Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2003; 40(6): 999-1014

Printed November 1, 2003

Copyright © The Korean Mathematical Society.

Boundedness and inversion properties of certain convolution transforms

Semyon B. Yakubovich

University of Porto

Abstract

For a fixed function $h$ we deal with a class of convolution transforms $f \to f*h$, where $$ (f*h)(x) = {1\over 2x}\int_{{\bf R}_+^2}e^{-{1\over 2}\left(x{u^2+y^2\over uy}+{yu\over x}\right)}f(u)h(y)dudy, x \in {\bf R}_+$$ as integral operators $L_p({\bf R}_+; xdx) \to L_r({\bf R}_+; xdx), \ p, r \ge 1$. The Young type inequality is proved. Boundedness properties are investigated. Certain examples of these operators are considered and inversion formulas in $L_2({\bf R}_+; xdx)$ are obtained.

Keywords: convolution transform, Kontorovich-Lebedev transform, Young inequality

MSC numbers: 44A15, 44A35

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