Related papers: Bounds on the Hermite spectral projection operator

We study asymptotic spectral properties of the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold $X$…

We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to two endpoints. The results can be applied to improve currently known bounds on sparse…

Let $f\in L^p(\mathbb{R}^d)$, $d\ge 3$, and let $A_t f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function $\sup_{t\in…

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian, and to the boundedness of $L^p$ norms of eigenfunctions of the…

We consider the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature…

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that…

We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all p between $2 and infinity, up to logarithmic losses…

Let $1\leq p \leq q <\infty$, and let $w \in \mathbb{C}$. Weissler conjectured that the Hermite operator $e^{w\Delta}$ is bounded as an operator from $L^{p}$ to $L^{q}$ on the Hamming cube $\{-1,1\}^{n}$ with the norm bound independent of…

We prove $L^p$-$L^q$-estimates for the Restriction-Extension operator acting on block-radial functions with the aid of new oscillatory integral estimates and interpolation results in mixed Lorentz spaces. Similar techniques apply to the…

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{\rho,\delta}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{\rho,\delta}^m) \subset…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we…

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger…

Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted…

On a compact manifold with boundary, consider the realization B of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say R_-. In the first part of the paper we define and…

In this paper we study maximal directional singular integral operators in $ \mathbb{R}^n $ given by a H\"ormander--Mihlin multiplier on an $ (n-1)$-dimensional subspace and acting trivially in the perpendicular direction. The subspace is…

The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry is studied. For any…

We study $L^p\to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier restriction…

This is a continuation of our previous research about an oscillatory integral operator $T_{\alpha, \beta}$ on compact manifolds $\mathbb{M}$. We prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator $T^{*}_{\alpha,…

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…