Strong type endpoint bounds for analytic families of fractional integrals

Grafakos, Loukas

doi:10.1090/S0002-9939-1993-1100652-3

Strong type endpoint bounds for analytic families of fractional integrals
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Proc. Amer. Math. Soc. 117 (1993), 653-663 Request permission

Abstract:

In ${\mathbb {R}^2}$ we consider an analytic family of fractional integrals, whose convolution kernel is obtained by taking some transverse derivatives of arclength measure on the parabola $(t,{t^2})$ multiplied by $|t{|^\gamma }$ and doing so in a homogeneous way. We determine the exact range of $p,\;q$ for which the analytic family maps ${L^p}$ to ${L^q}$. We also resolve a similar issue on the Heisenberg group.

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