1. Introduction

The purpose of this paper is to study the boundedness properties of the fractional integral operator $I_{\alpha }$ , $0<\alpha <n$ , and the fractional maximal operator $M_{\alpha }$ , $0\le \alpha <n$ , in the framework of Choquet integrals with respect to Hausdorff content.

Let $n\in {\mathbb N}$ and $0<d\le n$ . The symbol ${\mathcal Q}({\mathbb R}^n)$ denotes the family of all cubes with sides parallel to the coordinate axes in ${\mathbb R}^n$ . The d-dimensional Hausdorff content of $E\subset {\mathbb R}^n$ is defined by

$$ \begin{align*} H^d(E) = \inf\bigg\{ \sum_{j=1}^{{\infty}}\ell(Q_j)^d:\, E\subset\bigcup_{j=1}^{{\infty}}Q_j, Q_j\in{\mathcal Q}({\mathbb R}^n) \bigg\}, \end{align*} $$

where the infimum is taken over all coverings of the set E by countable families of cubes $Q_j$ and $\ell (Q)$ stands for the side length of the cube Q. It is easily seen that $H^n(E)$ is just the Lebesgue measure of E, which we will denote by $|E|$ . For any cube Q, one has $H^d(Q)=\ell (Q)^d$ .

For a nonnegative function f, the integral of f with the respect to $H^d$ is taken in the Choquet sense:

$$ \begin{align*} \int_{{\mathbb R}^n}f\,{ d}H^d = \int_0^{{\infty}} H^d(\{ x\in {\mathbb R}^n:\,f(x)>t \})\,{d}t. \end{align*} $$

For $0<p<{\infty }$ , the Choquet space $L^p(H^d)$ and the weak Choquet space $\mathrm {w}\hskip -0.6pt{L}^p(H^d)$ consist of all functions with the properties

$$ \begin{align*} \|f\|_{L^p(H^d)} &:= \bigg( \int_{{\mathbb R}^n}|f|^p\,{ d}H^d \bigg)^{1/p}\!<{\infty} \end{align*} $$

and

$$ \begin{align*}\|f\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)}&:=\sup_{t>0}tH^d(\{x\in{\mathbb R}^n:\,|f(x)|>t\})^{1/p}\!<{\infty},\end{align*} $$

respectively. For $0<q\le p<{\infty }$ , the Choquet–Morrey space ${\mathcal M}^p_q(H^d)$ consists of all functions with the property

$$ \begin{align*} \|f\|_{{\mathcal M}^p_q(H^d)} := \sup_{Q\in{\mathcal Q}} \ell(Q)^{d/p- d/q} \bigg( \int_{Q}|f|^q\,{ d}H^d \bigg)^{1/q}\!<{\infty}. \end{align*} $$

The fractional maximal operator of order $\alpha $ , $0\le \alpha <n$ , is defined by

$$ \begin{align*} M_{\alpha}f(x) = \sup_{Q\in{\mathcal Q}({\mathbb R}^n)} \chi_{Q}(x) \ell(Q)^{\alpha-n}\int_{Q}|f(y)|\,{ d}y, \quad x\in{\mathbb R}^n, \end{align*} $$

where $\chi _{E}$ is the characteristic function of the set E. For $\alpha =0$ , the operator $M_0$ is the usual Hardy–Littlewood maximal operator which is denoted simply by M. The fractional integral operator of order $\alpha $ , $0<\alpha <n$ , is defined by

$$ \begin{align*} I_{\alpha}f(x) = \int_{{\mathbb R}^n} \frac{f(y)}{|x-y|^{n-\alpha}} \,{ d}y,\quad x\in{\mathbb R}^n. \end{align*} $$

Orobitg and Verdera [Reference Orobitg and Verdera6] proved that, for $0<d<n$ and $p>d/n$ ,

$$ \begin{align*}\|Mf\|_{\mathrm{w}\hskip-0.6pt{L}^{d/n}(H^d)} \lesssim \|f\|_{L^{d/n}(H^d)} \quad\mbox{and}\quad \|Mf\|_{L^p(H^d)} \lesssim \|f\|_{L^p(H^d)}.\end{align*} $$

The tools and ideas that we will use are essentially contained in this classical paper.

We note a disadvantage of the Hausdorff content $H^d$ . It is not true that there exists a constant $C>0$ such that if $Q_1,\ldots ,Q_m$ are nonoverlapping dyadic cubes and $f\ge 0$ , then

$$ \begin{align*} \sum_{j=1}^m\int_{Q_j}f\,{ d}H^d \le C \int_{\bigcup_jQ_j}f\,{ d}H^d. \end{align*} $$

This can be shown by subdividing the interval $[0, 1]$ into $2^m$ (m large enough) equal intervals and taking $f\equiv 1$ .

However, an advantage of the Hausdorff content $H^d$ is that the spaces $L^p(H^d)$ , $0<p\le 1$ , have a block decomposition. As a corollary, (see [Reference Saito, Tanaka and Watanabe8, Theorem 2.3]), if T is a subadditive operator and $0<d,\delta <n$ , $0<p\le 1$ and $q\ge p$ , then the following statements are equivalent:

1. (a) the inequality $\displaystyle \|Tf\|_{L^q(H^{\delta })} \le C_1 \|f\|_{L^p(H^d)}$ holds;

2. (b) the testing inequality $\displaystyle \|T\chi _{Q}\|_{L^q(H^{\delta })} \le C_2\ell (Q)^{1/p}$ holds for any $Q\in {\mathcal Q}({\mathbb R}^n)$ .

Moreover, the least possible constants $C_1$ and $C_2$ are equivalent. Because of this advantage, by using an easy testing inequality, one can verify the following results from [Reference Adams2]. For $d/n<p<d/\alpha $ ,

(1.1) $$ \begin{align} \|M_{\alpha}f\|_{L^p(H^{d-\alpha p})} \lesssim \|f\|_{L^p(H^d)}. \end{align} $$

For $0<d\le n$ ,

(1.2) $$ \begin{align} \|M_{\alpha}f\|_{\mathrm{w}\hskip-0.6pt{L}^{d/n}(H^{d-\alpha d/n})} \lesssim \|f\|_{L^{d/n}(H^d)}. \end{align} $$

The reason why the advantage influences the cases $p>1$ is that, for the fractional maximal operator $M_{\alpha }$ , the pointwise estimate

$$ \begin{align*} M_{\alpha}f(x)^p\le M_{\alpha p}[f^p](x), \quad x\in{\mathbb R}^n, \end{align*} $$

gives a reduction to the case $p=1$ . This no longer works for the fractional integral operator $I_{\alpha }$ . The difficulty is overcome by using Hedberg’s trick (see Lemma 2.7), which is due to the first author (Hatano).

Theorem 1.1. Let $0<d\le n$ and $0\le \alpha <n$ . Suppose that $d/n\le r<p<d/\alpha $ and

$$ \begin{align*} \frac{d-\alpha r}{q} = \frac{d-\alpha p}{p}. \end{align*} $$

Then:

1. (i) $\|M_{\alpha }f\|_{\mathrm {w}\hskip -0.6pt{L}^q(H^{d-\alpha r})} \lesssim \|f\|_{\mathrm {w}\hskip -0.6pt{L}^p(H^d)}$ ;

2. (ii) $\|I_{\alpha }f\|_{\mathrm {w}\hskip -0.6pt{L}^q(H^{d-\alpha r})} \lesssim \|f\|_{\mathrm {w}\hskip -0.6pt{L}^p(H^d)}$ for $0<d<n$ , $0<\alpha <n$ and $d/n<r$ .

3. (iii) $\|I_{\alpha }f\|_{L^q(H^{d-\alpha r})} \lesssim \|f\|_{L^p(H^d)}$ for $0<d<n$ , $0<\alpha <n$ and $d/n<r$ .

The key ingredient in the proof of Theorem 1.1(i) is a Kolmogorov-type inequality: for any measurable set $E\subset {\mathbb R}^n$ ,

$$ \begin{align*} H^d(E)^{1/p-1/q} \|f\chi_{E}\|_{L^q(H^d)} \le \bigg(\frac{q}{p-q}\bigg)^{1/p} \|f\chi_{E}\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)}, \quad 0<q<p<{\infty}. \end{align*} $$

This means that weak $L^p$ integrability implies local $L^q$ integrability provided that ${q<p}$ . In the above inequality, taking the supremum over all measurable sets, we can obtain the reverse inequality and, as a consequence, the weak Choquet norm can be estimated as follows (see Proposition 2.5). For $0<q<p<{\infty }$ ,

$$ \begin{align*} \|f\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)} \le \sup_{0<H^d(E)<{\infty}} H^d(E)^{1/p-1/q} \|f\chi_{E}\|_{L^q(H^d)} \le \bigg(\frac{q}{p-q}\bigg)^{1/p} \|f\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)}. \end{align*} $$

We notice that the parameter q does not affect the set $\mathrm {w}\hskip -0.6pt{L}^p(H^d)$ . However, in the above supremum, if one restricts to the cube $Q\in {\mathcal Q}({\mathbb R}^n)$ instead of the general set $E\subset {\mathbb R}^n$ , then one gets Morrey spaces and one can no longer ignore the influence of the second parameter q. We establish the following results. The first part is from [Reference Tang10, Theorem 2].

Theorem 1.3. Let $0<d\le n$ and $0\le \alpha <n$ . Suppose that $d/n<r\le p<d/\alpha $ and

$$ \begin{align*} \frac{d-\alpha r}{q} = \frac{d-\alpha p}{p}. \end{align*} $$

Then:

1. (i) $\|M_{\alpha }f\|_{{\mathcal M}^q_r(H^{d-\alpha r})} \lesssim \|f\|_{{\mathcal M}^p_r(H^d)}$ ;

2. (ii) $\|I_{\alpha }f\|_{{\mathcal M}^q_r(H^{d-\alpha r})} \lesssim \|f\|_{{\mathcal M}^p_r(H^d)}$ for $0<d<n$ and $0<\alpha <n$ .

The paper is organised as follows. In Section 2, we give a proof of the Kolmogorov-type inequality and summarise some elementary properties. In Section 3, we prove the theorems. In Section 4, as an appendix, we gather some further results.

Throughout the paper, we use the following notation. If X and Y are normed spaces with $\|\cdot \|_{X} \lesssim \|\cdot \|_{Y}$ , then we write $X\hookleftarrow Y$ or $Y\hookrightarrow X$ (sometimes called the embedding). If $X\hookleftarrow Y$ and $X\hookrightarrow Y$ , then we write $X=Y$ .

For quantities A and B, if $A\le C B$ , then we write $A\lesssim B$ or $B\gtrsim A$ , and if $A\lesssim B$ and $A\gtrsim B$ , then we write $A\sim B$ .