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Singular integral operators with rough kernels supported by subvarieties. (English) Zbl 0899.42002

This paper is concerned with singular integral and maximal operators associated to rough kernels supported in polynomial subvarieties of Euclidean spaces. Suppose that \(P:\mathbb{R}^n\to\mathbb{R}^d\) is polynomial, that \(\omega:\mathbb{R}^n\to\mathbb{C}\) is homogeneous of degree \(0\), has mean \(0\) on the unit sphere \(S^{n-1}\), and belongs to the Hardy space \(H^1(S^{n-1})\), and that \(b:\mathbb{R}^+\to\mathbb{C}\) is measurable. Write \(K(y)=b(| y|)\Omega(y)/| y|^n\) for \(y\) in \(\mathbb{R}^n\setminus\{0\}\), and define the operators \(T_\varepsilon\), \(T\) and \(T_*\) on functions \(f\) on \(\mathbb{R}^d\) by the formulae \[ T_\varepsilon f(x)=\int_{| y|>\varepsilon} f(x-P(y)) K(y) dy, \quad Tf=\lim_{\varepsilon\to 0} T_\varepsilon f \quad\text{and}\quad T_*f=\sup_{\varepsilon >0}| T_\varepsilon f|. \] It is shown that, under various different extra hypotheses on \(P\) or \(b\), that these operators are bounded on \(L^p(\mathbb{R}^d)\). For example, if \(\int_0^R | b(t)|^\gamma dt \leq CR\), then, for example, if \(\int_0^R | b(t)|^\gamma dt \leq CR\), then \(T\) is bounded on \(L^p(\mathbb{R}^d)\) if \(| {1/p}-{1/2}| \leq \min\{{1/2},1-{1/\gamma}\}\), while if \(b\in L^\infty(\mathbb{R}^t)^+\), then \(T_*\) is bounded on \(L^p(\mathbb{R}^d)\) when \(1<p<\infty\).
Reviewer: M.Cowling (Sydney)

MSC:

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