First and second order Opial inequalities

Bloom, Steven

Bloom, Steven

Studia Mathematica, Tome 122 (1997), p. 27-50 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $T_{γ} f (x) = ʃ_{0}^{x} k {(x, y)}^{γ} f (y) d y$ , where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form $ʃ_{0}^{\infty} (\prod_{i = 1}^{n} | T_{γ_{i}} {f (x) |}^{q_{i}} {|) | f (x) |}^{q_{0}} w (x) d x \leq C (ʃ_{0}^{\infty} {| f (x) |}^{p} {v (x) d x)}^{(q_{0} + \dots + q_{n}) / p}$ . Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent $q_{0} = 0$ . When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold.

Publié le : 1997-01-01

Zbl 0890.26008

EUDML-ID : urn:eudml:doc:216442

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     author = {Steven Bloom},
     title = {First and second order Opial inequalities},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {27-50},
     zbl = {0890.26008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv126i1p27bwm}
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Bloom, Steven. First and second order Opial inequalities. Studia Mathematica, Tome 122 (1997) pp. 27-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv126i1p27bwm/

Bibliographie

[00000] [1] M. Artola, untitled and unpublished manuscript.

[00001] [2] P. R. Beesack, Elementary proofs of some Opial-type inequalities, J. Anal. Math. 36 (1979), 1-14. | Zbl 0437.26006

[00002] [3] P. R. Beesack and K. M. Das, Extensions of Opial's inequality, Pacific J. Math. 26 (1968), 215-232. | Zbl 0162.07901

[00003] [4] S. Bloom and R. A. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141. | Zbl 0753.42010

[00004] [5] D. W. Boyd, Inequalities for positive integral operators, Pacific J. Math. 38 (1971), 9-24. | Zbl 0206.41004

[00005] [6] D. W. Boyd, Best constants in a class of integral inequalities, ibid. 30 (1969), 367-383. | Zbl 0179.08603

[00006] [7] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. | Zbl 0402.26006

[00007] [8] W. S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), 136-142. | Zbl 0687.26004

[00008] [9] J. D. Li, Opial-type inequalities involving several higher order derivatives, J. Math. Anal. Appl. 167 (1992), 98-110. | Zbl 0821.26014

[00009] [10] F. J. Martín-Reyes and E. T. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), 727-733. | Zbl 0704.42018

[00010] [11] V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, 1985.

[00011] [12] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 34 (1972), 31-38. | Zbl 0236.26015

[00012] [13] Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32. | Zbl 0089.27403

[00013] [14] B. G. Pachpatte, On Opial type inequalities involving higher order derivatives, J. Math. Anal. Appl. 190 (1995), 763-773. | Zbl 0831.26010

[00014] [15] E. T. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), 329-337. | Zbl 0538.47020

[00015] [16] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. | Zbl 0508.42023

[00016] [17] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545. | Zbl 0665.42023

[00017] [18] G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445. | Zbl 0756.26011

[00018] [19] V. D. Stepanov, Two-weighted estimates for Riemann-Liouville integrals, Math. USSR-Izv. 36 (1991), 669-681. | Zbl 0724.26011

[00019] [20] V. D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, Report/Institute for Applied Mathematics, Khabarovsk, 1992.

[00020] [21] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185. | Zbl 0218.26011

[00021] [22] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 21 (1969), 622-631. | Zbl 0188.12103