Albert Mas

Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs

We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1<p finite. The analogous result holds for the n-dimensional Riesz transform on n-dimensional Lipschitz graphs, as well as for other singular integral operators with odd kernel. In particular, our results strengthen the classical theorem on the L^2 boundedness of the Cauchy transform on Lipschitz graphs by Coifman, McIntosh, and Meyer.

Variation for the Riesz transform and uniform rectifiability

For 0 2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). 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 L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.

An Isoperimetric-Type Inequality for Electrostatic Shell Interactions for Dirac Operators

In this article we investigate spectral properties of the coupling

Klein’s paradox and the relativistic δ-shell interaction in ℝ3

{H + V_\lambda}

A dual characterization of the

H+Vλ, where

harmonic capacity and applications

{H = -i\alpha \cdot \nabla+m\beta}