Conformally invariant inequalities in domains in Euclidean space
- Authors: Avkhadiev F.G.1
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Affiliations:
- Kazan (Volga Region) Federal University
- Issue: Vol 83, No 5 (2019)
- Pages: 3-26
- Section: Articles
- URL: https://bakhtiniada.ru/1607-0046/article/view/142285
- DOI: https://doi.org/10.4213/im8805
- ID: 142285
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Abstract
We study conformally invariant integral inequalities for real-valuedfunctions defined on domains $\Omega$ in $n$-dimensional Euclideanspace. The domains considered are of hyperbolic type, that is, they admita hyperbolic radius $R=R(x, \Omega)$ satisfying the Liouvillenon-linear differential equation and vanishing on the boundary of thedomain. We prove several inequalities which hold for all smooth compactlysupported functions $u$ defined on a given domain of hyperbolic type.Here are two of them:$$\int|\nabla u|^2R^{2-n} dx \geq n (n-2)\int|u|^2R^{-n} dx,$$$$\int|(\nabla u, \nabla R)|^p R^{p-s} dx\geq \frac{2^pn^p}{p^p}\int|u|^pR^{-s} dx,$$where $n\geq 2$, $1\leq p< \infty$ and $1+n/2 \leq s <\infty$.We also study the relations between Euclidean and hyperbolic characteristicsof domains.
About the authors
Farit Gabidinovich Avkhadiev
Kazan (Volga Region) Federal University
Email: avkhadiev47@mail.ru
Doctor of physico-mathematical sciences, Professor
References
- A. A. Balinsky, W. D. Evans, R. T. Lewis, The analysis and geometry of Hardy's inequality, Universitext, Springer, Cham, 2015, xv+263 pp.
- О. А. Ладыженская, Краевые задачи математической физики, Наука, М., 1973, 407 с.
- В. Г. Мазья, Пространства С. Л. Соболева, Изд-во Ленингр. ун-та, Л., 1985, 416 с.
- Г. Харди, Дж. И. Литтлвуд, Г. Полиа, Неравенства, ИЛ, М., 1948, 456 с.
- T. Matskewich, P. E. Sobolevskii, “The best possible constant in generalized Hardy's inequality for convex domain in ${R}^n$”, Nonlinear Anal., 28:9 (1997), 1601–1610
- M. Marcus, V. J. Mizel, Y. Pinchover, “On the best constant for Hardy's inequality in $mathbb{R}^n$”, Trans. Amer. Math. Soc., 350:8 (1998), 3237–3255
- F. G. Avkhadiev, “Hardy type inequalities in higher dimensions with explicit estimate of constants”, Lobachevskii J. Math., 21 (2006), 3–31
- Ф. Г. Авхадиев, И. К. Шафигуллин, “Точные оценки констант Харди для областей со специальными граничными свойствами”, Изв. вузов. Матем., 2014, № 2, 69–73
- G. Psaradakis, “$L^1$ Hardy inequalities with weights”, J. Geom. Anal., 23:4 (2013), 1703–1728
- J. L. Fernandez, “Domains with strong barrier”, Rev. Mat. Iberoamericana, 5:1-2 (1989), 47–65
- A. Ancona, “On strong barriers and an inequality of Hardy for domains in $mathbb{R}^n$”, J. London Math. Soc. (2), 34:2 (1986), 274–290
- Ch. Pommerenke, “Uniformly perfect sets and the Poincare metric”, Arch. Math. (Basel), 32:2 (1979), 192–199
- F. G. Avkhadiev, K.-J. Wirths, Schwarz–Pick type inequalities, Front. Math., Birkhäuser Verlag, Basel, 2009, viii+156 pp.
- L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Math., McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg, 1973, ix+157 pp.
- Л. Альфорс, Преобразования Мeбиуса в многомерном пространстве, Мир, М., 1986, 112 с.
- C. Bandle, M. Flucher, “Harmonic radius and concentration of energy; hyperbolic radius and Liouville's Equations $Delta U=e^U$ and $Delta U =U^{frac{n+2}{n-2}}$”, SIAM Rev., 38:2 (1996), 191–238
- C. Loewner, L. Nirenberg, “Partial differential equations invariant under conformal or projective transformations”, Contribution to analysis, A collection of papers dedicated to L. Bers, Academic Press, New York, 1974, 245–272
- R. Schoen, Shing-Tung Yau, “Conformally flat manifolds, Kleinian groups and scalar curvature”, Invent. Math., 92:1 (1988), 47–71
- J. L. Fernandez, J. M. Rodriguez, “The exponent of convergence of Riemann surfaces. Bass Riemann surfaces”, Ann. Acad. Sci. Fenn. Ser. A I Math., 15:1 (1990), 165–183
- Ф. Г. Авхадиев, “Интегральные неравенства в областях гиперболического типа и их применения”, Матем. сб., 206:12 (2015), 3–28
- Ф. Г. Авхадиев, “Конформно инвариантные неравенства”, Комплексный анализ, Итоги науки и техники. Сер. Соврем. мат. и ее прил. Темат. обз., 142, ВИНИТИ РАН, М., 2017, 28–41
- Ф. Г. Авхадиев, “Неравенства Реллиха для полигармонических операторов в областях на плоскости”, Матем. сб., 209:3 (2018), 4–33
- D. Sullivan, “Related aspects of positivity in Riemannian geometry”, J. Differential Geom., 25:3 (1987), 327–351
- F. Avkhadiev, A. Laptev, “Hardy inequalities for nonconvex domains”, Around the research of Vladimir Maz'ya, v. I, Int. Math. Ser. (N. Y.), 11, Springer, New York, 2010, 1–12
- Ф. Г. Авхадиев, “Геометрическое описание областей, для которых константа Харди равна $1/4$”, Изв. РАН. Сер. матем., 78:5 (2014), 3–26
- F. G. Avkhadiev, “Hardy–Rellich inequalities in domains of the Euclidean space”, J. Math. Anal. Appl., 442:2 (2016), 469–484
- Ф. Г. Авхадиев, “Интегральные неравенства Харди и Реллиха в областях, удовлетворяющих условию внешней сферы”, Алгебра и анализ, 30:2 (2018), 18–44
- M. P. Owen, “The Hardy–Rellich inequality for polyharmonic operators”, Proc. Roy. Soc. Edinburgh Sect. A, 129:4 (1999), 825–839
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