Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-13T16:17:42.647Z Has data issue: false hasContentIssue false
  • English
  • Français

Compactness of Hardy-Type Operators over Star-Shaped Regions in ${{\mathbb{R}}^{N}}$

Published online by Cambridge University Press:  20 November 2018

Pankaj Jain ,
Pawan K. Jain  and
Babita Gupta
Pankaj Jain
Affiliation:
Department of Mathematics Deshbandhu College University of Delhi Kalkaji, New Delhi 110 019 India, e-mail: pankajkrjain@hotmail.com
Pawan K. Jain
Affiliation:
Department of Mathematics University of Delhi Delhi 110 007 India, e-mail: pkjain104@vsnl.com
Babita Gupta
Affiliation:
Department of Mathematics Shivaji College University of Delhi Raja Garden, Delhi 110 027 India, e-mail: babita74@hotmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in ${{\mathbb{R}}^{N}}$, and (ii) when the average is taken over all dilations of star-shaped regions in ${{\mathbb{R}}^{N}}$. These cases include, respectively, the average over annuli and the average over balls centered at origin.

Keywords

46E3526D10Hardy operatorHardy-Steklov operatorcompactnessboundednessstar-shaped regions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 4 , 01 December 2004 , pp. 540 - 552
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Ando, T., On the compactness of integral operators. Proc. Ned. Akad.Wetensch. 62 (1962), 235–239.Google Scholar
[2] Diestel, J., Sequences and Series in Banach Spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1986.Google Scholar
[3] Dunford, N. and Schwartz, J. T., Linear Operators I. General Theory. Interscience Publishers, Inc., New York-London, 1958.Google Scholar
[4] Heinig, H. P. and Sinnamon, G.,Mapping properties of integral averaging operators. Studia Math. 129 (1998), 157–177.Google Scholar
[5] Jain, P. K., Ahuja, O. P. and Ahmad, K., Functional Analysis. New Age International, New Delhi, 1995.Google Scholar
[6] Jain, P. and Gupta, B., Compactness of Hardy-Steklov operator. J. Math. Anal. Appl. 228 (2003), 680–691.Google Scholar
[7] Kufner, A. and Persson, L. E.,Weighted Inequalities of Hardy Type. World Scientific, 2003.Google Scholar
[8] Opic, B. and Kufner, A., Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.Google Scholar
[9] Sinnamon, G., One dimensional Hardy-type inequalities in many dimensions. Proc. Roy. Soc. Edinburgh A 128 (1998), 833–848.Google Scholar
[10] Sinnamon, G., Hardy-type inequalities for a new class of integral operators. In: Analysis of Divergence, Control and Management of Divergent Processes (eds.,W. O. Bray and C. V. Stanojevic), Birkhauser, Boston, 1999, 297–307.Google Scholar