Related papers: Trasferring $L^p$ eigenfunction bounds from $S^{2n…

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times…

We prove that Heisenberg groups, a.k.a. the boundaries of Siegel domains, minimize the $L^p$ operator norm of the Szeg\H{o} projection in a large class of weighted CR manifolds of hypersurface type.

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…

For an operator generating a group on $L^p$ spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction…

We prove restriction type estimates for sub-Laplacians on general two-step stratified Lie groups. The core of our approach is to use spectral cluster estimates to effectively control the eigenvalue distribution of a family of anisotropic…

We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and…

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window…

The Stein-Tomas restriction theorem on Euclidean space says one can meaningfully restrict $\hat{f}$ to the unit sphere of $\mathbb{R}^n$ provided $f \in L^p(\mathbb{R}^n)$ with $1 < p < 2$. This result can be rewritten in terms of the…

Given a Riemannian manifold endowed with its Laplace-Beltrami operator, consider the associated spectral projector on a thin interval. As an operator from L2 to Lp, what is its operator norm? For a window of size 1, this question is fully…

Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.

A certain representation for the Heisenberg algebra in finite-difference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is proposed. Using $sl_2$-algebra based…

We investigate the concentration of eigenfunctions for the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^n$ by establishing local $L^p$ bounds over the compact sets with arbitrary dilations and translations. These new results extend the…

We prove a restricted projection theorem for an n-2 dimensional family of projections from $\mathbb R^n$ to $\mathbb R$. The family we consider arises naturally in the context of the adjoint representation of the maximal unipotent subgroup…

We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the K\'oranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding…

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $\mu_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*\mu_t|$. We prove…

The "projective lightcone limit" has been proposed as an alternative holographic dual of an AdS space. It is a new type of group contraction for a coset G/H preserving the isometry group G but changing H. In contrast to the usual group…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

In this article we prove the restriction theorem for Helgason-Fourier transform on homogeneous tree. Our proof is based on the duality argument and the norm estimates of Poisson transform. We also characterize all eigenfunctions of the…

We prove that, if \Delta_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-H\"ormander multiplier on the positive half-line, with L^2-order of smoothness greater than…

For $e \in S^{2}$, the unit sphere in $\mathbb{R}^3$, let $\pi_{e}$ be the orthogonal projection to $e^{\perp} \subset \mathbb{R}^{3}$, and let $W \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K…