Related papers: Higher order Hardy-Rellich identities

Let $(X,\mathbf{q},\mu)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $\mu$ satisfies an additional reverse doubling property. Let…

In this paper, we characterize the weighted local Hardy spaces $h^p_\rho(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{1}^{\rho,\,\infty}(\mathbb{R}^{n})$ by localized Riesz transforms $\widehat{R}_j$, in…

Let $\{K_t\}_{t>0}$ be the semigroup of linear operators generated by a Schr\"odinger operator $-L=\Delta - V(x)$ on $\mathbb R^d$, $d\geq 3$, where $V(x)\geq 0$ satisfies $\Delta^{-1} V\in L^\infty$. We say that an $L^1$-function $f$…

We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use $h$-harmonic expansions to reduce the problem in the…

In this dissertation we explore the $[L^{\mathrm{p}},\ L^{q}]$-boundedness of certain integral operators on weighted spaces on cones in ${\mathbb R}^{n}.$ These integral operators are of the type $\displaystyle \int_{V}k(x,\ y)f(y)dy$…

We investigate two related problems concerning the dimension of joint eigenspaces of the Laplace--Beltrami operator and a finite set of Hecke operators on $\mathbb{X}=\mathrm{PGL}_2(\mathbb{Z})\backslash \mathbb{H}$. First, we consider…

In this article, we study and settle several structural questions concerning the exact solvability of the Olshanetsky-Perelomov quantum Hamiltonians corresponding to an arbitrary root system. We show that these operators can be written as…

This paper explores Paley-Wiener type theorems within the framework of hypercomplex variables. The investigation focuses on a space-fractional version of the Dirac operator $\mathbf{D}_\theta^{\alpha}$ of order $\alpha$ and skewness…

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic…

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…

We show that some previous results concerning the boundedness of differentiation and integration operators on weighted spaces given by radial weights in the unit disk or the complex plane might fail without some natural additional…

We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. This operator appears very naturally in the theory of bilinear operators such as the bilinear…

In this paper we discuss cylindrical extensions of improved Hardy, Sobolev type and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants and identities in the spirit of Badiale-Tarantello [2]. All identities are obtained in the…

In this paper we present Hardy type inequalities for magnetic Dirichlet forms with singular integral weights. We analyze the local and global optimality of the integral weight and discuss several examples in details. An application of our…

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$.…

Let $L=-\Delta+V$ be a Schr\"{o}dinger operator, where $\Delta $ is the Laplacian operator on $\rz$, while the nonnegative potential $V$ belongs to certain reverse H\"{o}lder class. In this paper, we establish some weighted norm…

In this paper, we consider the first order Hardy inequalities using simple equalities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new…

We establish sharp Adams type inequalities on Sobolev spaces $W^{\alpha, n/\alpha}(X)$ of any fractional order $\alpha< n$ on Riemannian symmetric space $X$ of noncompact type with dimension $n$ and of arbitrary rank. We also establish…

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of…