Related papers: Dimension-free Maximal Inequalities for Spherical …

We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\mathrm{Im}\,\lambda \leq -\frac{1}{2}$ which is optimal in dimension $2$. In odd dimensions we also show that…

In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp-spaces, 1 < p < infinity, was established and, among other things, corresponding maximal ergodic inequalities and individual…

Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schr\"odinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between…

In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of…

Let $A_tf(x)=\int f(x+ty)d\sigma(y)$ denote the spherical means in $\Bbb R^d$ ($d\sigma$ is surface measure on $S^{d-1}$, normalized to $1$). We prove sharp estimates for the maximal function $M_E f(x)=\sup_{t\in E}|A_tf(x)|$ where $E$ is a…

We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schr\"odinger operators and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only if…

We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the…

In this article, we study the fractional spherical maximal function and its lacunary counterpart. We study the necessary and sufficient conditions for $L^p-L^q$ boundedness of both maximal functions. In particular, we prove the restricted…

Let $0<\alpha<d$ and $1\leq p<d/\alpha$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_\alpha f$ are weakly differentiable and $…

We consider here the free two step nilpotent Lie group, provided with the homogeneous Kor\'anyi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$.

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 $M$…

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous…

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ for which the…

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(|x|^\alpha)$ spaces. Aside from endpoint cases, a complete description of the allowable ranges of $p$, $\alpha$ is given in…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

We prove optimal spectral inequalities for Landau operators in full space and in arbitrary dimension. Spectral inequalities are lower bounds on the L 2 -mass of functions in spectral subspaces of finite energy when integrated over a…

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the…

We prove a uniform upper and lower bound for Delannoy numbers. This is achieved by using the representation of Delannoy numbers as the number of lattice points in high-dimensional cross-polytopes (also known as hyper-octahedrons or $\ell^1$…

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield…

Based on a proper hypothesis on the noncommutative Fourier integral operators, we establish in this paper the strong-type $(p,p)$ (with $2\leq p\leq \infty$) estimates for the operator-valued Stein's maximal spherical means.