

Title of Thesis
On Trace Inequalities And Compactness For
PotentialType 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,
PotentialType, Operators, boundedness, compactness, variable,
exponent, integral, equations 
Abstract Necessary and
sufficient conditions on a weight function v guaranteeing the
boundedness/compactness of the RiemannLiouville transform with
variable parameter R (x) from Lp(x) to Lq(x) v are found. The
measure of noncompactness of this operator 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 defined on cones of nilpotent groups are established.
Compactness criteria for these operators in classical weighted
Lebesgue spaces are also obtained. Twosided estimate for Schattenvon
Neumann ideal norms of weighted higher order kernel operators are
established. Asymptotic formulas for singular numbers for some
potentialtype operators are derived. Some of these results are
applied to the problem of the existence of nonnegative solution for
certain nonlinear integral equations.

