Weak and Strong Inequalities for Hardy Type Operators

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.