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A sharp interpolation between the Hölder and Gaussian Young inequalities

Paolo Da Pelo

Dipartimento di Matematica, Universita’ Degli Studi di Bari, Via E. Orabona, 4, 70125 Bari, Italia

E-mail Address: paolo.dapelo@uniba.it

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Alberto Lanconelli

Dipartimento di Matematica, Universita’ Degli Studi di Bari, Via E. Orabona, 4, 70125 Bari, Italia

E-mail Address: alberto.lanconelli@uniba.it

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Aurel I. Stan

Department of Mathematics, The Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion, OH 43302, USA

E-mail Address: stan.7@osu.edu

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https://doi.org/10.1142/S0219025716500016Cited by:1 (Source: Crossref)

Abstract

We prove a very general sharp inequality of the Hölder–Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong–Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson’s hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.

Communicated by H. H. Kuo

Keywords:

AMSC: 60H40, 60H10

References

1. W. Ayed and H.-H. Kuo, An extension of the Itô integral, Commun. Stoch. Anal. 2 (2008) 323–333. Google Scholar
2. W. Ayed and H.-H. Kuo, An extension of the Ito integral: Toward a general theory of stochastic integration, Theory Stoch. Process. 16 (2010) 17–28. Google Scholar
3. W. Beckner, Inequalities in fourier analysis, Ann. Math. 102 (1975) 159–182. Crossref, Web of Science, Google Scholar
4. W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989) 397–400. Web of Science, Google Scholar
5. D. R. Bell, The Malliavin Calculus (Dover, 2006). Google Scholar
6. J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp–Lieb inequalities: Finiteness, structure and extremals, arXiv:math/0505065. Google Scholar
7. H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976) 151–173. Crossref, Web of Science, Google Scholar
8. E. Carlen and D. Cordero-Erausquin, Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities, Geom. Funct. Anal. 19 (2009) 373–405. Crossref, Web of Science, Google Scholar
9. E. Carlen, E. Lieb and M. Loss, A sharp form of Young’s inequality on $S^{N}$ and related entropy inequalities, J. Geom. Anal. 14 (2004) 487–520. Crossref, Web of Science, Google Scholar
10. H. Chernoff, A note on an inequality involving the normal distribution, Ann. Probab. 9 (1981) 533–535. Crossref, Web of Science, Google Scholar
11. P. Da Pelo, A. Lanconelli and A. I. Stan, A Hölder–Young–Lieb inequality for norms of Gaussian Wick products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011) 375–407. Link, Web of Science, Google Scholar
12. P. Da Pelo and A. Lanconelli On a new probabilistic representation for the solution of the heat equation, Stochastics 84 (2012) 171–181. Crossref, Google Scholar
13. P. Da Pelo, A. Lanconelli and A. I. Stan, An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems, Stoch. Process. Appl. 123 (2013) 3183–3200. Crossref, Web of Science, Google Scholar
14. P. Da Pelo, A. Lanconelli and A. I. Stan, An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces, Stoch. Anal. Appl. 34 (2016) 47–64. Crossref, Web of Science, Google Scholar
15. L. Gross, Logarithmic sobolev inequality, Amer. J. Math. 97 (1975) 1061–1083. Crossref, Web of Science, Google Scholar
16. M. Hairer and E. Pardoux, A Wong–Zakai theorem for stochastic PDEs, arXiv:1409.3138. Google Scholar
17. Y. Hu and B. Øksendal, Wick approximation of quasilinear stochastic differential equations, Stochastic Analysis and Related Topics V, Progr. Probab., Vol. 38 (Birkhäuser, 1996), pp. 203–231. Crossref, Google Scholar
18. S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Math., Vol. 129 (Cambridge Univ. Press, 1997). Crossref, Google Scholar
19. H.-H. Kuo, White Noise Distribution Theory, Probability and Stochastic Series (CRC Press, 1996). Google Scholar
20. H.-H. Kuo, K. Saitô and A. I. Stan, A Hausdorff–Young inequality for white noise analysis, in Quantum Information IV, eds. T. Hida and K. Saitô (World Scientific, 2002), pp. 115–126, Google Scholar
21. A. Lanconelli and A. I. Stan, Hölder type inequalities for norms of Wick products, J. Appl. Math. Stoch. Anal. 2008 (2008), Article ID 254897, doi: 10.1155/2008/254897. Crossref, Google Scholar
22. A. Lanconelli and A. I. Stan, Some inequalities for norms of Gaussian Wick products, Stoch. An. Appl. 28 (2010) 523–539. Crossref, Web of Science, Google Scholar
23. E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990) 179–208. Crossref, Web of Science, Google Scholar
24. E. H. Lieb and M. Loss, Analysis 2nd ed., Graduate Studies in Mathematics, Vol. 14 (Amer. Math. Soc., 2001). Crossref, Google Scholar
25. J. Nash, Continuity of solutions of partial and elliptic equations, Amer. J. Math. 80 (1958) 931–954. Crossref, Web of Science, Google Scholar
26. E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973) 211–227. Crossref, Google Scholar
27. N. Obata, White Noise Calculus on Fock Space, Lecture Notes in Math., Vol. 1577 (Springer-Verlag, 1994). Crossref, Google Scholar
28. E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations. Intern. J. Engr. Sci. 3 (1965) 213–229. Crossref, Google Scholar