Weighted norm inequalities for maximal singular integrals with nondoubling measures

Abstract

Abstract. Let µ be a nonnegative Radon measure on R d which satisfies µ(B(x, r)) ≤ Cr n for any x ∈ R d and r > 0 and some positive constants C and n ∈ (0, d]. In this paper, some weighted norm inequalities with A p (µ) weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure µ, via certain weighted estimates with A ∞ (µ) weights of Muckenhoupt type involving the JohnStrömberg maximal operator and the John-Strömberg sharp maximal operator, where , p ∈ [1, ∞).

Maximal multilinear Calderón-Zygmund operators with non-doubling measures

“…In fact, we only change p ∈ [1, ∞) there into p ∈ (0, ∞). With minor changes in the proof of Theorem 2.1 in [7], we can obtain the present lemma. We omit the proof for brevity.…”

“…Lemma 4 is a variant of Theorem 2.1 in Hu and Yang [7]. In fact, we only change p ∈ [1, ∞) there into p ∈ (0, ∞).…”

“…We say that a cube Q is a (α, β)-doubling cube if μ(αQ) βμ(Q), where αQ denotes the cube with the same center as Q and l(αQ) = αl(Q). It was pointed out by Tolsa in [16] that if β > α n , then for any x ∈ supp (μ) and any R > 0, there exists some (α, β)-doubling [16] for the case of ρ = 1 and [7] for the case of ρ > 1. If ρ = 1, we denote Q ρ by Q and K ρ Q 1 ,Q 2 by K Q 1 ,Q 2 for simplicity.…”

“…It has been proved in [7] that if 0 < s < β −1 2ρ,d /4, then for any μ-locally integrable function f and any λ > 0, Case II: μ(R d ) < ∞. Choose p = 1/m.…”

“…The main tools for proving Theorem 4 is the JohnStrömberg maximal operator and the JohnStrömberg sharp maximal operator with non-doubling measures introduced by Hu and Yang in [7].…”