# Optimal couples of rearrangement invariant spaces for the Riesz potential on the bounded domain

## Abstract

We prove continuity of the Riesz potential operator in optimal couples of rearrangement invariant function spaces defined in ${\mathbf{R}}^{n}$ with the Lebesgue measure.

MSC:46E30, 46E35.

## 1 Introduction

Let be the space of all locally integrable functions f on $\mathrm{\Omega }\subset {\mathbf{R}}^{n}$ with the Lebesgue measure, finite almost everywhere, and let ${\mathcal{M}}^{+}$ be the space of all non-negative locally integrable functions on $\left(0,\mathrm{\infty }\right)$ with respect to the Lebesgue measure, finite almost everywhere. We shall also need the following two subclasses of ${\mathcal{M}}^{+}$. The subclass M consists of those elements g of ${\mathcal{M}}^{+}$ for which there exists an $m>0$ such that ${t}^{m}g\left(t\right)$ is increasing. The subclass ${M}_{0}$ consists of those elements g of ${\mathcal{M}}^{+}$ which are decreasing.

The Riesz potential operator ${R}_{\mathrm{\Omega }}^{s}$, $0, $n\ge 1$ is defined formally by

${R}_{\mathrm{\Omega }}^{s}f\left(x\right)={\int }_{\mathrm{\Omega }}f\left(y\right){|x-y|}^{s-n}\phantom{\rule{0.2em}{0ex}}dy,\phantom{\rule{1em}{0ex}}f\in {\mathcal{M}}^{+};\phantom{\rule{2em}{0ex}}|\mathrm{\Omega }|=1.$
(1.1)

We shall consider rearrangement invariant quasi-Banach spaces E, continuously embedded in ${L}^{1}\left({\mathbf{R}}^{n}\right)+{L}^{\mathrm{\infty }}\left({\mathbf{R}}^{n}\right)$, such that the quasi-norm ${\parallel f\parallel }_{E}$ in E is generated by a quasi-norm ${\rho }_{E}$, defined on ${\mathcal{M}}^{+}$ with values in $\left[0,\mathrm{\infty }\right]$, in the sense that ${\parallel f\parallel }_{E}={\rho }_{E}\left({f}^{\ast }\right)$. In this way equivalent quasi-norms ${\rho }_{E}$ give the same space E. We suppose that E is nontrivial. Here ${f}^{\ast }$ is the decreasing rearrangement of f, given by

${f}^{\ast }\left(t\right)=inf\left\{\lambda >0:{\mu }_{f}\left(\lambda \right)\le t\right\},\phantom{\rule{1em}{0ex}}t>0,$

where ${\mu }_{f}$ is the distribution function of f, defined by

${\mu }_{f}\left(\lambda \right)=|\left\{x\in {\mathbf{R}}^{n}:|f\left(x\right)|>\lambda \right\}{|}_{n},$

${|\cdot |}_{n}$ denoting the Lebesgue n-measure.

Note that ${f}^{\ast }\left(t\right)=0$, if $t>1$.

There is an equivalent quasi-norm ${\rho }_{p}$ that satisfies the triangle inequality ${\rho }_{p}^{p}\left({g}_{1}+{g}_{2}\right)\le {\rho }_{p}^{p}\left({g}_{1}\right)+{\rho }_{p}^{p}\left({g}_{2}\right)$ for some $p\in \left(0,1\right)$ that depends only on the space E (see ).

We say that the norm ${\rho }_{E}$ is K-monotone (cf. , p.84, and also , p.305) if

${\int }_{0}^{t}{g}_{1}^{\ast }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\le {\int }_{0}^{t}{g}_{2}^{\ast }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}{\rho }_{E}\left({g}_{1}^{\ast }\right)\le {\rho }_{E}\left({g}_{2}^{\ast }\right),\phantom{\rule{1em}{0ex}}{g}_{1},{g}_{2}\in {\mathcal{M}}^{+}.$
(1.2)

Then ${\rho }_{E}$ is monotone, i.e., ${g}_{1}\le {g}_{2}$ implies ${\rho }_{E}\left({g}_{1}\right)\le {\rho }_{E}\left({g}_{2}\right)$.

We use the notations ${a}_{1}\lesssim {a}_{2}$ or ${a}_{2}\gtrsim {a}_{1}$ for non-negative functions or functionals to mean that the quotient ${a}_{1}/{a}_{2}$ is bounded; also, ${a}_{1}\approx {a}_{2}$ means that ${a}_{1}\lesssim {a}_{2}$ and ${a}_{1}\gtrsim {a}_{2}$. We say that ${a}_{1}$ is equivalent to ${a}_{2}$ if ${a}_{1}\approx {a}_{2}$.

We say that the norm ${\rho }_{E}$ satisfies the Minkovski inequality if for the equivalent quasi-norm ${\rho }_{p}$,

${\rho }_{p}^{p}\left(\sum {g}_{j}\right)\lesssim \sum {\rho }_{p}^{p}\left({g}_{j}\right),\phantom{\rule{1em}{0ex}}{g}_{j}\in {\mathcal{M}}^{+}.$
(1.3)

For example, if E is a rearrangement invariant Banach function space as in , then by the Luxemburg representation theorem ${\parallel f\parallel }_{E}={\rho }_{E}\left({f}^{\ast }\right)$ for some norm ${\rho }_{E}$ satisfying (1.2) and (1.3). More general example is given by the Riesz-Fischer monotone spaces as in , p.305.

Recall the definition of the lower and upper Boyd indices ${\alpha }_{E}$ and ${\beta }_{E}$. Let

${h}_{E}\left(u\right)=sup\left\{\frac{{\rho }_{E}\left({g}_{u}^{\ast }\right)}{{\rho }_{E}\left({g}^{\ast }\right)}:g\in {\mathcal{M}}^{+}\right\},\phantom{\rule{2em}{0ex}}{g}_{u}\left(t\right):=g\left(t/u\right)$

be the dilation function generated by ${\rho }_{E}$. Then

${\alpha }_{E}:=\underset{0

If ${\rho }_{E}$ is monotone, then the function ${h}_{E}$ is submultiplicative, increasing, ${h}_{E}\left(1\right)=1$, ${h}_{E}\left(u\right){h}_{E}\left(1/u\right)\ge 1$, hence $0\le {\alpha }_{E}\le {\beta }_{E}$. If ${\rho }_{E}$ is K-monotone, then by interpolation (analogously to , p.148), we see that ${h}_{E}\left(s\right)\le max\left(1,s\right)$. Hence in this case we have also ${\beta }_{E}\le 1$.

Using the Minkovski inequality for the equivalent quasi-norm ${\rho }_{p}$ and monotonicity of ${f}^{\ast }$, we see that

(1.4)

where ${f}^{\ast \ast }\left(t\right)=\frac{1}{t}{\int }_{0}^{t}{f}^{\ast }\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$. The main goal of this paper is to prove continuity of the Riesz potential operator ${R}_{\mathrm{\Omega }}^{s}:E↦G$ in optimal couples of rearrangement invariant function spaces E and G, where ${\parallel f\parallel }_{G}:={\rho }_{G}\left({f}^{\ast }\right)$. It is convenient to introduce the following classes of quasi-norms, where the optimality of ${R}_{\mathrm{\Omega }}^{s}:E↦G$ is investigated. Let ${\mathcal{N}}_{d}$ stand for all domain quasi-norms ${\rho }_{E}$, which are monotone, rearrangement invariant, satisfying Minkowski’s inequality, ${\rho }_{E}\left({\chi }_{\left(0,1\right)}\right)<\mathrm{\infty }$ and

$E↪{L}^{1}\left(\mathrm{\Omega }\right).$
(1.5)

Let ${\mathcal{N}}_{t}$ consist of all target quasi-norms ${\rho }_{G}$ that are monotone, satisfy Minkowski’s inequality, ${\rho }_{G}\left({\chi }_{\left(0,1\right)}\right)<\mathrm{\infty }$, ${\rho }_{G}\left({\chi }_{\left(1,\mathrm{\infty }\right)}{t}^{s/n-1}\right)<\mathrm{\infty }$ and

$G↪{\mathrm{\Lambda }}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)\left({\mathcal{R}}^{n}\right),$
(1.6)

where ${\chi }_{\left(a,b\right)}$ is the characteristic function of the interval $\left(a,b\right)$, $0. Note that technically it is more convenient not to require that the target quasi-norm ${\rho }_{G}$ is rearrangement invariant. Of course, the target space G is rearrangement invariant, since ${\parallel f\parallel }_{G}={\rho }_{G}\left({f}^{\ast }\right)$. Finally, let $\mathcal{N}:={\mathcal{N}}_{d}×{\mathcal{N}}_{t}$.

We say that the couple $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$ is admissible for the Riesz potential if the following estimate is valid:

${\rho }_{G}\left({\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\right)\lesssim {\rho }_{E}\left({f}^{\ast }\right).$
(1.7)

Moreover, ${\rho }_{E}\left(E\right)$ is called domain quasi-norm (domain space), and ${\rho }_{G}$ (G) is called a target quasi-norm (target space).

For example, by Theorem 2.2 below (the sufficient part), the couple $E={\mathrm{\Lambda }}^{q}\left({t}^{s/n}w\right)\left(\mathrm{\Omega }\right)$, $G={\mathrm{\Lambda }}^{q}\left(v\right)$, $1\le q\le \mathrm{\infty }$, is admissible if ${\beta }_{E}<1$ and v is related to w by the Muckenhoupt condition :

${\left({\int }_{0}^{t}{\left[v\left(s\right)\right]}^{q}\phantom{\rule{0.2em}{0ex}}ds/s\right)}^{1/q}{\left({\int }_{t}^{\mathrm{\infty }}{\left[w\left(s\right)\right]}^{-r}\phantom{\rule{0.2em}{0ex}}ds/s\right)}^{1/r}\lesssim 1,\phantom{\rule{2em}{0ex}}1/q+1/r=1.$
(1.8)

Definition 1.2 (Optimal target quasi-norm)

Given the domain quasi-norm ${\rho }_{E}\in {\mathcal{N}}_{d}$, the optimal target quasi-norm, denoted by ${\rho }_{G\left(E\right)}$, is the strongest target quasi-norm, i.e.,

${\rho }_{G}\left({g}^{\ast }\right)\lesssim {\rho }_{G\left(E\right)}\left({g}^{\ast }\right),\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+},$
(1.9)

for any target quasi-norm ${\rho }_{G}\in {\mathcal{N}}_{t}$ such that the couple ${\rho }_{E}$, ${\rho }_{G}$ is admissible.

Definition 1.3 (Optimal domain quasi-norm)

Given the target quasi-norm ${\rho }_{G}\in {\mathcal{N}}_{t}$, the optimal domain quasi-norm, denoted by ${\rho }_{E\left(G\right)}$, is the weakest domain quasi-norm, i.e.,

${\rho }_{E\left(G\right)}\left({g}^{\ast }\right)\lesssim {\rho }_{E}\left({g}^{\ast }\right),\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+},$
(1.10)

for any domain quasi-norm ${\rho }_{E}\in {\mathcal{N}}_{d}$ such that the couple ${\rho }_{E}$, ${\rho }_{G}$ is admissible.

Definition 1.4 (Optimal couple)

The admissible couple ${\rho }_{E}$, ${\rho }_{G}$ is said to be optimal if ${\rho }_{E}={\rho }_{E\left(G\right)}$ and ${\rho }_{G}={\rho }_{G\left(E\right)}$.

The optimal quasi-norms are uniquely determined up to equivalence, while the corresponding optimal quasi-Banach spaces are unique.

Here we give a characterization of all admissible couples $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$. It is convenient to define the case ${\beta }_{E}=1$ as limiting and the case ${\beta }_{E}<1$ as sublimiting.

Theorem 2.1 (General case ${\beta }_{E}\le 1$)

The couple $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$ is admissible if and only if

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+}\phantom{\rule{0.25em}{0ex}}\mathit{\text{or}}\phantom{\rule{0.25em}{0ex}}g\in {M}_{0},$
(2.1)

where

${S}_{1}g\left(t\right):=\left\{\begin{array}{cc}{t}^{s/n-1}{\int }_{0}^{t}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du+{\int }_{t}^{1}{u}^{s/n}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du/u,\hfill & 01,0
(2.2)

Proof First we prove

${\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\lesssim {S}_{1}{f}^{\ast }.$
(2.3)

We are going to use real interpolation for quasi-Banach spaces. First we recall some basic definitions. Let $\left({A}_{0},{A}_{1}\right)$ be a couple of two quasi-Banach spaces (see [2, 5]) and let

$K\left(t,f\right)=K\left(t,f;{A}_{0},{A}_{1}\right)=\underset{f={f}_{0}+{f}_{1}}{inf}\left\{{\parallel {f}_{0}\parallel }_{{A}_{0}}+t{\parallel {f}_{1}\parallel }_{{A}_{1}}\right\},\phantom{\rule{1em}{0ex}}f\in {A}_{0}+{A}_{1}$

be the K-functional of Peetre (see ). By definition, the K-interpolation space ${A}_{\mathrm{\Phi }}={\left({A}_{0},{A}_{1}\right)}_{\mathrm{\Phi }}$ has a quasi-norm

${\parallel f\parallel }_{{A}_{\mathrm{\Phi }}}={\parallel K\left(t,f\right)\parallel }_{\mathrm{\Phi }},$

where Φ is a quasi-normed function space with a monotone quasi-norm on $\left(0,\mathrm{\infty }\right)$ with the Lebesgue measure and such that $min\left\{1,t\right\}\in \mathrm{\Phi }$. Then (see )

${A}_{0}\cap {A}_{1}↪{A}_{\mathrm{\Phi }}↪{A}_{0}+{A}_{1},$

where by $X↪Y$ we mean that X is continuously embedded in Y. If ${\parallel g\parallel }_{\mathrm{\Phi }}={\left({\int }_{0}^{\mathrm{\infty }}{t}^{-\theta q}{g}^{q}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt/t\right)}^{1/q}$, $0<\theta <1$, $0, we write ${\left({A}_{0},{A}_{1}\right)}_{\theta ,q}$ instead of ${\left({A}_{0},{A}_{1}\right)}_{\mathrm{\Phi }}$ (see ).

Using the Hardy-Littlewood inequality ${\int }_{{\mathbf{R}}^{n}}|f\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{0}^{\mathrm{\infty }}{f}^{\ast }\left(t\right){g}^{\ast }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$, we get the well-known mapping property

${R}_{\mathrm{\Omega }}^{s}:{\mathrm{\Lambda }}^{1}\left({t}^{s/n}\right)\left(\mathrm{\Omega }\right)↦{L}^{\mathrm{\infty }}\left({\mathcal{R}}^{n}\right)$

and by the Minkovski inequality for the norm ${f}^{\ast \ast }$ we get

${R}_{\mathrm{\Omega }}^{s}:{L}^{1}\left(\mathrm{\Omega }\right)↦{\mathrm{\Lambda }}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)\left({\mathcal{R}}^{n}\right).$

Hence

${t}^{1-s/n}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\left(t\right)\lesssim K\left({t}^{1-s/n},f;{L}^{1}\left(\mathrm{\Omega }\right),{\mathrm{\Lambda }}^{1}\left({t}^{s/n}\right)\left(\mathrm{\Omega }\right)\right),$

therefore (see , Section 5.7)

${t}^{1-s/n}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\left(t\right)\lesssim \left\{\begin{array}{cc}{\int }_{0}^{t}{f}^{\ast }\left(u\right)\phantom{\rule{0.2em}{0ex}}du+{t}^{1-s/n}{\int }_{t}^{1}{u}^{s/n}{f}^{\ast }\left(u\right)\phantom{\rule{0.2em}{0ex}}du/u,\hfill & 01,\hfill \end{array}$

implies

${\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\left(t\right)\lesssim {S}_{1}{f}^{\ast }\left(t\right).$

It is clear that (1.7) follows from (2.1) and (2.3).

Now we prove that (1.7) implies (2.1). To this end we choose the test function in the form $f\left(x\right)=g\left(c{|x|}^{n}\right)$, $g\in {\mathcal{M}}^{+}$, so that ${f}^{\ast }\left(t\right)={g}^{\ast }\left(t\right)$ for some positive constant c (cf. ). Then

${R}_{\mathrm{\Omega }}^{s}f\left(x\right)={\int }_{|y|<|x|}g\left(c{|y|}^{n}\right){|x-y|}^{s-n}\phantom{\rule{0.2em}{0ex}}dy+{\int }_{|y|>|x|}g\left(c{|y|}^{n}\right){|x-y|}^{s-n}\phantom{\rule{0.2em}{0ex}}dy,$

whence

$|{R}_{\mathrm{\Omega }}^{s}f\left(x\right)|\gtrsim {|x|}^{s-n}{\int }_{0}^{c{|x|}^{n}}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du+{\int }_{c{|x|}^{n}}^{|\mathrm{\Omega }|=1}{u}^{s/n-1}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du\gtrsim {\chi }_{\left(0,1\right)}\left({S}_{1}g\right)\left(c{|x|}^{n}\right).$

Note that ${\chi }_{\left(0,1\right)}{S}_{1}g\approx {\chi }_{\left(0,1\right)}{Q}_{1}{T}_{1}^{\prime }g+{\chi }_{\left(0,1\right)}{\int }_{0}^{1}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du$, where

${Q}_{1}g:={\int }_{t}^{1}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du/u,\phantom{\rule{1em}{0ex}}t<1,$

and

${T}_{1}^{\prime }g\left(t\right):=\left\{\begin{array}{cc}{t}^{s/n-1}{\int }_{0}^{t}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du,\hfill & 01,0

hence ${\chi }_{\left(0,1\right)}{S}_{1}g$ is decreasing, therefore

${|{R}_{\mathrm{\Omega }}^{s}f|}^{\ast }\left(t\right)\gtrsim {\chi }_{\left(0,1\right)}{S}_{1}g\left(t\right).$
(2.4)

Thus, if (1.7) is given, then (2.4) implies (2.1). □

In the sublimiting case ${\beta }_{E}<1$ we can simplify the condition (2.1), replacing ${S}_{1}$ by ${T}_{1}$. Here

${T}_{1}g\left(t\right):=\left\{\begin{array}{c}{t}^{s/n-1}{\int }_{t}^{1}{u}^{s/n}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du/u,\phantom{\rule{1em}{0ex}}01.\hfill \end{array}$
(2.5)

Theorem 2.2 (Sublimiting case ${\beta }_{E}<1$)

The couple $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$ is admissible if and only if

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{1em}{0ex}}g\in M,$
(2.6)

where we recall that

$M:=\left\{g\in {\mathcal{M}}^{+}\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}{t}^{m}g\left(t\right)\phantom{\rule{0.25em}{0ex}}\mathit{\text{is increasing for some}}\phantom{\rule{0.25em}{0ex}}m>0\right\}.$

Proof Let ${\rho }_{E}$, ${\rho }_{G}$ be an admissible couple, then

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)\lesssim {\rho }_{E}\left(g\right).$

Since ${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}g\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)$, it follows that ${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}g\right)\lesssim {\rho }_{E}\left(g\right)$, $g\in M$. Now we need to prove sufficiency of (2.6). We have

${\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\approx {\chi }_{\left(0,1\right)}{T}_{1}{g}^{\ast \ast }+{\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(1\right),$

so

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}{g}^{\ast \ast }\right)+{\rho }_{G}\left({\chi }_{\left(0,1\right)}\right){g}^{\ast \ast }\left(1\right)$

implies

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)\lesssim {\rho }_{E}\left({g}^{\ast }\right).$

□

In the subcritical case ${\alpha }_{E}>s/n$ we have another simplification of (2.1).

Theorem 2.3 (Case ${\alpha }_{E}>s/n$)

The couple $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$ is admissible if and only if

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{2em}{0ex}}g\in {M}_{0}:=\left\{g\in {\mathcal{M}}^{+},g\phantom{\rule{0.25em}{0ex}}\mathit{\text{is decreasing}}\right\},$
(2.7)

where

${T}_{1}^{\prime }g\left(t\right):=\left\{\begin{array}{cc}{t}^{s/n-1}{\int }_{0}^{t}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du,\hfill & 01,0

Proof Let $\left({\rho }_{E},{\rho }_{G}\right)\in \mathcal{N}$ be admissible, then

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{1em}{0ex}}g\in {M}_{0}.$

As

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right),$

we have

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\right)\lesssim {\rho }_{E}\left(g\right).$

For the reverse, it is enough to check that (2.7) implies (2.1) for $g\in {M}_{0}$, or

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{1em}{0ex}}g\in {M}_{0}.$

As

${\chi }_{\left(0,1\right)}{T}_{1}g\lesssim {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}g\right),$

so

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}g\right)\lesssim {\rho }_{E}\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}g\right)\approx {\rho }_{E}\left({t}^{-s/n}{Q}_{1}\left({t}^{s/n}g\right)\right)\lesssim {\rho }_{E}\left(g\right).$

Here we use

${\rho }_{E}\left({Q}_{1}\left({t}^{-s/n}g\right)\right)\lesssim {\rho }_{E}\left({t}^{-s/n}g\right),\phantom{\rule{1em}{0ex}}g\in {M}_{0},{\alpha }_{E}>s/n,t<1.$

□

### 2.1 Optimal quasi-norms

Here we give a characterization of the optimal domain and optimal target quasi-norms. We can define an optimal target quasi-norm by using Theorem 2.1.

Definition 2.4 (Construction of the optimal target quasi-norm)

For a given domain quasi-norm ${\rho }_{E}\in {\mathcal{N}}_{d}$ we set

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}g\right):=inf\left\{{\rho }_{E}\left(h\right):{\chi }_{\left(0,1\right)}g\le {\chi }_{\left(0,1\right)}{S}_{1}h,h\in {\mathcal{M}}^{+}\right\},\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+}.$
(2.8)

Then

${\rho }_{G\left(E\right)}\left(g\right):={\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}g\right)+\underset{t>1}{sup}{t}^{1-s/n}g.$

Theorem 2.5 Let ${\rho }_{E}\in {\mathcal{N}}_{d}$ be a given domain quasi-norm. Then ${\rho }_{G\left(E\right)}\in {\mathcal{N}}_{t}$, the couple ${\rho }_{E}$, ${\rho }_{G\left(E\right)}$ is admissible and the target quasi-norm is optimal. By definition,

$G\left(E\right):=\left\{f\in \mathcal{M}:\underset{t\to \mathrm{\infty }}{lim}{f}^{\ast }\left(t\right)=0,{\rho }_{G\left(E\right)}\left({f}^{\ast }\right)<\mathrm{\infty }\right\}.$
(2.9)

Proof To see that ${\rho }_{G\left(E\right)}$ is a quasi-norm, we first prove (1.6), for that we first prove

$\underset{0
(2.10)

Take $g\in {\mathcal{M}}^{+}$ and consider an arbitrary $h\in {\mathcal{M}}^{+}$ such that, for $t<1$, ${g}^{\ast }⩽{S}_{1}h$. By the Hardy inequality ${g}^{\ast }\lesssim {S}_{1}\left({h}^{\ast }\right)$. Then,

${t}^{1-s/n}{g}^{\ast }\le K\left({t}^{1-s/n},h;{L}^{1}\left(\mathrm{\Omega }\right),{\mathrm{\Lambda }}^{1}\left({t}^{s/n}\right)\left(\mathrm{\Omega }\right)\right).$

Hence

$\underset{0

Taking the infimum over all h such that ${g}^{\ast }⩽{S}_{1}h$, we get (2.10). Hence ${G}_{E}↪{\mathrm{\Lambda }}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)\left(0,1\right)$, also ${\rho }_{G}\left(\chi \left(1,\mathrm{\infty }\right)g\right)={sup}_{t>1}{t}^{1-s/n}g$. And these two together give (1.6). ${\rho }_{G\left(E\right)}$ is indeed a quasi-norm on ${\mathcal{M}}^{+}$. Since ${\chi }_{\left(0,1\right)}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{S}_{1}{f}^{\ast }$, which gives ${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast }\right)\lesssim {\rho }_{E}\left({f}^{\ast }\right)$. Also

$\underset{t>1}{sup}{t}^{1-s/n}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast }\lesssim \underset{t>1}{sup}{t}^{1-s/n}{S}_{1}{f}^{\ast }={\int }_{0}^{1}{f}^{\ast }\left(u\right)\phantom{\rule{0.2em}{0ex}}du\lesssim {\rho }_{E}\left({f}^{\ast }\right).$

Hence ${\rho }_{E}$, ${\rho }_{G\left(E\right)}$ is admissible couple. Now we are going to prove that ${\rho }_{G\left(E\right)}$ is optimal. For this purpose, suppose that the couple $\left({\rho }_{E},{\rho }_{{G}_{1}}\right)\in \mathcal{N}$ is admissible. Then by Theorem 2.1,

${\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)\lesssim {\rho }_{E}\left(g\right),\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+}.$

Therefore if ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{S}_{1}h$, $h\in {\mathcal{M}}^{+}$, then

${\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\le {\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{S}_{1}h\right)\lesssim {\rho }_{E}\left(h\right),$

so taking the infimum on the right-hand side, we get

${\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right),$

hence ${\rho }_{{G}_{1}}\left({g}^{\ast }\right)\lesssim {\rho }_{G\left(E\right)}\left({g}^{\ast }\right)$. □

In the sublimiting case ${\beta }_{E}<1$ we can simplify the optimal target quasi-norm.

Theorem 2.6 If ${\rho }_{E}\in {\mathcal{N}}_{d}$ be a given domain quasi-norm. Then for $g\in {\mathcal{M}}^{+}$,

$\begin{array}{r}{\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\approx \rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right),\\ \rho \left({\chi }_{\left(0,1\right)}g\right):=inf\left\{{\rho }_{E}\left(h\right):{\chi }_{\left(0,1\right)}g\le {\chi }_{\left(0,1\right)}{T}_{1}h,h\in M\right\},\end{array}$
(2.11)

i.e.,

${\rho }_{G\left(E\right)}\left(g\right)\approx \rho \left({\chi }_{\left(0,1\right)}g\right)+\underset{t>1}{sup}{t}^{1-s/n}g.$

Proof If ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{T}_{1}h$, $h\in M$, then ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{S}_{1}h$, therefore

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\le {\rho }_{E}\left(h\right)$

and taking the infimum, we get

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\le \rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right).$

Now for the reverse, let ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{S}_{1}h$, $h\in {\mathcal{M}}^{+}$.

Then

${\chi }_{\left(0,1\right)}{g}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{S}_{1}\left({h}^{\ast }\right)\approx {\chi }_{\left(0,1\right)}{T}_{1}\left({h}^{\ast \ast }\right)+{\chi }_{\left(0,1\right)}{f}^{\ast \ast }\left(1\right),$

so

${\chi }_{\left(0,1\right)}{g}^{\ast }-{\chi }_{\left(0,1\right)}{f}^{\ast \ast }\left(1\right)\lesssim {\chi }_{\left(0,1\right)}{T}_{1}\left({h}^{\ast \ast }\right),$

which gives, since ${h}^{\ast \ast }\in M$,

$\rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }-{\chi }_{\left(0,1\right)}{f}^{\ast \ast }\left(1\right)\right)\lesssim {\rho }_{E}\left({h}^{\ast \ast }\right)\approx {\rho }_{E}\left({h}^{\ast }\right)\approx {\rho }_{E}\left(h\right),$

and this implies

$\rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{E}\left(h\right)+{f}^{\ast \ast }\left(1\right),$

which gives

$\rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{E}\left(h\right).$

Taking the infimum, we get $\rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)$, hence $\rho \left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\approx {\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)$. □

A simplification of the optimal target quasi-norm is possible also in the subcritical case ${\alpha }_{E}>s/n$.

Theorem 2.7 Let ${\rho }_{E}\in {\mathcal{N}}_{d}$ be a given domain quasi-norm. Then for $g\in {\mathcal{M}}^{+}$,

$\begin{array}{r}{\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\approx {\rho }_{1}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right),\\ {\rho }_{1}\left({\chi }_{\left(0,1\right)}g\right):=inf\left\{{\rho }_{E}\left(h\right):{\chi }_{\left(0,1\right)}g\le {T}_{1}^{\prime }h,h\in {M}_{0}\right\},\end{array}$
(2.12)

i.e.,

${\rho }_{G\left(E\right)}\left(g\right)\approx {\rho }_{1}\left({\chi }_{\left(0,1\right)}g\right)+\underset{t>1}{sup}{t}^{1-s/n}g.$

Proof If ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }h$, $h\in {M}_{0}$, then

${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{S}_{1}h.$

Therefore

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\le {\rho }_{E}\left(h\right),$

and taking the infimum, we get

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\le {\rho }_{1}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right).$

For the reverse, let ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{S}_{1}h$. Then ${\chi }_{\left(0,1\right)}{g}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{T}_{1}\left({h}^{\ast }\right)+{\chi }_{\left(0,1\right)}{T}_{1}^{\prime }\left({h}^{\ast }\right)$. As

${\chi }_{\left(0,1\right)}{T}_{1}g\lesssim {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}g\right),$

we get

${\chi }_{\left(0,1\right)}{g}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }\left({h}^{\ast }+{t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}\left({h}^{\ast }\right)\right),$

whence

$\begin{array}{rl}{\rho }_{1}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)& \lesssim {\rho }_{E}\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}\left({h}^{\ast }\right)\right)+{\rho }_{E}\left(h\right)\\ \approx {\rho }_{E}\left({t}^{-s/n}{Q}_{1}\left({t}^{s/n}{h}^{\ast }\right)\right)+{\rho }_{E}\left(h\right)\\ \lesssim {\rho }_{E}\left(h\right),\end{array}$

where we use

${\rho }_{E}\left({Q}_{1}\left({t}^{-s/n}g\right)\right)\lesssim {\rho }_{E}\left({t}^{-s/n}g\right),\phantom{\rule{1em}{0ex}}g\in {M}_{0},{\alpha }_{E}>s/n,t<1.$

Therefore, taking the infimum we arrive at

${\rho }_{1}\left({g}^{\ast }\right)\lesssim {\rho }_{{G}_{E}}\left({g}^{\ast }\right).$

□

We can construct an optimal domain quasi-norm ${\rho }_{E\left(G\right)}$ by Theorem 2.1 as follows.

Definition 2.8 (Construction of an optimal domain quasi-norm)

For a given target quasi-norm ${\rho }_{G}\in {\mathcal{N}}_{t}$, we construct an optimal domain quasi-norm ${\rho }_{E\left(G\right)}$ by

${\rho }_{E\left(G\right)}\left(g\right):={\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right),\phantom{\rule{1em}{0ex}}g\in {\mathcal{M}}^{+}.$
(2.13)

Theorem 2.9 If ${\rho }_{G}\in {\mathcal{N}}_{t}$ is a given target quasi-norm, then the domain quasi-norm ${\rho }_{E\left(G\right)}$ is optimal. Moreover, if ${\beta }_{G}<1-s/n$, then the couple ${\rho }_{E\left(G\right)}$, ${\rho }_{G}$ is optimal.

Proof Since ${\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\approx {\chi }_{\left(0,1\right)}{T}_{1}{g}^{\ast \ast }+{\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(1\right)$, so

${\rho }_{E\left(G\right)}\left(g\right)\approx {\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}{g}^{\ast \ast }+{\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(1\right)\right),$

it follows that ${\rho }_{E\left(G\right)}$ is a quasi-norm. To prove the property (1.5), we notice that

$\begin{array}{rcl}{\rho }_{E\left(G\right)}\left({f}^{\ast }\right)& =& {\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{f}^{\ast }\right)\ge {\rho }_{G}\left({\chi }_{\left(0,1\right)}\right)\left(S{f}^{\ast }\right)\left(1\right)\\ \gtrsim & {\int }_{0}^{1}{f}^{\ast }\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\approx {\parallel f\parallel }_{{L}^{1}\left(\mathrm{\Omega }\right)}.\end{array}$

The couple ${\rho }_{E\left(G\right)}$, ${\rho }_{G}$ is admissible since ${\rho }_{E\left(G\right)}\left(g\right)={\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)\ge {\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}g\right)$. Moreover, ${\rho }_{E\left(G\right)}$ is optimal, since for any admissible couple $\left({\rho }_{{E}_{1}},{\rho }_{G}\right)\in \mathcal{N}$ we have ${\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}h\right)\lesssim {\rho }_{{E}_{1}}\left(h\right)$, $h\in {\mathcal{M}}^{+}$. Therefore,

${\rho }_{E\left(G\right)}\left({g}^{\ast }\right)\le {\rho }_{{E}_{1}}\left({g}^{\ast }\right).$

To check that if ${\beta }_{G}<1-s/n$, the couple ${\rho }_{E\left(G\right)}$, ${\rho }_{G}$ is optimal, we need only to prove that ${\rho }_{G}$ is an optimal target quasi-norm, i.e., $\rho \left({g}^{\ast }\right)\lesssim {\rho }_{G}\left({g}^{\ast }\right)$, where $\rho ={\rho }_{G\left(E\left(G\right)\right)}$ is defined by (2.11), since ${\beta }_{E\left(G\right)}<1$. We have ${\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(t\right)-{\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(1\right)={\chi }_{\left(0,1\right)}{T}_{1}h$, where $h\left(t\right)={t}^{-s/n}\left[{g}^{\ast \ast }\left(t\right)-{g}^{\ast }\left(t\right)\right]\in M$, $t<1$, therefore,

${\rho }_{{G}_{E\left(G\right)}}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(t\right)-{\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(1\right)\right)\le {\rho }_{E\left(G\right)}\left(h\right)={\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{h}^{\ast }\right)$

implies

${\rho }_{{G}_{E\left(G\right)}}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(t\right)\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{h}^{\ast }\right)+{g}^{\ast \ast }\left(1\right),$

since

${\chi }_{\left(0,1\right)}{S}_{1}{h}^{\ast }={\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }+{\chi }_{\left(0,1\right)}{T}_{1}{h}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }+{\chi }_{\left(0,1\right)}{T}_{1}{h}^{\ast \ast },$

so

${\rho }_{{G}_{E\left(G\right)}}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\left(t\right)\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }\right)+{\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}{h}^{\ast \ast }\right)+{g}^{\ast \ast }\left(1\right).$

Now we define

${P}_{1}g\left(t\right):=\frac{1}{t}{\int }_{0}^{t}g\left(u\right)\phantom{\rule{0.2em}{0ex}}du,\phantom{\rule{1em}{0ex}}t<1.$

For $t<1$, since ${h}^{\ast }\lesssim {Q}_{1}h$, we have ${h}^{\ast \ast }={P}_{1}{h}^{\ast }\lesssim {Q}_{1}{P}_{1}h$, therefore ${T}_{1}{h}^{\ast \ast }\lesssim {T}_{1}{Q}_{1}\left({P}_{1}h\right)\lesssim {T}_{1}\left({P}_{1}h\right)$. Also ${T}_{1}\left({P}_{1}h\right)\approx {T}_{1}h+{t}^{s/n}{P}_{1}h$ and ${P}_{1}h\le {h}^{\ast \ast }$. Therefore,

$\begin{array}{rl}{\rho }_{{G}_{E\left(G\right)}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)& \lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}h\right)+{\rho }_{G}\left({\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }\right)+{g}^{\ast \ast }\left(1\right)\\ \lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right)+{\rho }_{G}\left({\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }\right)+{g}^{\ast \ast }\left(1\right).\end{array}$

For $t<1$, since $h\left(t\right)\le {t}^{-s/n}{g}^{\ast \ast }\left(t\right)$ we have ${h}^{\ast }\left(t\right)\le {t}^{-s/n}{g}^{\ast \ast }$, therefore using ${\beta }_{G}<1-s/n$, Minkowski’s inequality, and monotonicity of ${\rho }_{G}$, we have

${\rho }_{G}\left({\chi }_{\left(0,1\right)}{t}^{s/n}{h}^{\ast \ast }\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right).$

Thus

${\rho }_{{G}_{E\left(G\right)}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right)\approx {\rho }_{G}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right),$

hence $\rho \left({g}^{\ast }\right)\lesssim {\rho }_{G}\left({g}^{\ast }\right)$. □

Example 2.10 If $G={C}_{0}$ consists of all bounded continuous functions such that ${f}^{\ast }\left(\mathrm{\infty }\right)=0$ and ${\rho }_{G}\left(g\right)={g}^{\ast }\left(0\right)={g}^{\ast \ast }\left(0\right)$, then ${\alpha }_{G}={\beta }_{G}=0$ and ${\rho }_{E\left(G\right)}\left(g\right)\approx {\int }_{0}^{1}{t}^{s/n}{g}^{\ast \ast }\phantom{\rule{0.2em}{0ex}}dt/t$, i.e., $E={\mathrm{\Gamma }}^{1}\left({t}^{s/n}\right)\left(\mathrm{\Omega }\right)$ and the couple E, G is optimal.

Example 2.11 Let $G={\mathrm{\Lambda }}^{\mathrm{\infty }}\left(v\right)$ with ${\beta }_{G}<1-s/n$ and let

${\rho }_{E}\left(g\right)=supv\left(t\right){\int }_{t}^{1}{u}^{s/n}{g}^{\ast \ast }\left(u\right)\phantom{\rule{0.2em}{0ex}}du/u.$

Then, the couple E, G is optimal and ${\beta }_{E}<1$. In particular, this is true if v is slowly varying since then ${\alpha }_{G}={\beta }_{G}=0$ and ${\alpha }_{E}={\beta }_{E}=s/n<1$.

### 2.2 Subcritical case

Here we suppose that $s/n<{\alpha }_{E}$.

Theorem 2.12 (Sublimiting case ${\beta }_{E}<1$)

For a given domain quasi-norm ${\rho }_{E}\in {\mathcal{N}}_{d}$ with ${\rho }_{E}\left({\chi }_{\left(0,1\right)}\left(t\right){t}^{-s/n}\right)<\mathrm{\infty }$, we have

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\approx {\rho }_{E}\left({t}^{-s/n}{g}^{\ast }\right)\approx {\rho }_{E}\left({t}^{-s/n}{g}^{\ast \ast }\right),$
(2.14)

i.e.,

${\rho }_{G\left(E\right)}\left({g}^{\ast }\right)\approx {\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)+\underset{t>1}{sup}{t}^{1-s/n}g.$

Moreover, the couple ${\rho }_{E}$, ${\rho }_{G\left(E\right)}$ is optimal.

Proof If ${\chi }_{\left(0,1\right)}{g}^{\ast }\le {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }h$, $h\in {M}_{0}$, then for $t<1$, ${t}^{-s/n}{g}^{\ast }\le {h}^{\ast \ast }$, whence

${\rho }_{E}\left({t}^{-s/n}{g}^{\ast }\right)\lesssim {\rho }_{E}\left({h}^{\ast \ast }\right)\approx {\rho }_{E}\left({h}^{\ast }\right)\approx {\rho }_{E}\left(h\right).$

Taking the infimum, we get

${\rho }_{E}\left({t}^{-s/n}{g}^{\ast }\right)\lesssim {\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right).$

For the reverse, we notice that ${\chi }_{\left(0,1\right)}{T}_{1}^{\prime }\left({t}^{-s/n}{g}^{\ast }\right)\gtrsim {\chi }_{\left(0,1\right)}{g}^{\ast }={g}^{\ast }$, hence ${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\lesssim {\rho }_{E}\left({t}^{-s/n}{g}^{\ast }\right)$.

It remains to prove that the domain quasi-norm ${\rho }_{E}$ is also optimal. Let ${\rho }_{{E}_{1}}$, ${\rho }_{G\left(E\right)}$ be an admissible couple in $\mathcal{N}$. Then

$\begin{array}{rl}{\rho }_{{E}_{1}}\left({g}^{\ast }\right)& \gtrsim {\rho }_{G\left(E\right)}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)\\ ={\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)+\underset{t>1}{sup}{t}^{1-s/n}{\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\\ \approx {\rho }_{E}\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{S}_{1}{g}^{\ast }\right)+0\\ \gtrsim {\rho }_{E}\left({t}^{-s/n}{\chi }_{\left(0,1\right)}{T}_{1}^{\prime }{g}^{\ast }\right)\\ \gtrsim {\rho }_{E}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right)\\ \approx {\rho }_{E}\left({\chi }_{\left(0,1\right)}{g}^{\ast }\right)\\ \approx {\rho }_{E}\left({g}^{\ast }\right).\end{array}$

□

Now we give an example.

Example 2.13 Let

$E={\mathrm{\Lambda }}^{q}\left({t}^{\alpha }{w}_{1}\right)\left(\mathrm{\Omega }\right)\cap {\mathrm{\Lambda }}^{r}\left({t}^{\beta }{w}_{2}\right)\left(\mathrm{\Omega }\right),\phantom{\rule{1em}{0ex}}s/n<\alpha <\beta <1,0

where ${w}_{1}$ and ${w}_{2}$ are slowly varying. Then we have ${\alpha }_{E}=\alpha$, ${\beta }_{E}=\beta$. Now by applying the previous theorem, we get

$G\left(E\right)={\mathrm{\Lambda }}_{0}^{q}\left({t}^{\alpha -s/n}{w}_{1}\right)\cap {\mathrm{\Lambda }}_{0}^{r}\left({t}^{\beta -s/n}{w}_{2}\right),$

and the couple $\left(E,G\left(E\right)\right)$ is optimal.

In the limiting case ${\beta }_{E}=1$, the formula for the optimal target quasi-norm is more complicated.

Theorem 2.14 (Limiting case)

Let

${\rho }_{E}\left(g\right):={\rho }_{H}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right),\phantom{\rule{2em}{0ex}}{\rho }_{{G}_{1}}\left(g\right):={\rho }_{H}\left({t}^{-1}\underset{0

where ${\rho }_{H}$ is a monotone quasi-norm with ${\alpha }_{H}={\beta }_{H}=1$, ${\rho }_{H}\left({\chi }_{\left(0,1\right)}\right)<\mathrm{\infty }$, ${\rho }_{H}\left({\chi }_{\left(1,\mathrm{\infty }\right)}{t}^{-1}\right)<\mathrm{\infty }$ and let

$\begin{array}{r}E:=\left\{f\in \mathcal{M}:t{f}^{\ast \ast }\left(t\right)\to 0\phantom{\rule{0.25em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.25em}{0ex}}t\to 0\phantom{\rule{0.25em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.25em}{0ex}}{\rho }_{E}\left({f}^{\ast }\right)<\mathrm{\infty }\right\},\\ {G}_{1}:=\left\{f\in \mathcal{M}:\underset{0

Define

${\rho }_{G}\left(g\right):={\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}g\right)+\underset{t>1}{sup}{t}^{1-s/n}g.$

Then the couple ${\rho }_{E}$, ${\rho }_{G}$ is optimal.

Proof Note that

$E↪{L}^{1}\left(\mathrm{\Omega }\right).$

Indeed, ${\rho }_{E}\left({f}^{\ast }\right)={\rho }_{H}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right)\gtrsim {f}^{\ast \ast }\left(1\right)={\int }_{0}^{1}{f}^{\ast }\left(u\right)\phantom{\rule{0.2em}{0ex}}du$. Hence the above embedding follows. Consequently, ${\rho }_{E}\in {\mathcal{N}}_{d}$. On the other hand,

$\begin{array}{rl}{\rho }_{G}\left({f}^{\ast }\right)& \ge {\rho }_{H}\left({\chi }_{\left(1,\mathrm{\infty }\right)}{t}^{-1}\underset{0

Hence ${G}_{1}↪{\mathrm{\Lambda }}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)\left(0,1\right)$. This together with ${\rho }_{G}\left({\chi }_{\left(1,\mathrm{\infty }\right)}\right)={sup}_{t>1}{t}^{1-s/n}g$ gives $G↪{\mathrm{\Lambda }}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)$. Then from the conditions on ${G}_{1}$ it follows that ${\rho }_{G}\in {\mathcal{N}}_{t}$. Also, ${\alpha }_{E}={\beta }_{E}=1$ and ${\alpha }_{G}={\beta }_{G}=1-s/n$. On the other hand, if $0, then

${u}^{1-s/n}{\left({R}_{\mathrm{\Omega }}^{s}f\right)}^{\ast \ast }\left(u\right)\lesssim {\int }_{0}^{u}{f}^{\ast }\left(v\right)\phantom{\rule{0.2em}{0ex}}dv+{u}^{1-s/n}{\int }_{u}^{1}{v}^{s/n-1}{f}^{\ast }\left(v\right)\phantom{\rule{0.2em}{0ex}}dv.$

For every $\epsilon >0$, we can find a $\delta >0$, such that $v{f}^{\ast \ast }\left(v\right)<\epsilon$ for all $0. Then for $0,

$\underset{0
(2.15)

Now it is easy to check that ${lim}_{t\to 0}{sup}_{0 if $f\in E$.

To prove that ${R}^{s}:E\to G$ we need to check that the couple ${\rho }_{E}$, ${\rho }_{G}$ is admissible. We write for $t<1$,

${T}_{1}^{\prime }g\left(t\right)={T}_{1}^{\prime }{g}^{\ast }\left(t\right)={t}^{s/n}{g}^{\ast \ast }\left(t\right),\phantom{\rule{1em}{0ex}}g\in {M}_{0}.$

Then

$\begin{array}{rl}{\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\right)& ={\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\right)+\underset{t>1}{sup}{t}^{1-s/n}{\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\\ ={\rho }_{H}\left({\chi }_{\left(0,1\right)}{t}^{-1}\underset{01}{sup}{t}^{1-s/n}{\chi }_{\left(0,1\right)}{T}_{1}^{\prime }g\\ ={\rho }_{H}\left({\chi }_{\left(0,1\right)}{g}^{\ast \ast }\right)\\ ={\rho }_{E}\left(g\right).\end{array}$

To prove that the target space is optimal, notice first that

$\underset{0

If $f\in G$, then by 

If $h\left(v\right)={h}_{1}\left({v}^{1-s/n}\right){v}^{-s/n}$ then obviously $h\in {M}_{0}$ and

$\underset{0

whence

${\rho }_{E}\left(h\right)\approx {\rho }_{H}\left({\chi }_{\left(0,1\right)}{h}^{\ast \ast }\right)\approx {\rho }_{H}\left({\chi }_{\left(0,1\right)}{t}^{-1}\underset{0

On the other hand,

$\underset{0

implies ${t}^{1-s/n}{f}^{\ast }\left(t\right)\lesssim t{h}^{\ast \ast }\left(t\right)$, which gives ${f}^{\ast }\lesssim {t}^{s/n}{h}^{\ast \ast }$, which implies ${\chi }_{\left(0,1\right)}{f}^{\ast }\lesssim {\chi }_{\left(0,1\right)}{T}_{1}^{\prime }h$, and therefore

${\rho }_{{G}_{E}}\left({\chi }_{\left(0,1\right)}{f}^{\ast }\right)\lesssim {\rho }_{E}\left(h\right)\lesssim {\rho }_{{G}_{1}}\left({\chi }_{\left(0,1\right)}{f}^{\ast }\right),$

proving optimality of G. To check optimality of E, we notice that

$\begin{array}{rl}{\rho }_{E\left(G\right)}\left(h\right)& ={\rho }_{G}\left({\chi }_{\left(0,1\right)}{S}_{1}{h}^{\ast }\right)\gtrsim {\rho }_{G}\left({\chi }_{\left(0,1\right)}{T}_{1}{h}^{\ast \ast }\right)\\ \approx {\rho }_{H}\left({t}^{-1}\underset{0

Hence

${\rho }_{E\left(G\right)}\left(h\right)\gtrsim {\rho }_{E}\left(h\right).$

□

Example 2.15 Let $E={\mathrm{\Gamma }}_{0}^{\mathrm{\infty }}\left(tw\right)\left(\mathrm{\Omega }\right)$, consisting of all $f\in {\mathrm{\Gamma }}^{\mathrm{\infty }}\left(tw\right)\left(\mathrm{\Omega }\right)$ such that $t{f}^{\ast \ast }\left(t\right)\to 0$ as $t\to 0$, w is slowly varying. Then ${\beta }_{E}=1$. If $G={\mathrm{\Gamma }}_{1}^{\mathrm{\infty }}\left({t}^{1-s/n}v\right)\cap {\mathrm{\Gamma }}^{\mathrm{\infty }}\left(tw\right)$, where $v\left(t\right)={sup}_{u>t}w\left(u\right)$ and

then this couple is optimal. In particular, if $w=1$, then $E={L}^{1}\left(\mathrm{\Omega }\right)$ and $G={\mathrm{\Gamma }}_{1}^{\mathrm{\infty }}\left({t}^{1-s/n}\right)$.

## References

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## Acknowledgements

This study was supported by research funds from Dong-A University.

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Correspondence to Young Chel Kwun.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

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Kang, S.M., Rafiq, A., Nazir, W. et al. Optimal couples of rearrangement invariant spaces for the Riesz potential on the bounded domain. J Inequal Appl 2014, 60 (2014). https://doi.org/10.1186/1029-242X-2014-60 