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Title of Thesis

On Trace Inequalities and Compactness for Potential Type Operators

Author(s)

Usman Ashraf

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2009
Subject
Mathematics
Number of Pages
100
Keywords (Extracted from title, table of contents and abstract of thesis)
Trace, Inequalities, Compactness, Potential, Type. Operators, weight, function, nilpotent, groups, Asymptotic, formulas

Abstract
Necessary and sufficient conditions on a weight function v guaranteeing the boundedness/compactness of the Riemann-Liouville transform with variable parameter R(x) from Lp(x) to Lq(x)v are found. The measure of non-compactness of this operation is also estimated from both sides in variable exponent spaces.
Necessary and sufficient conditions guaranteeing the trace inequality for positive kernel operators in classical Lebesgue spaces de ned on cones of nilpotent groups are established. Compactness criteria for these operators in classical weighted Lebesgue spaces are also obtained. Two-sided estimate for Schatten-von Neumann ideal norms of weighted higher order kernel operators are established. Asymptotic formulas for singular numbers for some potential-type operators are derived. Some of these results are applied to the problem of the existence of non-negative solution for certain nonlinear integral equation.

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402 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

v
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2

1

PRELIMINARIES

1.1 Lp(x) Spaces

1.2 Homogeneous Groups
1.3 Singular Numbers and Schattan-von Neumann Ideal
1.4 Hardy-type Inequalities

9
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3 2 ONE-SIDED POTENTIALS IN Lp(x) SPACES

2.1 Boundedness

2.2 Compactness
2.3 Measure of Non-compactness

21
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4 3 POTENTIALS ON CONES

3.1 Trace Inequality

3.2 Compactness

3.3 Two-weight Inequality

3.4 Schatten-von Neumann Ideal Estimates

3.5 Operators With Radial Kernels

3.6 Singular Numbers

3.7 Applications in Non-linear Integral Equations

41
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5 4 BIBLIOGRAPHY

83


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