Hardy's inequalities for monotone functions on partially ordered measure spaces

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} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+ ({\mathbb R}^n), \end{align*} and \begin{align*} \bigg\| \,\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} f \cdot \int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} ...

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq +\infty$, $u$, $w$ and $v$ are weight functions on $(0,\infty)$. It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restrict...

Boundedness of an abstract formulation of Hardy operators between Lebesgue spaces over general measure spaces is studied and, when the domain is L^1, shown to be equivalent to the existence of a Hardy inequality on the half line with general Borel measures. This is done by extending the greatest decreasing minorant construction to general measure spaces depending on a totally ordered collection of measurable sets, called an ordered core. A functional description of the greate...

We prove a general self-improvement property for a family of weighted pointwise inequalities on open sets, including pointwise Hardy inequalities with distance weights. For this purpose we introduce and study the classes of $p$-Poincar\'e and $p$-Hardy weights for an open set $\Omega\subset X$, where $X$ is a metric measure space. We also apply the self-improvement of weighted pointwise Hardy inequalities in connection with usual integral versions of Hardy inequalities.

In the present paper we shall establish n-dimensional Hardy's inequalities with non-doubling weight functions of the distance to the boundary, where the boundary is a $C^2$ class bounded domain of $R^N$. This work is essentially based on one dimensional weighted Hardy's inequalities with one-sided boundary condition and sharp remainders. As weights we admit rather general ones that may vanish or blow up in infinite order such as $e^{-1/t}$ or $e^{1/t}$ at $t=0$ in one dimensi...

In this paper, we prove a $p$-Hardy inequality on the discrete half-line with weights $n^{\alpha}$ for all real $p > 1$. Building on the work of Miclo for $p = 2$ and Muckenhoupt in the continuous settings, we develop a quantitative approach for the existence of a $p$-Hardy inequality involving two measures $\mu$ and $\nu$ on the discrete half-line. We also investigate the comparison between sharp constants in the discrete and continuous settings and explore the stability of ...

The main goal of this paper is to prove a two-weight criteria for multidimensio-nal Hardy type operator from weighted Lebesgue spaces into $p$-convex weighted Banach function spaces. Analogously problem for the dual operator is considered. As application we prove a two-weight criteria for boundedness of multidimensional geometric mean operator and sufficient condition on the weights for boundedness of certain sublinear operator from weighted Lebesgue spaces into weighted Musi...

We provide a general framework for fractional Hardy inequalities. Our framework covers, for instance, fractional inequalities related to the Dirichlet forms of some L\'evy processes, and weighted fractional inequalities on irregular open sets.

A simple normal form for Hardy operators is introduced that unifies and simplifies the theory of weighted Hardy inequalities. A straightforward transition to normal form is given that applies to the various Hardy operators and their duals, whether defined on Lebesgue spaces of sequences, of functions on the half-line, or of functions on $\mathbb R^n$ or more general metric spaces. This is done by introducing an abstract formulation of Hardy operators, more general than any of...

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent \lambda=n-1. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying stan...