Weak and Strong Inequalities for Hardy Type Operators

Strong and weak inequalities for the Hardy type
integral operator involving variable limits and a
kernel are studied.
A characterization of the weight functions for which
the strong type inequality of the operator from a
weighted L^p to a weighted L^q holds is established
in the case of 1 q p infinity and that the
involved kernel satisfies the GHO condition of Bloom
and Kerman. The Nearly Block Diagonal Decomposition
technique and the concept of Normalizing Measures are
introduced for this purpose.
Weak type inequalities for various instances of the
operator are studied. These include the case that the
operator has only one variable limit, the case that
the operator has a trivial kernel or a kernel
depending on only one variable, and the case the
operator has a kernel satisfying some special growth
conditions such as the GHO condition. A newly
introduced decomposition techinque, good lambda
inequalities, and the monotonicity of the kernel, are
used to characterize weak type inequalities in
different situations.
Strong type inequalities for some other special cases
and in higher dimensional spaces are also studied.

Autorentext

Ph.D. in Mathematics, The University of Western Ontario, Canada, 2001. Associate Professor, Mathematical Sciences Department, University of South Carolina Aiken, USA, 2008 - present. Assistant Professor, Mathematical Sciences Department, University of South Carolina Aiken, USA, 2002 - 2008

Klappentext

Strong and weak inequalities for the Hardy typeintegral operator involving variable limits and akernel are studied. A characterization of the weight functions for whichthe strong type inequality of the operator from aweighted L^p to a weighted L^q holds is establishedin the case of 1 q p infinity and that theinvolved kernel satisfies the GHO condition of Bloomand Kerman. The Nearly Block Diagonal Decompositiontechnique and the concept of Normalizing Measures areintroduced for this purpose. Weak type inequalities for various instances of theoperator are studied. These include the case that theoperator has only one variable limit, the case thatthe operator has a trivial kernel or a kerneldepending on only one variable, and the case theoperator has a kernel satisfying some special growthconditions such as the GHO condition. A newlyintroduced decomposition techinque, good lambdainequalities, and the monotonicity of the kernel, areused to characterize weak type inequalities indifferent situations. Strong type inequalities for some other special casesand in higher dimensional spaces are also studied.