Related papers: Norms of maximal functions between generalized and…

In this note we prove the estimate $M^{\sharp}_{0,s}(Tf)(x) \le c\,M_\gamma f(x)$ for general fractional type operators $T$, where $M^{\sharp}_{0,s}$ is the local sharp maximal function and $M_\gamma$ the fractional maximal function, as…

This article is the continuation of the work [DK] where we had proved maximal estimates $$\left\|\sup_{t > 0} |m(tA)f| \right\|_{L^p(\Omega,Y)} \leq C \|f\|_{L^p(\Omega,Y)}$$ for sectorial operators $A$ acting on $L^p(\Omega,Y)$ ($Y$ being…

In this paper, we discuss tangential limits for regular harmonic functions with respect to $\phi(\Delta):=-\phi(-\Delta)$ in the $C^{1,1}$ open set $D$ in $\mathbb{R}^d$, where $\phi$ is the complete Bernstein function and $d \ge 2$. When…

We prove the sharp mixed $A_{p}-A_{\infty}$ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely \[ \|M\|_{L^{p,q}(w)} \lesssim_{p,q,n}…

We devote this note to correct an estimate concerning mixed inequalities for the generalized maximal function $M_\Phi$, when certain properties of the associated Young function $\Phi$ are assumed. Although the obtained estimates turn out to…

Let $M$ be the Hardy-Littlewood maximal function. Denote by $M_b$ and $[b,M]$ the maximal and the nonlinear commutators of $M$ with a function $b$. The boundedness of $M_b$ and $[b,M]$ on weighted Lebesgue spaces are characterized when the…

Let $\Omega $ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in \Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. \emd for the…

For an Orlicz function $\varphi$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the K\"othe dual of an Orlicz-Lorentz function space $\Lambda_{\varphi,w}$ or a sequence space…

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\,…

Given an operator $\phi:X\rightarrow Y$ between Banach spaces, we consider its tensor powers $\phi^{\otimes k}$ as operators from the $k$-fold injective tensor product of $X$ to the $k$-fold projective tensor product of $Y$. We show that…

We give a new characterization of a continuous embedding between two function spaces of type $G\Gamma$. Such spaces are governed by functionals of type \begin{equation*} \|f\|_{G\Gamma(r,q;w,\delta)} := \left(\int_{0}^{L} \left(…

We prove mixed inequalities for the generalized maximal operator $M_\Phi$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^\beta$ with $\beta<-n$ and $r\geq 1$.…

Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^{\varphi}(\mathbb…

Let $\mu$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $\alpha>-1$, the generalized integral type Hilbert operator defined as follows: $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int^1_{0}…

In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),\lambda}$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\lambda}$ for radially decreasing…

In this paper we introduce the atomic Hardy space $\mathcal{H}^1((0,\infty),\gamma_\alpha)$ associated with the non-doubling probability measure $d\gamma_\alpha(x)=\frac{2x^{2\alpha+1}}{\Gamma(\alpha+1)}e^{-x^2}dx$ on $(0,\infty)$, for…

The purpose of the paper is to study the operators on the weighted Bergman spaces on the unit disk ${\mathbb{D}}$, denoted by $A^{p}_{\lambda,w}({\mathbb{D}})$, that are associated with a class of generalized analytic functions, named the…

In this paper we consider the generalized shift operator, generated by the Gegenbauer differential operator $$ G =\left(x^2-1\right)^{\frac{1}{2}-\lambda} \frac{d}{dx} \left(x^2-1\right)^{\lambda+\frac{1}{2}}\frac{d}{dx}. $$ Maximal…

A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta <\infty$, is defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which is a generalization of exponential class. An alternative definition of…

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the…