Two stars theorems for traces of the Zygmund space

Brudnyi, A.

doi:10.1090/spmj/1744

Two stars theorems for traces of the Zygmund space
HTML articles powered by AMS MathViewer

by A. Brudnyi

St. Petersburg Math. J. 34 (2023), 25-44

DOI: https://doi.org/10.1090/spmj/1744

Published electronically: December 16, 2022

PDF | Request permission

Abstract:

For a Banach space $X$ defined in terms of a big-$O$ condition and its subspace x defined by the corresponding little-$o$ condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to $X$. The property is known for pairs of many classical function spaces (such as $(\ell _\infty , c_0)$, (BMO, VMO), (Lip, lip), etc.) and plays an important role in the study of their geometric structure. The present paper is devoted to the biduality property for traces to closed subsets $S\subset \mathbb {R}^n$ of a generalized Zygmund space $Z^\omega (\mathbb {R}^n)$. The method of the proof is based on a careful analysis of the structure of geometric preduals of the trace spaces along with a powerful finiteness theorem for the trace spaces $Z^\omega (\mathbb {R}^n)|_S$.

References

Edward Bierstone, Pierre D. Milman, and Wiesław Pawłucki, Higher-order tangents and Fefferman’s paper on Whitney’s extension problem, Ann. of Math. (2) 164 (2006), no. 1, 361–370. MR 2233851, DOI 10.4007/annals.2006.164.361
Alexander Brudnyi and Yuri Brudnyi, Methods of geometric analysis in extension and trace problems. Volume 1, Monographs in Mathematics, vol. 102, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2882877
Alexander Brudnyi and Yuri Brudnyi, On the Banach structure of multivariate BV spaces, Dissertationes Math. 548 (2020), 52. MR 4078209, DOI 10.4064/dm801-7-2019
Alexander Brudnyi and Yuri Brudnyi, Multivariate bounded variation functions of Jordan-Wiener type, J. Approx. Theory 251 (2020), 105346, 70. MR 4040115, DOI 10.1016/j.jat.2019.105346
Ju. A. Brudnyĭ, A multidimensional analogue of a certain theorem of Whitney, Mat. Sb. (N.S.) 82 (124) (1970), 175–191 (Russian). MR 0267319
Alexander Brudnyi, On properties of geometric preduals of $C^{k,\omega }$ spaces, Rev. Mat. Iberoam. 35 (2019), no. 6, 1715–1744. MR 4029781, DOI 10.4171/rmi/1099
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
V. K. Dzyadyk and I. A. Shevchuk, Continuation of functions which, on an arbitrary set of the line, are traces of functions with a given second modulus of continuity, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 2, 248–267 (Russian). MR 697295
Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 101294, DOI 10.1007/BF02790231
Alf Jonsson, The duals of Lipschitz spaces defined on closed sets, Indiana Univ. Math. J. 39 (1990), no. 2, 467–476. MR 1089048, DOI 10.1512/iumj.1990.39.39025
Leonid G. Hanin, Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces, Proc. Amer. Math. Soc. 115 (1992), no. 2, 345–352. MR 1097344, DOI 10.1090/S0002-9939-1992-1097344-5
Leonid G. Hanin, Duality for general Lipschitz classes and applications, Proc. London Math. Soc. (3) 75 (1997), no. 1, 134–156. MR 1444316, DOI 10.1112/S0024611597000294
K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961/62), 55–66. MR 140927, DOI 10.4064/sm-21-1-55-66
M. A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, NUMDAM, [place of publication not identified], 1927 (French). MR 3532941
Karl-Mikael Perfekt, Duality and distance formulas in spaces defined by means of oscillation, Ark. Mat. 51 (2013), no. 2, 345–361. MR 3090201, DOI 10.1007/s11512-012-0175-7
Karl-Mikael Perfekt, On $M$-ideals and o-O type spaces, Math. Scand. 121 (2017), no. 1, 151–160. MR 3708971, DOI 10.7146/math.scand.a-96626
P. A. Shvartsman, Traces of functions of Zygmund class, Sibirsk. Mat. Zh. 28 (1987), no. 5, 203–215 (Russian). MR 924998
Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3
A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. MR 12691, DOI 10.1215/S0012-7094-45-01206-3