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Limiting weak-type behavior for singular integral and maximal operators

Authors:	Prabhu Janakiraman

Affiliation:	Department of Mathematics, Purdue University, West Lafayette, Indiana 47906

Abstract:	The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator in $mathbb{R}^n$ : for $fin L^1(mathbb{R}^n)$ , , $displaystyle lim_{lambdarightarrow 0} lambdahspace{1mm}m{xinmathbb{R}^... ...a} = frac{1}{n} int_{S^{n-1}}vertOmega(x)vert dsigma(x)Vert fVert _1.$ For the maximal operator , the corresponding result is $displaystyle lim_{lambdarightarrow 0} lambdahspace{1mm}m{xinmathbb{R}^n: vert Mf(x)vert>lambda} = Vert fVert _1.$

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