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We prove l p ( Z d ) bounds, for p ∈ ( 1 , ∞ ) , 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.

We study a class of operators on nilpotent Lie groups G given by convolution with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0)⊂V1⊂…⊂Vk⊂…⫋G. We show that product kernels can be written as finite sums of flag kernels, that flag kernels can be characterized in terms of their Fourier transforms, and that flag kernels have good regularity, restriction, and composition properties. We then apply this theory to the study of the □b-complex on certain quadratic CR submanifolds of C n. We obtain Lp regularity for certain derivatives of the relative fundamental solution of □b and for the corresponding Szego projections onto the null space of □b by showing that the distribution kernels of these operators are finite sums of flag kernels.

On etudie des operateurs de convolution singuliers sur des groupes de Lie nilpotents generaux, et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

This paper is devoted to the study of the operator given by Ω where K is a standard Calderon-Zygmund kernel, and P is a real polynomial on R n × R n. We show T is bounded on Lp to itself, when 1

In this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications. These spaces are the “tent spaces.” They provide the natural setting for the study of such things as maximal functions (the relevant space here is T∞p), and also square functions (where the space T2p is relevant). As such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis. Thus they are closely related to Lp and Hardy spaces, important parts of whose theory become corollaries of the description of tent spaces. Also, as (“Proc. Conf. Harmonic Analysis, Cortona,” Lect. Notes in Math. Vol. 992, Springer-Verlag, Berlin/New York,1983), already indicated where these spaces first appeared explicitly, the tent spaces can be used to simplify some of the results related to the Cauchy integral on Lipschitz curves, and multilinear analysis. In retrospect one can recognize that various ideas important for tent spaces had been used, if only implicitly, for quite some time. Here one should mention Carleson's inequality, its simplifications and extensions, the theory of Hardy spaces, and atomic decompositions.

In this paper we continue the analysis of integral operators on nilpotent Lie groups with kernels supported on submanifolds and/or containing an oscillatory factor of polynomial type. In our previous articles [13, 143 we have essentially considered singular integral operators, i.e., those defined by kernels having the critical degree of homogeneity. Here we focus our attention on kernels having a higher degree of homogeneity, so that locally they have a finite mass, but do not have enough decay at infinity. More precisely, if Q is the homogeneous dimension of the group G (which in this article we always assume endowed with automorphic dilations) we consider kernels K* having the same homogeneity as 1x1 -Q(l-a), where 1x1 denotes a homogeneous gauge and Re a > 0. These kernels are assumed to be supported on analytic homogeneous submanifolds V; we call them fractional integration kernels along V. We discuss three interrelated problems concerning fractional integration kernels