Scientic Supervisor / Contact Person
Localization & Research Area
MSCA & ERC experience
Research Team & Research Topic
The resolution of the Dirichlet problem, and other non-linear PDE's like the p-Laplacian, on certain discrete settings like simple graphs. Finding harmonic extensions with prescribed boundary values is a well-known problem, but our approach is new, based on iteration techniques that go back to the original works of Lebesgue. For this purpose we aim to develop the necessary techniques to tackle these results, like Gerschgorin circles lemma or Kirchhoff's theorem for counting the number of spanning trees.
We have recently studied the optimal norms for the Hardy operator H minus the identity and the norms of H and its adjoint H*, restricted to the cone of positive and decreasing functions, just for positive functions or all arbitrary functions in Lebesgue spaces L^p (0,∞). Our objective is to study this kind of questions for other types of weighted inequalities.
Also related to the questions above, very recently, end-point estimates of weak- type for p = 1 and p = ∞ have been obtained for the case or general decreasing weights or power weights, respectively. We plan to study the same type of end-point estimates without the restriction of the weight being necessarily decreasing or not being a power weight.
In 2019, we introduced the study of the characterization of the best doubling constant C(X) for general metric measure spaces (X,d), proving the optimal lower bound C(X) ≥ 2. The precise value of this constant is related to the different notions of metric dimensions and, motivated by the previous results, turns out to be of special interest in the discrete case of graphs. Finding C(X) for specific graphs is a challenge that requires the use of very deep results in optimization theory and new techniques are still needed, which is the main question to overcome in this setting.
Finally, in 2009 we were able to characterize the best constant in Minkowski's inequality for Lorentz spaces Lp,q in the case p < q, where the functional is not a norm. Similarly, in 2010 we extended this result for a general weighted Lorentz space, when the weight is monotone and in Ap. The general situation is still open, under the only requirement that the space is normable.
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