Some new results related to Lorentz G-gamma spaces and interpolation

TL;DR

This paper computes the K-functional for certain space couples, enabling the determination of interpolation spaces as G-gamma spaces, with applications to regularity estimates for weak solutions of linear equations.

Contribution

It provides explicit K-functional computations for Lebesgue and Lorentz-Marcinkiewicz spaces, extending previous results and characterizing interpolation spaces as G-gamma spaces.

Findings

Interpolation spaces are G-gamma spaces covering many classical spaces.

Explicit K-functional formulas are derived for specific space couples.

Results facilitate regularity estimates for weak solutions of PDEs.

Abstract

We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of the previous works of the authors. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a G-gamma space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of linear equation in a domain Omega with data in the space of the integrable function with respect to the distance function to the boundary of Omega.

Equations

$$|u{|}_{L^1(\Omega )}⩽c|f{|}_{L^1(\Omega ,\mathrm{dist}(x,\partial \Omega ))}$$

$$GD(\Omega)=\begin{cases}-\displaystyle\int_{\Omega}u\Delta\varphi\,dx=\int_{\Omega}f\varphi\,dx,\quad\forall\varphi\in C^{2}_{0}(\overline{\Omega}),\\ \hbox{with }C^{2}_{0}(\overline{\Omega})=\Big{\{}\varphi\in C^{2}({\overline{\Omega}}),\ \varphi=0\hbox{ on }\partial\Omega\Big{\}}.\end{cases}$$

$$-{\int }_{\Omega }u\Delta \phi dx={\int }_{\Omega }f\phi dx\forall \phi \in C^2(\overline{\Omega }),\phi =0on\partial \Omega .$$

$$u\in L^{(n^{\prime },n\alpha -n+1}(\Omega )=G\Gamma (n^{\prime },1;w_{\alpha }),w_{\alpha }(t)=t^{-1}(1-\mathrm{Log}t{)}^{\alpha -1-\frac{1}{n^{\prime }}}$$

$$||u|{|}_{G\Gamma (n^{\prime },1;w_{\alpha })}⩽K_0|f{|}_{L^1\left(\Omega ;\delta (1+|\mathrm{Log}\delta |{)}^{\alpha }\right)}$$

$$u\in L^{n^{\prime }}(\Omega )and\; similar\; estimate\; as\; (\text{ref}eqeq2)\; holds.$$

$$G\Gamma (p,m;w_1,w_2)=\left\{v:\Omega \to \mathrm{I}\mathrm{R}measurable{\int }_0^t{v}_{\ast }^p(\sigma )w_2(\sigma )d\sigma is\; in{L}^{\frac{m}{p}}(0,1;w_1)\right\}.$$

$$\rho (v)={\left[{\int }_0^1{w}_1(t){\left({\int }_0^t{v}_{\ast }^p(\sigma )w_2(\sigma )d\sigma \right)}^{\frac{m}{p}}dt\right]}^{\frac{1}{m}}$$

$$G\Gamma (p,m;w_1,1)=G\Gamma (p,m;w_1).$$

$$w_{2}\hbox{ satisfies condition c1) and $w_{1}$ and $w_{2}$ are in }L_{exp}^{\max(\gamma;\beta)}\big{(}]0,1[\big{)}\,.$$

$$|u{|}_{L^{n^{\prime },\infty }(\Omega )}⩽K_1|f{|}_{L^1(\Omega ;\delta )},$$

$$|u{|}_{(L^{n^{\prime },\infty },L^{(n^{\prime }}{)}_{\alpha ,1}}⩽K_2|f{|}_{L^1(\Omega ;\delta (1+|\mathrm{Log}\delta |{)}^{\alpha })}for\; 0<\alpha ⩽1.$$

$$\Big{(}L^{n^{\prime},\infty}(\Omega),L^{(n^{\prime}}(\Omega)\Big{)}_{\alpha,1}\subset\Big{(}L^{n^{\prime})}(\Omega),L^{(n^{\prime}}(\Omega)\Big{)}_{\alpha,1}$$

$$\Big{(}L^{n^{\prime})}(\Omega),L^{(n^{\prime}}(\Omega)\Big{)}_{\alpha,1}=G\Gamma(n^{\prime};1;w_{1};w_{2})\hbox{ with }w_{1}(t)=\dfrac{(1-{\rm Log\,}t)^{\alpha-1}}{t},\ w_{2}(t)=\dfrac{1}{1-{\rm Log\,}t}.$$

$$||u|{|}_{G\Gamma (n^{\prime };1;w_1;w_2)}={\int }_0^1(1-\mathrm{Log}t{)}^{\alpha }{\left({\int }_0^t\frac{u_{\ast }^{n^{\prime }}(x)dx}{1-\mathrm{Log}x}\right)}^{\frac{1}{n^{\prime }}}\frac{dt}{(1-\mathrm{Log}t)t}⩽K_4|f{|}_{L^1(\Omega ;\delta (1+|\mathrm{Log}\delta |{)}^{\alpha })}$$

$$\Vert f{\Vert }_{G\Gamma (p,r;w_1,w_2)\cap G\Gamma (\infty ,r;v_1,v_2)}\lesssim \Vert f{\Vert }_{(L^{p,\infty },L^{(p}{)}_{\theta ,1}},$$

$$(L^{n^{\prime }},L^{(n^{\prime }}{)}_{\theta ,r}=G(n^{\prime },r;w_1;1)with{w}_1(t)=t^{-1}(1-\mathrm{Log}t{)}^{r\frac{\theta }{n}-1}.$$

$$(L^{n^{\prime }},L^{(n^{\prime }}{)}_{\theta ,1}=L^{(n^{\prime },\theta }.$$

$$K_0(f;t)=t\sup \left\{{\left({\int }_E{f}_{\ast }^{n^{\prime }}(\sigma )d\sigma \right)}^{\frac{1}{n^{\prime }}};t^{-n^{\prime }}={\int }_E\frac{dx}{x}\right\}$$

$$K_{0}(f;t)=t\left(\int_{0}^{t^{-p}}\Big{(}\psi(s)\Big{)}^{p}_{*,\nu}(x)dx\right)^{\frac{1}{p}}.$$

$$(L^{p,\infty },L^p{)}_{\theta ,\frac{p}{\theta }}=L^{p,\frac{p}{\theta }}.$$

$${\int }_0^1{t}^{-\alpha }\underset{\{E:|E{|}_{\nu }=t^{-n^{\prime }}\}}{\sup }{\left({\int }_E{u}_{\ast }^{n^{\prime }}(x)dx\right)}^{\frac{1}{n^{\prime }}}dt⩽c|f{|}_{L^1(\Omega ;\delta (1+|\mathrm{Log}\delta |{)}^{\frac{\alpha }{n^{\prime }}})}.$$

$$||v|{|}_{L^{n^{\prime },\infty }}⩽c||v|{|}_{L^{n^{\prime })}}^{1-\alpha }||v|{|}_{L^{(n^{\prime }}}^{\alpha }.$$

$$||v|{|}_{(L^{n^{\prime },\infty },L^{(n^{\prime }}{)}_{\alpha ,1}}⩽c||v|{|}_{L^{n^{\prime }}}^{1-\alpha }||v|{|}_{L^{(n^{\prime }}}^{\alpha }\forall v\in L^{(n^{\prime }}.$$

$$D_{f}(t)=\hbox{measure }\Big{\{}x\in\Omega:|f(x)|>t\Big{\}}$$

$$f_{*}(s)=\inf\Big{\{}t:D_{f}(t)\leqslant s\Big{\}}\hbox{ with }s\in\big{(}0,|\Omega|\big{)},\ |\Omega|\hbox{ is the measure of }\Omega,$$

$$A_1\lesssim A_{2}if\; there\; existsc>0\; (independent\; of\; the\; parameters)\; such\; that{A}_1⩽c{A}_2$$

$$A_1\simeq A_{2}if\; and\; only\; if{A}_1\lesssim A_{2}and{A}_2\lesssim A_1.$$

$$K(g,t)=\inf_{g=g_{0}+g_{1}}\big{(}||g_{0}||_{0}+t||g_{1}||_{1}\big{)}.$$

$$(X_{0},X_{1})_{\theta,p;\alpha}=\Big{\{}g\in X_{0}+X_{1},\ ||g||_{\theta,p;\alpha}=||t^{-\theta-\frac{1}{p}}\big{(}1-{\rm Log\,}t\big{)}^{\alpha}K(g,t)||_{L^{p}(0,1)}\hbox{ is finite}\Big{\}}.$$

Full text

The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain $\Omega$ with data in the space of the integrable function with respect to the distance function to the boundary of $\Omega$.

1. Introduction

The present work finds its motivation in the recent results in [11, 8, 20]. The original question comes from an unpublished manuscript by H. Brezis (see comments in [8]) and later presented in [6] (see also the mention made in [24]) concerning the following problem : let $f$ be given in ${L^{1}(\Omega,{\rm dist\,}(x,\partial\Omega))}$ ($\Omega$ bounded smooth open set of ${\rm I\!R}^{n}$), then H. Brezis shows the existence and uniqueness of a function $u\in{L^{1}(\Omega)}$ satisfying

[TABLE]

with

[TABLE]

Therefore, the question of the integrability of the generalized derivative $v:\partial_{i}v=\displaystyle\frac{\partial v}{\partial x_{i}}$ arises in a natural way and was raised already in the note by H. Brezis and developed in [8], [20], [21]. More generally, the question of the regularity of $u$ is arised, according to $f$.

In [10, 11], we have shown the following theorem:

Theorem 1.1**.**

*Let $\Omega$ be a bounded open set of class $C^{2}$ of ${\rm I\!R}^{n}$, $|\Omega|=1$ and $\alpha\geqslant\dfrac{1}{n^{\prime}}$ where $n^{\prime}=\dfrac{n}{n-1},$ $f\in L^{1}(\Omega;\delta),$ with $\delta(x)={\rm dist\,}(x;\partial\Omega).$

*Consider $u\in L^{n^{\prime},\infty}(\Omega),$ the very weak solution (*v.w.s. ) of

[TABLE]

Then,

1. (1)

if f\in L^{1}\Big{(}\Omega;\delta(1+|{\rm Log\,}\delta|)^{\alpha}\Big{)} and $\alpha>\dfrac{1}{n^{\prime}}:$

[TABLE]

and

[TABLE] 2. (2)

if f\in L^{1}\Big{(}\Omega;\delta\big{(}1+|{\rm Log\,}\delta|\big{)}^{\frac{1}{n^{\prime}}}\Big{)} then

[TABLE]

Note that the assumption on the regularity of $\Omega$, needed in the proof of Theorem 1.1 is necessary, for the development of the theory of very weak solutions; we stress that the estimates in this paper will be obtained following arguments valid regardless of the regularity of $\Omega$, which will be definitively dropped in our statements.

The Lorentz $G\Gamma$-space is defined as follows :

Definition 1.2**.**

**of Generalized Gamma space with double weights (Lorentz-$G\Gamma$)

**Let $w_{1},\ w_{2}$ be two weights on $(0,1)$, $m\in[1,+\infty]$, $1\leqslant p<+\infty$. We assume the following conditions:

• c1)

There exists $K_{12}>0$ such that $w_{2}(2t)\leqslant K_{12}w_{2}(t)\ \forall\,t\in(0,1/2)$. The space $L^{p}(0,1;w_{2})$ is continuously embedded in $L^{1}(0,1)$.

• c2)

The function $\displaystyle\int_{0}^{t}w_{2}(\sigma)d\sigma$ belongs to $L^{\frac{m}{p}}(0,1;w_{1})$.

A generalized Gamma space with double weights is the set :

[TABLE]

A similar definition has been considered in [15]. They were interested in the embeddings between $G\Gamma$-spaces.

Property 1.3**.**

Let $G\Gamma(p,m;w_{1},w_{2})$ be a Generalized Gamma space with double weights and let us define for $v\in G\Gamma(p,m;w_{1},w_{2})$

[TABLE]

*with the obvious change for $m=+\infty$.

Then,*

1. (1)

$\rho$* is a quasinorm.* 2. (2)

$G\Gamma(p,m;w_{1},w_{2})$* endowed with $\rho$ is a quasi-Banach function space.* 3. (3)

If $w_{2}=1$

[TABLE]

Example 1.1**.**

**of weights

**Let $w_{1}(t)=(1-{\rm Log\,}t)^{\gamma},\quad w_{2}(t)=(1-{\rm Log\,}t)^{\beta}$ with $(\gamma,\beta)\in{\rm I\!R}^{2}$. Then

[TABLE]

Question 1 The natural question is how to extend of Theorem 1.1 for $\alpha<\dfrac{1}{n^{\prime}}$ and how to improve the estimate when $\alpha=\dfrac{1}{n^{\prime}}$?

Since the solution of (1.1) satisfies also

[TABLE]

the natural idea to obtain an estimate is to use the real interpolation method of Marcinkiewicz (see [4, 5, 7] ) to derive

[TABLE]

Note that $L^{(n^{\prime},1}=L^{(n^{\prime}}$ (see below for a full definition.)

Question 2 *How to characterize the space \Big{(}L^{n^{\prime},\infty}(\Omega),L^{(n^{\prime}}(\Omega)\Big{)}_{\alpha,1}?

We still have not an answer to this question. Therefore, we will provide a lower estimate for the norm of $u$ in relation (1.4), a particular overbound can be obtained from our work made in [12] :

Since $L^{n^{\prime},\infty}(\Omega)\subset L^{n^{\prime})}$, then we have

[TABLE]

and we have shown in [12] the following

Theorem 1.4**.**

**(characterization of the interpolation between Grand and Small Lebesgue space)

**

[TABLE]

(see next section for the definition of $G\Gamma$).

Therefore, we have the following non optimal result but valid for all $\alpha.$

Proposition 1.5**.**

Let $u$ be the solution of (1.1). Then,

[TABLE]

whenever $0<\alpha<1.$

To give a new improved statement for Proposition 1.5 namely, we will show the

Theorem 1.6**.**

Let $1<p<\infty$, $0<\theta<1$ and $1\leqslant r<\infty.$ Then

[TABLE]

where $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{r\theta-1}$, $w_{2}(t)=(1-{\rm Log\,}t)^{-1},$ $v_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{r\theta(1-1/p)-1}$ and $v_{2}(t)=t^{1/p}.$

Thanks to this last theorem, we deduce from relation (1.4) a new estimate of the solution $u$ valid also for $\alpha<\dfrac{1}{n^{\prime}}$ and better than Proposition 1.5 in that case. To complete the results of [12], we shall introduce different results on the interpolation spaces namely, between $(L^{n^{\prime}},L^{(n^{\prime}})_{\theta,r}$, $(L^{n^{\prime},\infty},L^{n^{\prime}})_{\theta,r}$, $(L^{p),\alpha},L^{p),\beta})_{\theta,r}$, $(L^{p)},L^{p})_{\theta,r}$, $(L^{p,\infty},L^{p})_{\theta,r}$. It happens all of these spaces are Lorentz G-gamma spaces. We state few of those results.

Theorem 1.7**.**

For $0<\theta<1,\ r\in[1,+\infty[$

[TABLE]

Corollary 1.8**.**

of Theorem 1.7* 

For $0<\theta<1$, one has*

[TABLE]

As in [12, 10], the proofs of the above results rely on the computation of the $K$-functional, as for the couple $(L^{n^{\prime},\infty},L^{n^{\prime}})$, we will show the following

Theorem 1.9**.**

The $K$-functional for $(L^{n^{\prime},\infty},L^{n^{\prime}})$ is given by, for $t\in]0,1[$, $f\geqslant 0$ in $L^{n^{\prime},\infty}$

[TABLE]

Remark 1.10**.**

*Setting $\displaystyle d\nu=\dfrac{dx}{x},\ |E|_{\nu}=\int_{E}d\nu,\quad f_{*,\nu}$ the decreasing rearrangement of a nonnegative function $f$ with respect to the measure $\nu$, then we can write the preceding theorem as : *

Theorem 1.11**.**

The $K$-functional for the couple $(L^{p,\infty},L^{p})$ is given, for $f\geqslant 0$ in $L^{p}+L^{p,\infty},\ t>0$

[TABLE]

Here $1\leqslant p<+\infty$, $\psi(s)=s^{\frac{1}{p}}f_{*}(s),\ s\in(0,1)$.

From this result, we can recover the following result due to Maligranda and Persson (see [19] :

Theorem 1.12**.**

Let $0<\theta<1,\ 1<p<+\infty$. Then

[TABLE]

Here $L^{p,\frac{p}{\theta}}$ is the usual Lorentz space.

Applying Theorem 1.9 with real interpolation method of Marcinkiewiecz, we then deduce the following partial answer for very weak solution :

Proposition 1.13**.**

For $0<\alpha\leqslant 1$, let $u$ be the solution of (1.1). Then one has a constant $c>0$ such that

[TABLE]

Other consequences of the above interpolation results are the interpolation inequalities, we state few of them.

Property 1.14**.**

(Interpolation inequalities for small and grand Lebesgue spaces)

1. (1)

Let $1\geqslant\alpha>\dfrac{1}{n^{\prime}}$ then $\forall\,v\in L^{(n^{\prime}}$

[TABLE] 2. (2)

For any $\alpha\in]0,1[$, one has

[TABLE]

2. Notation and Primary results

For a measurable function $f:\Omega\to{\rm I\!R}$, we set for $t\geqslant 0$

[TABLE]

and $f_{*}$ the decreasing rearrangement of $|f|$,

[TABLE]

that we shall assume to be equal to 1 for simplicity.

If $A_{1}$ and $A_{2}$ are two quantities depending on some parameters, we shall write

[TABLE]

[TABLE]

We recall also the following definition of interpolation spaces.

Let $(X_{0},||\cdot||_{0}),\ (X_{1},||\cdot||_{1})$ two Banach spaces contained continuously in a Hausdorff topological vector space (that is $(X_{0},X_{1})$ is a compatible couple).

For $g\in X_{0}+X_{1},\ t>0$ one defines the so called $K$ functional $K(g,t;X_{0},X_{1})\dot{=}K(g,t)$ by setting

[TABLE]

For $0\leqslant\theta\leqslant 1,\ 1\leqslant p\leqslant+\infty,\ \alpha\in{\rm I\!R}$ we shall consider

[TABLE]

Here $||\cdot||_{V}$ denotes the norm in a Banach space $V$. The weighted Lebesgue space $L^{p}(0,1;\omega)$, $0<p\leqslant+\infty$ is endowed with the usual norm or quasi norm, where $\omega$ is a weight function on $(0,1)$.

Our definition of the interpolation space is different from the usual one (see [4, 23]) since we restrict the norms on the interval $(0,1)$.

If we consider ordered couple, i.e. $X_{1}\hookrightarrow X_{0}$ and $\alpha=0,$

[TABLE]

is the interpolation space as it is defined by J. Peetre (see [4, 23, 5]).

2.1. Some remarkable $G\Gamma$-spaces

In this paragraph, we want to prove among other that $G\Gamma$-spaces cover many well-known spaces.

Proposition 2.1**.**

*Consider the classical Lorentz space $\displaystyle\Lambda^{p}(w_{2})$. Then it is equal to the set

\displaystyle\Big{\{}f:\Omega\to{\rm I\!R}\hbox{ measurable : }\left(\int_{0}^{1}f_{*}^{p}(\sigma)w_{2}(\sigma)d\sigma)\right)^{\frac{1}{p}}=||f||_{\Lambda^{p}(w_{2})}<+\infty\Big{\}}.

If $w_{1}$ and $w_{2}$ are integrable weights on $(0,1)$ and $w_{2}$ satisfies c1) then*

[TABLE]

**Proof

**If $v\in\Lambda^{p}(w_{2})$ then $\displaystyle\rho(v)\leqslant||v||_{\Lambda^{p}(w_{2})}\left[\int_{0}^{1}w_{1}(t)dt\right]^{\frac{1}{m}}<+\infty$.

Conversely, let $v$ be such that $\rho(v)<+\infty$. We have for some $a>0$, $\displaystyle\int_{a}^{1}w_{1}(t)dt>~{}0$. Then for all $t\geqslant a$

[TABLE]

from which we derive after multiplying by $w_{1}(t)$ and integrating from $a$ to 1,

[TABLE]

Between $(a,1)$, we have :

[TABLE]

The condition c1) implies

[TABLE]

So that relations (2.2) to (2.4) imply

[TABLE]

This shows

[TABLE]

$\diamondsuit$

Next we want to focus in a special case :

Proposition 2.2**.**

Assume that $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{\gamma},$ $w_{2}(t)=(1-{\rm Log\,}t)^{\beta},$ $(\gamma,\beta)\in{\rm I\!R}^{2},$ $m\in[1,+\infty[,$ $p\in[1,+\infty[$.

1. (1)

If $\gamma<-1$ then $G\Gamma(p,m;w_{1},w_{2})=\Lambda^{p}(w_{2})$. 2. (2)

If $\gamma>-1$ and $\gamma+\beta\dfrac{m}{p}+1\geqslant 0$ then

[TABLE]

Proof 

For the first statement, we observe that if $\gamma+1<0$, $\displaystyle\int_{0}^{1}(1-{\rm Log\,}t)^{\gamma}\dfrac{dt}{t}$ is finite. Then applying Proposition 2.1 we derive the first result.

For the case $\gamma+1>0$, we shall need the following lemma whose proof is in [12]:

Lemma 2.3**.**

Let $t_{k}=2^{1-2^{k}},\ k\in{\rm I\!N},\ \lambda>0,\ q>0,\ H$ a nonnegative locally integrable function on $(0,1)$ satisfying

[TABLE]

Then

1. (1)

$2^{k}\approx 1-{\rm Log\,}x$, $x\in[t_{k+1},t_{k}]$. 2. (2)

[TABLE]

We shall apply this Lemma with $\displaystyle H(x)=f_{*}^{p}(x)(1-{\rm Log\,}x)^{\beta}$. We have $\displaystyle\int_{0}^{1}H(x)dx\lesssim\int_{0}^{\frac{1}{2}}H(x)dx$ since $f_{*}^{p}$ is decreasing and $\displaystyle\int_{0}^{1}(1-{\rm Log\,}t)^{\gamma}dt<+\infty\quad\forall\,\gamma\in{\rm I\!R}$. Indeed

[TABLE]

Applying statement 2. of this Lemma 2.3, we derive

[TABLE]

[TABLE]

If $\gamma>-1,\ \gamma+\beta\dfrac{m}{p}+1\geqslant 0$, then the equality comes from the definition of $G\Gamma(p,m;\overline{w_{1}})$.

This ends of the proof of Proposition 2.2 $\diamondsuit$

Lemma 2.4**.**

Assume that $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{\gamma},$ $w_{2}(t)=(1-{\rm Log\,}t)^{\beta},$ $(\gamma,\beta)\in\mathbb{R}^{2}$, $m\in[1,\infty[,$ $p\in[1,\infty[.$ If $\gamma>-1$ and $\gamma+\beta\frac{m}{p}+1<0$, then

[TABLE]

Proof.

Put

[TABLE]

Let $t_{k}=2^{1-2^{k}},$ $k\in\mathbb{N}.$ Since $\gamma+\beta\frac{m}{p}+1<0$, we can apply the second assertion of Lemma $6.3$ in [12] to obtain

[TABLE]

then using the second assertion of Lemma $6.1$ in [12] gives

[TABLE]

by first assertion in Lemma $2.3$, we get

[TABLE]

since $\gamma>-1$ , we can apply the first assertion of Lemma $6.1$ in [12] to obtain

[TABLE]

finally an application of the second assertion in Lemma $2.3$ yields

[TABLE]

which completes the proof. ∎

We shall need in particular the Corollary 2.7, consequence of relation (2.7) and the following

Definition 2.5**.**

**of the small Lebesgue space [18, 9]

***The small Lebesgue space associated to the parameters $p\in]1,+\infty[$ and $\theta>0$ is the set

$L^{(p,\theta}(\Omega)=$*

[TABLE]

Definition 2.6**.**

**of the grand Lebesgue space [18, 9]

***The grand Lebesgue space is the associate space of the small Lebesgue space, with the parameters $p\in]1,+\infty[$ and $\theta>0$ is the set

$L^{p),\theta}(\Omega)=$*

[TABLE]

Corollary 2.7**.**

of Proposition 2.2* 

If $m=1,\gamma+1+\dfrac{\beta}{p}>0$, $\gamma>-1$ and $\beta\in{\rm I\!R}$, the functions $w_{i},\ i=1,2$ as in Proposition 2.2 then*

[TABLE]

3. Some $K$-functional computations and the associated interpolation spaces

3.1. The case of the couple $(L^{n^{\prime}},L^{(n^{\prime}})$

Theorem 3.1**.**

Let $\varphi(t)=e^{1-\frac{1}{t^{n}}},\ 0<t\leqslant 1$. Then

[TABLE]

for all $f\in L^{n^{\prime}}+L^{(n^{\prime}}$.

**Proof:

**First , let us show:

[TABLE]

Let $f=g+h\in L^{n^{\prime}}+L^{(n^{\prime}}$. Then, for all $x$, $f_{*}(x)\leqslant g_{*}\left(\dfrac{x}{2}\right)+h_{*}\left(\dfrac{x}{2}\right)$. Therefore, we have

[TABLE]

Taking the infinimum, one derives

[TABLE]

For the converse, we adopt the same decomposition as in [12]

[TABLE]

Then

[TABLE]

[TABLE]

[TABLE]

As in [12], we have

[TABLE]

Since

[TABLE]

we obtain for the first term $I_{1}$

[TABLE]

[TABLE]

and

[TABLE]

with relations (3.5) to (3.7), we derive

[TABLE]

Thus relations (3.3) and (3.8) infer :

[TABLE]

Thus

[TABLE]

The combination of the above relations (3.10), (3.1) gives Theorem 3.1. $\diamondsuit$

Corollary 3.2**.**

Theorem 3.1* 

One has, for $r\in[1,+\infty[$,   $0<\theta<1$,*

[TABLE]

**Proof:

**

One has for $f\in{L^{n^{\prime}}}+{L^{(n^{\prime}}},\ 1\leqslant r+\infty$

[TABLE]

Using Theorem 3.1 and making a change of variable $x=\varphi(t)$ that is

$t=(1-{\rm Log\,}x)^{-\frac{1}{n}}$, one derives from relation (3.11)

[TABLE]

[TABLE]

Applying Hardy’s inequality (taking into account that $\theta<1$), we have

[TABLE]

For the converse, since we have for all $x>0$

[TABLE]

we then have

[TABLE]

From this relation we deduce

[TABLE]

while to estimate the last integral, one has

[TABLE]

Since $({L^{n^{\prime}}},{L^{(n^{\prime}}})_{\theta,r}$ is continuously embedded in ${L^{n^{\prime}}}$, we then have

[TABLE]

Thus, we derive

[TABLE]

$\diamondsuit$

**Proof of Theorem 1.7

**We derive it from Corollary 3.2 of Theorem 3.1.    $\diamondsuit$

3.2. Interpolation between grand and classical Lebesgue spaces in the critical case

Lemma 3.3**.**

Let $1<p<\infty$, and let $f\in L^{p)}.$ Then, for all $0<t<1,$

[TABLE]

where $\varphi(t)=e^{1-\frac{1}{t^{p}}}.$

Proof.

Fix $f\in L^{p)}$ and $0<t<1.$ Set

[TABLE]

First we show that

[TABLE]

Let $f=g+h$ be an arbitrary decomposition with $g\in L^{p)}$ and $h\in L^{p}.$ Using the elementary inequality $f_{\ast}(x)\leqslant g_{\ast}(x/2)+h_{\ast}(x/2)$, we derive

[TABLE]

from which (3.15) follows. Next we establish the converse estimate

[TABLE]

To this end, we take the same particular decomposition $f=g+h$ as in Theorem 3.1 relation (3.2). Clearly,

[TABLE]

Next we note that

[TABLE]

which gives

[TABLE]

Now (3.16) follows from (3.17) and (3.18). The proof is complete. ∎

Theorem 3.4**.**

Let $1<p<\infty$, $0<\theta<1$, and $1\leqslant r<\infty.$ Then

[TABLE]

where $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{r\theta/p-1}$ and $w_{2}(t)=(1-{\rm Log\,}t)^{-1}.$

Proof.

Let $f\in L^{p)}.$ Then, using Lemma 3.3, we get at

[TABLE]

where

[TABLE]

Now, in view of Lemma 2.4 (applied with $\gamma=\frac{r\theta}{p}-1$, $\beta=-1$, $m=r$), it is sufficient to establish that

[TABLE]

The estimate $``\gtrsim"$ in (3.20) follows trivially from (3.19), while for the converse estimate we infer from Bennet-Rudnick Lemma ([3] Lemma $6.1$) that

[TABLE]

which, combined with (3.19), gives

[TABLE]

from which follows the desired estimate $``\lesssim"$ in (3.20) by Hardy inequality [3] Theorem $6.5$. The proof is complete. ∎

3.3. Interpolation between grand Lebesgue spaces in the critical case

Lemma 3.5**.**

Let $1<p<\infty$ and $0<\beta<\alpha.$ Let $f\in L^{p),\alpha}$. Then, for all $0<t<1,$

[TABLE]

where $\varphi(t)=e^{1-t^{\frac{p}{\beta-\alpha}}}.$

Proof.

Fix $f\in L^{p,\infty}$ and $0<t<1.$ Set

[TABLE]

and

[TABLE]

First we show that

[TABLE]

Let $f=g+h$ be an arbitrary decomposition with $g\in L^{\alpha),p}$ and $h\in L^{\beta),p}.$ Using the elementary inequality $f_{\ast}(x)\leqslant g_{\ast}(x/2)+h_{\ast}(x/2)$, we derive

[TABLE]

and

[TABLE]

Thus, we get

[TABLE]

from which (3.21) follows.

It remains to establish the converse estimate

[TABLE]

We again take the same particular decomposition $f=g+h$ as in Theorem 3.1. (relation (3.2). It is easy to check that

[TABLE]

Next we observe that

[TABLE]

where

[TABLE]

and

[TABLE]

Since

[TABLE]

we have

[TABLE]

Next we show that

[TABLE]

We have

[TABLE]

from which follows (3.26). Altogether from the relations (3.23)-(3.26), we get (3.22). The proof is complete. ∎

Theorem 3.6**.**

Let $1<p<\infty$, $0<\beta<\alpha,$ $0<\theta<1$, and $1\leqslant r<\infty.$ Then

[TABLE]

where $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{\frac{r\theta}{p}(\alpha-\beta)-1}$ and $w_{2}(t)=(1-{\rm Log\,}t)^{-\alpha}.$

Proof.

Let $X=(L^{p),\alpha},L^{p),\beta})_{\theta,r}$ and take $f\in L^{p),\alpha}.$ Then

[TABLE]

where

[TABLE]

and

[TABLE]

Put

[TABLE]

In view of Lemma 2.4 (applied with $\gamma=r\theta(\alpha-\beta)/p-1$, $\beta=-\alpha$, $m=r$), it is sufficient to show that

[TABLE]

Clearly, $I_{3}\lesssim I_{2}$. Thus, it remains to establish that $I_{1}\lesssim I_{3}$ and $I_{2}\lesssim I_{3}.$ Now

[TABLE]

by Bennet-Rudnick Lemma ([3] Lemma $6.1$), we get

[TABLE]

now applying Hardy inequality [3] Theorem $6.5$, we obtain $I_{1}\lesssim I_{3}.$ Next we again make use of Bennet-Rudnick Lemma ([3] Lemma $6.1$) to derive

[TABLE]

from which follows $I_{2}\lesssim I_{3}$ by Hardy inequality [3] Theorem $6.5$. The proof is complete. ∎

3.4. The $K$-functional for the couple $(L^{p,\infty},L^{p}),\ 1<p<+\infty$

Theorem 3.7**.**

For a measurable set $E\subset[0,1]$, we denote $|E|_{\nu}=\displaystyle\int_{E}\dfrac{dx}{x}$ and for $f\in L^{p,\infty}+L^{p}$, $1<p<+\infty$, we define

[TABLE]

Then

[TABLE]

and

[TABLE]

where $\psi(s)=s^{\frac{1}{s}}f_{*}(s)$, $\psi_{*,\nu}$ its decreasing rearrangement with respect to the measure $\nu$.

**Proof:

**Let $f=g+h\in L^{p,\infty}+L^{p}$. Then, $f_{*}(s)\leqslant g_{*}\left(\dfrac{s}{2}\right)+h_{*}\left(\dfrac{s}{2}\right)$, for $s\in]0,1]$

[TABLE]

The first term can be bound as follows :

[TABLE]

While the second term satisfies

[TABLE]

From the three last relations, we have

[TABLE]

From which we derive

[TABLE]

For the converse , let $t$ be fixed and set $\psi(s)=s^{\frac{1}{p}}f_{*}(s),\ \ s\in[0,1],\ \ \psi_{*,\nu}$ will denote its decreasing rearrangement with respect to $\nu$, $A_{t}=\{s:\psi(s)>\psi_{*,\nu}(t^{-p})\}.$

By equimesurability, we have

[TABLE]

Let us consider the measure preserving mapping $\sigma:{\rm I\!R}\to(0,+\infty)$ such that $f=f_{*}\circ\sigma$ and set $f_{i}=g_{i}\circ\sigma$, $i=1,2$ where, for $s\in(0,1)$

[TABLE]

and $A^{c}_{t}$ is the complement of $A_{t}$ in $(0,1)$, say A^{c}_{t}=\Big{\{}s:\psi(s)\leqslant\psi_{*,\nu}(t^{-p})\Big{\}}.

Since $\sigma$ is measure preserving we have

[TABLE]

From which we derive

[TABLE]

While for $f_{1}$, we have

[TABLE]

Since $f=f_{1}+f_{2}\in L^{p,\infty}+L^{p}$, we derive from relation (3.32) and (3.33) that

[TABLE]

Since the function $x\to\psi_{*,\nu}(x)$ is decreasing one has

[TABLE]

Thus, we derive from (3.34) and (3.35)

[TABLE]

Making use of the Hardy Littlewood (see[22]), we have

[TABLE]

Thus

[TABLE]

This equality with relation (3.36) leads to

[TABLE]

$\diamondsuit$

As we noticed at the beginning, we recover the Maligranda-Persson’s results stating that

[TABLE]

**Proof of Maligranda-Persson’s result

**One has, from the above result

[TABLE]

where we set temporarily $\Phi(x)=\psi_{*,\nu}(x)^{p}$.

Making the following change of variable $\sigma=t^{-p}$, we derive that

[TABLE]

but by the Hardy inequality this integral is equivalent to $\displaystyle\int_{0}^{+\infty}\Phi(x)^{\frac{1}{\theta}}dx$.

Therefore, we have

[TABLE]

This last quantity is equivalent to the norm of $f$ in $L^{p,\frac{p}{\theta}}$. $\diamondsuit$

We end this section by proving Theorem 1.6 so we start with the following lemma :

Lemma 3.8**.**

Let $1<p<\infty.$ Then for any $f\in L^{p,\infty}$ and all $0<t<1$,

[TABLE]

where $\rho(t)=(1-{\rm Log\,}t)^{-1+\frac{1}{p}}.$

Proof.

Fix $f\in L^{p,\infty}$ and $0<t<1.$ Set

[TABLE]

It is sufficient to show that the following estimate

[TABLE]

holds for an arbitrary decomposition $f=f_{0}+f_{1}$ with $f_{0}\in L^{p,\infty}$ and $f_{1}\in L^{(p}.$ In view of the elementary inequality $f_{\ast}(t)\leqslant f_{0_{\ast}}(t/2)+f_{1_{\ast}}(t/2)$, it follows that

[TABLE]

Clearly, $I(t/2,f_{0})\leqslant\|f_{0}\|_{L^{p,\infty}}.$ Therefore, it remains to show that

[TABLE]

Note that $\rho(t)\approx 1,\;1/2<t<1.$ Therefore, (3.39) holds for all $1/2<t<1$ in view of the fact that $L^{(p}\hookrightarrow L^{p,\infty}.$ Next let $0<t<1/2$ and take $0<v<t/2.$ Then

[TABLE]

whence we obtain (3.39) since $v$ was arbitrarily taken to be between [math] and $t/2.$ The proof is complete. ∎

Theorem 3.9**.**

Let $1<p<\infty$, $0<\theta<1$, and $1\leqslant r<\infty.$ Then, for any $f\in(L^{p,\infty},L^{(p})_{\theta,r},$ one has

[TABLE]

where $v_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{r\theta(1-1/p)-1}$ and $v_{2}(t)=t^{1/p}.$

Proof.

Put $\rho(t)=(1-{\rm Log\,}t)^{-1+\frac{1}{p}},\;0<t<1.$ It immediately follows from Lemma 3.8 that

[TABLE]

The simple observation

[TABLE]

completes the proof. ∎

Now the estimate resulting from Theorem 1.3 in [12] is:

Theorem 3.10**.**

Let $1<p<\infty$, $0<\theta<1$, and $1\leqslant r<\infty.$ Then

[TABLE]

where $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{r\theta-1}$ and $w_{2}(t)=(1-{\rm Log\,}t)^{-1}.$

The G$\Gamma$ spaces in Theorems 3.9 say $G\Gamma(\infty,r;v_{1},v_{2})$ and 3.10 say $G\Gamma(p,r,w_{1},w_{2})$ are not comparable. Thus we get Theorem 1.6.

4. Some interpolation inequalities for Small and Grand Lebesgue spaces

One may combine the above results with some standard results on interpolation spaces to deduce few inequalities as Property 1.14.

We recall the following result that can be found in [23].

Theorem 4.1**.**

*Let $E_{0}$ and $E_{1}$ be two Banach spaces continuously embedded into some topological vector space.

For $0\leqslant\theta\leqslant 1$, one has*

[TABLE]

if and only if

[TABLE]

where $||\cdot||_{i}$ denotes the norm in $E_{i}$, $i=0,1$.

**Proof of Property 1.14

**We apply the above Theorem 4.1 with $E_{0}=L^{n^{\prime})}$, $E_{1}=L^{(n^{\prime}}$.

Then from Theorem 1.4 and Corollary 2.7 of Proposition 2.2 one has

[TABLE]

with $\beta=n\alpha-n+1$ and $\alpha>\dfrac{1}{n^{\prime}}$.

Since $L^{(n^{\prime},\beta}\subset L^{n^{\prime},\infty}\subset L^{n^{\prime})}$, we deduce the result from Theorem 4.1 with $E=L^{n^{\prime},\infty},\ \theta=\alpha$.

The same argument holds for the second inequality, since

[TABLE]

$\diamondsuit$

5. A remark on the associate space of $G\Gamma(p,r;\omega)$

Theorem 5.1**.**

Let $1<p<\infty$, $1<r<\infty$ and $\delta>0.$ Put $w(t)=t^{-1}(1-{\rm Log\,}t)^{\delta-1}.$ Then

[TABLE]

where $w_{1}(t)=t^{-1}(1-{\rm Log\,}t)^{\frac{r^{\prime}\delta}{r}-1}$ and $w_{2}(t)=(1-{\rm Log\,}t)^{-\frac{2p^{\prime}\delta}{r}}$.

Proof.

Put $\eta=p^{\prime}\delta/r,$ and take $\alpha=2\eta,$ $\beta=\eta/2$ and $\theta=1/3.$ Then by [1] Theorem $7$, we have

[TABLE]

Thus, using duality relation of real interpolation spaces (see, for instance, [5] Theorem $3.7.1$), we get

[TABLE]

finally, an application of Theorem 3.6 completes the proof. ∎

Remark 5.2**.**

In view of Lemma $2.4$ , we get the following equivalent norm on $\left[G\Gamma(p,r;w)\right]^{\prime}$:

[TABLE]

*which is apparently simpler than the one which follows from [14] Theorem $1.1$ (vi). *

**Acknowledgment ** :

The third author, M.R. Formica has been partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the project ”sostegno alla Ricerca individuale”.

The work of fourth author, A.Gogatishvili , has been partially supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [grand number FR17-589], by Grant 18-00580S of the Czech Science Foundation and RVO:67985840. We address a grateful thanks to Professor Teresa Signes to point out an error in the first version of this work.