Rubio de Francia's extrapolation theory: estimates for the distribution function

Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invariant spaces are established too.Comment: 29 page

A Transmission Problem in the Scattering of Electromagnetic Waves by a Penetrable Object

This is the published version, also available here: http://dx.doi.org/10.1137/S0036141094267388.Layer-potential techniques are used to study a transmission problem arising in the scattering of electromagnetic waves by a penetrable object. The method proposed does not involve the use of the calculus of pseudodifferential operators and hence it can be applied in domains with very little regularity. The solutions are represented as a combination of a curl and a double curl of a single layer-potential operator. The work relies on the important harmonic-analysis tools developed in recent years to study boundary-value problems in domains with minimal regularity assumptions

The multilinear strong maximal function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal function, are also proved.Comment: appeared in J. of Geometric Ana

Compact Bilinear Operators and Commutators

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Caldeŕon-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

Minimal regularity conditions on the kernels of bilinear operators are identi- fied and shown to be sufficient for the existence of end-point estimates within the context of the bilinear Calderón-Zygmund theory.Ministerio de Ciencia e InnovaciónJunta de AndalucíaNational Science Foundatio