Function of the distance as a Muckenhoupt weight.

TOSCHI, MARISA; IAFFEI, BIBIANA; CARENA. MARILINA

Padova

Congreso; Mini-courses in Mathematical Analysis 2015; 2015

Dipartimento di Ingegneria Civile, Edile ed Ambientale of the University of Padova

The class of Muckenhoupt weights are extensively used in real and harmonic analysis, as well as, in the theory of partial differential equations. For example, the behavior of the source near the boundary of the domain of a Dirichlet boundary value problem, may cause non-solvability in a non-weighted Sobolev space. Nevertheless, the problem can be solved in an adequate weighted Sobolev space, in which the difficulties might be avoided. If the source has an unbounded growth near the boundary $F$ of the domain, we should search for a weight which vanishes there.This is the case of the power-type weights, which are of the form $d^eta(x,F)$, where $d(x,F)$ is the distance from the point $x$ to $F$.In the general framework of metric measure spaces $(X,d,mu)$, in cite{ACDT} the authors give sufficient conditions on a closed set $Fsubseteq X$ and on a real number $eta$ in such a way that $d(x,F)^eta$ becomes a Muckenhoupt weight.On the other hand, Kokilashvili and Samko study in cite{Samko} under which conditions $w(d(x,x_0))$ belongs to the Muckenhoupt class$A_p(X,d,mu)$, where $x_0in X$ and $w(t)$ is a function generalizing the powers $t^eta$.In this talk we will give an extension of these result. That is, sufficient conditions on $w(t)$ and on a subset $F$ of a space of homogeneous type $(X,d,mu)$, such that $w(d(x,F))in A_p(X,d,mu)$.