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Elias M. Stein

Publications and source records attributed to Elias M. Stein.

At least 19 recordsLinked to original sources
arXiv2023

On a multi-parameter variant of the Bellow-Furstenberg problem

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

math.DS↗
arXiv2019

Jump inequalities via real interpolation

Jump inequalities are the $r=2$ endpoint of Lépingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers.

math.CA↗
arXiv2019

Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$

Dimension-free bounds will be provided in maximal and $r$-variational inequalities on $\ell^p(\mathbb Z^d)$ corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in $\mathbb Z^d$. We will also construct an example of a symmetric convex body in $\mathbb Z^d$ for which maximal dimension-free bounds fail on $\ell^p(\mathbb Z^d)$ for all $p\in(1, \infty)$. Finally, some applications in ergodic theory will be discussed.

math.CA↗
arXiv2019

A bootstrapping approach to jump inequalities and their applications

The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of $r$-variational estimates, previously known for $r>2$, to end-point results for the jump quasi-seminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain and Wróbel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright.

math.CA↗
arXiv2018

$\ell^p\big(\mathbb Z^d\big)$-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

We prove $\ell^p\big(\mathbb Z^d\big)$ bounds, for $p\in(1, \infty)$, of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.

math.CA↗
arXiv2017

The role of an integration identity in the analysis of the Cauchy-Leray transform

The purpose of this paper is to complement the results in [LS-1] by showing the dense definability of the Cauchy-Leray transform for the domains that give the counterexamples of [LS-1], where $L^p$-boundedness is shown to fail when either the "near" $C^2$ boundary regularity, or the strong $\mathbb C$-linear convexity assumption is dropped.

math.CV↗
arXiv2017

The Cauchy-Leray integral: counter-examples to the $L^p$-theory

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong $\mathbb C$-linear convexity, together with regularity of order 2, are both necessary.

math.CV↗
arXiv2015

L^p(Z^d)-estimates for discrete operators of Radon type: Variational estimates

We prove $\ell^p\big(\mathbb Z^d\big)$ bounds for $p\in(1, \infty)$, of $r$-variations $r\in(2, \infty)$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of $p$ and $r$ which coincide with the ranges of their continuous counterparts.

math.CA↗
arXiv2015

Algebras of singular integral operators with kernels controlled by multiple norms

The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calderón-Zygmund operators with different homogeneities, such as operators that occur in sub-elliptic problems and those arising in elliptic problems. For example, one would like to describe the algebras containing the operators related to the Kohn-Laplacian for appropriate domains, or those related to inverses of Hörmander sub-Laplacians, when these are composed with the more standard class of pseudo-differential operators. The algebras we study can be characterized in a number of different but equivalent ways, and consist of operators that are pseudo-local and bounded on $L^{p}$ for $1<p<\infty$. While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators we study fall outside this class, and reflect a multi-parameter structure.

math.FA↗
arXiv2014

Pseudodifferential operators of mixed type adapted to distributions of $k$-planes

We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest.

math.CA↗
arXiv2013

Cauchy-type integrals in several complex variables

We present the theory of Cauchy-Fantappié integral operators, with emphasis on the situation when the domain of integration, $D$, has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter $z\in D$. The goal is to prove $L^p$ estimates for these operators and, as a consequence, to obtain $L^p$ estimates for the canonical Cauchy-Szegö and Bergman projection operators (which are not of Cauchy-Fantappié type).

math.CV↗