Fan, Dashan; Pan, Yibiao Singular integral operators with rough kernels supported by subvarieties. (English) Zbl 0899.42002 Am. J. Math. 119, No. 4, 799-839 (1997). 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) Cited in 8 ReviewsCited in 114 Documents MSC: 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B30 \(H^p\)-spaces Keywords:oscillatory integrals; maximal averages; singular integral operators; maximal operators × Cite Format Result Cite Review PDF Full Text: DOI Link