Related papers: Weighted inequalities for a superposition of the C…

We formulate and prove a generalization of Hardy's inequality (Hardy,1925) in terms of random variables and show that it contains the usual (or familiar) continuous and discrete forms of Hardy's inequality. Next we improve the recent…

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in $n$-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision…

Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the…

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)…

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to…

We investigate the weighted fractional order Hardy inequality $$ \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partial\Omega)^{-\alpha}\text{dist}(y,\partial\Omega)^{-\beta}\,dy\,dx\geq…

We present factorizations of weighted Lebesgue, Ce\-s\` aro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy's integral operator between weighted Lebesgue spaces. Our results enhance, among…

We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical…

Our goal in this paper is to find a characterization of $n$-dimensional bilinear Hardy inequalities \begin{align*} \bigg\| \,\int_{B(0,\cdot)} f \cdot \int_{B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} & \leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n}…

We study weighted Poincar\'e and Poincar\'e-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ \left…

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$…

Let $U:[0,\infty)^2 \to [0,\infty)$ be a~measurable kernel satisfying: (i) $U(x,y)$ is nonincreasing in $x$ and nondecreasing in $y$; (ii) there exists a~constant $\theta>0$ such that $U(x,z) \le \theta\left( U(x,y)+U(y,z) \right)$ for all…

In this paper embeddings between weighted Copson function spaces ${\operatorname{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Ces\`{a}ro function spaces ${\operatorname{Ces}}_{p_2,q_2}(u_2,v_2)$ are characterized. In particular, two-sided…

In this paper, we introduce a new weighted Hardy-Ces\`{a}ro operator defined by $U_{\psi,s}f(x)=\int\limits_0^1 f(s(t)\cdot x) \psi(t)dt$, which is associated to the parameter curve $s(t,x)=s(t)x$. Under certain conditions on $s(t)$ and on…

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove…

In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such…

Let $1\leq p<\infty$, $\alpha>-1$, and let $\varphi$ be a measurable function on $(0,\infty)$. The main purpose of this paper is to study the Hausdorff operator \[ \mathscr H_\varphi f(z)=\int_0^\infty f\left(\frac{z}{t}\right)…

In \cite{MR447956}, Muckenhoupt and Wheeden formulated a weighted weak $(p,p)$ inequality where the weight for the weak $L^p$ space is treated as a multiplier rather than a measure. They proved such inequalities for the Hardy-Littlewood…

We study $l^p$ operator norms of factorable matrices and related results. We give applications to $l^p$ operator norms of weighted mean matrices and Copson's inequalities. We also apply the method in this paper to study the best constant in…

We obtain Fourier inequalities in the weighted $L_p$ spaces for any $1<p<\infty$ involving the Hardy-Ces\`aro and Hardy-Bellman operators. We extend these results to product Hardy spaces for $p\le 1$. Moreover, boundedness of the…