Abstract
In this paper, the Kantorovich operators \(K_n, n\in \mathbb {N}\) are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference \(K_n(f)-f\) for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.
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References
Bernstein, S.: Demonstration du théorème de Weierstrass fondé sur le calcul des probabilités. Commun. Kharkov Math. Soc. 13, 1–2 (1912)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, London (1988)
Blasco, O., Ruiz, A., Vega, L.: Non-interpolation in Morrey–Campanato and block spaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1), 31–40 (1999)
Campiti, M., Metafune, G.: \( L^ p \)-convergence of Bernstein–Kantorovich-type operators. Ann. Polon. Math. 63(3), 273–280 (1996)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. 7(3–4), 273–279 (1987)
Chlodowsky, I.: Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M. S. Bernstein. Compos. Math. 4, 380–393 (1937)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303. Springer, Berlin (1993)
Ditzian, Z., Totik, V.: Moduli of Smoothness, Springer Series in Computational Mathematics, 9. Springer, New York (1987)
Hakim, D.I., Sawano, Y.: Interpolation property of generalized Morrey spaces. Rev. Mat. Complut. 29(2), 295–340 (2016)
Hakim, D.I., Sawano, Y.: Calderón’s First and second complex interpolations of closed subspaces of Morrey spaces. J. Fourier Anal. Appl. 23(5), 1195–1226 (2017)
Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing Co., New York (1986)
Lemarié-Rieusset, P.G.: Multipliers and Morrey spaces. Potential Anal. 38(3), 741–752 (2013)
Lemarié-Rieusset, P.G.: Erratum to: Multipliers and Morrey spaces. Potential Anal. 41, 1359–1362 (2014)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938)
Ruiz, A., Vega, L.: Corrigenda to Unique continuation for Schrödinger operators with potential in Morrey spaces and a remark on interpolation of Morrey spaces. Publ. Mat. 39, 405–411 (1995)
Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22, 663–683 (2015)
Totik, V.: Approximation by Bernstein polynomials. Am. J. Math. 116(4), 995–1018 (1994)
Triebel, H.: Hybrid function spaces, heat and Navier-Stokes equations. In: EMS tracts in Mathematics 24, European Mathematical Society (2014)
Yang, D., Yuan, W.: Dual properties of Triebel–Lizorkin-type spaces and their applications. Z. Anal. Anwend. 30, 29–58 (2011)
Yang, D., Yuan, W., Zhuo, C.: Complex interpolation on Besov-type and Triebel–Lizorkin-type spaces. Anal. Appl. (Singap.) 11(5), 1350021 (2013)
Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey–Campanato and related smoothness spaces. Sci. China Math. 58(9), 1835–1908 (2015)
Acknowledgements
The research of Victor Burenkov and Yoshihiro Sawano is partially supported by the Ministry of Education and Science of the Russian Federation (the Agreement Number: 02.a03.21.0008). Yoshihiro Sawano is partially supported by 16K05209 JSPS.
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Burenkov, V., Ghorbanalizadeh, A. & Sawano, Y. Uniform boundedness of Kantorovich operators in Morrey spaces. Positivity 22, 1097–1107 (2018). https://doi.org/10.1007/s11117-018-0561-x
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DOI: https://doi.org/10.1007/s11117-018-0561-x