Muckenhoupt weights with singularities on closed lower dimensional sets in spaces of homogeneous type.

AIMAR, HUGO; CARENA. MARILINA; TOSCHI, MARISA

Bahia Blanca

Congreso; Congreso Dr. Antonio A. R. Monteiro; 2013

INMABB organiza ?en forma conjunta con el Departamento de Matemática de la UNS

In this note we aim to produce weights with singularities on a closed set $F$, of the form \[w(x)=\mu(B(x,d(x,F)))^\beta,\] under certain dimensional conditions on $F$, for some positive and negative values of $\beta$. Here $d(x,F)=\inf\{d(x,y):y\in F\}$. We shall provided an interval about $0$ for $\beta$, such that $w(x)$ is an $A_p$-Muckenhoupt weight. We start by defining a particular type of $s$-dimensional set in a general space of homogeneous type. We prove that this concept coincides with the one of $s$-Ahlfors with respect to the normalized quasi-distance defined by Mac\'ias and Segovia in \cite{M-S}. We shall say that a closed subset $F$ of $X$ is \textbf{\emph{s-dimensional with respect to $\boldsymbol\mu$}}, $s0$ such that for every $x\in F$ and every $0