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On the maximal function for rotation invariant measures in ℝ n

Ana Vargas — 1994

Studia Mathematica

Given a positive measure μ in ℝ n , there is a natural variant of the noncentered Hardy-Littlewood maximal operator M μ f ( x ) = s u p x ∈ B 1 / μ ( B ) ʃ B | f | d μ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ℝ n . We give some necessary and sufficient conditions for M μ to be bounded from L 1 ( d μ ) to L 1 , ∞ ( d μ ) .

Checkerboards, Lipschitz functions and uniform rectifiability.

Peter W. Jones;  Nets Hawk Katz;  Ana Vargas — 1997

Revista Matemática Iberoamericana

In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof. Theorem. Suppose Ω is a bounded open set in Rn with n > 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M < ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε. Here Hk...

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