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## Studia Mathematica

1997 | 126 | 1 | 27-50
Tytuł artykułu

### First and second order Opial inequalities

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let $T_γ f(x) = ʃ_0^x k(x,y)^γ f(y)dy$, 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 $ʃ_0^∞ (∏_{i=1}^n |T_{γ_i}f(x)|^{q_i}|) |f(x)|^{q_0} w(x)dx ≤ C(ʃ_0^∞ |f(x)|^p v(x)dx)^{(q_0+…+q_n)/p}$. 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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
27-50
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-05-06
poprawiono
1997-03-24
Twórcy
autor
• Department of Mathematics, Siena College, Loudon Road, Loudonville, New York 12211, U.S.A.
Bibliografia
• [1] M. Artola, untitled and unpublished manuscript.
• [2] P. R. Beesack, Elementary proofs of some Opial-type inequalities, J. Anal. Math. 36 (1979), 1-14.
• [3] P. R. Beesack and K. M. Das, Extensions of Opial's inequality, Pacific J. Math. 26 (1968), 215-232.
• [4] S. Bloom and R. A. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141.
• [5] D. W. Boyd, Inequalities for positive integral operators, Pacific J. Math. 38 (1971), 9-24.
• [6] D. W. Boyd, Best constants in a class of integral inequalities, ibid. 30 (1969), 367-383.
• [7] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408.
• [8] W. S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), 136-142.
• [9] J. D. Li, Opial-type inequalities involving several higher order derivatives, J. Math. Anal. Appl. 167 (1992), 98-110.
• [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.
• [11] V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, 1985.
• [12] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 34 (1972), 31-38.
• [13] Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32.
• [14] B. G. Pachpatte, On Opial type inequalities involving higher order derivatives, J. Math. Anal. Appl. 190 (1995), 763-773.
• [15] E. T. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), 329-337.
• [16] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11.
• [17] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545.
• [18] G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445.
• [19] V. D. Stepanov, Two-weighted estimates for Riemann-Liouville integrals, Math. USSR-Izv. 36 (1991), 669-681.
• [20] V. D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, Report/Institute for Applied Mathematics, Khabarovsk, 1992.
• [21] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185.
• [22] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 21 (1969), 622-631.
Typ dokumentu
Bibliografia
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