0^{r} t^{-1}b_1^{q_1}(t) \: dt \right )^{\frac{1}{q_1}}} < \infty, \label{TELK0}
			\end{equation}
			\item
			$p_1 = p_2 = \infty$, $q_1 < \infty$, $q_2 = \infty$, and the functions $b_1, b_2$ satisfy
			\begin{equation}
			\sup_{r \in (0, \mu(R))} \frac{b_2(r)}{\left ( \int_0^{r} t^{-1}b_1^{q_1}(t) \: dt \right )^{\frac{1}{q_1}}} < \infty, \label{TELK0.1}
			\end{equation}
			\item
			$p_1 = p_2 = \infty$, $q_1 = q_2 = \infty$, and $\frac{b_2}{b_1}$ is bounded on $(0, \mu(R))$.
		\end{enumerate}
		\item
		Let $r$ satisfy $\frac{1}{r} = \frac{1}{q_2} - \frac{1}{q_1}$. If $p_1 = p_2$ and $q_1 > q_2$ then the embedding $L^{p_1, q_1, b_1} \hookrightarrow L^{p_2, q_2, b_2}$ holds if and only if one of the following condition holds:
		\begin{enumerate}[label=(\alph*)]
			\item
			$p_1=p_2<\infty$ and the functions $b_1, b_2$ satisfy
			\begin{equation}
			\int_0^{\mu(R)} t^{-1} \left ( \frac{b_2(t)}{b_1(t)} \right )^{r} \: dt < \infty, \label{TELK0.2}
			\end{equation}
			\item
			$p_1 = p_2 = \infty$, $q_1 < \infty$, and the functions $b_1, b_2$ satisfy
			\begin{equation} \label{TELK00}
			\int_0^{\mu(R)} \left ( \frac{\int_0^{t} s^{-1}b_2^{q_2}(s) \: ds}{\int_0^{t} s^{-1}b_1^{q_1}(s) \: ds} \right )^{\frac{r}{q_1}} t^{-1}b_2^{q_2}(t) \: dt < \infty,
			\end{equation}
			\item
			$p_1=p_2 = \infty$, $q_1 = \infty$, and the functions $b_1, b_2$ satisfy
			\begin{equation*}
			\int_0^{\mu(R)} t^{-1} \left ( \frac{b_2(t)}{b_1(t)} \right )^{r} \: dt < \infty. \label{TELK0.3}
			\end{equation*}
		\end{enumerate}
		\item
		If $p_1 < p_2$ then the embedding $L^{p_1, q_1, b_1} \hookrightarrow L^{p_2, q_2, b_2}$ never holds.
	\end{enumerate}
\end{theorem}

\begin{proof}
	We distinguish several cases.

	Suppose first that both $q_1, q_2 < \infty$. For this case, we will use \cite[Proposition~1]{Stepanov93} (see also \cite[Theorem~3.1]{CarroPick00} for further context). We further distinguish two subcases.

	Let us at first assume that $q_1 \leq q_2$. Then the first part of \cite[Proposition~1]{Stepanov93} tells us that for some two positive weights $v, w$ the inequality
	\begin{equation*}
	\lVert w f^* \rVert_{q_2} \lesssim \lVert v f^* \rVert_{q_1} \label{TELK1}
	\end{equation*}
	holds for all $f \in M$ if and only if
	\begin{equation}
	\sup_{r \in (0, \infty)} \lVert w \cdot \chi_{(0, r)} \rVert_{q_2} \cdot ( \lVert v \cdot \chi_{(0, r)} \rVert_{q_1} )^{-1} < \infty. \label{TELK2}
	\end{equation}
	Since the non-increasing rearrangement $f^*$ of any $f \in M(R, \mu)$ is supported on the set $[0, \mu(R)]$ and therefore $\lVert f \rVert_{p, q, b} = \lVert t^{\frac{1}{p}-\frac{1}{q}}b(t)\chi_{(0,\mu(R))}(t)f^*(t) \rVert_q$ for all $f \in M(R, \mu)$, we observe that the embedding in question is equivalent to \eqref{TELK2} with
	\begin{align*}
	v(t) &= t^{\frac{1}{p_1}-\frac{1}{q_1}}b_1(t)\chi_{(0,\mu(R))}(t), \\
	w(t) &= t^{\frac{1}{p_2}-\frac{1}{q_2}}b_2(t)\chi_{(0,\mu(R))}(t).
	\end{align*}
	We first rewrite it to a more explicit form and get
	\begin{equation}
	\sup_{r \in (0, \mu(R))} \frac{\bigg ( \int_0^{r} t^{\frac{q_2}{p_2}-1}b_2^{q_2}(t) \: dt \bigg )^{\frac{1}{q_2}}}{\bigg ( \int_0^{r} t^{\frac{q_1}{p_1}-1}b_1^{q_1}(t) \: dt \bigg )^{\frac{1}{q_1}}} < \infty. \label{TELK4}
	\end{equation}
	Now, by considering separately the four cases distinguished by finiteness or infiniteness of $p_1$ and $p_2$ and by using \ref{SV1}, \ref{SV2}, \ref{LEFF}, \ref{LTb}, and \ref{SV3}, one obtains that \eqref{TELK4} holds if and only if one of the following conditions holds:
	\begin{enumerate}
		\item $p_1 > p_2$ and $\mu(R) < \infty$,
		\item $p_1 = p_2 < \infty$ and $\frac{b_2(r)}{b_1(r)}$ is bounded on $(0, \mu(R))$,
		\item $p_1 = p_2 = \infty$ and \eqref{TELK0} holds.
	\end{enumerate}

	We have thus covered the subcase $q_1 \leq q_2$ and it remains to consider the subcase $q_1 > q_2$. The approach is similar, only we use the second part of \cite[Proposition~1]{Stepanov93} and thus deduce that the estimate in question is equivalent to the convergence of the integral
	\begin{equation} \label{TELK5}
	\int_0^{\mu(R)} \left ( \frac{\int_0^{t} s^{\frac{q_2}{p_2}-1}b_2^{q_2}(s) \: ds}{\int_0^{t} s^{\frac{q_1}{p_1}-1}b_1^{q_1}(s) \: ds} \right )^{\frac{r}{q_1}} t^{\frac{q_2}{p_2}-1}b_2^{q_2}(t) \: dt.
	\end{equation}
	By a similar approach to that above, we deduce that \eqref{TELK5} holds if and only if one of the following conditions holds:
	\begin{enumerate}
		\item $p_1 > p_2$ and $\mu(R) < \infty$,
		\item $p_1 = p_2 < \infty$ and \eqref{TELK0.2} holds,
		\item $p_1 = p_2 = \infty$ and \eqref{TELK00} holds.
	\end{enumerate}

	We now turn our attention to the cases when at least one of the parameters $q_1, q_2$ is infinite. Those cases do not follow from the more abstract theory of classical Lorentz spaces as using this theory would force us to add unnecessary assumptions on the s.v.~functions $b_1, b_2$. We first show that the presented conditions are sufficient.

	The sufficiency in the case when $p_1 = p_2$, $q_1 \leq q_2 = \infty$ and $\frac{b_2}{b_1}$ is bounded on $(0, \mu(R))$ is covered by Proposition~\ref{PELK} (except the case when also $q_1 = q_2$ which is trivial). That the weaker condition \eqref{TELK0.1} suffices in the case when $p_1 = p_2 = \infty$, $q_1 < q_2 = \infty$ follows by modifying the first step of the proof of Proposition~\ref{PELK}, namely \eqref{PELK4}:
	\begin{equation*} \label{TELK6}
	\begin{split}
	\lVert f \rVert_{\infty, \infty, b_2} &\lesssim \sup_{t \in (0, \infty)} \left ( \int_0^{t} s^{-1} b_1^{q_1}(s) \: ds \right )^{\frac{1}{q_1}} f^*(t)  \\
	&\leq \lVert f \rVert_{\infty, q_1, b_1}.
	\end{split}
	\end{equation*}

	As for the case when $p_1 = p_2$, $q_2 < q_1 = \infty$, the sufficiency follows from the calculation
	\begin{equation*}
	\begin{split}
	\lVert f \rVert_{p_2, q_2, b_2} &\leq \left ( \int_0^{\mu(R)} s^{-1} b_2^{q_2}(s) b_1^{-q_2}(s) \: ds \right )^{\frac{1}{q_2}} \cdot \sup_{s \in (0, \mu(R))} s^{\frac{1}{p_2}} b_1(s) f^*(s) \\
	&\lesssim \lVert f \rVert_{p_1, q_1, b_1},
	\end{split}
	\end{equation*}
	where the last estimate uses \eqref{TELK0.2} and which holds for any $f \in M$.

	It remains to show the sufficiency in the case when $p_1 > p_2$ and $\mu(R) < \infty$. We may assume that $q_1 = \infty$, since we know from Proposition~\ref{PELK} that $L^{p_1, q_1, b_1} \hookrightarrow L^{p_1, \infty, b_1}$. If $q_2 < \infty$, then the desired embedding follows from the calculation
	\begin{equation*}
	\begin{split}
	\lVert f \rVert_{p_2, q_2, b_2} &\leq \left ( \int_0^{\mu(R)} s^{\frac{q_2}{p_2}-\frac{q_2}{p_1}-1} b_2^{q_2}(s) b_1^{-q_2}(s) \: ds \right )^{\frac{1}{q_2}} \cdot \sup_{s \in (0, \mu(R))} s^{\frac{1}{p_1}} b_1(s) (f^*(s)) \\
	&\lesssim \lVert f \rVert_{p_1, \infty, b_1},
	\end{split}
	\end{equation*}
	where the last estimate holds because the function $b_2^{q_2} b_1^{-q_2}$ is s.v.~and which holds for any $f \in M$. The case when $q_2 = \infty$ is similar.

	Finally, we show that the presented conditions for the cases when at least one of the parameters $q_1, q_2$ is infinite are also necessary.

	The case when $p_1 = p_2$ is quite simple. In the subcase when also $q_1 \leq q_2$ it suffices to test the required inequality
	\begin{equation} \label{TELK7}
	\lVert \cdot \rVert_{p_2, q_2, b_2} \lesssim \lVert \cdot \rVert_{p_1, q_1, b_1}
	\end{equation}
	by functions $\chi_{E_t}$, where $t \in (0, \mu(R))$, $E_t \subseteq R$ and $\mu(E_t) = t$, and applying either \ref{LEFF} or monotonicity of $b_i$ as appropriate, depending on the finiteness or infiniteness of the parameters $p_i$ and $q_i$ (the monotonicity in the appropriate cases can be assumed thanks to Proposition~\ref{PbND}). In the remaining subcase when $q_1 > q_2$ we test the inequality \eqref{TELK7} by a function $f \in M(R, \mu)$ satisfying $f^*(t) \approx t^{-\frac{1}{p_1}}b_1(t)^{-1} \chi_{(0, \mu(R))}(t)$.

	In the case when $p_1 < p_2$ one can observe that the embedding does not hold by considering some $f \in M(R, \mu)$ such that  $f^*(t) \approx t^{-\frac{1}{p_1}} \lvert \log(t) \rvert^{-\frac{2}{q_1}} b_1^{-1}(t) \chi_{(0, \min\{\mu(R), 1 \})}(t)$.

	Similarly, in the case when $p_1 > p_2$ and $\mu(R) = \infty$ it is sufficient to consider some $f \in M(R, \mu)$ such that $f^*(t) \approx \chi_{(0, 1)} + t^{-\frac{1}{p_1}} \lvert \log(t) \rvert^{-\frac{2}{q_1}} b_1^{-1}(t) \chi_{[1, \infty)}(t)$.
\end{proof}

We conclude this section by presenting a useful corollary of Proposition~\ref{PELK} and Proposition~\ref{LP5}, the latter of which it extends.

\begin{corollary} \label{CP5b}
	Let $b$ be an s.v.~function. Then the following two statements hold:
	\begin{enumerate}
		\item
		 Let $q \leq 1$ and assume that there are some constants $K, t_0$ such that $b(t) \geq K$ for all $t \in (0, t_0)$ Then $\lVert \cdot \rVert_{1, q, b}$ satisfies \ref{P5}.
		 \item
		 Let $q \geq 1$ and assume that $\lim_{t \to 0^+} b(t) = 0$. Then $\lVert \cdot \rVert_{1, q, b}$ does not satisfy \ref{P5}.
	 \end{enumerate}
\end{corollary}

\subsection{Relations between \texorpdfstring{$L^{p, q, b}$}{L\^p,q,b}  and \texorpdfstring{$L^{(p, q, b)}$}{L\^(p,q,b)}} \label{SSQN}

In this section we study when $L^{p, q, b}=L^{(p, q, b)}$, which is a property that often plays an important role in applications, see for example \cite{BasteroMilman03}, \cite{MilmanPustylnik04}, and \cite{Turcinova19}.

The first observation follows from Propositions~\ref{LQ} and \ref{LN} since they tell us that, for $p<1$, $L^{(p,q,b)}$ is always trivial while $L^{p,q,b}$ never is. So we obtain $p \geq 1$ as a necessary condition.

Note that because we always have for any $f \in M$ that $f^* \leq f^{**}$, we get trivially from properties of $\lVert \cdot \rVert_q$ that $\lVert \cdot \rVert_{p, q, b} \lesssim \lVert \cdot \rVert_{(p, q, b)}$ for all $p, q, b$ allowed by Definition~\ref{DLK}. Thus, we ``only'' have to examine what are the parameters for which $\lVert \cdot \rVert_{(p, q, b)} \lesssim \lVert \cdot \rVert_{p, q, b}$.

We will first show in the next theorem that $p>1$ is sufficient. To this end, we will employ weighted Hardy inequalities.

\begin{theorem}\label{TEQN}
	Let $p, q, b$ be as in Definition~\ref{DLK} and let $q \in (0, \infty]$ and $p \in (1, \infty]$. Then $\lVert \cdot \rVert_{(p, q, b)} \lesssim \lVert \cdot \rVert_{p, q, b}$.
\end{theorem}

\begin{proof}
	Consider first the case when $q \geq 1$.	We employ a weighted Hardy inequality, which can be found for example in \cite[Section~1.3, Theorem~2]{Maz'ya11} and which yields that the desired conclusion is in fact equivalent to the following condition:
	\begin{equation} \label{TEQN1}
	\sup_{r \in (0, \infty)} \left \lVert t^{\frac{1}{p}-\frac{1}{q} -1}b(t)\chi_{(r, \infty)}(t) \right \rVert_q \cdot \left \lVert \left (t^{\frac{1}{p}-\frac{1}{q}}b(t) \right )^{-1}\chi_{(0, r)}(t) \right \rVert_{q'} < \infty.
	\end{equation}

	Firstly, note that both the norms inside the supremum are, thanks to either \ref{SVC} and \ref{SV3} or \ref{SV4}, in dependence on the value of $q$ and $q'$, always finite by our assumption $p \in (1, \infty]$.

	Secondly, since we assume $p>1$ we can use \ref{LEFF} to show that product inside the supremum in \eqref{TEQN1} is on $(0, \infty)$ equivalent to a constant. The condition is thus satisfied.

	Consider now the remaining case $q < 1$. This time we utilise a weighted Hardy-type inequality which has been obtained independently by Carro and Soria in \cite[Proposition~2.6b]{CarroSoria93}, Lai in \cite[Theorem~2.2]{Lai93}, and Stepanov in \cite[Theorem~3b]{Stepanov93} (see also \cite[Theorem~4.1]{CarroPick00} for context as well as a formulation more similar to the problem at hand). From this result, we obtain that the desired inequality holds if and only if the following two conditions hold:
	\begin{align}
		\sup_{r \in (0, \infty)} \bigg ( \int_0^r t^{\frac{q}{p}-1}b^q(t) \: dt \bigg )^{\frac{1}{q}} \cdot \bigg ( \int_0^r t^{\frac{q}{p}-1}b^q(t) \: dt \bigg )^{-\frac{1}{q}} &< \infty, \label{T2EQN1} \\
	\sup_{r \in (0, \infty)} r \cdot \bigg ( \int_r^{\infty} t^{\frac{q}{p}-q-1}b^q(t) \: dt \bigg )^{\frac{1}{q}} \cdot \bigg ( \int_0^r t^{\frac{q}{p}-1}b^q(t) \: dt \bigg )^{-\frac{1}{q}}  &< \infty. \label{T2EQN2}
	\end{align}

	As for the condition \eqref{T2EQN1}, we may assume that the integrals inside the supremum are finite. Indeed, if $p \in (1, \infty)$ this follows from \ref{SV4} while in the remaining case we can assume it since otherwise $L^{p,q,b} = L^{(p,q,b)} = \{0\}$ and there is nothing to prove. Therefore, the function we take supremum of in \eqref{T2EQN1} is identically equal to $1$ on $(0, \infty)$ and the condition is satisfied.

	It remains to verify \eqref{T2EQN2} which follows from \ref{LEFF} when $p< \infty$ and from \ref{LEFF} and \ref{LTb} when $p=\infty$.
\end{proof}

What now remains is to consider the case $p=1$. We shall show that in this case the equality never holds. The reason behind this behaviour is \eqref{LTb1}, more precisely the relation between $b^q$ and $\widehat{b^q}$ (for $q< \infty$) and between $b^{-1}$ and $\widetilde{b^{-1}}$ (for $q = \infty$), where $\widehat{b^q}$ and $\widetilde{b^{-1}}$ are the functions defined in \ref{LHb} (for $b^q$ and $b^{-1}$, respectively), that is
\begin{align*}
\widehat{b^q}(t) &= \int_t^{\infty} s^{-1} b^q(s) \: ds, & \widetilde{b^{-1}}(t) &= \int_0^{t} s^{-1} b^{-1}(s) \: ds.
\end{align*}

We begin with the case $q=1$ which is worth singling out.

\begin{proposition} \label{PEbbH}
	Let $p, q, b$ be as in Definition \ref{DLK} and suppose $q = p = 1$ and that $b$ satisfies $\lVert t^{- 1} b(t) \chi_{(1, \infty)}(t) \rVert_1 < \infty$. Then
	\begin{equation}
	L^{(1,1,b)} = L^{1, 1, \hat{b}}, \label{PEbbH1}
	\end{equation}
	where $\hat{b}$ is the function defined in \ref{LHb}. Consequently, $L^{(1,1,b)} \neq L^{1,1,b}$ for any s.v.~function $b$.
\end{proposition}

\begin{proof}
	The conclusion \eqref{PEbbH1} follows from Definition~\ref{DLK} by simple calculation, using only the classical Fubini's theorem.

	For the remaining part, it follows from Theorem~\ref{TELK} that $L^{1, 1, \hat{b}} = L^{1,1,b}$ if and only if $b \approx \hat{b}$ on $(0, \infty)$, which can never hold as follows from \eqref{LTb1}.
\end{proof}

For the remaining cases we will have to use some abstract theory.

\begin{theorem}\label{T3EQN}
	Let $p, q, b$ be as in Definition \ref{DLK} and suppose $p = 1$. Then $L^{(1,1,b)} \neq L^{1,1,b}$.
\end{theorem}

\begin{proof}
	The case $q=1$ has already been covered in the previous proposition.

	If $q \in (1, \infty)$, then it follows from \cite[Theorem~4]{Sawyer90} that $L^{(1,q,b)} = L^{1,q,b}$ if and only if
	\begin{equation} \label{T3EQN1}
		\int_t^{\infty} s^{-1} b^q(s) \: ds \lesssim \frac{1}{t^q} \int_0^t s^{q-1} b^q(s) \: ds
	\end{equation}
	on $(0,\infty)$. However, by \ref{LEFF}, the right-hand side of \eqref{T3EQN1} is equivalent to $b^q(t)$ while the left-hand side is just $\widehat{b^q}$. Hence, \eqref{T3EQN1} holds if and only if $b^q \approx \widehat{b^q}$ on $(0, \infty)$, which can never hold as follows from \eqref{LTb1}.

	For the case $q \in (0,1)$ we use the same result that we used when proving the second part of Theorem~\ref{TEQN} and that has been obtained independently in \cite[Proposition~2.6b]{CarroSoria93}, \cite[Theorem~2.2]{Lai93}, and \cite[Theorem~3b]{Stepanov93} and which can be also found in \cite[Theorem~4.1]{CarroPick00}. It follows from this result that in order for $L^{(1,q,b)} = L^{1,q,b}$ to hold it is necessary and sufficient that the conditions \eqref{T2EQN1} and \eqref{T3EQN1} hold. Now, while \eqref{T2EQN1} always holds by the same argument as in the proof of Theorem~\ref{TEQN}, the second condition \eqref{T3EQN1} is never satisfied as follows from \eqref{LTb1}.

	Finally if $q=\infty$, it follows from \cite[Theorem~5.1]{GogatishviliSoudsky14} and \ref{LEFF} that $L^{(1,\infty,b)} = L^{1,\infty,b}$ if and only if the function $b \widetilde{b^{-1}}$ is bounded on $(0, \infty)$, which is in turn equivalent to $b^{-1} \approx \widetilde{b^{-1}}$ on $(0, \infty)$ and thus not true for any s.v.~function $b$ (again, by \ref{LTb1}).
\end{proof}

The following corollary collects these results into a single statement.

\begin{corollary} \label{CEQNa}
	Let $p, q, b$ be as in Definition \ref{DLK}. Then $L^{(p,q,b)} = L^{p,q,b}$ if and only if $p>1$.
\end{corollary}

In conclusion of this section, we present two interesting consequences of Corollary~\ref{CEQNa} which extend some properties of $\lVert \cdot \rVert_{(p, q, b)}$ to $\lVert \cdot \rVert_{p, q, b}$.

\begin{corollary} \label{CEQN}
	Let $p, q, b$ be as in Definition~\ref{DLK}, suppose that $p>1$ and that $q \in [1, \infty]$. Then $\lVert \cdot \rVert_{p, q, b}$ is equivalent to an r.i.\ Banach function norm $\lVert \cdot \rVert_{(p, q, b)}$ and the space $L^{p, q, b} = L^{(p, q, b)}$ equipped with this norm is an r.i.\ Banach function space.
\end{corollary}

\begin{corollary} \label{CP5}
	Let $p, q, b$ be as in Definition~\ref{DLK} and suppose that $p>1$. Then $\lVert \cdot \rVert_{p, q, b}$ satisfies \ref{P5}.
\end{corollary}

\subsection{Associate spaces of Lorentz--Karamata spaces} \label{SSLKAS}

We first single out those $p$ for which $(L^{p,q,b})' = \{ 0 \}$ independently of $b$.
\begin{proposition} \label{PAS}
	Let $p, q, b$ be as in Definition \ref{DLK}. If $p \in (0, 1)$ then $(L^{p,q,b})' = \{ 0 \}$.
\end{proposition}

\begin{proof}
	Denote $r= \min \{ 1, \mu(R) \}$ and let $g_0$ be such a function from $M(R, \mu)$ that satisfies $g_0^*(t) = t^{-1} \chi_{(0, r)}(t)$. That such a function exists follows for example from \cite[Chapter~2, Corollary~7.8]{BennettSharpley88}. Then $\lVert g_0 \rVert_{p, q, b} < \infty$, as can be easily verified using \ref{LEFF}. We may assume that $\lVert g_0 \rVert_{p, q, b} \leq 1$, otherwise we would multiply $g_0$ by an appropriate number. On the other hand, $g_0^*$ is certainly not integrable on any deleted neighbourhood of zero and thus for any function $f \in M(R, \mu)$ other than the zero function we can use the resonance of $(R, \mu)$ to obtain
	\begin{equation*}
		\lVert f \rVert_{(L^{p,q,b})'} = \sup_{\substack{g \in M \\ \lVert g \rVert \leq 1}} \int_R \lvert fg \rvert \: d\mu \geq \sup_{\substack{g \in M \\ g^* = g_0^*}} \int_R \lvert fg \rvert \: d\mu = \int_0^{\infty} f^*g_0^*  = \infty.
	\end{equation*}
\end{proof}

This also shows that if $p<1$, then $\lVert \cdot \rVert_{p, q, b}$ does not satisfy \ref{P5}, and thus is never equivalent to a Banach function norm, because its associate functional does not satisfy the condition \ref{P4}.

For proofs of the theorems bellow, we use an appropriate modification of the approach used by Opic and Pick in \cite{OpicPick99} for describing associate spaces of generalised Lorentz-Zygmund spaces.

\begin{theorem}  \label{TAS}
	Let $p, q, b$ be as in Definition~\ref{DLK} and suppose $q \in (1, \infty)$ and that one of the following conditions holds:
	\begin{enumerate}
		\item \label{TASi}
		$p \in [1, \infty)$,
		\item \label{TASii}
		$p= \infty$, $\lVert t^{- \frac{1}{q}} b(t) \chi_{(1, \infty)}(t) \rVert_q = \infty$, and it holds that
		\begin{equation*}
		\sup_{r \in (0, \infty)} \left ( \int_0^{t} s^{-1} b^q(s) \: ds \right )^{\frac{1}{q} } \cdot \left (\int_t^{\infty} s^{-1} b^{-q'}(s) \: ds \right )^{\frac{1}{q'} } < \infty.
		\end{equation*}
	\end{enumerate}
	Then
	\begin{align}
	(L^{p,q,b})' = L^{(p',q',b^{-1})} \label{TAS1}
	\end{align}
	up to equivalence of the defining functionals.
\end{theorem}

\begin{proof}
	We begin by proving that
	\begin{equation*}
	(L^{p, q, b})'~\hookrightarrow~L^{(p', q', b^{-1})}
	\end{equation*}
	or, equivalently, that it holds for all $f \in M$ that
	\begin{equation}
	\lVert f \rVert_{(p', q', b^{-1})}~\lesssim~\lVert f \rVert_{(L_{p, q, b})'}. \label{TAS2}
	\end{equation}
	To this end, let us fix some arbitrary $f \in M$ and denote, for $t \in (0, \infty)$,
	\begin{align*}
	\varrho(t) &= (f^{**}(t))^{q'-1} t^{\frac{q'}{p'} -1}b^{-q'}(t) \\
	g(t) &= \int_t^{\infty} \frac{\varrho(s)}{s} \: ds.
	\end{align*}
	Since $g$ is non-increasing we can find a function $ \hat{g} \in M(R, \mu)$ such that $\hat{g}^* = g$. We may now employ classical Fubini's theorem and the H{\"o}lder inequality \eqref{THAS} to compute
	\begin{equation*}
	\begin{split}
		\lVert f \rVert_{(p', q', b^{-1})}^{q'} &= \int_0^{\infty} \frac{\varrho(t)}{t} \int_0^{t}   f^*(s) \: ds \: dt \\
		&= \int_0^{\infty} f^*(s) g(s) \: ds \\
		&\leq \lVert \hat{g} \rVert_{p, q, b} \cdot \lVert f \rVert_{(L_{p, q, b})'}.
	\end{split}
	\end{equation*}
	Thus, it suffices to show that $\lVert \hat{g} \rVert_{p, q, b} \lesssim \lVert f \rVert_{(p', q', b^{-1})}^{q'-1}$. Rewriting this expression in a more explicit form and using the substitution
	\begin{equation*}
	h(t) = (f^{**}(t))^{q'-1} t^{\frac{q'}{p'} -2} b^{-q'}(t)
	\end{equation*}
	we obtain that it is equivalent to a weighted Hardy inequality
	\begin{equation*}
		\left \lVert t^{\frac{1}{p} - \frac{1}{q}} b(t) \int_t^{\infty} h(s) \: ds \right \lVert_q \lesssim \left \lVert t^{\frac{1}{p} + \frac{1}{q'}} b(t) h(t) \right \rVert_q,
	\end{equation*}
	which, as it is characterised for example in \cite[Section~1.3, Theorem~3]{Maz'ya11}, holds if and only if the following condition holds:
	\begin{equation*}
	\sup_{r \in (0, \infty)} \bigg (\int_0^{r}t^{\frac{q}{p}-1} b^q(t) \:dt \bigg )^{\frac{1}{q}} \cdot \bigg (\int_r^{\infty} t^{-\frac{q'}{p}-1} b^{-q'}(t) \:dt \bigg )^{\frac{1}{q'}} < \infty.
	\end{equation*}

	In the case \ref{TASii} this condition is assumed to hold, while in the remaining case \ref{TASi} we have $p<\infty$ and thus we can use \ref{LEFF} to check that the condition holds in this case too.

	Since $f \in M$ was arbitrary, we have thus proven \eqref{TAS2}. We now turn our attention to the embedding
	\begin{equation*}
	L^{(p', q', b^{-1})}~\hookrightarrow~(L^{p, q, b})',
	\end{equation*}
	which holds if and only if
	\begin{equation}
	\lVert f \rVert_{(L_{p, q, b})'}~\lesssim~\lVert f \rVert_{(p', q', b^{-1})} \label{TAS8}
	\end{equation}
	for all $f \in M$.
	It follows from the Hardy-Littlewood inequality \eqref{THLI} that in order to get \eqref{TAS8} it is sufficient to show that
	\begin{equation*}
	\int_0^{\infty} f^*(t)g^*(t) dt \lesssim \lVert g \rVert_{p, q, b} \cdot \lVert f \rVert_{(p', q', b^{-1})}
	\end{equation*}
	holds for all $f, g \in M$. Now, to get this estimate we use the weighted inequality of Sawyer, proved in \cite[Theorem~1]{Sawyer90}, which for a weight $v$ gives us following:
	\begin{equation}
	\begin{split}
	\int_0^{\infty} f^*(t)g^*(t) dt \leq& \bigg ( \int_0^{\infty} (g^*(t))^q v(t) \: dt \bigg )^{\frac{1}{q}} \\
	&\cdot \Bigg [ \bigg ( \int_0^{\infty} (f^{**}(t))^{q'} \tilde{v}(t) \: dt \bigg )^{\frac{1}{q'}} + \frac{\int_0^{\infty} f^*(t) \: dt}{\big ( \int_0^{\infty} v(t) \: dt \big )^{\frac{1}{q}}} \Bigg ], \label{TAS10}
	\end{split}
	\end{equation}
	where $\tilde{v}(s)$ is given for $t \in (0, \infty)$ by
	\begin{equation}
	\tilde{v}(t) = \frac{t^{q'}v(t)}{\bigg ( \int_0^{t} v(s) \: ds \bigg )^{q'}}. \label{TAS11}
	\end{equation}
	For our purpose, we put, for $t \in (0, \infty)$,
	\begin{equation}
	v(t) = t^{\frac{q}{p} - 1} b^q(t) \label{TAS12}
	\end{equation}
	and see immediately that $v$ is not integrable on $(0, \infty)$, as follows from \ref{SV4} in the case \ref{TASi} and our assumptions in the case \ref{TASii}, and thus by Sawyer's convention $``\frac{\infty}{\infty} = 0 " $ we get that the second summand at the second line of \eqref{TAS10} is zero. Therefore, to obtain \eqref{TAS8} it suffices to show that
	\begin{equation*}
	\tilde{v}(t) \lesssim t^{\frac{q'}{p'}-1} b^{-q'}(t)
	\end{equation*}
	holds for all $t \in (0, \infty)$, which can be done with the help of either \ref{LEFF} or \ref{LTb}, depending on the finiteness of infiniteness of $p$.
\end{proof}

\begin{theorem} \label{T2AS}
	Let $p, q, b$ be as in Definition \ref{DLK} and suppose $q \in (0, 1]$. Then the associate spaces of $L^{p,q,b}$, up to equivalence of the defining functionals, can be described as follows:
	\begin{enumerate}
		\item
		If either $p \in (1, \infty)$, or $p=1$ and $b$ is equivalent to a non-increasing function, then
		\begin{equation} \label{T2AS1}
		(L^{p,q,b})' = L^{(p',\infty,b^{-1})}.
		\end{equation}
		\item
		If $p=\infty$, $q=1$ and $b$ satisfies $\lVert t^{-1} b(t) \chi_{(0, 1)}(t) \rVert_1 < \infty$ then
		\begin{equation} \label{T2AS2}
		(L^{\infty,q,b})' = L^{(1,\infty,\tilde{b}^{-1})}
		\end{equation}
		where $\tilde{b}$ is defined as in Lemma \ref{LTb}, that is
		\begin{equation*}
		\tilde{b}(t) = \int_0^{t} s^{-1} b(s) \: ds.
		\end{equation*}
	\end{enumerate}
\end{theorem}

Note that $\tilde{b}^{-1}$ is non-increasing and therefore $\lVert \tilde{b}^{-1} \chi_{(1, \infty)}(t) \rVert_{\infty} < \infty$.

\begin{proof}
	Suppose first $q = 1$ and either $p \in (1, \infty]$ or both $p=1$  and $b$ is equivalent to a non-increasing function. Those assumptions ensure that the fundamental function $\varphi$ of the r.i.\ quasi-Banach function spaces $L^{p, 1, b}$ is equivalent to a concave function and that it holds for all $f \in M$ that
	\begin{equation*}
	\lVert f \rVert_{\Lambda_{\varphi}} = \int_0^\infty f^* \: d\varphi \approx \int_0^{\infty} t^{\frac{1}{p}-1} b(t) f^*(t) \: dt = \lVert f \rVert_{p, q, b}.
	\end{equation*}
	Therefore, we have that $L^{p,q,b}=\Lambda_{\varphi}$, from which it follows (see Section~\ref{SSFF}) that
	\begin{equation}
	(L^{p,1,b})' = M_{\bar{\varphi}(t)}, \label{T2AS2.5}
	\end{equation}
	where $\bar{\varphi}$ is defined on $[0, \infty)$ by
	\begin{equation*}
	\bar{\varphi}(t) = \frac{t}{\varphi(t)}
	\end{equation*}
	that is, it holds for all $f \in M$ that
	\begin{equation*}
	\lVert f \rVert_{(L^{p,1,b})'} = \sup_{t \in (0, \infty)} \frac{t}{\int_0^{t} s^{\frac{1}{p}-1} b(s) \: ds} f^{**}(t).
	\end{equation*}
	As follows from \ref{LEFF}, the expression on the right-hand side is for $p \in [1, \infty)$ equivalent to $\lVert f \rVert_{(p',\infty,b^{-1})}$. On the other hand, for $p = \infty$ it is exactly the definition of $\lVert \cdot \rVert_{(1, \infty, \tilde{b}^{-1})}$.

	We now turn our attention to the case $q \in (0, 1)$ and either $p \in (1, \infty)$ or $p = 1$ and $b$ is equivalent to a non-increasing function. It follows from Proposition \ref{PELK} that $L^{p,q,b} \hookrightarrow L^{p,1,b}$ which implies $(L^{p,1,b})' \hookrightarrow (L^{p,q,b})'$. This, when put together with \eqref{T2AS2.5}, yields
	\begin{equation}
	M_{\bar{\varphi}(t)} \hookrightarrow (L^{p,q,b})'. \label{T2AS3}
	\end{equation}
	Now, if follows from Proposition \ref{LQ} that $\lVert \cdot \rVert_{p, q, b}$ satisfies \ref{P4}, since we have $p < \infty$. It also satisfies \ref{P5}, because in case $p>1$ we may use Corollary~\ref{CP5}, while in the remaining case our assumptions guarantee that we may use Corollary~\ref{CP5b}. Hence, $\lVert \cdot \rVert_{p, q, b}$ satisfies the conditions from Theorem~\ref{TFA} and we thus have that $(L^{p,q,b})'$ is a Banach function space. The remark after Definition~\ref{DMES} therefore implies that
	\begin{equation}
	(L^{p,q,b})' \hookrightarrow M_{\bar{\psi}}, \label{T2AS4}
	\end{equation}
	where $\psi$ denotes the fundamental function of the r.i.\ quasi-Banach function space $L^{p,q,b}$ and $\bar{\psi}$ denotes the fundamental function of $(L^{p,1,b})'$, which implies $\bar{\psi}(t) \geq \frac{t}{\psi(t)}$ on $(0, \infty)$. Thus, if we had
	\begin{equation}
	\varphi \approx \psi, \label{T2AS5}
	\end{equation}
	then it would follow that $\bar{\varphi}(t) \approx\frac{t}{\psi(t)} \leq \bar{\psi}(t)$ on $(0, \infty)$ and consequently
	\begin{equation}
	M_{\bar{\psi}} \hookrightarrow M_{\bar{\varphi}}. \label{T2AS6}
	\end{equation}
	We could then combine \eqref{T2AS6} with \eqref{T2AS3} and \eqref{T2AS4} to get $(L^{p,q,b})' = M_{\bar{\varphi}(t)}$ and the conclusion of the theorem would follow. So it remains only to verify \eqref{T2AS5}, which follows from \ref{LEFF} by trivial calculation.
\end{proof}

\begin{theorem} \label{T3AS}
	Let $p, q, b$ be as in Definition \ref{DLK} and suppose $q = \infty$ and $p \in [1, \infty]$. If $p=\infty$ then we assume that $b$ is non-decreasing. Then
	\begin{equation*}
	(L^{p,\infty,b})' = L^{p',1,b^{-1}}
	\end{equation*}
	up to equivalence of the defining functionals.
\end{theorem}

We would like to remind the leader that the assumption that $b$ is non-decreasing in the case when $p=q=\infty$ causes no loss of generality thanks to Proposition~\ref{PbND}.

\begin{proof}
	Suppose first that $p \in (1, \infty)$. Then by our conditions on $p, q, b$ we have, by the virtue of Corollary~\ref{CEQNa} and \ref{LEFF} with ($\alpha = \frac{1}{p'}$) that it holds for every $f \in M$ that
	\begin{equation*}
	\lVert f \rVert_{p, \infty, b} \approx \sup_{t \in (0, \infty)} \frac{t}{\int_0^{t} s^{\frac{1}{p'}-1} b^{-1}(s) \: ds} f^{**}(t). \label{T3AS2}
	\end{equation*}
	Therefore, we have from definition that $L^{p,q,b}=M_{\varphi}$, where
	\begin{equation*}
	\varphi(t) = \frac{t}{\int_0^{t} s^{\frac{1}{p'}-1} b^{-1}(s) \: ds},
	\end{equation*}
	from which it follows (see Section~\ref{SSFF}) that
	\begin{equation*}
	(L^{p,q,b})' = \Lambda_{\bar{\varphi}(t)},
	\end{equation*}
	where
	\begin{equation*}
	\bar{\varphi}(t) = \frac{t}{\varphi(t)} = \int_0^{t} s^{\frac{1}{p'}-1} b^{-1}(s) \: ds.
	\end{equation*}
	But since $p' > 1$, as follows from our assumptions, we already know from the proof of Theorem \ref{T2AS} that
	\begin{equation*}
	\Lambda_{\bar{\varphi}(t)} = L^{p',1,b^{-1}}.
	\end{equation*}

	Now let us turn our attention to the remaining cases. We suppose that either $p=1$ or $p=\infty$, in which case we assume that $b$ is non-decreasing. The embedding
	\begin{equation}
	L^{p',1,b^{-1}} \hookrightarrow (L^{p,\infty,b})' \label{T3AS4}
	\end{equation}
	follows from the classical H{\"o}lder inequality for the Lebesgue spaces over $(0, \infty)$, since it yields
	\begin{equation*}
	\begin{split}
	\int_0^{\infty} f^*(t) g^*(t) \: dt &\leq \left \lVert t^{\frac{1}{p} }b(t)g^*(t) \right \rVert_{\infty} \cdot \left \lVert t^{\frac{1}{p'} -1}b^{-1}(t)f^*(t) \right \rVert_1 \\
	&= \lVert g \rVert_{p, \infty, b} \cdot \lVert f \rVert_{p', 1, b^{-1}}.
	\end{split}
	\end{equation*}
	The embedding \eqref{T3AS4} then follows by combining this estimate with the Hardy--Littlewood inequality~\eqref{THLI} and taking the supremum over all $g \in M$ such that $ \lVert g \rVert_{p, \infty, b} \leq 1$.

	For the converse embedding
	\begin{equation}
	(L^{p,\infty,b})' \hookrightarrow L^{p',1,b^{-1}}, \label{T3AS5}
	\end{equation}
	define
	\begin{equation*}
	g(t) = t^{\frac{1}{p'} -1} b^{-1}(t).
	\end{equation*}
	In both of the cases, our assumptions guarantee that $g \approx g^*$, from which it follows that any function $\tilde{g} \in M(R, \mu)$ such that $\tilde{g}^* = g^*$ satisfies $\lVert \tilde{g} \rVert_{p, \infty, b} < \infty$. Furthermore, we may use the fact that $(R, \mu)$ is resonant and the H{\"o}lder inequality \eqref{THAS} to get
	\begin{equation*}
	\begin{split}
	\lVert f \rVert_{p', 1, b^{-1}} &\approx \sup_{\substack{\tilde{g} \in M \\ \tilde{g}^*=g^*}} \int_R \lvert \tilde{g} f \rvert \\
	&\leq \lVert \tilde{g_0} \rVert_{p, \infty, b} \cdot \lVert f \rVert_{(L^{p',\infty,b})'}
	\end{split}
	\end{equation*}
	where $\tilde{g_0}$ is any fixed function from $M(R, \mu)$ such that $\tilde{g_0}^* = g^*$.  The embedding \eqref{T3AS5} follows.
\end{proof}

\subsection{Lorentz--Karamata spaces as Banach function spaces} \label{SSBFS}
In this section we study when a given Lorentz--Karamata space is equivalent to a Banach function space. We provide a full characterisation in the following theorem, most of which is already proved throughout the paper. For the remaining cases, we turn to the more abstract theory of classical Lorentz spaces, relevant aspects of which have been covered in \cite{CarroSoria96}, \cite{CarroRaposo07}, \cite{Sawyer90} and even quite recently in \cite{GogatishviliSoudsky14}.

\begin{theorem} \label{TLKBFS}
	Let $p,q,b$ be as in Definition~\ref{DLK}. The space $L^{p,q,b}$ can be equipped with a Banach function norm equivalent to $\lVert \cdot \rVert_{p, q, b}$ if and only if $q \in [1, \infty]$ and one of the following conditions holds:
	\begin{enumerate}
		\item \label{TLKBFS_1i}
		$p\in (1, \infty)$,
		\item \label{TLKBFS_1ii}
		$p=\infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q < \infty$,
		\item \label{TLKBFS_1iii}
		$p=1$, $q = 1$, and $b$ is equivalent to a non-increasing function.
	\end{enumerate}
\end{theorem}

\begin{proof}
	As for the sufficiency, parts~\ref{TLKBFS_1i} and \ref{TLKBFS_1ii} constitute the content of Corollary~\ref{CEQN} while the part~\ref{TLKBFS_1iii} follows the characterisation of normability of the classical Lorentz space $\Lambda^1$ obtained in \cite[Theorem~2.3]{CarroSoria96} and \ref{LEFF}. Alternatively, one can prove that \ref{TLKBFS_1iii} is sufficient by using Theorem~\ref{T3AS}, Corollary~\ref{CP5}, Theorem~\ref{TFA}, and Theorem~\ref{TELK}.

	Let us now turn our attention to the necessity of those conditions. The case when $q \in (0,1)$ follows from \cite[Theorem~2.5.8]{CarroRaposo07}, the case when $p \in (0,1)$ from Proposition~\ref{PAS}, and the case when $p=\infty$ and $\lVert t^{- \frac{1}{q}} b(t) \chi_{(0, 1)}(t) \rVert_q = \infty$ from Proposition~\ref{LN}. Moreover, to cover all the cases when $p=1$ and $q \in (1, \infty]$ one only has to combine Corollary~\ref{CEQNa} with \cite[Theorem~4]{Sawyer90} (for $q \in (1, \infty)$) and \cite[Theorem~5.1]{GogatishviliSoudsky14} (for $q=\infty$).

	Finally, to obtain the necessity for the case when $p=q=1$ we turn again to \cite[Theorem~2.3]{CarroSoria96} which, when combined with \ref{LEFF}, provides as a necessary condition for normability that there must exist a constant $C > 0$ such that it holds for all $0 < s \leq r < \infty$ that
	\begin{equation} \label{TLKBFS1}
		b(r) \leq C b(s).
	\end{equation}
	We now assert that $b$ is not equivalent to any non-increasing function, as a special case of which we get that $b$ is not equivalent to $B$ defined for $s \in (0, \infty)$ by
	\begin{equation*}
		B(s) = \sup_{t \in [s, \infty)} b(t).
	\end{equation*}
	This property together with the fact that $B \geq b$ on $(0,\infty)$ allows us to find, for any given $C>0$, the required pair of numbers $r, s$ that violates \eqref{TLKBFS1}.
\end{proof}

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TITLE: Cheap calculation/estimation for $\text{tr}(\Sigma^2)$ with samples having covariance matrix $\Sigma$
QUESTION [1 upvotes]: Suppose a sample $\boldsymbol{x}_1,\dots, \boldsymbol{x}_N \in \mathbb{R}^d$ is given with sample matrix $\boldsymbol{X}\in\mathbb{R}^{N\times d}$ and covariance matrix $\boldsymbol{\Sigma}=\frac{1}{N-1}\boldsymbol{X}^T\boldsymbol{X}$. It is know, that for the trace of the matrix $\text{tr}(\boldsymbol{\Sigma})=\frac{1}{N-1}\sum_{j=1}^N ||\boldsymbol{x}_j||^2$, that means, $\text{tr}(\boldsymbol{\Sigma})$ can be calculated without calculating the covariance matrix or its eigenvalues.
My question: Can $\text{tr}(\boldsymbol{\Sigma}^2)$ be calculated (or good estimated) out of the data without calculating explicitly $\boldsymbol{\Sigma}$ or its eigenvalues?
$\text{tr}(\boldsymbol{\Sigma}^2)=\sum_{i=1}^d \lambda_i^2$ for the eigenvalues $\lambda_i$ of $\boldsymbol{\Sigma}$, so $\text{tr}(\boldsymbol{\Sigma})^2$ is obiously an upper bound.
I am grateful for any ideas.

REPLY [1 votes]: Let $M$ be a matrix for which I have a magic oracle which tells me $Mx$ when presented with $x$. Let $\omega$ be a random vector which isotropic in the sense that $\mathbb{E}[\omega\omega^*] = I$. (For instance, $\omega$ can be a vector of IID standard normal random variables.) Then one can easily see that $\mathbb{E}[\omega^*(M\omega)] = \operatorname{tr}(M)$. (This follows from the cyclic property of the trace.) Thus, averaging the quantity $\omega^*(M\omega)$ for $k$ random instances of $\omega$ yields an unbiased estimate of the trace. Like other sampling-based approximations the root-mean-square error of this approximation with $k$ samples scales like $k^{-1/2}$, not great but not terrible if you only need an estimate.
In your case for $M = \Sigma^2$, the oracle proceeds by multiplying $x$ by $\Sigma$ twice. Each multiplication by $\Sigma$ is done as $\Sigma y = (N-1)^{-1} X^T (Xy)$, which takes $O(Nd)$ time.
This idea for trace estimation is very well-studied. See Chapter 4 of the following review for more analysis. There may be more sophisticated approaches for this particular problem, but this technique is a fully general technique which works for any $M$ with a matrix-vector multiplication oracle.


TITLE: Method of characteristics for quasilinear PDE $u_x+u_y=2\sqrt{u}$
QUESTION [2 upvotes]: I'm having trouble with solving the quasilinear PDE
$$\begin{cases} u_x+u_y&=2\sqrt{u}, \\ u(x,x)&=g(x) \end{cases}$$
via method of characteristics as in this paper.
My attempt:
First I started by formulating the ODEs
$$\begin{cases} \dot{x}&=1 \\ \dot{y}&=1 \\ \dot{z}&=2\sqrt{z}. \end{cases}$$
Solving those and applying the initial conditions gives me
$$\begin{cases} x&=t+x_0 \\ y&=t+x_0 \\ z&=\frac{1}{4}(2t+z_0)^2=\frac{1}{4}(2t+2\sqrt{g(x_0)})^2. \end{cases}$$
But how do I eliminate $t$ and $x_0$ if $x=y$?

REPLY [2 votes]: The system
$$\left\lbrace
\begin{aligned}
t+x_0 = x\\
t+x_0=y
\end{aligned}\right.
$$
has no solution $(t, x_0)$ if $x≠y$, and it has infinitely many solutions if $x=y$. Therefore, we can't express the solution uniquely in terms of $x$, $y$. Situation is somewhat similar to this post and related ones.

The Lagrange-Charpit equations
$$
\frac{dx}{1} = \frac{dy}{1} = \frac{du}{2\sqrt u}
$$
provides the characteristic families $x-y = c_1$ and $x - \sqrt{u} = c_2$. The second characteristic family might be rewritten $u = (x - c_2)^2$). General solutions read
$$
u = \big(x - f(x-y)\big)^2
$$
where $c_2 = f(c_1)$ involves an arbitrary function $f$. If $x=y$, we find $u = (x - f(0))^2$ which needs to equal $g(x)$ according to the boundary condition. Two cases arise:

if $g(x) = (x - c_3)^2$, then we obtain an infinity of solutions under the constraint $f(0) = c_3$;
else, no function $f$ matches the boundary condition: there is no solution.

Again, we can't express the solution uniquely in terms of $x$, $y$.


TITLE: $p$-adic valuation of $P_n(p)$ if $P_n$ is the $n$th Legendre polynomial
QUESTION [5 upvotes]: In 3-adic valuation of a sum involving binomial coefficients  the $p$-adic valuation of $P_n(p)$ has been obtained.
Computations indicate that  $M_n(p)=\sum\binom{n}{k}^2(p-1)^k$ for odd $p$ has the same $p$-adic valuation:
$$\nu_p (P_n(p)) = \nu_p (M_n(p)).$$
Is this true for all odd $p$?
(Note that $P_n(3)=M  _n(3)$).

REPLY [1 votes]: This is a suggestion, in case it helps.
The two sequences satisfy the following recurrences, respectively,
\begin{align}
nP_n(p)&=p(2n-1)P_{n-1}(p)-(n-1)P_{n-2}(p) \\
nM_n(p)&=p(2n-1)M_{n-1}(p)-(n-1)M_{n-2}(p)\cdot(p-2)^2.
\end{align}


TITLE: ${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal
QUESTION [0 upvotes]: Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties.
Now from $F:V \to W$ using Hilbert Nullstelensatz we have the pullback $F^*: \Bbb C[W] \to \Bbb C[V]$. Now indeed given a point $p \in V$ we think it as a maximal ideal $m_v \in maxSpec \Bbb C[V]$. Then the image of $p$ under $F$ corresponds to the maximal ideal $(F^*)^{-1}(m_v) \in maxSpec \Bbb C[W]$ This is where I have the problem in showing that $(F^*)^{-1}(m_v)$ is a maximal ideal in $\Bbb C[W]$ which is not true in general.
My attempt: Suppose $(F^*)^{-1}(m_v)$ is not maximal then $\exists m_w \subset \Bbb C[W]$ maximal s.t $(F^*)^{-1}(m_v) \subsetneq m_w$ then $m_w$ corresponds to a point $q \in W$.
Let us comeback in the pullback part as we should have all the ingredients ready in our hand: for any $g \in \Bbb C[W]$ $(F^*)(g)=g \circ F$ so for any $g \in (F^*)^{-1}(m_v)$ we have $(F^*)(g)=g \circ F \in m_v$.
Now can we proceed from here, I think if we can prove $F^*$ is surjective then also we are done.

REPLY [1 votes]: Rings and algebras are assumed to be commutative.
Let $f\colon A\to B$ be a $k$-algebra homomorphism. If $J$ is a maximal ideal of $B$ and $I=f^{-1}(J)$, then there is an injective $k$-algebra homomorphism
$$
A/I\to B/J
$$
where the codomain is a field. If $B$ is finite-dimensional over $k$, then so are $B/J$ and $A/I$. Hence $A/I$ is an artinian ring which is a domain, hence a field. So $I$ is maximal.