

Title of Thesis
On The Boundedness And Measure Of NonCompactness 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, NonCompactness, 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 L^{p}_{w} to L^{q}_{v}.
It is proved that the measure of noncompactness 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 L^{p(x)}_{w} to L^{p(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 logHolder continuity such that
the Hilbert transform is compact from L^{p(x)}_{w} to L^{p(x)}_{v}.
Necessary and sufficient conditions on a weight pair (v;w) governing
the boundedness of generalized fractional maximal functions and
potentials on the halfspace from L^{p}_{w}(R^{n}) to L^{q(x)}_{v} (R^{n+1}_{+} ) are
derived. As a corollary, we have criteria for the trace inequality
for these operators in variable exponent Lebesgue spaces.

