Volume 4,  Issue 3 (GI8), 2003

Article 50

CONVOLUTION INEQUALITIES AND APPLICATIONS

SABUROU SAITOH, VU KIM TUAN AND MASAHIRO YAMAMOTO

DEPARTMENT OF MATHEMATICS,
FACULTY OF ENGINEERING, 
GUNMA UNIVERSITY, 
KIRYU 376-8515, JAPAN
E-Mail: ssaitoh@math.sci.gunma-u.ac.jp

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE,
FACULTY OF SCIENCE,
KUWAIT UNIVERSITY, 
P.O. BOX 5969, 
SAFAT 13060 KUWAIT
E-Mail: vu@mcs.sci.kuniv.edu.kw

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO,
3-8-1 KOMABA MEGURO
TOKYO 153-8914, JAPAN
E-Mail: myama@ms.u-tokyo.ac.jp

Received 09 December, 2002; Accepted 15 April, 2003.
Communicated by: L.-E. Persson

ABSTRACT.    We introduce various convolution inequalities obtained recently and at the same time, we give new type of reverse convolution inequalities and their important applications to inverse source problems. We consider the inverse problem of determining $f(t)$, $0 < t < T$, in the heat source of the heat equation $\partial_t u(x,t) = \Delta u(x,t) + f(t)\varphi(x)$, $% x\in R^n$, $t > 0$ from the observation $u(x_0,t)$, $0 < t < T$, at a remote point $x_0$ away from the support of $\varphi$. Under an a priori assumption that $f$ changes the signs at most $N$-times, we give a conditional stability of Hölder type, as an example of applications.
Key words:
Convolution, Heat source, Weighted convolution inequalities, Young's inequality, Hölder's inequality, Reverse Hölder's inequality, Green's function, Stability in inverse problems, Volterra's equation, Conditional stability of Hölder type, Analytic semigroup, Interpolation inequality, Sobolev inequality.

2000 Mathematics Subject Classification:
Primary 44A35; Secondary 26D20.

Download this article (PDF):

Suitable for a printer:    

Suitable for a monitor:        

 To view these files we recommend you save them to your file system and then view by using the Adobe Acrobat Reader. 

That is, click on the icon using the 2nd mouse button and select "Save Target As..." (Microsoft Internet Explorer) or "Save Link As..." (Netscape Navigator).

See our PDF pages for more information.
 

Other papers in this issue

Report of the General Inequalities 8 Conference; September 15-21, 2002, Noszvaj, Hungary
Compiled by Zsolt Páles

Convolution Inequalities and Applications
Saburou Saitoh, Vu Kim Tuan and Masahiro Yamamoto

The Hardy-Landau-Littlewood Inequalities with Less Smoothness
Constantin P. Niculescu and Constantin Buse

Continuity Properties of Convex-type Set-Valued Maps
Kazimierz Nikodem

Carleman's Inequality - History, Proofs and Some New Generalizations
Maria Johansson, Lars-Erik Persson and Anna Wedestig

Andersson's Inequality and Best Possible Inequalities
A.M. Fink  

On Some Results Involving the Cebysev Functional and its Generalisations
P. Cerone

Separation and Disconjugacy
R.C. Brown

New Norm Type Inequalities for Linear Mappings
Saburou Saitoh

An Integral Approximation in Three Variables
A. Sofo

On Some Spectral Results Relating to the Relative Values of Means
C.E.M. Pearce

On Zeros of Reciprocal Polynomials of Odd Degree
Piroska Lakatos and László Losonczi

Some New Hardy Type Inequalities and their Limiting Inequalities
Anna Wedestig

Generalizations of the Triangle Inequality
Saburou Saitoh

A Survey on Cauchy-Bunyakovsky-Schwarz Type Discrete Inequalities
S.S. Dragomir
 

Other issues
 

© 2000 School of Communications and Informatics, Victoria University of Technology. All rights reserved.
JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science, located in Melbourne, Australia. All correspondence should be directed to the editorial office.

Copyright/Disclaimer