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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.

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
Full Text: DOI
[1] G. Hardy, ?Notes on some points in the integral calculus (LXIV),? Messenger of Math.,57, 12-16 (1928).
[2] P. R. Beesack, Hardy’s inequality and its extensions,? Pac. J. Math.,11, No. 1, 39-62 (1961). · Zbl 0103.03503
[3] V. R. Portnov, ?Two imbedding theorems for the spacesL p,b 1 (?{\(\times\)}R +) and their applications,? Dokl. Akad. Nauk SSSR,155, No. 4, 761-764 (1964). · Zbl 0139.07103
[4] G. Talenti, ?Osservazioni sopra una classe di Disuguaglianze,? Rend. Sem. Mat. Fiz. Milano,39, 171-185 (1969). · Zbl 0218.26011
[5] G. Tomaselli, ?A class of inequalities,? Boll. Un. Mat. Ital.,2, No. 6, 622-631 (1969). · Zbl 0188.12103
[6] B. Muckenhoupt, ?Hardy’s inequality with weights,? Stud. Math.,44, No. 1, 31-38 (1972). · Zbl 0236.26015
[7] J. S. Bradley, ?Hardy inequalities with mixed norms,? Can. Math. Bull.,21, No. 4, 405-408 (1978). · Zbl 0402.26006
[8] V. G. Maz’ya (V. G. Mazja), Einbettungssätze für Sobolewsche Raume. Teil 1 [Translation from Russian], Teubner, Leipzig (1979).
[9] V. M. Kokilashvili, ?On Hardy’s inequalities in weighted spaces,? Soobshch. Akad. Nauk Gruz. SSR,96, No. 1, 37-40 (1979). · Zbl 0434.26007
[10] A. Kufner and H. Tribel, ?Generalizations of Hardy’s inequality,? in: Conf. Sem. Mat. Univ. Bari, No. 156 (1978).
[11] P. Gurka, ?Generalized Hardy’s inequality,? ?asopis P?st. Mat.,109, No. 2, 194-203 (1984). · Zbl 0537.26009
[12] V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).
[13] B. Muckenhoupt, ?Weighted norm inequalities for the Hardy maximal functions,? Trans. Am. Math. Soc.,165, 207-226 (1972). · Zbl 0236.26016
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