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.

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