Extension of Lipschitz-type operators on Banach function spaces

Wasthenny V. Cavalcante, Pilar Rueda, Enrique A. Sánchez-Pérez

DOI: http://dx.doi.org/10.12775/TMNA.2020.026


We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced. We analyze Lipschitz-type inequalities in two fundamental cases. The first concerns almost everywhere pointwise inequalities, while the second considers dominations involving integrals. These Lipschitz-type inequalities provide a suitable frame to work with operators that take values on Banach function spaces. In the last part of the paper we use some interpolation procedures to extend our study to interpolated Banach function spaces.


Lipschitz operator; Banach function space; integration; measure; metric space

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C. Bennett and R.C. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, vol. 129, Academic Press, New York, 1988.

Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Colloquium Publications, vol. 48, American Mathematical Society, Providence, 1998.

O. Blasco, J.M. Calabuig and E.A. Sánchez Pérez, p-Variations of vector measures with respect to vector measures and integral representation of operators, Banach J. Math. Anal. 9 (2015), 273–285.

J.M. Calabuig, O. Delgado and E.A. Sánchez Pérez, Factorizing operators on Banach function spaces through spaces of multiplication operators, J. Math. Anal. Appl. 364 (2010), 88–103.

Ş. Cobzaş, R. Miculescu and A. Nicolae, Lipschitz Functions, Lecture Notes in Mathematics, vol. 2241, Springer, Berlin, 2019.

D.H. Fremlin, Kirszbraun’s theorem, https://www1.essex.ac.uk/maths/people/fremlin/ n11706.pdf

G. Godefroy, N. Kalton and G. Lancien, Subspaces of c0 (N) and Lipschitz isomorphisms, Geom. Func. Anal. 10 (2000), 798–820.

N.J. Kalton, Extension of linear operators and Lipschitz maps into C(K)-spaces, New York J. Math. 13 (2007), 317–381.

N.J. Kalton, Extending Lipschitz maps into C(K)-spaces, Israel J. Math. 162 (2007), 275–315.

N.J. Kalton, Lipschitz and uniform embeddings into `∞ , Fund. Math. 212 (2011), 53–69.

M.D. Kirszbraun, Über die zusammenziehenden und Lipschitzian Transformationen, Fund. Math. 22 (1934), 77–108.

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, A Series of Modern Surveys in Mathematics, no. 97, Springer, Berlin, 1979.

S. Okada, W.J. Ricker and E.A. Sánchez Pérez, Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory: Adv. Appl., vol. 180, Birkhäuser, Basel, 2008.

J.T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach Science, New York, 1969.

N. Weaver, Lipschitz Algebras, second edition, World Scientific, Hackensack, 2018.


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