Garsia-Rodemich spaces: Local Maximal Functions and Interpolation

TL;DR

This paper characterizes Garsia-Rodemich spaces using local maximal functions, showing they serve as interpolation spaces between L^1 and BMO, and explores their applications in inequalities and functional analysis.

Contribution

It introduces a new characterization of Garsia-Rodemich spaces via local maximal operators and establishes their role as interpolation spaces between L^1 and BMO.

Findings

Garsia-Rodemich spaces are characterized by local maximal functions.

These spaces are interpolation spaces between L^1 and BMO.

New expressions for the K-functional of (L^1, BMO) are obtained.

Abstract

We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $R^{n}$. We show that there exists $s_{0}\in(0,1),$ such that for all $0<s<s_{0},$ and for all r.i. spaces $X(Q_{0}),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_{0}}^{\#}$ is the Str\"{o}mberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows that the $GaRo_{X}$ spaces are interpolation spaces between $L^{1}$ and $BMO.$ Among the applications, we prove, using real interpolation, the monotonicity under rearrangements of Garsia-Rodemich type functionals. We also give an approach to Sobolev-Morrey inequalities via Garsia-Rodemich norms, and prove necessary and sufficient conditions for $GaRo_{X}(Q_{0})=X(Q_{0}).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^{1},BMO)$.

Equations

$$GaR{o}_X(Q_0)=\{f\in L^1(Q_0):\Vert f{\Vert }_{GaR{o}_X}\simeq \Vert M_{s,Q_0}^{\#}f{\Vert }_X<\infty \},$$

$${\Vert f\Vert }_{J{N}_p}=\underset{\{Q_i{\}}_{i\in N}\in P}{\sup }{\left\{\sum _i\left|Q_i\right|{\left(\frac{1}{\left|Q_i\right|}{\int }_{Q_i}\left|f-f_{Q_i}\right|dx\right)}^p\right\}}^{1/p}<\infty ,$$

$$P:=P(Q_0)=\{\{Q_i{\}}_{i\in N}:Q_i\text{mbox}aresubcubesof{Q}_0\text{mbox}withpairwisedisjointinteriors\}.$$

$${\Vert f\Vert }_{J{N}_p}=\underset{\pi \in P}{\sup }{\Vert f_{\pi }\Vert }_{L^p(Q_0)}.$$

$${\Vert f\Vert }_{J{N}_{\infty }}=\underset{\pi \in P}{\sup }{\Vert f_{\pi }\Vert }_{L^{\infty }(Q_0)}.$$

$$J{N}_{\infty }(Q_0)=BMO(Q_0).$$

$$JN_{p}{(Q_{0})}\subset\left\{\begin{array}[c]{cc}L(p,\infty){(Q_{0})}&1<p<\infty\\ e^{L}{(Q_{0})}&p=\infty.\end{array}\right.,$$

$$\sum _i\frac{1}{\left|Q_i\right|}{\int }_{Q_i}{\int }_{Q_i}\left|f(x)-f(y)\right|dxdy\le C{\left(\sum _i\left|Q_i\right|\right)}^{1/p^{\prime }},$$

$${\Vert f\Vert }_{G_p}=\inf \{C:\text{ such that (refdada) holds}\}.$$

$${\int }_Q\left|f(x)-f_Q\right|dx\le \frac{1}{\left|Q\right|}{\int }_Q{\int }_Q\left|f(x)-f(y)\right|dxdy\le 2{\int }_Q\left|f(x)-f_Q\right|dx.$$

$$\sum _i\frac{1}{\left|Q_i\right|}{\int }_{Q_i}{\int }_{Q_i}\left|f(x)-f(y)\right|dxdy$$

$$=2\sum _i{\left|Q_i\right|}^{1/p^{\prime }}({\left|Q_i\right|}^{1/p}\frac{1}{\left|Q_i\right|}{\int }_Q\left|f(x)-f_Q\right|dx)$$

$$\le 2{\left\{\sum _i\left|Q_i\right|\right\}}^{1/p^{\prime }}{\left\{\sum _i\left|Q_i\right|{\left(\frac{1}{\left|Q_i\right|}{\int }_{Q_i}\left|f-f_{Q_i}\right|dx\right)}^p\right\}}^{1/p}.$$

$${\Vert f\Vert }_{G_p}\le 2{\Vert f\Vert }_{J{N}_p}.$$

$$G_p=L(p,\infty ),1<p<\infty .$$

$$G_{\infty }=BMO.$$

$${\int }_Q\left|f(x)-f_Q\right|dx\le \frac{1}{\left|Q\right|}{\int }_Q{\int }_Q\left|f(x)-f(y)\right|dxdy\le {\Vert f\Vert }_{G_{\infty }}\left|Q\right|,$$

$${\Vert f\Vert }_{BMO}\le {\Vert f\Vert }_{G_{\infty }}.$$

$$\sum _{Q\in \pi }\frac{1}{\left|Q\right|}{\int }_Q{\int }_Q\left|f(x)-f(y)\right|dxdy$$

$$\le 2{\Vert f\Vert }_{BMO}\sum _{Q\in \pi }\left|Q\right|.$$

$${\Vert f\Vert }_{G_{\infty }}\le 2{\Vert f\Vert }_{BMO}.$$

$${\Vert f\Vert }_{G_1}:=\underset{\pi \in P}{\sup }\sum _{Q\in \pi }\frac{1}{\left|Q\right|}{\int }_Q{\int }_Q\left|f(x)-f(y)\right|dxdy.$$

$$G_1=L^1.$$

$${\Vert f\Vert }_{G_1}\le 2{\Vert f\Vert }_{L^1}.$$

$${\Vert f\Vert }_{\widetilde{L^1}}$$

$$\le \frac{1}{\left|Q_0\right|}{\int }_{Q_0}{\int }_{Q_0}\left|f(x)-f(y)\right|dxdy\text{ (by (refextradulfa0))}$$

$$\le {\Vert f\Vert }_{G_1}.$$

$$G_{\infty }\subset L(\infty ,\infty ),$$

$$L(\infty ,\infty )=\{f\in L^1(Q_0):{\Vert f\Vert }_{L(\infty ,\infty )}=\underset{t}{\sup }(f^{\ast \ast }(t)-f^{\ast }(t)\}<\infty \}$$

$$f^{\ast \ast }(t)-f^{\ast \ast }(\left|Q_0\right|)$$

$$\le {\Vert f\Vert }_{L(\infty ,\infty )}\log \frac{\left|Q_0\right|}{t}.$$

$$BMO\subset L(\infty ,\infty ).$$

$$\mathbf{B}\subset L(n^{\prime },\infty ).$$

Full text

[TABLE]

where $M_{s,Q_{0}}^{\#}$ is the Strömberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows that the $GaRo_{X}$ spaces are interpolation spaces between $L^{1}$ and $BMO.$ Among the applications, we prove, using real interpolation, the monotonicity under rearrangements of Garsia-Rodemich type functionals. We also give an approach to Sobolev-Morrey inequalities via Garsia-Rodemich norms, and prove necessary and sufficient conditions for $GaRo_{X}(Q_{0})=X(Q_{0}).$ Using packings, we obtain a new expression for the $K-$functional of the pair $(L^{1},BMO)$.

SA was supported by the Ministry of Education and Science of the Russian Federation, project 1.470.2016/1.4, and by the RFBR grant 18-01-00414a.

MM was partially supported by a grant from the Simons Foundation ($\backslash$#207929 to Mario Milman)

This paper is in final form and no version of it will be submitted for publication elsewhere.

1. Introduction

The starting point of this research is the celebrated John-Nirenberg Lemma which we now recall. Let $Q_{0}\subset\mathbb{R}^{n}$ be a fixed cube111A “cube” in this paper will always mean a cube with sides parallel to the coordinate axes. We normalize $Q_{0}$ to have measure $1.$, $1<p<\infty,$ the John-Nirenberg spaces $JN_{p}:=JN_{p}(Q_{0})$ consist of all functions $f\in L^{1}(Q_{0})$ such that (cf. [18], [32])

[TABLE]

where $f_{Q}=\frac{1}{\left|Q\right|}\int_{Q}fdx$ and

[TABLE]

To understand the correct definition for $p=\infty,$ we proceed as follows. For $\pi={\{Q_{i}\}_{i\in N}}\in P,$ let $f_{\pi}(x)={\displaystyle\sum\limits_{i}}\left(\frac{1}{\left|Q_{i}\right|}\int_{Q_{i}}\left|f-f_{Q_{i}}\right|dx\right)\chi_{Q_{i}}(x),$ then

[TABLE]

This justifies the definition: $JN_{\infty}:=JN_{\infty}(Q_{0})$ consists of all functions $f\in L^{1}(Q_{0})$ such that

[TABLE]

It follows readily that

[TABLE]

The John-Nirenberg Lemma [18] implies the following embeddings

[TABLE]

where $L(p,\infty){(Q_{0})}$ is the ”weak” $L_{p}$-space and $e^{L}{(Q_{0})}$ is the Orlicz space of exponentially integrable functions. This result, and the spaces involved, has been the object of intensive study over the years and, in particular, the space $BMO$ now plays a very important role in harmonic analysis. We refer to [18], [7], [14], [32], for background, different proofs and extensive bibliographies.

In their paper, Garsia-Rodemich [16] proposed a very original approach222While [16] contains many interesting results, and indeed, has been widely quoted in the literature, their proposed approach to (1.2) has remained largely unnoticed until very recently (cf. [24]). to (1.2). It is based on the following idea: To effectively compare $JN_{p}$ with $L(p,\infty)$, $1<p<\infty,$ a new class of spaces was introduced in [16]. We shall say that333To describe the original results we shall use a temporary notation. $f\in G_{p}:=G_{p}(Q_{0})$ if and only if $f\in L^{1}(Q_{0}),$ and $\exists C>0$ such that for all $\{Q_{i}\}_{i\in N}$ $\in P$ we have

[TABLE]

where $1/p^{\prime}=1-1/p$, and we let

[TABLE]

The connection between the $JN_{p}$ and $G_{p}$ spaces can be seen from the readily verified computation

[TABLE]

Indeed, combining (1.4) with Hölder’s inequality, we find that for each $\{Q_{i}\}_{i\in N}$ $\in P,$ we have

[TABLE]

Consequently,

[TABLE]

The remarkable fact is that we actually have (cf. [16] for the one dimensional case and [24] in general) that as sets

[TABLE]

It is easy to see that the definition of $G_{p}$ also makes sense for $p=1$ and $p=\infty.$ Consider the case $p=\infty,$ then we let $p^{\prime}=1,$ and we see that the definition (1.3) makes sense in this case and we have

[TABLE]

In fact, since for any cube $Q,$ we have $\{Q\}\in P,$ it follows from (1.4) that

[TABLE]

yielding

[TABLE]

On the other hand, if $f\in BMO,$ then, once again using (1.4), we see that for any $\pi\in P,$

[TABLE]

Consequently,

[TABLE]

For $p=1,$ we let $p^{\prime}=\infty,$ and

[TABLE]

Then444When comparing $G_{p}$ spaces with  other function spaces we must take into account that for any constant $c,$ $\left\|f-c\right\|_{G_{p}}=\left\|f\right\|_{G_{p}}.$ (modulo constants)

[TABLE]

Indeed, it is easy to see that

[TABLE]

Conversely, let $\widetilde{L^{1}}$ be the space $L^{1}$ modulo constants, then since $\{Q_{0}\}\in P,$

[TABLE]

For $p=\infty,$ the method of proof of (1.5) that was given in [24] also yields

[TABLE]

where $L(\infty,\infty)$ is the Bennett-DeVore-Sharpley space

[TABLE]

(here, $f^{\ast}$ is the decreasing rearrangement of $f$ and $f^{\ast\ast}(t):=\frac{1}{t}\int_{0}^{t}f^{\ast}(s)\,ds$). Together, (1.6) and (1.7) therefore provide the improvement of the John-Nirenberg inequality obtained by Bennett-DeVore-Sharpley [4]555It is well known and easy to see that if $f\in L(\infty,\infty)$ then $f\in e^{L}.$ This can be seen from

$\displaystyle f^{\ast\ast}(t)-f^{\ast\ast}(\left|Q_{0}\right|)$ $\displaystyle=\int_{t}^{\left|Q_{0}\right|}(f^{\ast\ast}(s)-f^{\ast}(s))\frac{ds}{s}$

$\displaystyle\leq\left\|f\right\|_{L(\infty,\infty)}\log\frac{\left|Q_{0}\right|}{t}.$

Moreover, as shown by Bennett-DeVore-Sharpley [4], $L(\infty,\infty)$ is the rearrangement invariant hull of $BMO.$, namely

[TABLE]

In [25] it was shown how the Garsia-Rodemich spaces fit in the theory of Sobolev embeddings and in [26] the Garsia-Rodemich characterization of the weak $L^{p}$ spaces was used to provide a streamlined proof of the embedding theorem for the Bourgain-Brezis-Mironescu space $\mathbf{B}$ (cf. [8]),

[TABLE]

In short, the Garsia-Rodemich spaces provide a framework that can be used to study a number of classical problems in analysis. It was then natural to consider the problem of extending the Garsia-Rodemich construction. In particular, in view of the characterization of $L(p,\infty)$ provided by (1.5), we ask: what other rearrangement invariant spaces can be characterized via a suitable extension of the Garsia-Rodemich conditions? In this direction the following generalization of the condition (1.3) was proposed in [26].

Let $X:=X(Q_{0})$ be a rearrangement invariant space; for a given integrable function $f$ we consider the class $\Gamma_{f}$ of integrable functions $\gamma$ such that for all $\{Q_{i}\}_{i\in\mathbb{N}}\in P$ it holds

[TABLE]

To describe the corresponding enlarged class of spaces associated with these conditions, it will be convenient to replace our temporary notation for the $G-$spaces as follows. We let

[TABLE]

where666We let $\left\|f\right\|_{GaRo_{X}}=\infty,$ if $\Gamma_{f}=\varnothing,$ moreover we shall use the convention, $\left\|\gamma\right\|_{X}=\infty$ if $\gamma\notin X.$

[TABLE]

It is shown in [26, Corollary 1 and Remark 2] that, in this new notation, we have

[TABLE]

Moreover, at the end points it is easy to verify that

[TABLE]

and

[TABLE]

Thus, (1.5) and (1.7) now read

[TABLE]

More generally, the following generalization holds for any r.i. space $X$ (cf. [26]),

[TABLE]

where $\alpha_{X},\beta_{X}$ are the Boyd indices of $X$ (cf. Section 2.3).

The characterization (1.10) is very satisfactory since it captures all the main results at the level of $L^{p}$ spaces, $1<p<\infty$. However, the methods of [26] are not adequate to understand what happens when the Boyd indices are zero or one. In fact, the analysis of the end point cases of (1.10) seems to require a new set of ideas. In this paper we obtain a new characterization of the Garsia-Rodemich spaces via the Strömberg-Jawerth-Torchinsky local maximal operators (cf. [31], [17]). Let $s\in(0,1)$, then,

[TABLE]

where the supremum is taken over all the cubes $Q$ contained in $Q_{0}$ such that $x\in Q.$ One of our main results in this paper (cf. Theorem 1 below) states that there exists $s_{0}\in(0,1)$ such that, for all $0<s<s_{0},$ and for all r.i. spaces $X,$

[TABLE]

where the implied constants are independent of $f$777The expression $F\preceq G$ means that $F\leq c\cdot G$ for some constant $c>0$ independent of all or of a part of arguments $F$ and $G$. If $F\preceq G$ and $G\preceq F$ we write: $F\simeq G$..

This result not only allows us to study the limiting cases of (1.10) but at the same time provides a connection of the $GaRo_{X}$ spaces and classical harmonic analysis. In particular, in Theorem 2 we show a significant improvement over (1.10)

[TABLE]

In fact, for a large class of r.i. spaces of **fundamental type (**cf. Section 4, Definition 1) (1.12) is best possible. In Section 4 we prove Corollary 2: for every r.i. space $X$ of fundamental type

[TABLE]

Another consequence of (1.11) is the fact that the $GaRo_{X}$ spaces are real interpolation spaces between $L^{1}$ and $BMO.$ For example, this can be seen as a consequence of (1.11) and the formula of the $K-$functional for the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky [17],

[TABLE]

The characterization (1.11) connects $GaRo_{X}$ spaces with classical harmonic analysis. Let

[TABLE]

and for a r.i. space $X$ we define

[TABLE]

with

[TABLE]

We show that (cf. Theorem 6 below),

[TABLE]

Moreover, we consider generalized Fefferman-Stein inequalities of the form888The classical inequalities of Fefferman-Stein [15] correspond to $X=L^{p},1<p<\infty.$

[TABLE]

and prove that this inequality holds if $\alpha_{X}>0$ (cf. Theorem 5)999For a different approach to Fefferman-Stein inequalities in the more general setting of Banach function spaces we refer to Lerner [21]..

It is of interest to remark here that the conditions on the indices that appear in the results described above are connected with considerations arising from interpolation theory. For example, to compare the spaces $GaRo_{X}$ and $X^{\#}$ one needs to understand the relationship between the sharp maximal operator $f_{Q_{0}}^{\#}$and the local maximal operator $M_{s,Q_{0}}^{\#}f,$ and one way to achieve this is via the formula for the $K-$functional for the pair $L^{1}$ and $BMO$ provided by (1.14), and the formula obtained by Bennett-Sharpley (cf. Example 2 below)

[TABLE]

From (1.14) and (1.18) we see that the relationship between the sharp maximal operator and the local maximal operator is analogous to the classical relationship between $f^{\ast}$ and $f^{\ast\ast}$. Moreover, if we write $GaRo_{L(p,\infty)}=L(p,\infty)=(L^{1},BMO)_{1/p^{\prime},\infty}$ we see that as $p\rightarrow\infty$ we approach the space $BMO$, thus we expect to lose “rearrangement invariance”, and this may help to explain the requirement $\alpha_{X}>0,$ to be able to attain results of the form $GaRo_{X}=X.$

The connection between Garsia-Rodemich spaces and interpolation goes deeper. In fact, the ideas associated with the construction of Garsia-Rodemich spaces lead us to find a new formula for the $K-$functional associated with the pair $(L^{1},BMO),$ using packings (cf. Section 7 below), which we believe should be of interest when comparing pointwise averages, as one often does in the theory of weighted norm inequalities. As a concrete application of this circle of ideas we show how one can use interpolation methods to prove the monotonicity under rearrangements of certain Garsia-Rodemich type functionals (our approach should be compared with the one provided in [16]).

Finally, returning to some of the original results of Garsia and his collaborators, we show a simple proof of a Sobolev-Morrey embedding in Section 8.

We refer the reader to Section 2 and to the monographs [6], [22], [19], [10] and [32] for background information and notation.

Acknowledgement. We are very grateful to the referee for detailed and constructive criticism that helped us improve the presentation of the paper.

2. Background Information

2.1. Rearrangements

Let $\left(\Omega,\mu\right)$ be a resonant Borel probability space (cf. [6, pag. 64]). For a measurable function $f:{\Omega}\rightarrow\mathbb{R},$ the distribution function of $f$ is given by

[TABLE]

The decreasing rearrangement $f^{\ast}$ of a measurable function $f$ is the right-continuous non-increasing function, mapping $(0,1]$ into $\mathbb{[}0,\infty),$ which is equimeasurable with $f,$ i.e., satisfying

[TABLE]

where $\left|\cdot\right|$ denotes the Lebesgue measure on $[0,1].$ It can be defined by the formula

[TABLE]

The maximal average $f^{\ast\ast}(t)$ is defined by

[TABLE]

2.2. Rearrangement invariant spaces

We recall briefly the basic definitions and conventions we use from the theory of rearrangement invariant (r.i.) spaces, and refer the reader to the books [6], [19] and [22] for a complete treatment. In the next definition we follow [22].

Let $X:=X({\Omega})$ be a Banach function space on $({\Omega},\mu)$, which is either separable or has the Fatou property (the latter means that if $f_{n}\geq 0,$ $f_{n}\uparrow f,$ and $f\in X$, then $\left\|f_{n}\right\|_{X}\uparrow\left\|f\right\|_{X}$). We shall say that $X$ is a rearrangement invariant (r.i.) space, if $g\in X$ implies that all $\mu-$measurable functions $f$ with $f^{\ast}=g^{\ast}$ also belong to $X$ and, moreover, $\|f\|_{X}=\|g\|_{X}$. For any r.i. space $X$ we have

[TABLE]

with continuous embeddings. Many of the familiar spaces we use in analysis are examples of r.i. spaces, e.g. the $L^{p}$-spaces, Orlicz spaces, Lorentz spaces, Marcinkiewicz spaces, etc.

Let $M$ be an increasing convex function on $[0,\infty)$ such that $M(0)=0$. The Orlicz space $L_{M}$ consists of all measurable functions $x(t)$ on $[0,1]$ such that the function $M\left({|x(t)|}/{\lambda}\right)\in L^{1}$ for some $\lambda>0$. It is equipped with the Luxemburg norm

[TABLE]

In particular, if $M(u)=u^{p}$, $1\leqslant p<\infty$, we obtain usual $L^{p}$–spaces.

Let $\varphi$ be an increasing concave function on $[0,1]$, with $\varphi(0)=0$. The Marcinkiewicz space ${\mathcal{M}}(\varphi)$ consists of all measurable functions $x(t)$ such that

[TABLE]

The space $L(p,\infty)$, $1<p<\infty$, corresponds to taking $\varphi(s)=s^{1/p}$.

Let $X({\Omega})$ be a r.i. space, then there exists a unique r.i. space (cf. [6, pag. 64]) (the representation space of $X({\Omega})),$ $\bar{X}=\bar{X}(0,1)$ on $\left(\left(0,1\right),\left|\cdot\right|\right)$, such that

[TABLE]

In what follows if there is no possible confusion we shall not distinguish between $X$ and $\bar{X}$.

The following majorization principle, usually associated to the names Hardy-Littlewood-Pólya-Calderón (cf. [11], [22, Proposition 2.a.8]), holds for r.i. spaces: if

[TABLE]

then, for any r.i. space $\bar{X},$

[TABLE]

or equivalently,

[TABLE]

The **fundamental function **of $X$ is defined by

[TABLE]

We can assume without loss of generality that $\phi_{X}$ is concave (cf. [6]). For example, for an Orlicz space $L_{N}$ (cf. (2.1) above) we have, $\phi_{L_{N}}(t)=1/N^{-1}(1/t)$ and for a Marcinkiewicz space ${\mathcal{M}}(\varphi),$ the corresponding fundamental function is given by $\phi_{{\mathcal{M}}(\varphi)}(t)=\varphi(t)$.

2.3. Boyd indices and Hardy operators

Let $X=X({\Omega})$ be an arbitrary r.i. space. Then the compression/dilation operator $\sigma_{s}$ on $\bar{X}$, defined by

[TABLE]

is bounded on $\bar{X},$ and moreover (cf. [19, § 2.4])

[TABLE]

The Boyd indices (cf. [9]**) **are defined by

[TABLE]

For each r.i. space $X$ we have $0\leq\alpha_{X}\leq\beta_{X}\leq 1$. For example, it follows readily that $\alpha_{L^{p}}=\beta_{L^{p}}=\frac{1}{p}$ for all $1\leq p\leq\infty$.

It is known that the Boyd indices control the boundedness of the Hardy operators, which are defined by

[TABLE]

In fact, it is well known that (cf. [9], [19, Theorems 2.6.6 and 2.6.8]):

[TABLE]

2.4. K-functionals and real interpolation

Let $(A_{0},A_{1})$ be a compatible pair of Banach spaces. For all $f\in A_{0}+A_{1},t>0,$ we define the Peetre $K-$functional as follows

[TABLE]

Let $\theta\in(0,1),$ $1\leq q\leq\infty.$ The interpolation spaces $(A_{0},A_{1})_{\theta,q}$ are defined by

[TABLE]

where

[TABLE]

Example 1**.**

(Peetre-Oklander formula (cf. [6, (1.28) pag. 298], [27]): For the pair $(L^{1},L^{\infty})$ the $K-$functional is given by

[TABLE]

Let $M_{Q_{0}}$ be the maximal operator of Hardy-Littlewood,

[TABLE]

The maximal operator $M_{Q_{0}}$ is connected with $K(t,\cdot;L^{1},L^{\infty})$ via the Herz-Stein inequalities (cf. [6, Theorem $3.8,$ pag. 122]):

[TABLE]

Example 2**.**

For the pair $(L^{1},BMO)$ (we consider classes of equivalence modulo constants), we have the following formula due to Bennett-Sharpley (cf. [6, (8.11) pag. 393]):

[TABLE]

Comparing this with the Jawerth-Torchinsky formula (1.14) we see the equivalence (1.18).

In what follows any constant appearing in inequalities and depending only on the dimension $n$ will be referred to as absolute.

3. A new description of the Garsia-Rodemich spaces

In this section we give a new characterization of the Garsia-Rodemich spaces using local maximal operators. To motivate our result it will be useful to reformulate somewhat the definition of the $\Gamma_{f}$ classes (cf. (1.8) above).

It follows from inequalities (1.4) that for an integrable function $\gamma$ to belong to $\Gamma_{f}$ it is equivalent to verify the following condition: there exists a constant $C>0$ such that, for all subcubes $Q\subset Q_{0},$ we have

[TABLE]

whence

[TABLE]

The idea behind our main result can be now summarized as follows: for every $f\in L^{1},$ the Strömberg-Jawerth-Torchinsky maximal function $M_{s,Q_{0}}^{\#}f$ is an “optimal” choice of $\gamma$ from $\Gamma_{f}.$

Theorem 1**.**

There exists $s_{0}\in(0,1)$, depending only on dimension $n$, such that, for all $s\in(0,s_{0}),$ and every r.i. space $X,$ we have

[TABLE]

Moreover, with constants of equivalence, depending on $n\in\mathbb{N}$ and $s\in(0,s_{0})$,

[TABLE]

For the proof we shall need the following

Lemma 1**.**

(i) For every cube $Q\subset Q_{0}$, all $0<s<1$, $c\in\mathbb{R}$, and each $f\in L^{1}(Q_{0})$ we have

[TABLE]

(ii) There exists $0<s_{0}<1$ such that for all $s\in(0,s_{0})$and all $f\in L^{1}(Q_{0})$

[TABLE]

where $M_{Q_{0}}$ is the maximal function of Hardy-Littlewood (cf. (2.6)).

Proof.

(i) Let $Q\subset Q_{0}$ be an arbitrary cube. If $M_{s,Q}^{\#}f(y)>\lambda$ for $y\in Q$, then there is a cube $Q^{\prime}\subset Q$ such that $y\in Q^{\prime}$ and for all $c\in\mathbb{R}$

[TABLE]

Therefore, we have

[TABLE]

(here, $M_{Q}$ is the maximal operator of Hardy-Littlewood, corresponding to the cube $Q$). Hence,

[TABLE]

Combining this estimate with the fact that $M_{Q}$ is of weak type $(1,1)$ (cf. [6, Theorem 3.3.3]), we see that

[TABLE]

(ii) Let $x\in Q_{0}$ and $Q\subset Q_{0}$ be an arbitrary cube such that $x\in Q$. Denote by $2Q$ the cube with the same center as the cube $Q$ and with double side length. Clearly, there is a cube $\tilde{Q}$ such that $Q_{0}\cap(2Q)\subset\tilde{Q}\subset Q_{0}$ and $|\tilde{Q}|\leq|2Q|$. In particular, if $2Q\subset Q_{0}$, we take $\tilde{Q}=2Q$. Note that $\tilde{Q}\supset Q$.

Further, for all $y\in Q$ we have

[TABLE]

where the operator $R_{s,\tilde{Q}}^{\#}$ is defined in just the same way as $M_{s,\tilde{Q}}^{\#}$ except that the supremum is now taken over all cubes having non-empty intersection with the set $Q_{0}\setminus\tilde{Q}$. From the preceding inequality it follows that

[TABLE]

Applying part (i) of this lemma to the cube $\tilde{Q}$ and using the properties of the latter cube, we estimate the first integral from the right-hand side of (3.4) as follows:

[TABLE]

for any $c\in\mathbb{R}$. On the other hand, since the cube $\tilde{Q}$ is fixed, for each $\varepsilon>0$ we can choose a constant $c$ such that

[TABLE]

Combining these inequalities with the definition of $f_{Q_{0}}^{\#}(x)$, we infer that

[TABLE]

To estimate the second integral from the right-hand side of (3.4), we will use the following observation. For each cube $Q^{\prime}$ such that $Q^{\prime}\subset Q_{0}$ from $Q^{\prime}\cap(Q_{0}\setminus\tilde{Q})\neq\varnothing$ it follows that $Q^{\prime}\cap(\mathbb{R}^{n}\setminus(2Q))\neq\varnothing$. Therefore, then there is a cube $Q^{\prime\prime}\subset Q_{0}$ such that $Q^{\prime\prime}\supset Q$ and $|Q^{\prime\prime}|\leq 3^{n}|Q^{\prime}|$ and so from the definition of the operators $M_{s,Q}^{\#}$ and $R_{s,\tilde{Q}}^{\#}$ we see that

[TABLE]

where $s^{\prime}=s3^{-n}$. Now since $x\in Q$, we obtain,

[TABLE]

where the last inequality follows from Chebyshev’s inequality. Combining our findings with (3.4) and (3.5), we obtain

[TABLE]

Taking the supremum over all cubes $Q\subset Q_{0}$ such that $x\in Q$, and letting $\varepsilon\rightarrow 0$ we achieve the desired inequality (3.3). ∎

Proof of Theorem 1.

Suppose that $f\in L^{1}$ is such that $\left\|M_{s,Q_{0}}^{\#}f\right\|_{X}<\infty$ for some $s\in(0,1)$. Recall that by [20, Lemma 2.4], there exists $s_{0}=s_{0}(n)>0$ such that, for all $0<s<s_{0},$ and for every cube $Q\subset Q_{0}$, we have

[TABLE]

Consequently, by (1.4), $16M_{s,Q_{0}}^{\#}f\,\in\Gamma_{f}.$ Thus, for each $s\in(0,s_{0})$

[TABLE]

Conversely, let $f\in GaRo_{X}.$ Given $\varepsilon>0$ we can select $\gamma\in\Gamma_{f}$ $\cap X$ such that

[TABLE]

From the fact that $\gamma\in\Gamma_{f}$ it follows that (see the observation in the beginning of this section)

[TABLE]

Consequently, by (3.3), for all $0<s<1$

[TABLE]

Taking rearrangements in (3.9), and using Herz’s rearrangement inequality for the Hardy-Littlewood maximal operator (cf. (2.7)), for each $0<s<1$ we can find a constant $c=c(n,s)$ such that

[TABLE]

Hence, using successively the Hardy-Littlewood-Pólya-Calderón majorization principle (cf. (2.2)) and inequality (3.7), we get

[TABLE]

At this point we can let $\varepsilon\rightarrow 0$ to obtain the desired converse inequality. ∎

From Theorem 1, and its proof, we readily obtain the following alternative description of the Garsia-Rodemich spaces. Denote by $\Gamma_{f}^{\prime}$ the set of all functions $\gamma\in L^{1}(Q_{0})$ satisfying (3.8).

Corollary 1**.**

Let $X$ be a r.i. space. Then the Garsia-Rodemich space $GaRo_{X}$ consists of all functions $f\in L^{1}(Q_{0})$ for which $\Gamma_{f}^{\prime}\cap X\neq\varnothing$. Moreover, there exists an absolute constant $c=c(n)$ such that,

[TABLE]

4. A characterization of rearrangement invariant spaces via

Garsia-Rodemich conditions

The main result of this section is the following characterization of r.i. spaces which improves on (1.10) above.

Theorem 2**.**

Let $X$ be a r.i. space such that $\alpha_{X}>0.$ Then,

[TABLE]

Proof.

Let $f\in X.$ Since for all cubes $Q\subset Q_{0}$ we have

[TABLE]

it follows from (1.8) that $4\left|f\right|\in\Gamma_{f}.$ Consequently, the embedding $X\subset GaRo_{X}$ holds for every r.i. space $X$ and moreover

[TABLE]

We now show that if $\alpha_{X}>0,$ then $GaRo_{X}\subset X.$ Let $f\in GaRo_{X},$ and let $\gamma$ be an arbitrary element of $\Gamma_{f}.$ Then, we have (3.8), which combined with (2.7) implies

[TABLE]

Thus, from (2.5) and (2.8), we get

[TABLE]

where the implied constants are independent of $f$ and $\gamma.$ Fix $p>1/\alpha_{X}$. It is well known that (cf. [6, Theorem $8.11,$ pag 398])

[TABLE]

Therefore, by Holmstedt’s reiteration formula (cf. [6, Corollary $2.3,$ pag 310]), we have

[TABLE]

and

[TABLE]

with constants that depend only on $p$ (and hence on $X$). Combining these estimates with (4.1) yields

[TABLE]

with constants that depend only on $X$ and $n$. Since the pair $(L^{1},L^{p})$ is $K$-monotone (cf. [29], [12, Theorem 4])101010A different formulation of this result is given in [23, Theorem 3]., it follows that there exists a bounded linear operator $T$ acting on the pair $(L^{1},L^{p}),$ such that $f=T\gamma$. Moreover, from the fact that $p>1/\alpha_{X}$, we can deduce that $X$ is an interpolation space with respect to the pair $(L^{1},L^{p})$ (cf. [2, Theorem 2]). Consequently, by the $K$-monotonicity of $(L^{1},L^{p})$, there exists a Banach lattice $(\Phi,\left\|.\right\|_{\Phi})$ of Lebesgue measurable functions on $(0,\infty),$ such that the norm of $X$ can be represented as follows (cf. [10, Theorems 4.4.5 and 4.4.38])

[TABLE]

It follows that the operator $T$ is bounded on $X$ and, consequently,

[TABLE]

for some constant $c=c(n,X)$. Taking the infimum over all $\gamma\in\Gamma_{f}$, yields

[TABLE]

as we wished to show. ∎

Theorem 2 has a partial converse. To state the result we introduce the class of r.i. spaces of fundamental type.

Definition 1**.**

Let $X=X(Q_{0})$ be a r.i. space on $Q_{0}$, and let $\bar{X}=X(0,1)$ be its Luxemburg representation on $(0,1)$ (cf. Section 2.2). We shall say that $X$ is of fundamental type if there exists a constant $C>0,$ such that (cf. Section 2.3 above)

[TABLE]

Remark 1**.**

It is easy to verify that Orlicz, Lorentz, Marcinkiewicz spaces, etc., are all of fundamental type.

Definition 2**.**

A median value111111Note that $m_{f}(Q)$ is not uniquely defined. of $f$ on $Q$ is a number $m_{f}(Q)$ such that

[TABLE]

and

[TABLE]

It is well known that $m_{f}(Q)$ is one of the constants $c$ minimizing some functionals depending on the deviation $|f-c|$. In particular, we have (cf. [20, § 2, p. 2450])

[TABLE]

From this inequality one can easily deduce that for every r.i. space $X$ the following inequality holds:

[TABLE]

Theorem 3**.**

Let $X$ be a r.i. space of fundamental type, and let $0<s\leq 1/2$. If there exists a constant $C>0$ such that

[TABLE]

holds for all $f\in L^{1}(Q_{0})$, then we must have $\alpha_{X}>0$.

Proof.

To the contrary, suppose that $\alpha_{X}=0$. Since $X$ is of fundamental type we can find two numerical sequences $\{u_{k}\}_{k\in\mathbb{N}},\{a_{k}\}_{k\in\mathbb{N}}$ contained in $(0,1)$, converging to zero, and such that

[TABLE]

Without loss of generality we can assume that $Q_{0}=[0,1]^{n}$. Moreover, if $b>0$ we set $bQ_{0}:=[0,b]^{n}$. For $a\in(0,1),$ let $f_{a}(x):=n\ln(\frac{a^{1/n}}{|x|_{\infty}})\chi_{a^{1/n}Q_{0}}(x)$, $x\neq 0$, denoting $|x|_{\infty}:=\max_{i=1,2,\dots,n}|x_{i}|$ for every $x=(x_{i})_{i=1}^{n}\in\mathbb{R}^{n}$. One can readily verify that there exists a constant $D\geq 1,$ that depends only on the dimension and $s$, such that $M_{s,Q_{0}}^{\#}f_{a}(x)\leq D$ if $|x|\leq Da$ and $M_{s,Q_{0}}^{\#}f_{a}(x)=0$ if $|x|>Da$. Thus, using the concavity of the fundamental function $\phi_{X}$ (see Section 2.2), we get

[TABLE]

Moreover, it can be easily checked that $f_{a}^{\ast}(t)=\ln(a/t)\chi_{(0,a)}(t)$ and $m_{f_{a}}(Q_{0})=0$ if $a$ is sufficiently small. Thus, using (4.6), (4.4), (4.3) and (4.5), for sufficiently large $k\in\mathbb{N}$, we have

[TABLE]

This leads to a contradiction since $\lim_{k\rightarrow\infty}(\ln(u_{k}^{-1}))=\infty$. ∎

Applying Theorems 1 — 3, we immediately obtain the following result.

Corollary 2**.**

Let $X$ be an r.i. space of fundamental type. Then the following conditions are equivalent:

(a) $GaRo_{X}=X$;

(b) $GaRo_{X}\subset X$;

(c) $\alpha_{X}>0$.

5. $K-$functionals and rearrangement inequalities

In this section we consider some examples of the interaction of the Garsia-Rodemich functionals with rearrangements, that are connected with our development in this paper.

Our first application deals with a new proof of an inequality due to Bennett-Sharpley (cf. [6, Theorem $7.3,$ pag. 377]).

Example 3**.**

There exists an absolute constant $c>0,$ such that for all $f\in L^{1}(Q_{0}),$ we have

[TABLE]

Proof.

We recall the following fact from [26]: There exists an absolute constant $c_{1}$ such that for all $f\in L^{1}(Q_{0}),$ and all $\gamma\in\Gamma_{f}$, we have

[TABLE]

On the other hand, from (3.6), we know that for sufficiently small $s>0$ we have $16M_{s,Q_{0}}^{\#}f\in\Gamma_{f}.$ Consequently, by (5.2),

[TABLE]

Combining the last inequality with the fact that there exists an absolute constant $c_{2}$ such that (cf. (1.18))

[TABLE]

we obtain (5.1). ∎

Our second result shows how the continuity of rearrangements on Garsia-Rodemich spaces can be easily established using their description obtained in Theorem 1 and interpolation (compare with the methods to establish related rearrangement inequalities that were developed in [16] and [3])).

Theorem 4**.**

There exists an absolute constant $c>0$ such that for all $f\in GaRo_{X},$

[TABLE]

Proof.

From [16], [4] (cf. also [13]), we know that there exists an absolute constant $c_{1}\geq 1,$ such that

[TABLE]

On the other hand, it is well known that (cf. [16], [19, Theorem 2.3.1]) for all $f,g\in L^{1}(Q_{0}),$

[TABLE]

Consequently, for every $f\in L^{1}(Q_{0}),$

[TABLE]

In particular, in view of (1.14), there exists an absolute constant $c_{2}>0,$ such that

[TABLE]

By the Hardy-Littlewood-Pólya-Calderón principle, it follows that

[TABLE]

Applying (3.2) we finally obtain

[TABLE]

as we wished to show. ∎

Remark 2**.**

Essentially the same argument shows that if $T$ is a bounded operator on the pair $(L^{1},BMO)$, then $T$ is a bounded operator in the space $GaRo_{X}.$

Proof.

Indeed, for such operators we have

[TABLE]

which, in view of (1.14), implies

[TABLE]

Therefore, we get (2.2) and, as above, for any r.i. space $X$ we have

[TABLE]

The desired result now follows from Theorem 1. ∎

Remark 3**.**

As we have seen before (cf. (1.18)),$\left(M_{s,(0,1)}^{\#}f^{\ast}\right)^{\ast\ast}(t)\simeq\left(\left(f^{\ast}\right)_{(0,1)}^{\#}\right)^{\ast}(t),$ and $\left(M_{s,Q_{0}}^{\#}f\right)^{\ast\ast}(t)\simeq\left(f_{Q_{0}}^{\#}\right)^{\ast}(t),$ thus for a suitable constant $C>0,$ from (5.3) it follows that

[TABLE]

which should be compared with Theorem 3.

Remark 4**.**

The $K-$functional for the pair $(L^{\infty},BMO)$ was computed by several authors including Janson, Jawerth-Torchinsky, Shvartsman (cf. [17], [28] and the references therein). It would be of interest to connect the interpolation spaces with respect to the pair $(L^{\infty},BMO)$ and the Garsia-Rodemich constructions.

6. Fefferman-Stein inequality via Garsia-Rodemich

spaces

The original Fefferman-Stein inequality (cf. [15] and also [31] and the references therein) concerns with the embedding (cf. (1.15) and (1.16) above)

[TABLE]

In [31], Strömberg extended this result to an appropriate class of Orlicz spaces.

The connection between $X^{\#}$ and $GaRo_{X}$ can be seen from the fact that

[TABLE]

Indeed, we can easily show that from $f\in X^{\#}$ it follows $2f_{Q_{0}}^{\#}\in\Gamma_{f}$. This follows directly from (1.4) since for each $Q\subset Q_{0}$ we have

[TABLE]

and so $\gamma:=2f_{Q_{0}}^{\#}$ satisfies inequality (1.8). Consequently, (6.1) holds for all r.i. spaces $X$, and, moreover, we have

[TABLE]

Using the above observation, one can extend the Fefferman-Stein-Strömberg result121212However, note that unlike [31] we consider functions defined on a fixed cube $Q_{0}.$ to the setting of r.i. spaces.

Theorem 5**.**

If the lower Boyd index $\alpha_{X}$ of the r.i. space $X$ is positive, then $X^{\#}\subset X$.

Proof.

From the condition $\alpha_{X}>0$ and Theorem 2 we infer that $GaRo_{X}=X$. We conclude by combining this fact with (6.1). ∎

The next result establishes necessary and sufficient conditions, under which the opposite embedding $X\subset X^{\#}$ holds.

Theorem 6**.**

Let $X$ be an r.i. space on $[0,1]$. The following conditions are equivalent:

(i) $\beta_{X}<1$;

(ii) $GaRo_{X}\subset X^{\#}$;

(iii) $X\subset X^{\#}$.

Proof.

$(i)\rightarrow(ii)$. Let $f\in GaRo_{X}.$ As we have seen above for every $\gamma\in\Gamma_{f}$, we have $f_{Q_{0}}^{\#}(x)\leq M_{Q_{0}}\gamma(x)$. Since we are assuming that $\beta_{X}<1$, the Hardy-Littlewood operator $M_{Q_{0}}$ is bounded on $X$. Hence,

[TABLE]

Taking infimum over all $\gamma\in\Gamma_{f}$, we get

[TABLE]

whence $f\in X^{\#}.$

$(ii)\rightarrow(iii)$ The implication is trivial since the embedding $X\subset GaRo_{X}$ holds for all r.i. spaces $X$ (see the beginning of the proof of Theorem 2).

$(iii)\rightarrow(i)$. By [6, Theorem 5.7.3] (cf. also Example 3 in Section 5), we have

[TABLE]

for some absolute constant $c^{\prime}$. Therefore,

[TABLE]

From the latter inequality, (2.3), and our current assumption, it follows that

[TABLE]

This shows that the Hardy operator $P$ is bounded on $X$, and therefore, by (2.4), $\beta_{X}<1$. ∎

7. A packing formula for the $K-$functional of $(L^{1},BMO)$

The new characterization of the Garsia-Rodemich spaces discussed in the introduction (cf. (1.11) above) suggested a new formula for the $K-$functional of the pair $(L^{1},BMO)$ (see Section 2.4).

Remark 5**.**

In order to properly interpret the pair $(L^{1},BMO)$ as a compatible pair of Banach spaces, it is necessary to factor out the constant functions. Equivalently, we can restrict ourselves to consider functions with zero mean, i.e. $\int_{Q_{0}}f(x)dx=0.$

For any family of cubes $\pi=\{Q_{i}\}\in P:=P(Q_{0}),$ we define

[TABLE]

and let

[TABLE]

Theorem 7**.**

There exist absolute constants, such that for all $f\in L^{1}$ we have

[TABLE]

Proof.

It is plain that

[TABLE]

Consequently, by equivalence (2.8) (the implied constants depend only on the dimension), we have

[TABLE]

Thus, the desired result will follow if we show that

[TABLE]

with some absolute constant.

Given $t\in(0,1],$ we consider the set

[TABLE]

It follows that for each $x\in\Omega(t)$ there exists a cube $Q_{x}$ such that $Q_{x}\subset Q_{0}$, $x\in Q_{x}$, and

[TABLE]

Note that, by the definition of the set $\Omega(t)$, we have $Q_{x}\subset\Omega(t)$ for every $x\in\Omega(t)$. Consider the family of cubes $\{Q_{x}\}_{x\in\Omega(t)}.$ Using a Vitaly type covering lemma (cf. [30, p. 9]), we can select a subfamily of pairwise disjoint cubes $\{Q_{k}\}$ (which may contain a finite number of elements) such that

[TABLE]

Clearly $\pi=\{Q_{k}\}\in P$ and, moreover, by (7.2),

[TABLE]

Therefore, combining (7.3) and the fact that $|\Omega(t)|\geq t$, we obtain

[TABLE]

Thus, by the definition of the decreasing rearrangement of a measurable function, it follows that,

[TABLE]

Equivalently,

[TABLE]

From the latter inequality, (2.8) and the fact that $K(t):=K(t,f;L^{1},BMO)$ is an increasing function, we have

[TABLE]

Suppose now that $5^{-n}<t\leq 1$. Let us first remark that $K(1)\leq\|f\|_{L^{1}}.$ Indeed, we may assume that $\int_{Q_{0}}f(x)\,dx=0$ (see Remark 5) and therefore to compute $K(1)$ we can use the decomposition $f=f+0,$ and the assertion follows since

[TABLE]

Let us also note that, since $\pi=\{Q_{0}\}\in P,$ we have $F_{f,\sharp}(1)\geq\|f\|_{L^{1}}$. Consequently, using successively that $K(t)$ is increasing, $F_{f,\sharp}(t)$ is decreasing, and $5^{n}t>1,$ we get

[TABLE]

Thus, inequality (7.1) holds for all $0<t\leq 1$ with constant $c=5^{n}$. ∎

Remark 6**.**

Let $p\in(0,1)$. For any family of cubes $\pi=\{Q_{i}\}\in P(Q_{0})$ we let

[TABLE]

[TABLE]

Then, by a slight modification of the proof of Theorem 7 we see that the following equivalence holds

[TABLE]

(cf. [5, Remark 6.3]).

8. Extensions of the Garsia-Rodemich construction

We very briefly illustrate some of the results discussed in this paper showing how adding a parameter to the Garsia-Rodemich construction leads to a connection with the theory of Campanato spaces and the Morrey-Sobolev theorem. We refer to [1] for more information and background.

Definition 3**.**

Let $\lambda\in(-n,0],1<p\leq\infty.$ We shall say that $f\in L^{1}$ belongs to $GaRo_{p,\lambda}$ if there exists a constant $C>0$ such that for all $\{Q_{i}\}\in P,$

[TABLE]

and let

[TABLE]

Recall the definition of the homogeneous Campanato space $\mathcal{\dot{L}}^{1,\lambda}$ (cf. [1, Section 2.2, pag 8])$:$

Definition 4**.**

$\mathcal{\dot{L}}^{1,\lambda}=\{f:\left\|f\right\|_{\mathcal{\dot{L}}^{1,\lambda}}:=\sup_{Q\subset Q_{0}}\left|Q\right|^{-\frac{\lambda}{n}}(\frac{1}{\left|Q\right|}\int_{Q}\left|f-f_{Q}\right|)<\infty\}.$**

Theorem 8**.**

GaRo_{\infty,\lambda}=\left\{\begin{array}[c]{c}=\mathcal{\dot{L}}^{1,\lambda}\text{ },\text{ if }\lambda\in(-n,0)\\ =BMO,\text{ if }\lambda=0\end{array}\right..**

Proof.

Clearly, it is sufficiently to consider the case when $\lambda\in(-n,0)$.

We will use repeatedly the fact that (see (1.4))

[TABLE]

Consequently, we can write,

[TABLE]

Suppose that $f\in GaRo_{\infty,\lambda}.$ Then, since for each $Q\subset Q_{0}$ we have $\{Q\}\in$ $P,$ we see that

[TABLE]

Hence,

[TABLE]

Conversely, suppose that $f\in\mathcal{\dot{L}}^{1,\lambda}$ and let $\{Q_{i}\}$ be an arbitrary element of $P.$ We compute,

[TABLE]

Consequently,

[TABLE]

∎

The import of the Campanato spaces stems from a well known result by Campanato and Meyers (cf. [1, (2.3), pag. 9]) showing that for $\lambda\in(-1,0)$

[TABLE]

Let $\alpha\in(0,1),$ $p\geq 1.$ Define,

[TABLE]

Then, we have the classical

Theorem 9**.**

Let $p>\frac{n}{\alpha}$. Then

[TABLE]

Proof.

Note that $-1<\frac{n}{p}-\alpha<0$. In view of  Theorem 8, (8.1) and (8.2) for any cube $Q\subset Q_{0}$ we need to estimate from above the quantity

[TABLE]

We proceed as follows,

[TABLE]

∎