99 research outputs found
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights
This is the fourth article of our series. Here, we study weighted norm
inequalities for the Riesz transform of the Laplace-Beltrami operator on
Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the
doubling volume property and Gaussian upper bounds.Comment: 12 pages. Fourth of 4 papers. Important revision: improvement of main
result by eliminating use of Poincar\'e inequalities replaced by the weaker
Gaussian keat kernel bound
Local behavior of p-harmonic Green's functions in metric spaces
We describe the behavior of p-harmonic Green's functions near a singularity
in metric measure spaces equipped with a doubling measure and supporting a
Poincar\'e inequality
Regularity of maximal operators: recent progress and some open problems
This is an expository paper on the regularity theory of maximal operators,
when these act on Sobolev and BV functions, with a special focus on some of the
current open problems in the topic. Overall, a list of fifteen research
problems is presented. It summarizes the contents of a talk delivered by the
author at the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure
Theory and Applications, in Buenos Aires, Argentina.Comment: 19 pages. Expository paper with the contents of a lecture given at
the in the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure
Theory and Applications, in Buenos Aires, Argentin
Pointwise estimates to the modified Riesz potential
In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.Peer reviewe
Regularity of harmonic discs in spaces with quadratic isoperimetric inequality
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hölder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs
Area minimizing discs in metric spaces
We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces
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