Pointwise Conditions for Membership of Functions in Weighted Sobolev Classes

V. I. Bogachev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 10-28.

Résumé

According to a known characterization, a function $f$ belongs to the Sobolev space $W^{p,1}(\mathbb{R}^n)$ of functions contained in $L^p(\mathbb{R}^n)$ along with their generalized first-order derivatives precisely when there is a function $g\in L^p(\mathbb{R}^n)$ such that $$ |f(x)-f(y)|\le |x-y|(g(x)+g(y)) $$ for almost all pairs $(x,y)$. An analogue of this estimate is also known for functions from the Gaussian Sobolev space $W^{p,1}(\gamma)$ in infinite dimension. In this paper the converse is proved; moreover, it is shown that the above inequality implies membership in appropriate Sobolev spaces for a large class of measures on finite-dimensional and infinite-dimensional spaces.

Détail

Keywords: Sobolev space, Gaussian measure, differentiable measure, quasi-invariant measure.

Bibliographie

[1] B. Bojarski, “Remarks on some geometric properties of Sobolev mapping”, Functional Analysis and Related topics (Sapporo, 1990), World Sci. Publ., River Edge, NJ, 1991, 65–76 | MR | Zbl

[2] P. Hajłasz, “Sobolev spaces on an arbitrary metric space”, Potential Anal., 5:4 (1996), 403–415 | MR | Zbl

[3] S. K. Vodopyanov, “Monotonnye funktsii i kvazikonformnye otobrazheniya na gruppakh Karno”, Sib. matem. zhurn., 37:6 (1996), 1269–1295 | MR | Zbl

[4] S. V. Pavlov, S. K. Vodopyanov, “Grand prostranstva Soboleva na metricheskikh prostranstvakh s meroi”, Sib. matem. zhurn., 2022 (to appear)

[5] J. Heinonen, “Nonsmooth calculus”, Bull. Amer. Math. Soc., 44:2 (2007), 163–232 | DOI | MR | Zbl

[6] L. Ambrosio, E. Bruè, D. Trevisan, “Lusin-type approximation of Sobolev by Lipschitz functions, in: Gaussian and $RCD(K,\infty)$ spaces”, Adv. Math., 339 (2018), 426–452 | DOI | MR | Zbl

[7] V. I. Bogachev, Gaussian Measures, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[8] V. I. Bogachev, Differentiable Measures and the Malliavin Calculus, Amer. Math. Soc., Providence, RI, 2010 | MR | Zbl

[9] V. I. Bogachev, “Sobolev classes on infinite-dimensional spaces”, Geometric Measure Theory and Real Analysis, Publications of the Scuola Normale Superiore, 17, ed. L. Ambrosio ed., Edizioni della Normale, Pisa, 2014, 1–56 | MR

[10] V. I. Bogachev, “Operatory i polugruppy Ornshteina–Ulenbeka”, UMN, 73:2 (2018), 3–74 | DOI | MR | Zbl

[11] A. V. Shaposhnikov, “A note on Lusin-type approximation of Sobolev functions on Gaussian spaces”, J. Funct. Anal., 280:6 (2021), 108917 | DOI | MR | Zbl

[12] V. I. Averbukh, O. G. Smolyanov, S. V. Fomin, “Obobschennye funktsii i differentsialnye uravneniya v lineinykh prostranstvakh. I. Differentsiruemye mery”, Tr. MMO, 24 (1971), 133–174 | Zbl

[13] V. I. Bogachev, O. G. Smolyanov, “Analiticheskie svoistva beskonechnomernykh veroyatnostnykh raspredelenii”, UMN, 45:3 (1990), 3–83 | MR | Zbl

[14] V. I. Bogachev, Measure Theory, v. 2, Springer, New Your, 2007 | MR | Zbl

[15] A. Lunardi, M. Miranda, D. Pallara, “BV functions on convex domains in Wiener spaces”, Potential Anal., 43:1 (2015), 23–48 | DOI | MR | Zbl

[16] V. I. Bogachev, A. Yu. Pilipenko, A. V. Shaposhnikov, “Sobolev functions on infinite-dimensional domains”, J. Math. Anal. Appl., 419:2 (2014), 1023–1044 | DOI | MR | Zbl

[17] D. Addona, G. Menegatti, M. Miranda, “BV functions on open domains: the Wiener case and a Fomin differentiable case”, Commun. Pure Appl. Anal., 19:5 (2020), 2679–2711 | DOI | MR | Zbl

[18] I. M. Gelfand, “Nekotorye voprosy analiza i differentsialnykh uravnenii”, UMN, 14:3(87) (1959), 3–19 | Zbl

[19] M. P. Kats, “Kvaziinvariantnost i differentsiruemost mer”, UMN, 33:3 (1978), 175 | MR | Zbl