Characterization of nonlinear Besov spaces

Liu, Chong; Prömel, David; Teichmann, Josef

doi:10.1090/tran/7968

Characterization of nonlinear Besov spaces
HTML articles powered by AMS MathViewer

by Chong Liu, David J. Prömel and Josef Teichmann PDF

Trans. Amer. Math. Soc. 373 (2020), 529-550 Request permission

Abstract:

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of nonlinear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how nonlinear Besov spaces embed in nonlinear $p$-variation spaces, and vice versa. We emphasize that we assume neither the unconditional martingale difference property of the involved spaces nor their separability.

References

Israel Aharoni, Every separable metric space is Lipschitz equivalent to a subset of $c^{+}_{0}$, Israel J. Math. 19 (1974), 284–291. MR 511661, DOI 10.1007/BF02757727
Herbert Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56. MR 1461211, DOI 10.1002/mana.3211860102
Herbert Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III 35(55) (2000), no. 1, 161–177. Dedicated to the memory of Branko Najman. MR 1783238
Mats Bodin, Discrete characterisations of Lipschitz spaces on fractals, Math. Nachr. 282 (2009), no. 1, 26–43. MR 2473129, DOI 10.1002/mana.200610720
Z. Ciesielski, G. Kerkyacharian, and B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), no. 2, 171–204 (French, with English summary). MR 1244574, DOI 10.4064/sm-107-2-171-204
Peter Friz and Nicolas Victoir, A variation embedding theorem and applications, J. Funct. Anal. 239 (2006), no. 2, 631–637. MR 2261341, DOI 10.1016/j.jfa.2005.12.021
Peter Friz and Nicolas Victoir, Multidimensional stochastic processes as rough paths: Theory and applications, Cambridge University Press, Cambridge, England, 2010.
Martin Hairer and Cyril Labbé, The reconstruction theorem in Besov spaces, J. Funct. Anal. 273 (2017), no. 8, 2578–2618. MR 3684891, DOI 10.1016/j.jfa.2017.07.002
Maryia Kabanava, Characterization of Besov spaces on nested fractals by piecewise harmonic functions, Z. Anal. Anwend. 31 (2012), no. 2, 183–201. MR 2914970, DOI 10.4171/ZAA/1454
Anna Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. (N.S.) 13 (1997), no. 2, 63–77. MR 1750304
Anna Kamont, Discrete characterization of Besov spaces and its applications to stochastics, Multivariate approximation and splines (Mannheim, 1996) Internat. Ser. Numer. Math., vol. 125, Birkhäuser, Basel, 1997, pp. 89–98. MR 1484997
Kazimierz Kuratowski, Introduction to set theory and topology, International Series of Monographs in Pure and Applied Mathematics, vol. 101, PWN—Polish Scientific Publishers, Warsaw; Pergamon Press, Oxford-New York-Toronto, Ont., 1977. Containing a supplement, “Elements of algebraic topology” by Ryszard Engelking; Translated from the Polish by Leo F. Boroń. MR 0643829
Terry J. Lyons, Michael Caruana, and Thierry Lévy, Differential equations driven by rough paths, Lecture Notes in Mathematics, vol. 1908, Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; With an introduction concerning the Summer School by Jean Picard. MR 2314753, DOI 10.1007/978-3-540-71285-5
Giovanni Leoni, A first course in Sobolev spaces, 2nd ed., Graduate Studies in Mathematics, vol. 181, American Mathematical Society, Providence, RI, 2017. MR 3726909, DOI 10.1090/gsm/181
Chong Liu, David J. Prömel, and Josef Teichmann, Optimal extension to Sobolev rough paths, arXiv:1811.05173 (2018).
Terry Lyons and Nicolas Victoir, An extension theorem to rough paths, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 5, 835–847 (English, with English and French summaries). MR 2348055, DOI 10.1016/j.anihpc.2006.07.004
E. R. Love and L. C. Young, On Fractional Integration by Parts, Proc. London Math. Soc. (2) 44 (1938), no. 1, 1–35. MR 1575481, DOI 10.1112/plms/s2-44.1.1
Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, DOI 10.4171/RMI/240
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
Mathieu Rosenbaum, First order $p$-variations and Besov spaces, Statist. Probab. Lett. 79 (2009), no. 1, 55–62. MR 2483397, DOI 10.1016/j.spl.2008.07.019
Mathieu Rosenbaum, A new microstructure noise index, Quant. Finance 11 (2011), no. 6, 883–899. MR 2806970, DOI 10.1080/14697680903514352
Vyacheslav S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2) 60 (1999), no. 1, 237–257. MR 1721827, DOI 10.1112/S0024610799007723
Yoshihiro Sawano, Theory of Besov spaces, Developments in Mathematics, vol. 56, Springer, Singapore, 2018. MR 3839617, DOI 10.1007/978-981-13-0836-9
Benjamin Scharf, Local means and atoms in vector-valued function spaces, arXiv:1103.6159 (2011).
Jacques Simon, Sobolev, Besov and Nikol′skiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl. (4) 157 (1990), 117–148. MR 1108473, DOI 10.1007/BF01765315
Benjamin Scharf, Hans-Jürgen Schmeißer, and Winfried Sickel, Traces of vector-valued Sobolev spaces, Math. Nachr. 285 (2012), no. 8-9, 1082–1106. MR 2928400, DOI 10.1002/mana.201100011
Cornelia Schneider and Jan Vybíral, Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces, J. Funct. Anal. 264 (2013), no. 5, 1197–1237. MR 3010019, DOI 10.1016/j.jfa.2012.12.005
Hans Triebel, Spaces of distributions of Besov type on Euclidean $n$-space. Duality, interpolation, Ark. Mat. 11 (1973), 13–64. MR 348483, DOI 10.1007/BF02388506
Hans Triebel, A note on wavelet bases in function spaces, Orlicz centenary volume, Banach Center Publ., vol. 64, Polish Acad. Sci. Inst. Math., Warsaw, 2004, pp. 193–206. MR 2099469, DOI 10.4064/bc64-0-15
Hans Triebel, Theory of function spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010. Reprint of 1983 edition [MR0730762]; Also published in 1983 by Birkhäuser Verlag [MR0781540]. MR 3024598
Mark Veraar, Correlation inequalities and applications to vector-valued Gaussian random variables and fractional Brownian motion, Potential Anal. 30 (2009), no. 4, 341–370. MR 2491457, DOI 10.1007/s11118-009-9118-8
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI 10.1007/BF02401743