Related papers: Discrete multilinear spherical averages

We show results on $L^p$-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lam\'e system. Here we rely on…

We consider the multilinear pseudo-differential operators with symbols in a generalized $S_{0,0}$-type class and prove the boundedness of the operators from $(L^2,\ell^{q_1}) \times \dots \times (L^2,\ell^{q_N})$ to $(L^2,\ell^{r})$, where…

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…

We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture…

We consider a type of maximal operators associated to moment curves in $\mathbb R^d, d\geq 3.$ We derive $L^p$ mapping properties for these operators. In a special case, the estimate is sharp.

Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$, where $p\not=q$, we investigate the…

The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view…

We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-\beta}$ for $\beta <…

We characterize the boundedness properties on the spaces $L^p(\mathbb{H}^2)$ of the maximal operator $M_\mathcal{B}$ where $\mathcal{B}$ is an arbitrary family of hyperbolic triangles stable by isometries.

We use microlocal and paradifferential techniques to obtain $L^8$ norm bounds for spectral clusters associated to elliptic second order operators on two-dimensional manifolds with boundary. The result leads to optimal $L^q$ bounds, in the…

In the present paper we characterize the existence and uniqueness of maximal Lp-regular solutions of high order convolution operator equations. Particularly, we get coercive uniform estimates with respect to spectral parameter and we show…

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…

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the…

We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no…

We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of $L^p$ spaces, with $p$ close enough to 1. We also prove that the trilinear Hilbert transform is unbounded in a similar…

The aim of the article is to prove $L^{p}-L^{q}$ off-diagonal estimates and $L^{p}-L^{q}$ boundedness for operators in the functional calculus of certain perturbed first order differential operators of Dirac type for with $p\le q$ in a…

We provide a new direct proof of the $\ell^2$-boundedness of the Discrete Spherical Maximal Function that neither relies on abstract transference theorems (and hence Stein's Spherical Maximal Function Theorem) nor on delicate asymptotics…

We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_\alpha(\mathbb{B}_ n)$ to the Lebesgue spaces $L^q(\mathbb{S}_ n)$ for all $0<p,q<\infty$. For the case $n=1$, some partial results were…

Given a hypersurface $S\subset \mathbb{R}^{2d}$, we study the bilinear averaging operator that averages a pair of functions over $S$, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular…

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1,$$ where derivative of the function $Q$ is understood in the sense of…