×

Singular integral operators with rough convolution kernels. (English) Zbl 0858.42008

The author treats singular integral operators \(T\) with convolution kernel \(K\). He introduces the two quantities \[ V(\theta)= \sup_{R>0} \int^{2R}_R |K(r\theta)|r^{n-1} dr,\quad \theta\in S^{n-1}, \] and \[ \eta(\alpha)= \sup_{R\geq as} \int_{S^{n-1}} \int^{2R}_R |K((r- s)\theta)- K(r\theta)|r^{n-1} dr d\theta. \] Assuming \(T\) is bounded on \(L^2\), \(\int^\infty_2\eta(a)a^{-1}da<\infty\) and \(V\in L\log L(S^{n-1})\), he shows the weak (1,1) boundedness of \(T\) and hence \(L^2\) boundedness for \(1<p<\infty\). His theorem contains the known results in the case where \(K\) is homogeneous of degree \(-n\) by A. P. Calderón and A. Zygmund [Am. J. Math. 78, 289-309 (1956; Zbl 0072.11501)], M. Christ [Ann. Math., II. Ser. 128, No. 1, 19-42 (1988; Zbl 0666.47027)], M. Christ and J. L. Rubio de Francia [Invent. Math. 93, No. 1, 225-237 (1988; Zbl 0695.47052)] and S. Hofmann [Proc. Am. Math. Soc. 103, No. 1, 260-264 (1988; Zbl 0677.42013)].
Reviewer: K.Yabuta (Nara)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309, . · Zbl 0072.11501
[2] Michael Christ, Weak type (1,1) bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19 – 42. · Zbl 0666.47027 · doi:10.2307/1971461
[3] Michael Christ and José Luis Rubio de Francia, Weak type (1,1) bounds for rough operators. II, Invent. Math. 93 (1988), no. 1, 225 – 237. · Zbl 0695.47052 · doi:10.1007/BF01393693
[4] F. M. Christ and C. D. Sogge, The weak type \?\textonesuperior convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), no. 2, 421 – 453. · Zbl 0678.35096 · doi:10.1007/BF01394331
[5] Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9 – 36. · Zbl 0188.42601 · doi:10.1007/BF02394567
[6] Steve Hofmann, Weak (1,1) boundedness of singular integrals with nonsmooth kernel, Proc. Amer. Math. Soc. 103 (1988), no. 1, 260 – 264. · Zbl 0677.42013
[7] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.