# Optimality of constants in power-weighted Birman–Hardy–Rellich-Type inequalities with logarithmic refinements

### Abstract

The principal aim of this paper is to establish the optimality (*i.e.*, sharpness) of the constants \(A(m, \alpha)\) and \(B(m, \alpha)\), \(m \in \mathbb N\), \(\alpha \in \mathbb R\), of the form \begin{align*} &A(m, \alpha) = 4^{-m} \prod_{j=1}^{m} (2j - 1 -\alpha)^2, \\ &B(m, \alpha) = 4^{-m} \sum_{k=1}^{m} \ \prod_{\substack{j = 1\\ j \ne k}}^{m} ( 2j - 1 - \alpha )^{2}, \end{align*} in the power-weighted Birman--Hardy--Rellich-type integral inequalities with logarithmic refinement terms recently proved in [41], namely, \begin{align*} &\int_0^{\rho} dx \, x^{\alpha} \big| f^{(m )}(x) \big|^{2} \geq A(m, \alpha) \int_0^{\rho} dx \, x^{\alpha - 2m} \big|f(x)\big|^{2} \\ &\quad+ B(m, \alpha) \sum_{k=1}^{N} \int_0^{\rho} dx \, x^{\alpha - 2m}\prod_{p=1}^{k} [\ln_{p}(\gamma/x)]^{-2} \big|f(x)\big|^{2}, \\ & \, f \in C_{0}^{\infty}((0, \rho)), \; m, {N} \in \mathbb N, \; \alpha \in \mathbb R, \; \rho, \gamma \in (0,\infty), \; \gamma \geq e_{N} \rho. \end{align*} Here the iterated logarithms are given by \[ \ln_{1}( \, \cdot \,) = \ln(\, \cdot \,), \quad \ln_{j+1}( \, \cdot \,) = \ln( \ln_{j}(\, \cdot \,)), \quad j \in \mathbb N, \] and the iterated exponentials are defined via \[e_{0} = 0, \quad e_{j+1} = e^{e_{j}}, \quad j \in \mathbb N_{0} = \mathbb N \cup \{0\}. \] Moreover, we prove the analogous sequence of inequalities on the exterior interval \((r,\infty)\) for \(f \in C_{0}^{\infty}((r,\infty))\), \(r \in (0,\infty)\), and once again prove optimality of the constants involved.

### References

Adimurthi, N. Chaudhuri and M. Ramaswamy, “An improved Hardy-Sobolev inequality and its application”, Proc. Amer. Math. Soc., vol. 130, no. 2, pp. 489–505, 2002.

Adimurthi and M. J. Esteban, “An improved Hardy-Sobolev inequality in W^(1,p) and its application to Schrödinger operators”, Nonlinear Differential Equations Appl., vol. 12, no. 2, pp. 243–263, 2005.

Adimurthi, S. Filippas and A. Tertikas, “On the best constant of Hardy–Sobolev inequalities”, Nonlinear Anal., vol. 70, no. 8, pp. 2826–2833, 2009.

Adimurthi, M. Grossi and S. Santra, “Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem”, J. Funct. Anal., vol. 240, no. 1, pp. 36–83, 2006.

Adimurthi and K. Sandeep, “Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator”, Proc. Roy. Soc. Edinburgh Sect. A, vol. 132, no. 5, pp. 1021–1043, 2002.

Adimurthi and S. Santra, “Generalized Hardy–Rellich inequalities in critical dimension and its applications”, Commun. Contemp. Math., vol. 11, no. 3, pp. 367–394, 2009.

Adimurthi and A. Sekar, “Role of the fundamental solution in Hardy-Sobolev-type inequalities”, Proc. Roy. Soc. Edinburgh Sect. A, vol. 136, no. 6, pp. 1111–1130, 2006.

W. Allegretto, “Nonoscillation theory of elliptic equations of order 2n”, Pacific J. Math., vol. 64, no. 1, pp. 1–16, 1976.

A. Alvino, R. Volpicelli and B. Volzone, “On Hardy inequalities with a remainder term”, Ric. Mat., vol. 59, no. 2, pp. 265–280, 2010.

H. Ando and T. Horiuchi, “Missing terms in the weighted Hardy–Sobolev inequalities and its application”, Kyoto J. Math., vol. 52, no. 4, pp. 759–796, 2012.

W. Arendt, G. R. Goldstein and J. A. Goldstein, “Outgrowths of Hardy’s inequality”, in Recent Advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Providence, RI: Amer. Math. Soc., 2006, pp. 51–68.

F. G. Avkhadiev, “The generalized Davies problem for polyharmonic operators”, Sib. Math. J., vol. 58, no. 6, pp. 932–942, 2017.

A. A. Balinsky and W. D. Evans, Spectral analysis of relativistic operators, London: Imperial College Press, 2011.

A. A. Balinsky, W. D. Evans and R. T. Lewis, The analysis and geometry of Hardy’s inequality, Universitext, Cham: Springer, 2015.

G. Barbatis, “Best constants for higher-order Rellich inequalities in L^(p)(Ω)”, Math Z., vol. 255, no. 4, pp. 877–896, 2007.

G. Barbatis, S. Filippas and A. Tertikas, “Series expansion for L^p Hardy inequalities”, Indiana Univ. Math. J., vol. 52, no. 1, pp. 171–190, 2003.

G. Barbatis, S. Filippas and A. Tertikas, “Sharp Hardy and Hardy–Sobolev inequalities with point singularities on the boundary”, J. Math. Pures Appl. (9), vol. 117, pp. 146–184, 2018.

D. M. Bennett, “An extension of Rellich’s inequality”, Proc. Amer. Math. Soc., vol. 106, no. 4, pp. 987–993, 1989.

M. Š. Birman, “On the spectrum of singular boundary-value problems”, (Russian), Mat. Sb. (N.S.), vol. 55 (97), no. 2, pp. 125–174, 1961. Engl. transl. in Amer. Math. Soc. Transl., Ser. 2, vol. 53, pp. 23–80, 1966.

H. Brezis and M. Marcus, “Hardy’s inequalities revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 25, no. 1-2, pp. 217–237, 1997.

P. Caldiroli and R. Musina, “Rellich inequalities with weights”, Calc. Var. Partial Differential Equations, vol. 45, no. 1-2, pp. 147–164, 2012.

R. S. Chisholm, W. N. Everitt and L. L. Littlejohn, “An integral operator inequality with applications”, J. Inequal. Appl., vol. 3, no. 3, pp. 245–266, 1999.

C. Cowan, “Optimal Hardy inequalities for general elliptic operators with improvements”, Commun. Pure Appl. Anal., vol. 9, no. 1, pp. 109–140, 2010.

E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge: Cambridge University Press, 1995.

E. B. Davies and A. M. Hinz, “Explicit constants for Rellich inequalities in L_p(Ω)”, Math. Z., vol. 227, no. 3, pp. 511–523, 1998.

A. Detalla, T. Horiuchi and H. Ando, “Missing terms in Hardy-Sobolev inequalities and its application”, Far East J. Math. Sci., vol. 14, no. 3, pp. 333–359, 2004.

A. Detalla, T. Horiuchi and H. Ando, “Missing terms in Hardy-Sobolev inequalities”, Proc. Japan Acad. Ser. A Math. Sci., vol. 80, no. 8, pp. 160–165, 2004.

A. Detalla, T. Horiuchi and H. Ando, “Sharp remainder terms of Hardy–Sobolev inequalities”, Math. J. Ibaraki Univ., vol. 37, pp. 39–52, 2005.

A. Detalla, T. Horiuchi and H. Ando, “Sharp remainder terms of the Rellich inequality and its application”, Bull. Malays. Math. Sci. Soc. (2), vol. 35, no. 2A, pp. 519–528, 2012.

D. K. Dimitrov, I. Gadjev, G. Nikolov and R. Uluchev, “Hardy’s inequalities in finite dimensional Hilbert spaces”, Proc. Amer. Math. Soc., vol. 149, no. 6, pp. 2515–2529, 2021.

Y. A. Dubinskiĭ, “Hardy inequalities with exceptional parameter values and applications”, Dokl. Math., vol. 80, no. 1, pp. 558–562, 2009.

Y. A. Dubinskiĭ, “A Hardy-type inequality and its applications”, Proc. Steklov Inst. Math., vol. 269, no. 1, pp. 106–126, 2010.

Y. A. Dubinskiĭ, “Bilateral scales of Hardy inequalities and their applications to some problems of mathematical physics”, J. Math. Sci., vol. 201, no. 6, pp. 751–795, 2014.

T. D. Nguyen, N. Lam-Hoang and A. T. Nguyen, “Hardy-Rellich identities with Bessel pairs”, Arch. Math. (Basel), vol. 113, no. 1, pp. 95–112, 2019.

T. D. Nguyen, N. Lam-Hoang and A. T. Nguyen, “Improved Hardy and Hardy-Rellich type inequalities with Bessel pairs via factorizations”, J. Spectr. Theory, vol. 10, no. 4, pp. 1277–1302, 2020.

S. Filippas and A. Tertikas, Corrigendum to: “Optimizing improved Hardy inequalities” [J. Funct. Anal., vol. 192, no. 1, pp. 186—233, 2002], J. Funct. Anal., vol. 255, no. 8, 2095, 2008. see also [73].

F. Gazzola, H.-C. Grunau and E. Mitidieri, “Hardy inequalities with optimal constants and remainder terms”, Trans. Amer. Math. Soc., vol. 356, no. 6, pp. 2149–2168, 2004.

F. Gesztesy, “On non-degenerate ground states for Schrödinger operators”, Rep. Math. Phys., vol. 20, no. 1, pp. 93–109, 1984.

F. Gesztesy and L. L. Littlejohn, “Factorizations and Hardy-Rellich-type inequalities”, in Non-linear partial differential equations, mathematical physics, and stochastic analysis. A Volume in Honor of Helge Holden’s 60th Birthday, EMS Ser. Congr. Rep., F. Gesztesy, H. Hanche- Olsen, E. Jakobsen, Y. Lyubarskii, N. Risebro and K. Seip (eds.), Zürich: Eur. Math. Soc., 2018, pp. 207–226.

F. Gesztesy, L. L. Littlejohn, I. Michael and M. M. H. Pang, “Radial and logarithmic refinements of Hardy’s inequality”, St. Petersburg Math. J., vol. 30, no. 3, pp. 429–436, 2019.

F. Gesztesy, L. L. Littlejohn, I. Michael and M. M. H. Pang, “A sequence of weighted Birman-Hardy-Rellich inequalities with logarithmic refinements”, Integral Eq. Operator Th., vol. 94, no. 2, paper no. 13, 2022 (to appear). doi: 10.1007/s00020-021-02682-0

F. Gesztesy, L. L. Littlejohn, I. Michael and R. Wellman, “On Birman’s sequence of Hardy–Rellich-type inequalities”, J. Differential Equations, vol. 264, no. 4, pp. 2761–2801, 2018.

F. Gesztesy and M. Ünal, “Perturbative oscillation criteria and Hardy-type inequalities”, Math. Nachr., vol. 189, pp. 121–144, 1998.

K. T. Gkikas and G. Psaradakis, “Optimal non-homogeneous improvements for the series expansion of Hardy’s inequality”, arXiv:1805.10935, Comm. Contemp. Math., to appear.

I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators., New York: Daniel Davey & Co., Inc., 1966.

N. Ghoussoub and A. Moradifam, “On the best possible remaining term in the Hardy inequality”, Proc. Natl. Acad. Sci. USA, vol. 105, no. 37, pp. 13746–13751, 2008.

N. Ghoussoub and A. Moradifam, “Bessel pairs and optimal Hardy and Hardy-Rellich inequalities”, Math. Ann., vol. 349, no. 1, pp. 1–57, 2011.

N. Ghoussoub and A. Moradifam, Functional inequalities: new perspectives and new applications, Mathematical Surveys and Monographs, 187, Providence, RI: Amer. Math. Soc., 2013.

G. Ruiz Goldstein, J. A. Goldstein, R. M. Mininni and S. Romanelli, “Scaling and variants of Hardy’s inequality”, Proc. Amer. Math. Soc., vol. 147, no. 3, pp. 1165–1172, 2019.

G. Grillo, “Hardy and Rellich-type inequalities for metrics defined by vector fields”, Potential Anal., vol. 18, no. 3, pp. 187–217, 2003.

G. H. Hardy, “Notes on some points in the integral calculus LX: An inequality between integrals”, Messenger Math., vol. 54, pp. 150–156, 1925.

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Reprint of the 1952 edition. Cambridge Mathematical Library, Cambridge: Cambridge University Press, 1988.

P. Hartman, “On the linear logarithmic-exponential differential equation of the second-order”, Amer. J. Math., vol. 70, pp. 764–779, 1948.

P. Hartman, Ordinary Differential Equations, Classics Appl. Math., 38, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2002.

I. W. Herbst, “Spectral theory of the operator (p^2 + m^2)^1/2 − Ze^2/r”, Comm. Math. Phys., vol. 53, no. 3, pp. 285–294, 1977.

A. M. Hinz, “Topics from spectral theory of differential operators”, in Spectral theory of Schrödinger operators, Contemp. Math., 340, Providence, RI: Amer. Math. Soc., 2004, pp. 1–50.

N. Ioku and M. Ishiwata, “A scale invariant form of a critical Hardy inequality”, Int. Math. Res. Not. IMRN, no. 18, pp. 8830–8846, 2015.

H. Kalf, U.-W. Schmincke, J. Walter and R. Wüst, “On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials”, in Spectral theory and differential equations, Lecture Notes in Math., vol. 448, Berlin: Springer, 1975, pp. 182–226.

V. F. Kovalenko, M. A. Perel’muter and Y. A. Semenov, “Schrödinger operators with L_w^1/2(ℝ^l)-potentials”, J. Math. Phys., vol. 22, no. 5, pp. 1033–1044, 1981.

A. Kufner, Weighted Sobolev spaces, New York: John Wiley & Sons, Inc., 1985.

A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality: About its history and some related results, Plzeň: Vydavatelský servis, 2007.

A. Kufner, L.-E. Persson and N. Samko, Weighted inequalities of Hardy type, 2nd ed., Hackensack, NJ: World Scientific Publishing Co., 2017.

A. Kufner and A. Wannebo, “Some remarks on the Hardy inequality for higher order derivatives”, General inequalities 6, Internat. Ser. Numer. Math., vol. 103, Basel: Birkhäuser, 1992, pp. 33–48.

E. Landau, “A note on a theorem concerning series of positive terms: extract from a letter of Prof. E. Landau to Prof. I. Schur”, J. London Math. Soc., vol. 1, no. 1, pp. 38–39, 1926.

S. Machihara, T. Ozawa and H. Wadade, “Hardy type inequalities on balls”, Tohoku Math. J. (2), vol. 65, no. 3, pp. 321–330, 2013.

S. Machihara, T. Ozawa and H. Wadade, “Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space”, J. Inequal. Appl., 2015:281, 13 pages, 2015.

S. Machihara, T. Ozawa and H. Wadade, “Remarks on the Hardy type inequalities with remainder terms in the framework of equalities”, Asymptotic analysis for nonlinear dispersive and wave equations, Adv. Stud. Pure Math., vol. 81, Tokyo: Math. Soc. Japan, 2019, pp. 247–258.

S. Machihara, T. Ozawa and H. Wadade, “Remarks on the Rellich inequality”, Math. Z., vol. 286, no. 3-4, pp. 1367–1373, 2017.

G. Metafune, M. Sobajima and C. Spina, “Weighted Calderón–Zygmund and Rellich inequalities in L^p”, Math. Ann., vol. 361, no. 1-2, pp. 313–366, 2015.

È. Mitidieri, “A simple approach to Hardy inequalities”, Math. Notes, vol. 67, no. 3-4, pp. 479–486, 2000.

A. Moradifam, “Optimal weighted Hardy-Rellich inequalities on H^2 ∩ H_0^1”, J. London Math. Soc. (2), vol. 85, no. 1, pp. 22–40, 2012.

B. Muckenhoupt, “Hardy’s inequality with weights”, Studia Math., vol. 44, pp. 31–38, 1972.

R. Musina, “A note on the paper “Optimizing improved Hardy inequalities” by S. Filippas and A. Tertikas”, J. Funct. Anal., vol. 256, no. 8, pp. 2741–2745, 2009.

R. Musina, “Weighted Sobolev spaces of radially symmetric functions”, Ann. Mat. Pura Appl. (4), vol. 193, no. 6, pp. 1629–1659, 2014.

R. Musina, “Optimal Rellich–Sobolev constants and their extremals”, Differential Integral Equations, vol. 27, no. 5-6, pp. 579–600, 2014.

Q. A. Ngô and V. H. Nguyen, “A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems”, J. Differential Equations, vol. 268, no. 10, pp. 5996–6032, 2020.

E. S. Noussair and N. Yoshida, “Nonoscillation criteria for elliptic equations of order 2m”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), vol. 59, no. 1-2, pp. 57–64, 1976.

N. Okazawa, H. Tamura and T. Yokota, “Square Laplacian perturbed by inverse fourth-power potential. I: self-adjointness (real case)”, Proc. Roy. Soc. Edinburgh Sect. A, vol. 141, no. 2, pp. 409–416, 2011.

B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Harlow: Longman Scientific & Technical, 1990.

L.-E. Persson and S. G. Samko, “A note on the best constants in some Hardy inequalities”, J. Math. Inequal., vol. 9, no. 2, pp. 437–447, 2015.

F. Rellich, Perturbation theory of eigenvalue problems, New York-London-Paris: Gordon and Breach Science Publishers, Inc., 1969.

M. Ruzhansky and D. Suragan, “Hardy and Rellich inequalities, identities and sharp remainders on homogeneous groups”, Adv. Math., vol. 317, pp. 799–822, 2017.

M. Ruzhansky and N. Yessirkegenov, “Factorizations and Hardy-Rellich inequalities on stratified groups”, J. Spectr. Theory, vol. 10, no. 4, pp. 1361–1411, 2020.

M. Sano, “Extremal functions of generalized critical Hardy inequalities”, J. Differential Equations, vol. 267, no. 4, pp. 2594–2615, 2019.

M. Sano and F. Takahashi, “Sublinear eigenvalue problems with singular weights related to the critical Hardy inequality”, Electron. J. Differential Equations, paper no. 212, 12 pages, 2016.

U.-W. Schmincke, “Essential selfadjointness of a Schrödinger operator with strongly singular potential”, Math. Z., vol. 124, pp. 47–50, 1972.

B. Simon, “Hardy and Rellich inequalities in nonintegral dimension”, J. Operator Theory, vol. 9, no. 1, pp. 143–146, 1983.

M. Solomyak, “A remark on the Hardy inequalities”, Integral Equations Operator Theory, vol. 19, no. 1, pp. 120–124, 1994.

F. Takahashi, “A simple proof of Hardy’s inequality in a limiting case”, Arch. Math. (Basel), vol. 104, no. 1, pp. 77–82, 2015.

A. Tertikas and N. B. Zographopoulos, “Best constants in the Hardy-Rellich inequalities and related improvements”, Adv. Math., vol. 209, no. 2, pp. 407–459, 2007.

D. Yafaev, “Sharp constants in the Hardy-Rellich inequalities”, J. Funct. Anal., vol. 168, no. 1, pp. 121–144, 1999.

*CUBO*, vol. 24, no. 1, pp. 115–165, Apr. 2022.

Copyright (c) 2022 F. Gesztesy et al.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.