Uniform rectifiability and singular sets
Uniqueness theorems for Cauchy integrals
Variation for the Riesz transform and uniform rectifiability
For integers and , we prove that an -dimensional Ahlfors-David regular measure in is uniformly -rectifiable if and only if the -variation for the Riesz transform with respect to is a bounded operator in . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the boundedness of the Riesz transform to the uniform rectifiability of .
Variational inequalities for singular integral operators
In these notes we survey some new results concerning the -variation for singular integral operators defined on Lipschitz graphs. Moreover, we investigate the relationship between variational inequalities for singular integrals on AD regular measures and geometric properties of these measures. An overview of the main results and applications, as well as some ideas of the proofs, are given.
Vector Potential Theory on Nonsmooth Domains in R... and Applications to Electromagnetic Scattering.
Vector valued inequalities for strongly singular Calderón-Zygmund operators.
In this article we consider a theory of vector valued strongly singular operators. Our results include Lp, Hp and BMO continuity results. Moreover, as is well known, vector valued estimates are closely related to weighted norm inequalities. These results are developed in the first four sections of our paper. In section 5 we use our vector valued singular integrals to estimate the corresponding maximal operators. Finally in section 6 we discuss applications to weighted norm inequalities for pseudo-differential...
Vector-valued Calderón-Zygmund theory applied to tent spaces
Vector-valued inequalities for the commutators of fractional integrals with rough kernels
Some conditions implying vector-valued inequalities for the commutator of a fractional integral and a fractional maximal operator are established. The results obtained are substantial improvements and extensions of some known results.
Vector-valued inequalities with weights.
This paper deals with the following problem:Let T be a given operator. Find conditions on v(x) (resp. u(x)) such that∫ |Tf(x)|pu(x) dx ≤ C ∫ |f(x)|pv(x) dxis satisfied for some u(x) (resp. v(x)).Using vector-valued inequalities the problem is solved for: Carleson's maximal operator of Fourier partial sums, Littlewood-Paley square functions, Hilbert transform of functions valued in U.M.D. Banach spaces and operators in the upper-half plane.
Vector-Valued Singular Integrals Revisited-with Random Dyadic Cubes
The vector-valued T(1) theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a simplified approach to these results is presented, exploiting Nazarov, Treil and Volberg's method of random dyadic cubes, which allows one to circumvent the most subtle parts of the original arguments.
Wavelet Frames on Lipschitz Curves and Applications.
Wavelet transform for functions with values in UMD spaces
We extend the classical theory of the continuous and discrete wavelet transform to functions with values in UMD spaces. As a by-product we obtain equivalent norms on Bochner spaces in terms of g-functions.
Weak type estimates for certain Calderón-Zygmund singular integral operators
We prove weak type (1,1) estimates for a special class of Calderón-Zygmund homogeneous kernels represented as l¹ sums of "equidistributed" H¹ atoms on 𝕊¹.
Weak type estimates for operators of potential type
We derive two-weight weak type estimates for operators of potential type in homogeneous spaces. The conditions imposed on the weights are testing conditions of the kind first studied by E. T. Sawyer [4]. We also give some applications to strong type estimates as well as to operators on half-spaces.
Weak-strong convolution operators on certain disconnected groups
Weak-Type (1-1) Inequality for the Monge-Ampère SIO's.
Weak-type estimates for the Riesz transforms associated with the Gaussian measure.
In this paper we will study the behavior of the Riesz transform associated with the Gaussian measure γ(x)dx = e-|x|2dx in the space Lγ1 (Rn).
Weight inequalities for singular integrals defined on spaces of homogeneous and nonhomogeneous type.
Weighted boundedness of multilinear Littlewood-Paley operators for the extreme cases of .
Weighted boundedness of Toeplitz type operators related to singular integral operators with non-smooth kernel
Some weighted sharp maximal function inequalities for the Toeplitz type operator are established, where are a fixed singular integral operator with non-smooth kernel or ±I (the identity operator), are linear operators defined on the space of locally integrable functions, k = 1,..., m, and . The weighted boundedness of on Morrey spaces is obtained by using sharp maximal function inequalities.