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Some new Hardy type inequalities with general kernels. (English) Zbl 1177.26038

Summary: We state and prove some new weighted Hardy type inequalities with an integral operator A k defined by

A k f(x):=1 K(x) Ω 2 k(x,y)f(y)dμ 2 (y),

where k:Ω 1 ×Ω 2 is a general nonnegative kernel, (Ω 1 ,μ 1 ) and (Ω 2 ,μ 2 ) are measure spaces and

K(x):= Ω 2 k(x,y)dμ 2 (y),xΩ 1 ·

In particular, the obtained results unify and generalize most of the results of this type (including the classical ones by Hardy, Hilbert and Godunova).

26D15Inequalities for sums, series and integrals of real functions