CORRECTION TO ‘CHOQUET INTEGRALS, HAUSDORFF CONTENT AND FRACTIONAL OPERATORS’

Reference [1] studied the boundedness of the fractional maximal operator $M_{\alpha }$ and the fractional integral operator $I_{\alpha }$ on the Choquet–Morrey space ${\mathcal M}^p_q(H^d)$ and the weak Choquet space $\mathrm {w}\hskip -0.6pt{L}^p(H^d)$ . The purpose of this note is to correct the bound for $I_\alpha $ in [1, Theorem 1.3(ii)] by restricting the range of the parameters and to correct a minor error in the proofs of [1, Theorems 1.1(ii) and 1.3(ii)].

Let $n\in {\mathbb N}$ and $0<d\le n$ . For $0<p<\infty $ , the Choquet space $L^p(H^d)$ and the weak Choquet space $\mathrm {w}\hskip -0.6pt{L}^p(H^d)$ comprise the functions such that the quasi-norms

$$ \begin{align*} \|f\|_{L^p(H^d)} = \bigg(\int_{{\mathbb R}^n}|f|^p\,{\, d}H^d\bigg)^{1/p} \quad\mbox{and}\quad \|f\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)} = \sup_{t>0} t H^d(\{x\in{\mathbb R}^n:\,|f(x)|>t\})^{1/p} \end{align*} $$

are finite, where $H^d$ denotes the d-dimensional Hausdorff content, and the integral with respect to $H^d$ is taken in the Choquet sense. For $0<q\le p<\infty $ , the Choquet–Morrey space ${\mathcal M}^p_q(H^d)$ is the set of all functions such that the quasi-norm

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

is finite, where ${\mathcal Q}$ denotes the family of cubes Q with sides parallel to the coordinate axes in $\mathbb {R}^n$ and $\ell (Q)$ is the side length of the cube Q.

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

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

where $\chi _{E}$ is the characteristic function of the set E. The fractional integral operator of order $\alpha $ , $0<\alpha <n$ , is defined by

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

We restate the relevant results from [1] with the correction to [1, Theorem 1.3(ii)].

Theorem 1 [1, Theorem 1.1].

If $0<d\le n$ , $0\le \alpha <n$ , $d/n\le r<p<d/\alpha $ and

(1) $$ \begin{align} \frac{d-\alpha r}{q}=\frac{d-\alpha p}{p}, \end{align} $$

then:

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

• (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<p<d/\alpha $ .

Theorem 2 [1, Theorem 1.3 corrected].

If $0<d\le n$ , $0\le \alpha <n$ , $d/n<r\le p<d/\alpha $ and (1) holds, then:

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

• (ii) $\|I_{\alpha }f\|_{{\mathcal M}^q_r(H^{d-\alpha r})}\lesssim \|f\|_{{\mathcal M}^p_s(H^d)}$ for $0<d<n$ , $0<\alpha <n$ , $d/n<r<s<p<d/\alpha $ and $q/p\le n/d$ .

Remark 3. In Theorem 2(ii), our proof requires $r<s$ . We have not been able to prove the result when $s=r$ .

Proof of Theorem 1(ii).

Choose $\theta $ and $\beta $ so that

(2) $$ \begin{align} \theta(d-\beta p)=d-\alpha r=d-(r/p) \alpha p. \end{align} $$

By (1), for $r\alpha /p<\beta <\alpha $ , this defines $\theta $ as an increasing function of $\beta $ and $1<\theta < q/p$ . Choose $\theta \le n/d$ and set $\delta =\theta d$ and $u=\theta p$ . Since $\beta <\alpha $ , we have $p<u<q$ . By (1),

(3) $$ \begin{align} \frac{\delta-\beta u}{q}=\frac{d-\alpha r}{q}=\frac{d-\alpha p}{p}=\frac dp-\alpha=\frac{\delta}{u}-\alpha=\frac{\delta-\alpha u}{u}. \end{align} $$

From (3), we can apply [1, Lemma 2.7] with the parameters $d,p$ replaced by $\delta ,u$ (noting that $\delta \le n$ ), to obtain

$$ \begin{align*} |I_{\alpha}f(x)| \lesssim \|f\|_{{\mathcal M}^u_{\delta/n}(H^{\delta})}^{1-u/q} M_{\beta}f(x)^{u/q}, \quad x\in{\mathbb R}^n. \end{align*} $$

Let $s=r\alpha /\beta $ so that $r<s<p$ . From (2),

$$ \begin{align*} \frac{d-\beta s}{u}=\frac{d-\alpha r}{u}=\frac{\delta-\beta u}{u}=\frac dp-\beta=\frac{d-\beta p}{p}, \end{align*} $$

so we can apply Theorem 1(i) with the parameters $\alpha , q, \alpha r$ replaced by $\beta , u, \beta s$ to obtain

$$ \begin{align*} \|(M_{\beta}f)^{u/q}\|_{\mathrm{w}\hskip-0.6pt{L}^q(H^{d-\alpha r})} = \|M_{\beta}f\|_{\mathrm{w}\hskip-0.6pt{L}^u(H^{d-\beta s})}^{u/q} \lesssim \|f\|_{\mathrm{w}\hskip-0.6pt{L}^p(H^d)}^{u/q}. \end{align*} $$

Since we always have $ \|f\|_{{\mathcal M}^u_{\delta /n}(H^{\delta })} \lesssim \|f\|_{{\mathcal M}^p_{d/n}(H^d)} \lesssim \|f\|_{\mathrm {w}\hskip -0.6pt{L}^p(H^d)}, $ this completes the proof.

Proof of Theorem 2(ii).

Set $\beta =r\alpha /s$ and define $\theta $ by (2). By the hypotheses, $r\alpha /p<\beta <\alpha $ and $1<\theta < q/p \le n/d$ , using the definition of $\theta $ and the assumption $q/p\le n/d$ . Again, set $\delta =\theta d$ and $u=\theta p$ , so that $p<u<q$ . Just as in the proof of Theorem 1(ii), [1, Lemma 2.7] yields

$$ \begin{align*} |I_{\alpha}f(x)| \lesssim \|f\|_{{\mathcal M}^u_{\delta/n}(H^{\delta})}^{1-u/q} M_{\beta}f(x)^{u/q}, \quad x\in{\mathbb R}^n. \end{align*} $$

Since

$$ \begin{align*} \frac{d-\beta s}{u} = \frac{d-\beta p}{p}, \end{align*} $$

Theorem 2(i) yields

$$ \begin{align*} \|(M_{\beta}f)^{u/q}\|_{{\mathcal M}^q_r(H^{d-\alpha r})} \le \|M_{\beta}f\|_{{\mathcal M}^u_r(H^{d-\alpha r})}^{u/q} \le \|M_{\beta}f\|_{{\mathcal M}^u_s(H^{d-\beta s})}^{u/q} \lesssim \|f\|_{{\mathcal M}^p_s(H^d)}^{u/q}. \end{align*} $$

Since we always have $ \|f\|_{{\mathcal M}^u_{\delta /n}(H^{\delta })} \lesssim \|f\|_{{\mathcal M}^p_{d/n}(H^d)} \le \|f\|_{{\mathcal M}^p_s(H^d)}, $ this completes the proof.