# First and second order Opial inequalities

Studia Mathematica (1997)

• Volume: 126, Issue: 1, page 27-50
• ISSN: 0039-3223

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## Abstract

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Let ${T}_{\gamma }f\left(x\right)={ʃ}_{0}^{x}k{\left(x,y\right)}^{\gamma }f\left(y\right)dy$, 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 ${ʃ}_{0}^{\infty }\left({\prod }_{i=1}^{n}|{T}_{{\gamma }_{i}}{f\left(x\right)|}^{{q}_{i}}{|\right)|f\left(x\right)|}^{{q}_{0}}w\left(x\right)dx\le C\left({ʃ}_{0}^{\infty }{|f\left(x\right)|}^{p}{v\left(x\right)dx\right)}^{\left({q}_{0}+\dots +{q}_{n}\right)/p}$. 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.

## How to cite

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Bloom, Steven. "First and second order Opial inequalities." Studia Mathematica 126.1 (1997): 27-50. <http://eudml.org/doc/216442>.

@article{Bloom1997,
abstract = {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.},
author = {Bloom, Steven},
journal = {Studia Mathematica},
keywords = {weighted norm inequality; second-order reduced Opial inequalities},
language = {eng},
number = {1},
pages = {27-50},
title = {First and second order Opial inequalities},
url = {http://eudml.org/doc/216442},
volume = {126},
year = {1997},
}

TY - JOUR
AU - Bloom, Steven
TI - First and second order Opial inequalities
JO - Studia Mathematica
PY - 1997
VL - 126
IS - 1
SP - 27
EP - 50
AB - 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.
LA - eng
KW - weighted norm inequality; second-order reduced Opial inequalities
UR - http://eudml.org/doc/216442
ER -

## References

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1. [1] M. Artola, untitled and unpublished manuscript.
2. [2] P. R. Beesack, Elementary proofs of some Opial-type inequalities, J. Anal. Math. 36 (1979), 1-14. Zbl0437.26006
3. [3] P. R. Beesack and K. M. Das, Extensions of Opial's inequality, Pacific J. Math. 26 (1968), 215-232. Zbl0162.07901
4. [4] S. Bloom and R. A. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141. Zbl0753.42010
5. [5] D. W. Boyd, Inequalities for positive integral operators, Pacific J. Math. 38 (1971), 9-24. Zbl0206.41004
6. [6] D. W. Boyd, Best constants in a class of integral inequalities, ibid. 30 (1969), 367-383. Zbl0179.08603
7. [7] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
8. [8] W. S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), 136-142. Zbl0687.26004
9. [9] J. D. Li, Opial-type inequalities involving several higher order derivatives, J. Math. Anal. Appl. 167 (1992), 98-110. Zbl0821.26014
10. [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. Zbl0704.42018
11. [11] V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, 1985.
12. [12] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 34 (1972), 31-38. Zbl0236.26015
13. [13] Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32. Zbl0089.27403
14. [14] B. G. Pachpatte, On Opial type inequalities involving higher order derivatives, J. Math. Anal. Appl. 190 (1995), 763-773. Zbl0831.26010
15. [15] E. T. Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), 329-337. Zbl0538.47020
16. [16] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. Zbl0508.42023
17. [17] E. T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545. Zbl0665.42023
18. [18] G. J. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445. Zbl0756.26011
19. [19] V. D. Stepanov, Two-weighted estimates for Riemann-Liouville integrals, Math. USSR-Izv. 36 (1991), 669-681. Zbl0724.26011
20. [20] V. D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, Report/Institute for Applied Mathematics, Khabarovsk, 1992.
21. [21] G. Talenti, Osservazioni sopra una classe di disuguaglianze, Rend. Sem. Mat. Fis. Milano 39 (1969), 171-185. Zbl0218.26011
22. [22] G. Tomaselli, A class of inequalities, Boll. Un. Mat. Ital. (4) 21 (1969), 622-631. Zbl0188.12103

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