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A limited-range Calderón-Zygmund theorem. (English) Zbl 1440.42054

Summary: For a limited range of indices \(p\), we obtain \(L^p(\mathbb{R}^n)\) boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical Hörmander smoothness estimate. These operators are assumed to be bounded (or weakly bounded) on \(L^s(\mathbb{R}^n)\) for some index \(s\). Our estimates are obtained via interpolation from the appropriate weak-type estimates. We provide two proofs of this result. One proof is based on the Calderón-Zygmund decomposition, while the other uses ideas of F. Nazarov et al. [Int. Math. Res. Not. 1998, No. 9, 463–487 (1998; Zbl 0918.42009)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0918.42009
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References:

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