Abstract
We study the mapping properties of singular integral operators defined by mappings of finite type. We prove that such singular integral operators are bounded on the Lebesgue spaces under the condition that the singular kernels are allowed to be in certain block spaces.
Highlights
Introduction and resultsFor n ∈ N, n ≥ 2, let K(·) be a Calderón-Zygmund kernel defined on Rn, that is, KΩ(y) = Ω(y)|y|−n, (1.1)where Ω ∈ L1(Sn−1) is a homogeneous function of degree zero that satisfies Ω(y)dσ (y) = 0 Sn−1 (1.2)with dσ (·) being the normalized Lebesgue measure on the unit sphere
Let Bn(0, 1) be the unit ball centered at the origin in Rn
It is known that if Φ is of finite type at 0 and Ω ∈ Ꮿ1(Sn−1)
Summary
We study the mapping properties of singular integral operators defined by mappings of finite type. Regarding the condition Ω ∈ Bq0,1(Sn−1 × Sm−1) in Theorem 1.3, we should remark here that in a recent paper [5], Al-Salman was able to obtain a similar result to that in [2]. It is worth pointing out that, as in the one-parameter setting, we can show that the Lp boundedness of the operators PΩ,Φ,Ψ and PΩ∗,Φ,Ψ may fail for any p if at least one of the mappings Φ and Ψ is not of finite type at 0. For 1 < q ≤ ∞, it is said that a measurable function b(x, y) on Sn−1 × Sm−1 is a q-block if it satisfies the following:. M (l,s)(f ) p ≤ B2 f p for all (l, s) ∈ {(1, 2), (2, 1), (1, 1)}, 1 < p ∞, and f ∈ Lp(Rn × Rm)
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