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On the \(H^p\)-\(L^p\)-boundedness of some integral operators. (English) Zbl 1230.42030

Summary: We obtain the \(H^p(\mathbb{R}^n)\to L^p(\mathbb{R}^n)\) boundedness, \(0<p\leq 1\), of integral operators of the form
\[ Tf(x)=\int |x-a_1y|^{-\alpha_1}\cdots |x-a_my|^{-\alpha_m}f(y)dy, \]
\(\alpha_1+\dots \alpha_m=n\) and \(a_i\{0\}\), \(a_i\neq a_j\) for \(i\neq j\), \(1\leq i, j\leq m\). We also show that these operators are not bounded on \(H^p(\mathbb{R})\).

MSC:

42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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References:

[1] Godoy T., Rev. Un. Mat. Argentina 38 pp 3– (1993)
[2] DOI: 10.1023/A:1026437621978 · Zbl 0937.47032 · doi:10.1023/A:1026437621978
[3] Mikhailov L. G., Dokl. Akad. Nauk SSSR 176 pp 263– (1967)
[4] DOI: 10.1007/BF01161645 · Zbl 0638.42019 · doi:10.1007/BF01161645
[5] DOI: 10.1007/s10587-005-0032-y · Zbl 1081.42018 · doi:10.1007/s10587-005-0032-y
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