Variants of Littlewood-Paley theory.
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 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 and Carleson measures on spaces of homogeneous nature
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.