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 y 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) = μ(B(x, d(x, F )))^β , under certain dimensional conditions on F , for some positive and negative values of β. Here d(x,F) = inf{d(x, y) : y ∈ F }. We shall provided an interval about 0 for β, 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 Macías and Segovia in [MS79]. We shall say that a closed subset F of X is s-dimensional with respect to μ, s < 1, if there exist a Borel measure ν supported on F and three constants c1 , c2 , c3 > 0 such that for every x ∈ F and every 0 < r < diam(F )the following two conditions are satisfied; (1) if t is a positive number for which μ(B(x, t)) < r, then ν(B(x, t)) ≤ c1 r^s ; (2) there exists a d-ball B containing x with μ(B) < c2 r and ν(B) ≥ c3 r^s . If F is unbounded and the above conditions hold for every 0 < r < r_0 , where r_0 is a positive number less than diam(F ), we say that F is locally s-dimensional with respect to μ. The main result in this note is contained in the next statement. Theorem 1. Let (X, d, μ) be a space of homogeneous type and let F ⊆ X be s-dimensional with respect to μ, with 0 ≤ s < 1. If no atoms of X belongs to F , then w(x) = μ (B(x, d(x, F )))γ(s−1) belongs to A_1 (X, d, μ) for every 0 ≤ γ < 1. Consequently μ (B(x, d(x, F)))^β ∈ A_p (X, d, μ) for −(1 − s) < β < (1 − s)(p − 1) and 1 ≤ p < ∞. In order to prove