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In this article we obtain an "off-diagonal" version of the Fefferman-Stein vector-valued maximal inequality on weighted Lebesgue spaces with variable exponents. As an application of this result and the atomic decomposition developed in [12]…

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \geq 1$, so that the following inequalities hold for all $u \in C_{0}^{\infty}(B)$: $\int_{B}V(x)|\nabla u |^{2}dx…

The ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish a new inequality using weight function which generalizes the inequalities of Dragomir, Wang and Cerone…

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-\Delta)^s v = \mu \quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure…

We show that the uncentered Hardy-Littlewood maximal operators associated with the Radon measure $\mu$ on $\mathbb{R}^d$ have the uniform lower $L^p$-bounds (independent of $\mu$) that are strictly greater than $1$, if $\mu$ satisfies a…

A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces (R^n,mu) with mu(B(x,r))<Cr^d, in which non-doubling harmonic analysis has…

Adams inequalities with exact growth conditions are derived for Riesz-like potentials on metric measure spaces. The results extend and improve those obtained recently on $\mathbb R^n$ by the second author, for Riesz-like convolution…

We give an alternative characterization of the class of Muckenhoupt weights $A_{\infty, \mathfrak B}$ for homothecy invariant Muckenhoupt bases $\mathfrak B$ consisting of convex sets. In particular we show that $w\in A_{\infty, \mathfrak…

In this paper, we establish a class of Stein-Weiss type inequality with partial variable weight functions on the upper half space using a weighted Hardy type inequality. Overcoming the impact of weighted functions, the existence of extremal…

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$ on $\mathbb R^n$, where $n\geqslant2$, $\alpha\in(0,2]$, and $\alpha<n$, we consider the problem of minimizing the Gauss functional…

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1} \beta_{2,\mu}^{2}(3Q)\frac{\ell(Q)}{\mu(Q)}…

In this paper we characterize the validity of the inequalities $\|g\|_{p,(a,b),\lambda} \le c \|u(x) \|g\|_{\infty,(x,b),\mu}\|_{q,(a,b),\nu}$ and $\label{eq.0.1.2} \|g\|_{p,(a,b),\lambda} \le c \|u(x)…

We answer a special case of a question of T. Hytonen regarding the two weight norm inequality for the maximal function M in the affirmative, namely that there is a constant D > 1, depending only on dimension n, such that the two weight norm…

We obtain convolution inequalities in Lebesgue and Lorentz spaces with power weights when the functions involved are assumed to be radially symmetric. We also present applications of these results to inequalities for Riesz potentials of…

Given $\alpha >0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (\alpha_n + |x|^\alpha) |u|^{\frac{n}{n-1}} \big)…

We prove that, for totally irregular measures $\mu$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,\mu}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d \mu(y) $$ adapted to the…

If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen hold. We…

Inspired by applications in weighted polynomial approximation problems, we study an optimal mass distribution problem. Given a gauge function $h$ and a positive "roof" function $R$ compactly supported in $\mathbb{R}^n$, we are interested in…

We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi…

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int…