Inclusion relations between modulation and Triebel-Lizorkin spaces

Guo, Weichao; Wu, Huoxiong; Zhao, Guoping

doi:10.1090/proc/13614

Inclusion relations between modulation and Triebel-Lizorkin spaces
HTML articles powered by AMS MathViewer

by Weichao Guo, Huoxiong Wu and Guoping Zhao

Proc. Amer. Math. Soc. 145 (2017), 4807-4820

DOI: https://doi.org/10.1090/proc/13614

Published electronically: May 30, 2017

PDF | Request permission

Abstract:

In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces $M_{p,q}^s$ and Triebel-Lizorkin spaces $F_{p,r}$ for $p\leq 1$, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].

References

Árpád Bényi, Karlheinz Gröchenig, Kasso A. Okoudjou, and Luke G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal. 246 (2007), no. 2, 366–384. MR 2321047, DOI 10.1016/j.jfa.2006.12.019
H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Technical Report, University of Vienna, 1983. Published in: Proc. Internat. Conf. on Wavelet and Applications, 99-140. New Delhi Allied Publishers, India, 2003.
Hans G. Feichtinger, Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006), no. 2, 109–140. MR 2233968, DOI 10.1007/BF03549447
H. G. Feichtinger, Banach convolution algebras of Wiener type, Functions, series, operators, Vol. I, II (Budapest, 1980) Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 509–524. MR 751019
P. Gröbner, Banachrä ume Glatter Funktionen and Zerlegungsmethoden, Doctoral Thesis, University of Vienna, 1992.
Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
David Marc Goldberg, A LOCAL VERSION OF REAL HARDY SPACES, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–Princeton University. MR 2627942
Weichao Guo and Jiecheng Chen, Stability of nonlinear Schrödinger equations on modulation spaces, Front. Math. China 9 (2014), no. 2, 275–301. MR 3171501, DOI 10.1007/s11464-014-0346-x
W. Guo, J. Chen, D. Fan, G. Zhao, Characterization of some properties on weighted modulation spaces, arXiv:1602.02871 (2016).
WeiChao Guo, DaShan Fan, HuoXiong Wu, and GuoPing Zhao, Sharpness of some properties of weighted modulation spaces, Sci. China Math. 59 (2016), no. 1, 169–190. MR 3437002, DOI 10.1007/s11425-015-5067-4
Jinsheng Han and Baoxiang Wang, $\alpha$-modulation spaces (I) scaling, embedding and algebraic properties, J. Math. Soc. Japan 66 (2014), no. 4, 1315–1373. MR 3272601, DOI 10.2969/jmsj/06641315
Masaharu Kobayashi, Akihiko Miyachi, and Naohito Tomita, Embedding relations between local Hardy and modulation spaces, Studia Math. 192 (2009), no. 1, 79–96. MR 2491791, DOI 10.4064/sm192-1-7
Masaharu Kobayashi and Mitsuru Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal. 260 (2011), no. 11, 3189–3208. MR 2776566, DOI 10.1016/j.jfa.2011.02.015
Akihiko Miyachi, Fabio Nicola, Silvia Rivetti, Anita Tabacco, and Naohito Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3869–3883. MR 2529896, DOI 10.1090/S0002-9939-09-09968-7
Kasso A. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1639–1647. MR 2051124, DOI 10.1090/S0002-9939-04-07401-5
Michael Ruzhansky, Mitsuru Sugimoto, and Baoxiang Wang, Modulation spaces and nonlinear evolution equations, Evolution equations of hyperbolic and Schrödinger type, Progr. Math., vol. 301, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 267–283. MR 3014810, DOI 10.1007/978-3-0348-0454-7_{1}4
Michael Ruzhansky, Mitsuru Sugimoto, Joachim Toft, and Naohito Tomita, Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr. 284 (2011), no. 16, 2078–2092. MR 2844679, DOI 10.1002/mana.200910199
Mitsuru Sugimoto and Naohito Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal. 248 (2007), no. 1, 79–106. MR 2329683, DOI 10.1016/j.jfa.2007.03.015
Joachim Toft and Patrik Wahlberg, Embeddings of $\alpha$-modulation spaces, Pliska Stud. Math. Bulgar. 21 (2012), 25–46. MR 3114337
Joachim Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal. 207 (2004), no. 2, 399–429. MR 2032995, DOI 10.1016/j.jfa.2003.10.003
H. Triebel, Modulation spaces on the Euclidean $n$-space, Z. Anal. Anwendungen 2 (1983), no. 5, 443–457 (English, with German and Russian summaries). MR 725159, DOI 10.4171/ZAA/79
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. MR 1163193, DOI 10.1007/978-3-0346-0419-2
Wang Baoxiang, Zhao Lifeng, and Guo Boling, Isometric decomposition operators, function spaces $E^\lambda _{p,q}$ and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), no. 1, 1–39. MR 2204673, DOI 10.1016/j.jfa.2005.06.018
Baoxiang Wang and Chunyan Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations 239 (2007), no. 1, 213–250. MR 2341554, DOI 10.1016/j.jde.2007.04.009
Baoxiang Wang, Zhaohui Huo, Chengchun Hao, and Zihua Guo, Harmonic analysis method for nonlinear evolution equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR 2848761, DOI 10.1142/9789814360746
Baoxiang Wang and Henryk Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations 232 (2007), no. 1, 36–73. MR 2281189, DOI 10.1016/j.jde.2006.09.004
Guoping Zhao, Jiecheng Chen, Dashan Fan, and Weichao Guo, Sharp estimates of unimodular multipliers on frequency decomposition spaces, Nonlinear Anal. 142 (2016), 26–47. MR 3508056, DOI 10.1016/j.na.2016.04.003