The work of José Luis Rubio de Francia (III).
The aim of this paper is to review a set of articles ([6], [10], [11], [13], [16], [25]) of which José Luis Rubio de Francia was author and co-author written between 1985 and 1987.
200 años de convergencia de las series de Fourier.
Directional operators and mixed norms.
We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calderón and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting questions remain unanswered. [Proceedings of the 6th International Conference on Harmonic...
Multiple singular integrals and maximal functions along hypersurfaces
Maximal functions written as convolution with a multiparametric family of positive measures, and singular integrals whose kernel is decomposed as a multiple series of measures, are shown to be bounded in , . The proofs are based on the decomposition of the operators according to the size of the Fourier transform of the measures, assuming some regularity at zero and decay at infinity of these Fourier transforms. Applications are given to homogeneous singular integrals in product spaces with size...
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 Hölder’s inequality. Moreover, sharpness of the constants is usually easy to prove by studying the equality cases of Hölder’s inequality. Comparison of averages, extension to weighted integrals and -dimensional results are also given.
Weighted inequalities for square and maximal functions in the plane
We prove weighted inequalities for square functions of Littlewood-Paley type defined from a decomposition of the plane into sectors of lacunary aperture and for the maximal function over a lacunary set of directions. Some applications to multiplier theorems are also given.
Weighted inequalities for rough square functions through extrapolation
Weighted inequalities for some square functions are studied. L² results are proved first using the particular structure of the operator and then extrapolation of weights is applied to extend the results to other spaces. In particular, previous results for square functions with rough kernel are obtained in a simpler way and extended to a larger class of weights.
Hardy-type estimates for Dirac operators
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