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Title of Thesis

On The Boundedness And Measure Of Non-Compactness For Maximal And Potential Operators

Author(s)

MUHAMMAD ASIF

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2009
Subject
Mathematics
Number of Pages
94
Keywords (Extracted from title, table of contents and abstract of thesis)
Boundedness, Measure, Non-Compactness, Maximal, Potential, Operators, Homogeneous, Poisson, integrals, Banach

Abstract
The essential norm of maximal and potential operators defined on homogeneous groups is estimated in terms of weights. The same problem is discussed for partial sums of Fourier series, Poisson integrals and Sobolev embeddings. In some cases we conclude that there is no a weight pair (v;w) for which the given operator is compact from Lpw to Lqv.
It is proved that the measure of non-compactness of a bounded linear operator from a Banach space into a weighted Lebesgue space with variable parameter is equal to the distance between this operator and the class of finite rank operators. The essential norm of the Hilbert transform acting from Lp(x)w to Lp(x)v is estimated from below. As a corollary we have that there is no a weight pair (v;w) and a function p from the class of log-Holder continuity such that the Hilbert transform is compact from Lp(x)w to Lp(x)v.
Necessary and sufficient conditions on a weight pair (v;w) governing the boundedness of generalized fractional maximal functions and potentials on the half-space from Lpw(Rn) to Lq(x)v (Rn+1+ ) are derived. As a corollary, we have criteria for the trace inequality for these operators in variable exponent Lebesgue spaces.

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

 

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2

1

PRELIMINARIES

1.1 Homogeneous Groups

1.2 Measure of Non-Compactness
1.3 Lp(x) Spaces
1.4 Carleson-Hormander Type Inequality

8
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3 2 MEASURE OF NON-COMPACTNESS

2.1 Maximal Functions

2.2 Riesz Potentials
2.3 Truncated Potentials
2.4 Partial Sums of Fourier Series
2.5 Poisson Integrals
2.6 Sobolev Imbeddings
2.7 Hilbert Transforms in Lp(x) spaces

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4 3 POTENTIALS IN LP(X) SPACES

3.1 Generalized Maximal Functions on the Half-space

3.2 Riesz Potentials on the Half-space

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5 4 BIBLIOGRAPHY

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