Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T12:21:41.518Z Has data issue: false hasContentIssue false
  • English
  • Français

FRACTIONAL INTEGRAL OPERATORS ON $\unicode[STIX]{x1D6FC}$-MODULATION SPACES IN THE FULL RANGE

Part of: Harmonic analysis in several variables Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  28 March 2018

GUOPING ZHAO ,
WEICHAO GUO Open the ORCID record for WEICHAO GUO [Opens in a new window]  and
XIAO YU
GUOPING ZHAO
Affiliation:
School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, PR China email guopingzhaomath@gmail.com
WEICHAO GUO*
Affiliation:
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, PR China email weichaoguomath@gmail.com
XIAO YU
Affiliation:
Department of Mathematics, Shangrao Normal University, Shangrao, 334001, PR China email yx2000s@163.com
*
✉weichaoguomath@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use a unified approach to study the boundedness of fractional integral operators on $\unicode[STIX]{x1D6FC}$-modulation spaces and find sharp conditions for boundedness in the full range.

Keywords

fractional integral operatorsmodulation space𝛼-modulation spaceBesov spacesharp conditions

MSC classification

Primary: 47G40: Potential operators
Secondary: 42B15: Multipliers 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 499 - 512
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work is partially supported by the National Natural Science Foundation of China (nos. 11601456, 11701112, 11671414, 11771388, 11371316), China Postdoctoral Science Foundation (no. 2017M612628) and the Natural Science Foundation of Jiangxi Province (no. 20151BAB211002).

References

Bényi, A., Grafakos, L., Gröchenig, K. and Okoudjou, K. A., ‘A class of Fourier multipliers for modulation spaces’, Appl. Comput. Harmon. Anal. 19(1) (2005), 131–139.Google Scholar
Bényi, A., Gröchenig, K., Okoudjou, K. A. and Rogers, L. G., ‘Unimodular Fourier multiplier for modulation spaces’, J. Funct. Anal. 246 (2007), 366–384.Google Scholar
Brenner, P., Thomee, V. and Wahlbin, L. B., Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Mathematics, vol. 434 (Springer, Berlin–Heidelberg, 1975).Google Scholar
Borup, L., ‘Pseudodifferential operators on 𝛼-modulation spaces’, J. Funct. Spaces Appl. 2 (2004), 107–123.Google Scholar
Borup, L. and Nielsen, M., ‘Nonlinear approximation in 𝛼-modulation spaces’, Math. Nachr. 279 (2006), 101–120.Google Scholar
Borup, L. and Nielsen, M., ‘Banach frames for multivariate 𝛼-modulation spaces’, J. Math. Anal. Appl. 321 (2006), 880–895.CrossRefGoogle Scholar
Borup, L. and Nielsen, M., ‘Boundedness for pseudodifferential operators on multivariate𝛼-modulation spaces’, Ark. Mat. 44 (2006), 241–259.CrossRefGoogle Scholar
Chen, J., Fan, D. and Sun, L., ‘Asymptotic estimates for unimodular Fourier multipliers on modulation spaces’, Discrete Contin. Dyn. Syst. 32 (2012), 467–485.CrossRefGoogle Scholar
Feichtinger, H. G., ‘‘Modulation spaces on locally compact Abelian group’, Technical Report, University of Vienna, 1983’, in: Proc. Internat. Conf. on Wavelet and Applications (New Delhi Allied Publishers, India, 2003), 99–140.Google Scholar
Galperin, Y. V. and Samarah, S., ‘Time-frequency analysis on modulation spaces M m p, q , 0 < p, q⩽∞ ’, Appl. Comput. Harmon. Anal. 16(1) (2004), 1–18.CrossRefGoogle Scholar
Gröbner, P., ‘Banachräume Glatter Funktionen und Zerlegungsmethoden’, Doctoral Thesis, University of Vienna, 1992.Google Scholar
Guo, W. and Chen, J., ‘Strichartz estimates on 𝛼-modulation spaces’, Electron. J. Differ. Equ. 118 (2013), 1–13.Google Scholar
Guo, W., Fan, D., Wu, H. and Zhao, G., ‘Sharpness of complex interpolation on 𝛼-modulation spaces’, J. Fourier Anal. Appl. 22(2) (2016), 427–461.Google Scholar
Guo, W., Fan, D., Wu, H. and Zhao, G., ‘Full characterization of embedding relations between𝛼-modulation spaces’, Science China Math. (in press).Google Scholar
Han, J. and Wang, B., ‘𝛼-modulation spaces (I): embedding, interpolation and algebra properties’, J. Math. Soc. Japan 66 (2014), 1315–1373.Google Scholar
Miyachi, A., Nicola, F., Rivetti, S., Tabacco, A. and Tomita, N., ‘Estimates for unimodular Fourier multipliers on modulation spaces’, Proc. Amer. Math. Soc. 137 (2009), 3869–3883.CrossRefGoogle Scholar
Ruzhansky, M., Sugimoto, M., Tomita, N. and Toft, J., ‘Changes of variables in modulation and Wiener amalgam spaces’, Math. Nachr. 284 (2011), 2078–2092.Google Scholar
Stein, E. M., Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970).Google Scholar
Sugimoto, M. and Tomita, N., ‘A remark on fractional integrals on modulation spaces’, Math Nachr. 281 (2008), 1372–1379.CrossRefGoogle Scholar
Toft, J., ‘The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators’, J. Pseudo-Differ. Oper. Appl. 3 (2012), 145–227.Google Scholar
Toft, J., ‘Gabor analysis for a broad class of quasi-Banach modulation spaces’, in: Pseudo-Differential Operators, Generalized Functions, Operator Theory: Advances and Applications, 245 (eds. Pilipović, S. and Toft, J.) (Birkhäuser, Cham, 2015), 249–278.Google Scholar
Toft, J., ‘Images of function and distribution spaces under the Bargmann transform’, J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.CrossRefGoogle Scholar
Toft, J. and Wahlberg, P., ‘Embeddings of 𝛼-modulation spaces’, Pliska Stud. Math. Bulgar. 21 (2012), 25–46.Google Scholar
Tomita, N., ‘Fractional integrals on modulation spaces’, Math. Nachr. 279 (2006), 672–680.Google Scholar
Triebel, H., Theory of Function Spaces, Monographs in Mathematics, vol. 78 (Birkhäuser, Basel–Boston–Stuttgart, 1983).CrossRefGoogle Scholar
Wang, B. and Hudzik, H., ‘The global Cauchy problem for the NLS and NLKG with small rough data’, J. Differ. Equ. 232 (2007), 36–73.CrossRefGoogle Scholar
Wu, X. and Chen, J., ‘Boundedness of fractional integral operators on 𝛼-modulation spaces’, Appl. Math. J. Chinese Univ. Ser. A 29(3) (2014), 339–351.CrossRefGoogle Scholar
Zhong, Y. and Chen, J., ‘Modulation space estimates for the fractional integral operators’, Sci. China Math. 54 (2011), 1–11.CrossRefGoogle Scholar
Zhao, G., Chen, J., Fan, D. and Guo, W., ‘Sharp estimates of unimodular multipliers on frequency decomposition spaces’, Nonlinear Anal. 142 (2016), 26–47.CrossRefGoogle Scholar
Zhao, G., Fan, D. and Guo, W., ‘Fractional integral operators on 𝛼-modulation spaces’, Math. Nachr. 289 (2016), 1288–1300.Google Scholar