Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 72, No. 1, pp. 87-110, 2022

Hardy and Rellich type inequalities with remainders

Ramil Nasibullin

Received July 29, 2020. Published online May 12, 2021.

Abstract: Hardy and Rellich type inequalities with an additional term are proved for compactly supported smooth functions on open subsets of the Euclidean space. We obtain one-dimensional Hardy type inequalities and their multidimensional analogues in convex domains with the finite inradius. We use Bessel functions and the Lamb constant. The statements proved are a generalization for the case of arbitrary $p\geq2$ of the corresponding inequality proved by F. G. Avkhadiev, K.-J. Wirths (2011) for $p = 2$. Also we establish Rellich type inequalities on arbitrary domains, regular sets, on domains with $\theta$-cone condition and on convex domains.

Keywords: Hardy inequality; Rellich type inequality; Bessel function; Lamb constant; distance function; Laplace operator

Classification MSC: 26D10, 26D15

DOI: 10.21136/CMJ.2021.0325-20

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] F. G. Avkhadiev: Hardy type inequalities in higher dimensions with explicit estimate of constants. Lobachevskii J. Math. 21 (2006), 3-31. MR 2220697 | Zbl 1120.26008
[2] F. G. Avkhadiev: Hardy-type inequalities on planar and spatial open sets. Proc. Steklov Inst. Math. 255 (2006), 2-12. DOI 10.1134/S008154380604002X | MR 2301606 | Zbl 1351.42024
[3] F. G. Avkhadiev: A geometric description of domains whose Hardy constant is equal to 1/4. Izv. Math. 78 (2014), 855-876. DOI 10.1070/IM2014v078n05ABEH002710 | MR 3308642 | Zbl 1315.26012
[4] F. G. Avkhadiev: Integral inequalities in domains of hyperbolic type and their applications. Sb. Math. 206 (2015), 1657-1681. DOI 10.1070/SM2015v206n12ABEH004508 | MR 3438572 | Zbl 1359.30004
[5] F. G. Avkhadiev: Hardy-Rellich inequalities in domains of the Euclidean space. J. Math. Anal. Appl. 442 (2016), 469-484. DOI 10.1016/j.jmaa.2016.05.004 | MR 3504010 | Zbl 1342.26046
[6] F. G. Avkhadiev: Rellich inequalities for polyharmonic operators in plane domains. Sb. Math. 209 (2018), 292-319. DOI 10.1070/SM8739 | MR 3769213 | Zbl 1395.35003
[7] F. G. Avkhadiev: Hardy-Rellich integral inequalities in domains satisfying the exterior sphere condition. St. Petersbg. Math. J. 30 (2019), 161-179. DOI 10.1090/spmj/1536 | MR 3790730 | Zbl 1408.26017
[8] F. G. Avkhadiev, R. G. Nasibullin: Hardy-type inequalities in arbitrary domains with finite inner radius. Sib. Math. J. 55 (2014), 191-200. DOI 10.1134/S0037446614020013 | MR 3237329 | Zbl 1315.26016
[9] F. G. Avkhadiev, I. K. Shafigullin: Sharp estimates of Hardy constants for domains with special boundary properties. Russ. Math. 58 (2014), 58-61. DOI 10.3103/S1066369X14020091 | MR 3254462 | Zbl 1317.46020
[10] F. G. Avkhadiev, K.-J. Wirths: Unified Poincaré and Hardy inequalities with sharp constants for convex domains. ZAMM, Z. Angew. Math. Mech. 87 (2007), 632-642. DOI 10.1002/zamm.200710342 | MR 2354734 | Zbl 1145.26005
[11] F. G. Avkhadiev, K.-J. Wirths: Sharp Hardy-type inequalities with Lamb's constants. Bull. Belg. Math. Soc.-Simon Stevin 18 (2011), 723-736. DOI 10.36045/bbms/1320763133 | MR 2907615 | Zbl 1237.26014
[12] A. A. Balinsky, W. D. Evans, R. T. Lewis: The Analysis and Geometry of Hardy's Inequality. Universitext. Springer, Cham (2015). DOI 10.1007/978-3-319-22870-9 | MR 3408787 | Zbl 1332.26005
[13] G. Barbatis: Improved Rellich inequalities for the polyharmonic operator. Indiana Univ. Math. J. 55 (2006), 1401-1422. DOI 10.1512/iumj.2006.55.2752 | MR 2269418 | Zbl 1225.31006
[14] H. Brezis, M. Marcus: Hardy's inequality revisited. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25 (1997), 217-237. MR 1655516 | Zbl 1011.46027
[15] E. B. Davies: Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics 42. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511623721 | MR 1349825 | Zbl 0893.47004
[16] E. B. Davies: The Hardy constant. Q. J. Math., Oxf. II. Ser. 46 (1995), 417-431. DOI 10.1093/qmath/46.4.417 | MR 1366614 | Zbl 0857.26005
[17] W. D. Evans, R. T. Lewis: Hardy and Rellich inequalities with remainders. J. Math. Inequal. 1 (2007), 473-490. DOI 10.7153/jmi-01-40 | MR 2408402 | Zbl 1220.47024
[18] S. Filippas, V. Maz'ya, A. Tertikas: On a question of Brezis and Marcus. Calc. Var. Partial Differ. Equ. 25 (2006), 491-501. DOI 10.1007/s00526-005-0353-6 | MR 2214621 | Zbl 1121.26014
[19] G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities. Cambridge University Press, Cambridge (1952). MR 0944909 | Zbl 0047.05302
[20] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev: A geometrical version of Hardy's inequality. J. Funct. Anal. 189 (2002), 539-548. DOI 10.1006/jfan.2001.3859 | MR 1892180 | Zbl 1012.26011
[21] R. V. Makarov, R. G. Nasibullin: Hardy type inequalities and parametric Lamb equation. Indag. Math., New Ser. 31 (2020), 632-649. DOI 10.1016/j.indag.2020.06.004 | MR 4126759 | Zbl 1452.26017
[22] M. Marcus, V. J. Mizel, Y. Pinchover: On the best constant for Hardy's inequality in $\mathbb R^n$. Trans. Am. Math. Soc. 350 (1998), 3237-3255. DOI 10.1090/S0002-9947-98-02122-9 | MR 1458330 | Zbl 0917.26016
[23] T. Matskewich, P. E. Sobolevskii: The best possible constant in a generalized Hardy's inequality for convex domains in $\mathbb R^n$. Nonlinear Anal., Theory Methods Appl. 28 (1997), 1601-1610. DOI 10.1016/S0362-546X(96)00004-1 | MR 1431208 | Zbl 0876.46025
[24] V. G. Maz'ya: Sobolev spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985). DOI 10.1007/978-3-662-09922-3 | MR 0817985 | Zbl 0692.46023
[25] R. G. Nasibullin: Hardy type inequalities with weights dependent on the Bessel functions. Lobachevskii J. Math. 37 (2016), 274-283. DOI 10.1134/S1995080216030185 | MR 3512705 | Zbl 1350.26036
[26] R. G. Nasibullin: Sharp Hardy type inequalities with weights depending on Bessel function. Ufa Math. J. 9 (2017), 89-97. DOI 10.13108/2017-9-1-89 | MR 3646148
[27] R. G. Nasibullin: A geometrical version of Hardy-Rellich type inequalities. Math. Slovaca 69 (2019), 785-800. DOI 10.1515/ms-2017-0268 | MR 3985017 | Zbl 07289558
[28] R. G. Nasibullin: Brezis-Marcus type inequalities with Lamb constant. Sib. Èlektron. Mat. Izv. 16 (2019), 449-464. DOI 10.33048/semi.2019.16.027 | MR 3938782 | Zbl 1411.26021
[29] R. G. Nasibullin: Multidimensional Hardy type inequalities with remainders. Lobachevskii J. Math. 40 (2019), 1383-1396. DOI 10.1134/S1995080219090166 | MR 4021527 | Zbl 1439.26046
[30] R. G. Nasibullin, A. M. Tukhvatullina: Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains. Ufa Math. J. 5 (2013), 43-55. DOI 10.13108/2013-5-2-43 | MR 3430775
[31] M. P. Owen: The Hardy-Rellich inequality for polyharmonic operators. Proc. R. Soc. Edinb., Sect. A, Math. 129 (1999), 825-839. DOI 10.1017/S0308210500013160 | MR 1718522 | Zbl 0935.46032
[32] D. T. Shum: On a class of new inequalities. Trans. Am. Math. Soc. 204 (1975), 299-341. DOI 10.1090/S0002-9947-1975-0357715-3 | MR 0357715 | Zbl 0302.26010
[33] J. Tidblom: A geometrical version of Hardy's inequality for \accent23W$^{1,p}(\Omega)$. Proc. Am. Math. Soc. 132 (2004), 2265-2271. DOI 10.1090/S0002-9939-04-07526-4 | MR 2052402 | Zbl 1062.26010
[34] A. M. Tukhvatullina: Hardy type inequalities for a special family of non-convex domains. Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 153 (2011), 211-220. (In Russian.) MR 3151535 | Zbl 1259.26032

Affiliations: Ramil Nasibullin, Kazan Federal University, Kremlyovskaya Street 18, Kazan 420008, Russia, e-mail: NasibullinRamil@gmail.com

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