arXiv subjects
Publications and source records attributed to Walter A. Ortiz.
An $RD$ space is a doubling measure metric space $Ω$ with the additional property that it has a reverse doubling property. In this paper we introduce a new class of Hajłasz-Besov spaces on $Ω$ and extend several results from classical theory, such as embeddings and Sobolev-type embeddings.
We introduce Poincaré type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.
In this paper we will focus on understanding the relation between Sobolev embedding theorems for Hajłasz-Besov spaces defined on a doubling metric measure space $(Ω,d,μ)$ and the non-collapsing condition of the measure, i.e. \[ \inf_{x\inΩ}μ(B(x,1))>0. \] We will also obtain embedding results for Hajłasz-Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.
We obtain symmetrization inequalities in the context of Fractional Hajlasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure.
We obtain symmetrization inequalities on probability metric spaces with convex isoperimetric profile which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincaré and Nash inequalities.
We show the density of smooth Sobolev functions $W^{k,\infty}(Ω)\cap C^\infty(Ω)$ in the Orlicz-Sobolev spaces $L^{k,Ψ}(Ω)$ for bounded simply connected planar domains $Ω$ and doubling Young functions $Ψ$.