Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a fixed exponent $p$, $1\le p<\infty$, and suitable nonnegative weight functions $v_j$, $j=1,\dots,m$, an optimal associated weight function $\omega _m$ is constructed for which the iterated convolution product satisfies

\begin{displaymath}\int _0^{\infty}\bigg|\bigg[\prod _{j=1}^m*F_j\bigg](x)\bigg|^p\, \dfrac{dx}{\omega _m(x)}\le \prod _{j=1}^m\int _0^{\infty}|F_j(t)|^p\, \dfrac{dt}{v_j(t)}\end{displaymath}

for all complex valued measurable functions $F_j$ with $\int _0^{\infty}|F_j(t)|^p\,dt/v_j(t)<\infty$. Here $[\prod _{j=1}^2*F_j](x)=[F_1*F_2](x)= \int _0^xF_1(t)F_2(x-t)\,dt$ and for each $m>2$, $\prod _{j=1}^m*F_j=\bigg[\prod _{j=1}^{m-1}*F_j \bigg]*F_m$. Analogous results are given when $R^+=(0,\infty)$ is replaced by $R^n$ and also when the convolution $F_1*F_2$ on $R^+$ is taken instead to be $\int _0^{\infty}F(t)G(x/t)\,dt/t$. The extremal functions are also discussed.


References [Enhancements On Off] (What's this?)

  • 1. J. Burbea, Inequalities for weighted $L^2$-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 $L_2$ functions on the half line, Proc. Amer. Math. Soc. 91 (1984), pp. 2643-2651.MR 85j:30010
  • 5. -, On the convolution of $L_2$ functions, Kodai Math. J. 9 (1986), pp. 2643-2651.MR 87e:42016
  • 6. -, Inequalities in the most simple Sobolev space and convolutions of $L_2$ 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26D15, 44A35, 42A85, 26D10

Retrieve articles in all journals with MSC (1991): 26D15, 44A35, 42A85, 26D10


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

American Mathematical Society