Inequalities of Rellich type
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 65-76.

Voir la notice de l'article provenant de la source Math-Net.Ru

Hardy inequalities have been important topics of research for a century, and in the past twenty years or so, there has been a deluge of important papers on various versions, including discrete and fractional forms and extensions to Rellich and higher-order inequalities. This paper is a brief survey of known fractional and nonfractional forms of the Rellich inequality together with some new results and remarks concerning the Rellich inequality, including the case of the $p$-Laplacian for $p \in (2,\infty)$.
@article{FAA_2021_55_2_a5,
     author = {W. D. Evans and D. E. Edmunds and R. T. Lewis},
     title = {Inequalities of {Rellich} type},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {65--76},
     publisher = {mathdoc},
     volume = {55},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a5/}
}
TY  - JOUR
AU  - W. D. Evans
AU  - D. E. Edmunds
AU  - R. T. Lewis
TI  - Inequalities of Rellich type
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2021
SP  - 65
EP  - 76
VL  - 55
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a5/
LA  - ru
ID  - FAA_2021_55_2_a5
ER  - 
%0 Journal Article
%A W. D. Evans
%A D. E. Edmunds
%A R. T. Lewis
%T Inequalities of Rellich type
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2021
%P 65-76
%V 55
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a5/
%G ru
%F FAA_2021_55_2_a5
W. D. Evans; D. E. Edmunds; R. T. Lewis. Inequalities of Rellich type. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 65-76. http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a5/

[1] A. Balinsky, W. D. Evans, R. T. Lewis, The Analysis and Geometry of Hardy's Inequality, Springer, Cham, 2015 | MR | Zbl

[2] R. Baňuelos, G. Wang, “Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations”, Duke Math. J., 80:3 (1995), 575–600 | MR

[3] E. B. Davies, “Some norm bounds and quadratic form inequalities for Schrödinger operators. II”, J. Operator Theory, 12:1 (1984), 177–196 | MR | Zbl

[4] E. B. Davies, A. M. Hinz, “Explicit constants for Rellich inequalities in $L_p(\Omega)$”, Math. Z., 227:3 (1998), 511–523 | DOI | MR | Zbl

[5] D. E. Edmunds, W. D. Evans, “The Rellich inequality”, Rev. Mat. Complut., 29:3 (2016), 511–530 | DOI | MR | Zbl

[6] D. E. Edmunds, W. D. Evans, Elliptic Differential Operators and Spectral Analysis, Springer Monographs in Mathematics, Springer, Cham, 2018 | DOI | MR | Zbl

[7] D. E. Edmunds, W. D. Evans, R. T. Lewis, “Fractional inequalities of Rellich type” (to appear)

[8] W. D. Evans, R. T. Lewis, “Hardy and Rellich inequalities with remainders”, J. Math. Inequal., 1:4 (2007), 473–490 | DOI | MR | Zbl

[9] R. L. Frank, E. Lieb, R. Seiringer, “Hardy–Lieb-Thirring inequalities for fractional Schrödinger operators”, J. Amer. Math. Soc., 21:4 (2008), 925–950 | DOI | MR | Zbl

[10] I. Herbst, “Spectral theory of the operator $(p^2 +m^2)^{1/2}-Ze^2\!/r$”, Comm. Math. Phys., 53 (1977), 285–294 | DOI | MR | Zbl

[11] T. Iwaniec, G. Martin, “Riesz transforms and related singular integrals”, J. Reine Angew. Math., 473 (1996), 25–57 | MR | Zbl

[12] M. Loss, C. Sloane, “Hardy inequalities for fractional integrals on general domains”, J. Funct. Anal., 259:6 (2010), 1369–1379 | DOI | MR | Zbl

[13] M. P. Owen, “The Hardy–Rellich inequality for polyharmonic operators”, Proc. Roy. Soc. Edinburgh, A, 129:4 (1999), 825–839 | DOI | MR | Zbl

[14] J. Tidblom, “A geometric version of Hardy's inequality in $ W_0^{1,p}(\Omega)$”, Proc. Amer. Math. Soc., 132:8 (2004), 2265–2271 | DOI | MR | Zbl