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On Birman’s sequence of Hardy-Rellich-type inequalities. (English) Zbl 1384.26043

In this nice paper, Birman’s sequence of inequalities has been reconsidered on some Hilbert space of functions defined on the closed unbounded interval of real numbers. The new proof of this sequence of inequalities is presented. Also it is shown that “Birman constants” in the inequalities considered are sharp and optimal. The Hardy and Rellich inequalities are special cases of the Birman’s one. The paper is useful both for experts as well as for all who want to find interesting information on mathematical analysis and functional analysis (function spaces, Hilbert spaces, linear operators, Cesaro operator, Mellin transform, spectral properties of Cesaro operator).

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
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[1] Albanese, A. A.; Bonet, J.; Ricker, W. J., On the continuous Cesàro operator in certain function spaces, Positivity, 19, 659-679 (2015) · Zbl 1354.47010
[2] Amann, H., Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, vol. 89 (1995), Birkhäuser: Birkhäuser Basel · Zbl 0819.35001
[3] Arendt, W.; Batty, C. K.; Hieber, M.; Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Transforms, Monographs in Mathematics, vol. 96 (2001), Birkhäuser: Birkhäuser Basel · Zbl 0978.34001
[4] Balinsky, A. A.; Evans, W. D.; Lewis, R. T., The Analysis and Geometry of Hardy’s Inequality, Universitext (2015), Springer · Zbl 1332.26005
[5] Barbatis, G., Best constants for higher-order Rellich inequalities, Math. Z., 255, 877-896 (2007) · Zbl 1130.46015
[6] Baumgärtel, H.; Wollenberg, M., Mathematical Scattering Theory, Operator Theory: Advances and Applications, vol. 9 (1983), Birkhäuser: Birkhäuser Boston · Zbl 0536.47007
[7] Berchio, E., On some higher order Hardy-Rellich type inequalities with boundary terms, Nonlinear Anal., 70, 2834-2841 (2009) · Zbl 1162.26311
[8] Birman, M. S., The spectrum of singular boundary problems, Mat. Sb.. Mat. Sb., Amer. Math. Soc. Transl. Ser. 2, 53, 97, 23-80 (1966), (Russian) Engl. transl. in: · Zbl 0174.42502
[9] Birman, M. S.; Solomjak, M. Z., Spectral Theory of Self-Adjoint Operators in Hilbert Space (1987), Reidel: Reidel Dordrecht
[10] Boyd, D. W., The spectrum of the Cesàro operator, Acta Sci. Math. (Szeged), 29, 31-34 (1968) · Zbl 0157.45501
[11] Brown, A.; Halmos, P. R.; Shields, A. L., Cesàro operators, Acta Sci. Math. (Szeged), 26, 125-137 (1965) · Zbl 0149.09902
[12] Burenkov, V. I., Sobolev Spaces on Domains (1998), Teubner: Teubner Stuttgart · Zbl 0893.46024
[13] Chisholm, R. S.; Everitt, W. N., On bounded integral operators in the space of integrable-square functions, Proc. Roy. Soc. Edinburgh Sect. A, 69, 199-204 (1970/71) · Zbl 0259.47044
[14] Chisholm, R. S.; Everitt, W. N.; Littlejohn, L. L., An integral operator inequality with applications, J. Inequal. Appl., 3, 245-266 (1999) · Zbl 0936.47029
[15] Davies, E. B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42 (1995), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0893.47004
[16] Davies, E. B.; Hinz, A. M., Explicit constants for Rellich inequalities in \(L_p(\Omega)\), Math. Z., 227, 511-523 (1998) · Zbl 0903.58049
[17] Diestel, J.; Uhl, J. J., Vector Measures, Mathematical Surveys, vol. 15 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0369.46039
[18] Edmunds, D. E.; Evans, W. D., Spectral Theory and Differential Operators (1989), Clarendon Press: Clarendon Press Oxford · Zbl 0664.47014
[19] Galaktionov, V. A., On extensions of higher-order Hardy’s inequalities, Diff. Integral Eq., 19, 327-344 (2006) · Zbl 1212.35209
[20] Gesztesy, F.; Littlejohn, L. L., Factorizations and Hardy-Rellich-type inequalities, (Gesztesy, F.; Hanche-Olsen, H.; Jakobsen, E.; Lyubarskii, Y.; Risebro, N.; Seip, K., Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. A Volume in Honor of Helge Holden’s 60th Birthday. Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. A Volume in Honor of Helge Holden’s 60th Birthday, EMS Congress Reports (2017)), in press
[21] Gesztesy, F.; Weikard, R.; Zinchenko, M., Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials, Oper. Matrices, 7, 241-283 (2013) · Zbl 1283.34023
[22] Ghoussoub, N.; Moradifam, A., Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349, 1-57 (2011) · Zbl 1216.35018
[23] Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (1966), Israel Program for Scientific Translations: Israel Program for Scientific Translations Jerusalem: Daniel Davey & Co., Inc.: Israel Program for Scientific Translations: Israel Program for Scientific Translations Jerusalem: Daniel Davey & Co., Inc. New York
[24] González, M.; León-Saavedra, F., Cyclic behavior of the Cesàro operator on \(L_2(0, \infty)\), Proc. Amer. Math. Soc., 137, 2049-2055 (2009) · Zbl 1166.47014
[25] Hardy, G. H., Notes on some points in the integral calculus, XLI. On the convergence of certain integrals and series, Messenger Math., 45, 163-166 (1915) · JFM 45.1289.03
[26] Hardy, G. H., Notes on some points in the integral calculus, LI. On Hilbert’s double-series theorem, and some connected theorems concerning the convergence of infinite series and integrals, Messenger Math., 48, 107-112 (1919) · JFM 46.0410.02
[27] Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6, 314-317 (1920) · JFM 47.0207.01
[28] Hardy, G. H., Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math., 54, 150-156 (1925) · JFM 51.0192.01
[29] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge, UK, reprinted · Zbl 0634.26008
[30] Hille, E.; Phillips, R. S., Functional Analysis and Semi-Groups, Colloquium Publications, vol. 31 (1985), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[31] Kufner, A.; Maligranda, L.; Persson, L.-E., The Hardy Inequality: About Its History and Some Related Results (2007), Vydavatelský Servis: Vydavatelský Servis Pilsen · Zbl 1213.42001
[32] Kufner, A.; Persson, L.-E., Weighted Inequalities of Hardy Type (2003), World Scientific: World Scientific Singapore · Zbl 1065.26018
[33] Kuroda, S. T., An Introduction to Scattering Theory, Aarhus University Lecture Notes Series, vol. 51 (1978) · Zbl 0407.47003
[34] Lacruz, M.; León-Saavedra, F.; Petrovic, S.; Zabeti, O., Extended eigenvalues for Cesàro operators, J. Math. Anal. Appl., 429, 623-657 (2015) · Zbl 1312.47041
[35] Leibowitz, G., The Cesàro operators and their generalizations: examples in infinite-dimensional linear analysis, Amer. Math. Monthly, 80, 654-661 (1973) · Zbl 0271.40014
[36] Lieb, E. H.; Loss, M., Analysis, Graduate Studies in Math., vol. 14 (2001), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0966.26002
[37] Meskhi, A., Solution of some weight problems for the Riemann-Liouville and Weyl operators, Georgian Math. J., 5, 565-574 (1998) · Zbl 0931.42008
[38] Mikusiński, J., The Bochner Integral (1978), Academic Press: Academic Press New York · Zbl 0369.28010
[39] Muckenhoupt, B., Hardy’s inequality with weights, Studia Math., 44, 31-38 (1972) · Zbl 0236.26015
[40] Newman, J.; Solomyak, M., Two-sided estimates on singular values for a class of integral operators on the semi-axis, Integral Equations Operator Theory, 20, 335-349 (1994) · Zbl 0817.47024
[41] Opic, B.; Kufner, A., Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219 (1990), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0698.26007
[42] Owen, M. P., The Hardy-Rellich inequality for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A, 129, 825-839 (1999) · Zbl 0935.46032
[43] Pettis, B. J., On integration in vector spaces, Trans. Amer. Math. Soc., 44, 277-304 (1938) · JFM 64.0371.02
[44] Prokhorov, D. V., On the boundedness and compactness of a class of integral operators, J. Lond. Math. Soc. (2), 61, 617-628 (2000) · Zbl 0956.47019
[45] Ruzhansky, M.; Suragan, D., Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math., 317, 799-822 (2017) · Zbl 1376.22009
[46] Schmincke, U.-W., Essential self-adjointness of a Schrödinger operator with strongly singular potential, Math. Z., 124, 47-50 (1972) · Zbl 0225.35037
[47] Talenti, G., Osservazioni sopra una classe di disuguaglianze, Rend. Semin. Mat. Fis. Milano, 39, 171-185 (1969) · Zbl 0218.26011
[48] Tertikas, A.; Zographopoulos, N. B., Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 209, 407-459 (2007) · Zbl 1160.26010
[49] Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (1986), Chelsea: Chelsea New York · JFM 63.0367.05
[50] Tomaselli, G., A class of inequalities, Boll. Unione Mat. Ital., 4, 622-631 (1969) · Zbl 0188.12103
[51] Yafaev, D., Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168, 121-144 (1999) · Zbl 0981.26016
[52] Yakubovich, S. B., Index Transforms (1996), World Scientific: World Scientific Singapore · Zbl 0845.44001
[53] Yosida, K., Functional Analysis (1980), Springer: Springer Berlin · Zbl 0217.16001
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