Weighted inequalities for iterated convolutions

Author:
Kenneth F. Andersen

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2643-2651

MSC (1991):
Primary 26D15, 44A35, 42A85; Secondary 26D10

DOI:
https://doi.org/10.1090/S0002-9939-99-05271-5

Published electronically:
May 4, 1999

MathSciNet review:
1657742

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a fixed exponent , , and suitable nonnegative weight functions , , an optimal associated weight function is constructed for which the iterated convolution product satisfies

for all complex valued measurable functions with . Here and for each , . Analogous results are given when is replaced by and also when the convolution on is taken instead to be . The extremal functions are also discussed.

**1.**J. Burbea,*Inequalities for weighted -functions on the half-line*, Arch. Math.**47**(1986), pp. 2643-2651.MR**88c:30015****2.**M. Cwikel and R. Kerman,*On a convolution inequality of Saitoh*, Proc. Amer. Math. Soc.**124**(1996), pp. 2643-2651.MR**96g:26027****3.**E. Lieb and M. Loss,*Analysis*, Graduate Studies in Mathematics, vol. 14, Amer. Math. Soc., Providence, RI, 1996.MR**98b:00004****4.**S. Saitoh,*A fundamental inequality in the convolution of functions on the half line*, Proc. Amer. Math. Soc.**91**(1984), pp. 2643-2651.MR**85j:30010****5.**-,*On the convolution of functions*, Kodai Math. J.**9**(1986), pp. 2643-2651.MR**87e:42016****6.**-,*Inequalities in the most simple Sobolev space and convolutions of functions with weights*, Proc. Amer. Math. Soc.**118**(1993), pp. 2643-2651.MR**93g:46029****7.**J. Tabor,*Cauchy and Jensen equations in a restricted domain almost everywhere*, Publ. Math. Debrecen**39**(1991), pp. 2643-2651.MR**93b:39007**

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Additional Information

**Kenneth F. Andersen**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Email:
kanderse@vega.math.ualberta.ca

DOI:
https://doi.org/10.1090/S0002-9939-99-05271-5

Keywords:
Convolution,
weights,
inequalities

Received by editor(s):
June 24, 1997

Published electronically:
May 4, 1999

Additional Notes:
This research was supported in part by NSERC research grant #OGP-8185.

Communicated by:
Frederick W. Gehring

Article copyright:
© Copyright 1999
American Mathematical Society