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1997•arXiv (Cornell University)Open access

$L^p$ bounds for singular integrals and maximal singular integrals with\n rough kernels

Loukas Grafakos, Atanas Stefanov

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Abstract

Convolution type Calder\\'on-Zygmund singular integral operators with rough\nkernels $\\pv \\Om(x)/|x|^n$ are studied. A condition on $\\Om$ implying that the\ncorresponding singular integrals and maximal singular integrals map $L^p \\to\nL^p$ for $1<p<\\nf$ is obtained. This condition is shown to be different from\nthe condition $\\Om\\in H^1(\\sn)$.\n

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Convolution type Calder\\'on-Zygmund singular integral operators with rough\nkernels $\\pv \\Om(x)/|x|^n$ are studied. A condition on $\\Om$ implying that the\ncorresponding singular integrals and maximal singular integrals map $L^p \\to\nL^p$ for $1<p<\\nf$ is obtained. This condition is shown to be different from\nthe condition $\\Om\\in H^1(\\sn)$.\n

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Available abstract

Convolution type Calder\\'on-Zygmund singular integral operators with rough\nkernels $\\pv \\Om(x)/|x|^n$ are studied. A condition on $\\Om$ implying that the\ncorresponding singular integrals and maximal singular integrals map $L^p \\to\nL^p$ for $1<p<\\nf$ is obtained. This condition is shown to be different from\nthe condition $\\Om\\in H^1(\\sn)$.\n

Key concepts: Singular integral, Mathematics, Convolution (computer science), Mathematical analysis, Singular integral operators, Singular solution, Type (biology), Pure mathematics

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