We show that the fractional integral operator $I_{\alpha }$, $0<\alpha <n$, and the fractional maximal operator $M_{\alpha }$, $0\le \alpha <n$, are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_\alpha $ are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, Georgian Math. J. 18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{\alpha }$ are essentially new.

Block decomposition of $L^{p}$ spaces with weighted Hausdorff content is established for $0<p\leqslant 1$ and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

Let $\unicode[STIX]{x1D6FD}\geqslant 0$, let $e_{1}=(1,0,\ldots ,0)$ be a unit vector on $\mathbb{R}^{n}$, and let $d\unicode[STIX]{x1D707}(x)=|x|^{\unicode[STIX]{x1D6FD}}dx$ be a power weighted measure on $\mathbb{R}^{n}$. For $0\leqslant \unicode[STIX]{x1D6FC}<n$, let $M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}$ be the centered Hardy-Littlewood maximal function and fractional maximal functions associated with measure $\unicode[STIX]{x1D707}$. This paper shows that for $q=n/(n-\unicode[STIX]{x1D6FC})$, $f\in L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})$,

$$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}(\{x\in \mathbb{R}^{n}:M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)>\unicode[STIX]{x1D706}\})=\frac{\unicode[STIX]{x1D714}_{n-1}}{(n+\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D707}(B(e_{1},1))}\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}^{q}, & & \displaystyle \nonumber\\ \displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}\left(\left\{x\in \mathbb{R}^{n}:\left|M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)-\frac{\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}}{\unicode[STIX]{x1D707}(B(x,|x|))^{1-\unicode[STIX]{x1D6FC}/n}}\right|>\unicode[STIX]{x1D706}\right\}\right)=0, & & \displaystyle \nonumber\end{eqnarray}$$

$\unicode[STIX]{x1D6FD}=0$

$\unicode[STIX]{x1D707}$

In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$ .

We give weighted norm inequalities for the maximal fractional operator ${{\mathcal{M}}_{q}},\beta $ of Hardy–Littlewood and the fractional integral ${{I}_{\gamma }}$. These inequalities are established between ${{\left( {{L}^{q}},\,{{L}^{p}} \right)}^{\alpha }}\left( X,\,d,\,\mu \right)$ spaces (which are superspaces of Lebesgue spaces ${{L}^{\alpha }}\left( X,\,d,\,\mu \right)$ and subspaces of amalgams $\left( {{L}^{q}},\,{{L}^{p}} \right)\left( X,d,\mu \right)$) and in the setting of space of homogeneous type $\left( X,d,\mu \right)$. The conditions on the weights are stated in terms of Orlicz norm.

The relation between the fractional integral operator and the fractional maximal operator is investigated in the framework of Morrey spaces. Applications to the Fefferman–Phong and the Olsen inequalities are also included.