Related papers: Variational inequalities for generalized spherical…

Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$.

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d.…

We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb R^{1+d}$. In the range for which…

We show that, under general conditions, the operator $\bigl (-\nabla \cdot \mu \nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(\Omega)$ and $L^p(\Omega)$, for $p \in {]1,2[}$ if…

We investigate mapping properties of non-centered Hardy-Littlewood maximal operators related to the exponential measure $d\mu(x) = \exp(-|x_1|-\ldots-|x_d|)dx$ in $\mathbb{R}^d$. The mean values are taken over Euclidean balls or cubes…

We study the operator $L=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and…

We present a simple geometric approach to studying the $L^p$ boundedness properties of Stein's spherical maximal operator, which does not rely on the Fourier transform. Using this, we recover a weak form of Stein's spherical maximal…

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function…

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is…

We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class…

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

We find sharp conditions for the maximal operator associated with generalized spherical mean Radon transform on radial functions $M^{\a,\b}_t$ to be bounded on power weighted Lebesgue spaces. Moreover, we also obtain the corresponding…

Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$…

We obtain a necessary and sufficient condition on an exponent $p(\cdot)$ for which the Hardy--Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. It is formulated in terms of the Muckenhoupt-type condition…

Given an elliptic diffusion operator $L$ defined on a compact and connected manifold (possibly with a convex boundary in a suitable sense) with an $L$-invariant measure $m$, we introduce the non-linear $p-$operator $L_p$, generalizing the…

In this paper we prove $L^p$-estimates for H\"ormander classes of pseudo-differential operators on the torus $\mathbb{T}^n$. The results are presented in the context of the global symbolic calculus of Ruzhansky and Turunen on…

We study uniform Sobolev inequalities for the second order differential operators $P(D)$ of non-elliptic type. For $d\ge3$ we prove that the Sobolev type estimate $\|u\|_{L^q(\mathbb{R}^d)}\le C \|P(D)u\|_{L^p(\mathbb{R}^d)}$ holds with $C$…

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator $P$ with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that $P$ is globally hypoelliptic.…

We prove that wave operators of scattering theory for fourth order Schr\"odinger operators $H = \Delta^2 + V (x)$ on $\mathbb{R}^2$ with real potentials $V(x)$ such that $\langle x \rangle^3 V(x) \in L^{\frac43}(\mathbb{R}^2)$ and $\langle…