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Weighted inequalities of Hardy type. (English. Russian original) Zbl 0729.42007
Sib. Math. J. 30, No. 1, 8-16 (1989); translation from Sib. Mat. Zh. 30, No. 1(173), 13-22 (1989).
In Section 1 of this paper we present a new method for obtaining weighted Hardy inequalities for integration operators. The indicated method allows us to simplify and unify the known results of B. Muckenhoupt, V. G. Maz’ya, V. M. Kokilashvili and other authors on integration operators. In Section 2 we obtain weighted estimates for the Steklov operators. A particular case of these estimates is Muckenhoupt’s theorem on the boundedness of the maximal Hardy-Littlewood operator in weighted spaces. As an application of the statements regarding the Steklov operators, we give an a priori estimate in weighted norms (necessary and sufficient conditions) of the solutions of the Cauchy problem for the wave equation.

MSC:
42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B45 A priori estimates in context of PDEs
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