Javier Duoandikoetxea

SELF-ADJOINT EXTENSIONS OF DIRAC OPERATORS WITH COULOMB TYPE SINGULARITY

In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.

Directional operators and radial functions on the plane

LetEçS1 be a set with Minkowski dimensiond(E)1. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions ofE. The main result of this paper shows that these operators are bounded onLradp (R2) forp>1+d(E) and unbounded whenp<1+d(E). We also give some end-point results.

Mixed Norm Inequalities for Directional Operators Associated to Potentials

We study mixed norm inequalities for directional operators which appear applying the method of rotations to homogeneous operators with variable kernel and with the homogeneity of Riesz potentials. The results are sharp for a range of values of the parameter and for all its values when the inequalities are restricted to radial functions.

A unified approach to several inequalities involving functions and derivatives

There are many inequalities measuring the deviation of the average of a function over an interval from a linear combination of values of the function and some of its derivatives. A general setting is given from which the desired inequalities are obtained using Holders inequality. Moreover, sharpness of the constants is usually easy to prove by studying the equality cases of Holders inequality. Comparison of averages, extension to weighted integrals and n-dimensional results are also given.

MIXED NORM ESTIMATES FOR POTENTIAL OPERATORS RELATED TO THE RADON TRANSFORM

(November 14, 2007)AbstractWe de ne potential operators on hyperplanes and give sharp mixed norm inequalities forthem. One of the operators coincides with the Radon transform for which mixed normestimates are known but in reverse order. Those inequalities will be crucial in our proofs.Keywords and phrases: Potential operators, Radon transform, mixed norm estimates.

Homogeneous Fourier multipliers in the plane

Given a homogeneous of degree zero function on the plane, we study conditions on the first derivative of its restriction to the unit circle in order to deduce that it is an Lp-multiplier.

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt

The bilinear Hilbert transform acting on Hermite and Laguerre functions

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