We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions

\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}

\begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}

We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$.

The aim of this note is twofold. First, we prove an abstract version of the Calderón transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does not increase the constants in the transferred inequalities. Second, we use the last information to study a certain dichotomy arising in problems of finding the best constants in the weak type $(1,1)$ and strong type $(p,p)$ inequalities for one-parameter ergodic maximal operators.

On a compact Lie group $G$ of dimension $n$, we study the Bochner–Riesz mean $S_{R}^{\unicode[STIX]{x1D6FC}}(f)$ of the Fourier series for a function $f$. At the critical index $\unicode[STIX]{x1D6FC}=(n-1)/2$, we obtain the convergence rate for $S_{R}^{(n-1)/2}(f)$ when $f$ is a function in the block-Sobolev space. The main theorems extend some known results on the $m$-torus $\mathbb{T}^{m}$.

In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^{p}(\mathbb{R}^{d},w)$ space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.

We investigate a dichotomy property for Hardy–Littlewood maximal operators, noncentred $M$ and centred $M^{c}$, that was noticed by Bennett et al. [‘Weak-$L^{\infty }$ and BMO’, Ann. of Math. (2) 113 (1981), 601–611]. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^{c}$ in the context of nondoubling metric measure spaces $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$. In addition, if $X=\mathbb{R}^{d}$, $d\geq 1$, and $\unicode[STIX]{x1D70C}$ is the metric induced by an arbitrary norm on $\mathbb{R}^{d}$, then we give the exact characterisation (in terms of $\unicode[STIX]{x1D707}$) of situations in which $M^{c}$ possesses the dichotomy property provided that $\unicode[STIX]{x1D707}$ satisfies some very mild assumptions.

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

Let ${{T}_{\Omega }}$ be the singular integral operator with kernel $\left( \Omega \left( x \right) \right)/{{\left| x \right|}^{n}}$, where $\Omega $ is homogeneous of degree zero, has mean value zero, and belongs to ${{L}^{q}}\left( {{S}^{n-1}} \right)$ for some $q\,\in \,\left( 1,\,\infty \right]$. In this paper, the authors establish the compactness on weighted ${{L}^{p}}$ spaces and the Morrey spaces, for the commutator generated by $\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ function and ${{T}_{\Omega }}$. The associated maximal operator and the discrete maximal operator are also considered.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

We prove weak-type $\left( 1,\,1 \right)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta *\Psi $ where $\Delta *$ is Bourgain’s maximal multiplier operator and $\Psi $ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the ${{L}^{q}}$ operator norm when $1\,<\,q\,<\,2$. We also consider associated variation-norm estimates.

Let $\left( \text{ }\!\!\chi\!\!\text{ ,}\,d,\,\mu \right)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu \left( \left\{ x \right\} \right)\,=\,0$ for all $x\,\in \,\text{ }\!\!\chi\!\!\text{ }$. In this paper, we show that the boundedness of a Calderón–Zygmund operator $T$ on ${{L}^{2}}\left( \mu \right)$ is equivalent to that of $T$ on ${{L}^{p}}\left( \mu \right)$ for some $p\,\in \,\left( 1,\,\infty \right)$, and that of $T$ from ${{L}^{1}}\left( \mu \right)$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. As an application, we prove that if $T$ is a Calderón–Zygmund operator bounded on ${{L}^{2}}\left( \mu \right)$, then its maximal operator is bounded on ${{L}^{p}}\left( \mu \right)$ for all $p\,\in \,\left( 1,\,\infty \right)$ and from the space of all complex-valued Borel measures on $\text{ }\!\!\chi\!\!\text{ }$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Let $\mu $ be a nonnegative Radon measure on ${{\mathbb{R}}^{d}}$ that satisfies the growth condition that there exist constants ${{C}_{0}}\,>\,0$ and $n\,\in \,(0,\,d]$ such that for all $x\,\in \,{{\mathbb{R}}^{d}}$ and $r\,>\,0$, $\mu \left( B\left( x,\,r \right) \right)\,\le \,{{C}_{0}}{{r}^{n}}$, where $B(x,\,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\text{BMO}$-type space $\text{RBMO(}\mu \text{)}$ of Tolsa, then the homogeneous maximal function ${{\dot{\mathcal{M}}}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is not an initial cube) and the inhomogeneous maximal function ${{\overset{{}}{\mathop{\mathcal{M}}}\,}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is an initial cube) associated with a given approximation of the identity $S $ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, ${{\dot{\mathcal{M}}}_{s}}$ and ${{\mathcal{M}}_{s}}$ are bounded from $\text{RBMO(}\mu \text{)}$ to the $\text{BLO}$-type space $\text{RBMO(}\mu \text{)}$. The authors also prove that the inhomogeneous maximal operator ${{\mathcal{M}}_{s}}$ is bounded from the local $\text{BMO}$-type space $\text{rbmo(}\mu \text{)}$ to the local $\text{BLO}$-type space $\text{rblo(}\mu \text{)}$.

For a class of convex curves in ℝd we prove that the corresponding maximal operator and Hilbert transform are of weak type Llog L. The point of interest here is that this class admits curves which are infinitely flat at the origin. We also prove an analogous weak type result for a class of nonconvex hypersurfaces.

Let 𝒳 be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, two weighted estimates related to weights are established for singular integral operators with nonsmooth kernels via a new sharp maximal operator associated with a generalized approximation to the identity. As applications, the weighted Lp(𝒳) and weighted endpoint estimates with general weights are obtained for singular integral operators with nonsmooth kernels, their commutators with BMO (𝒳) functions, and associated maximal operators. Some applications to holomorphic functional calculi of elliptic operators and Schrödinger operators are also presented.