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JournalISSN: 0214-1493

Publicacions Matematiques 

Autonomous University of Barcelona
About: Publicacions Matematiques is an academic journal published by Autonomous University of Barcelona. The journal publishes majorly in the area(s): Bounded function & Holomorphic function. It has an ISSN identifier of 0214-1493. Over the lifetime, 1151 publications have been published receiving 13139 citations.


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TL;DR: In this article, a survey of results on Dirichlet problems with nonlocal operators of the form Lu (x) = PV Z Rn u(x) u (x + y) K(y)dy is presented.
Abstract: In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65.

227 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that weak solutions of the following fractional Laplacian equation are also continuous solutions in the viscosity sense up to the boundary of a bounded, open subset of R n n with C 2 -boundary.
Abstract: Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in the viscosity sense. Here s2 (0; 1) is a xed parameter, is a bounded, open subset of R n (n > 1) with C 2 -boundary, and ( ) s is the fractional Laplacian operator, that may be dened as

218 citations

Journal ArticleDOI
TL;DR: In this paper, the authors compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients.
Abstract: In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Holder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We also give applications of these properties. In Part 2 some results on the microlocal spaces contained in [B2] will be recalled. Theorems 3 and 4 are also essentially contained in [B2]. The starting point for this paper was a note ([J1]) the author had written on a comparison between the Holder criterion of regularity at a given point x0 and a corresponding property defined on the wavelet coefficients. Some easy proofs are omitted or abridged and can be found in [J2].

213 citations

Journal ArticleDOI
TL;DR: In this paper, the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions was studied and theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces were proved.
Abstract: We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

186 citations

Journal ArticleDOI
TL;DR: In this article, the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight was studied and sharp weighted L p estimates were obtained for the Hilbert, Beurling, and martingale transforms.
Abstract: We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r 1 p 1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.

160 citations

Performance
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No. of papers from the Journal in previous years
YearPapers
202318
202224
202125
202027
201921
201823