## Weighted Hardy and Opial-type inequalities.(English)Zbl 0756.26011

A characterization of weights $$u,v$$ is given for which the Hardy operator $$Hf(x)=\int^ x_ af(t)dt$$ is bounded from $$L^ p((a,b);u dx)$$ into $$L^ q((a,b);v dx)$$ with $$0<q<1<p<\infty$$ (Section 2). In Section 3 the author shows that Opial-type inequalities easily follow from inequalities of Hardy type.
Reviewer: B.Opic (Praha)

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals
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### References:

 [1] Beesack, P. R., On an integral inequality of Z. Opial, Trans. Amer. Math. Soc., 104, 470-475 (1962) · Zbl 0122.30102 [2] Beesack, P. R., Elementary proofs of some Opial-type integral inequalities, J. Analyse Math., 36, 1-14 (1979) · Zbl 0437.26006 [3] Boyd, D. W.; Wong, J. S.W, An extension of Opial’s inequality, J. Math. Anal. Appl., 19, 100-102 (1967) · Zbl 0173.05701 [4] Bradley, J. S., Hardy inequalities with mixed norms, Canad. Math. Bull., 21, 405-408 (1978) · Zbl 0402.26006 [5] Das, K. M.; Beesack, P. R., Extensions of Opial’s inequality, Pacific J. Math., 26, 215-232 (1968) · Zbl 0162.07901 [6] Halperin, I., Function spaces, Canad. J. Math., 5, 273-288 (1953) · Zbl 0052.11303 [7] Levinson, N., On an inequality of Opial and Beesack, (Proc. Amer. Math. Soc., 15 (1964)), 565-566 · Zbl 0134.27902 [8] Mallows, C. L., An even simpler proof of Opial’s inequality, (Proc. Amer. Math. Soc., 16 (1965)), 173 · Zbl 0152.05002 [9] Maz’ja, V. G., Sobolev Spaces (1985), Springer-Verlag: Springer-Verlag Berlin/Heidelberg [10] Muckenhoupt, B., Hardy’s inequality with weights, Studia Math., 44, 31-38 (1972) · Zbl 0236.26015 [11] Olech, C., A simple proof of a certain result of Z. Opial, Ann. Polon. Math., 8, 61-63 (1960) · Zbl 0089.27404 [12] Opial, Z., Sur une inégalité, Ann. Polon. Math., 8, 29-32 (1960) · Zbl 0089.27403 [13] Pachpatte, B. G., On Opial-type integral inequalities, J. Math. Anal. Appl., 120, 547-556 (1986) · Zbl 0608.26009 [14] Pederson, R. N., On an inequality of Opial, Beesack and Levinson, (Proc. Amer. Math. Soc., 16 (1965)), 174 · Zbl 0125.03102 [15] Sinnamon, G. J., Operators on Lebesgue Spaces with General Measures, (Doctoral Thesis (1987), McMaster University) [16] Yang, G. S., On a certain result of Z. Opial, (Proc. Japan Acad., 42 (1966)), 78-83 · Zbl 0151.05202
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