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Remarks on a theorem by N. Yu. Antonov

Per SjölinFernando Soria — 2003

Studia Mathematica

We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh-Fourier system, for which we prove a.e. convergence in the class L log L log log log L. Other applications are given in the theory of differentiation of integrals.

Muckenhoupt-Wheeden conjectures in higher dimensions

Alberto CriadoFernando Soria — 2016

Studia Mathematica

In recent work by Reguera and Thiele (2012) and by Reguera and Scurry (2013), two conjectures about joint weighted estimates for Calderón-Zygmund operators and the Hardy-Littlewood maximal function were refuted in the one-dimensional case. One of the key ingredients for these results is the construction of weights for which the value of the Hilbert transform is substantially bigger than that of the maximal function. In this work, we show that a similar construction is possible for classical Calderón-Zygmund...

Rough maximal functions and rough singular integral operators applied to integrable radial functions.

Peter SjögrenFernando Soria — 1997

Revista Matemática Iberoamericana

Let Ω be homogeneous of degree 0 in R and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor Ω in the definition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels Ω(y) / |y|, provided that the mean value of Ω vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary finite...

Regularity of solutions of the fractional porous medium flow

Luis CaffarelliFernando SoriaJuan Luis Vázquez — 2013

Journal of the European Mathematical Society

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = · ( u ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x n , t } with nonnegative initial data u ( x , 0 ) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and C α regularity of such weak solutions. Finally, we extend the existence...

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