Sobolev spaces on compact groups
-
Manoj Kumar
Abstract
In this note, we introduce Sobolev spaces over (not necessarily abelian) compact groups and study their properties. In particular, we focus on Sobolev embedding and compactness theorems. As an application, we prove the existence of solutions for the generalized Bosonic equations posed over compact groups.
Funding statement: The author Manoj Kumar is supported by the NBHM post-doctoral fellowship with reference number 0204/3/2021/R&D-II/7356.
Acknowledgements
The authors would like to thank the referee for his perspicuous comments and suggestions which helped in improving the quality of the paper.
References
[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Suche in Google Scholar
[2] R. G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley Class. Libr., John Wiley & Sons, New York, 1995. 10.1002/9781118164471Suche in Google Scholar
[3] G. Calcagni, M. Montobbio and G. Nardelli, Localization of nonlocal theories, Phys. Lett. B 662 (2008), no. 3, 285–289. 10.1016/j.physletb.2008.03.024Suche in Google Scholar
[4] A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Universitext, Springer, New York, 2009. Suche in Google Scholar
[5] H. G. Feichtinger, S. S. Pandey and T. Werther, Minimal norm interpolation in harmonic Hilbert spaces and Wiener amalgam spaces on locally compact abelian groups, J. Math. Kyoto Univ. 47 (2007), no. 1, 65–78. 10.1215/kjm/1250281068Suche in Google Scholar
[6] H. G. Feichtinger and T. Werther, Robustness of regular sampling in Sobolev algebras, Sampling, Wavelets, and Tomography, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston (2004), 83–113. 10.1007/978-0-8176-8212-5_4Suche in Google Scholar
[7] G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., Textb. Math., CRC Press, Boca Raton, 2016. 10.1201/b19172Suche in Google Scholar
[8] B. Franchi, P. Hajł asz and P. Koskela, Definitions of Sobolev classes on metric spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 6, 1903–1924. 10.5802/aif.1742Suche in Google Scholar
[9] P. Górka, H. Prado and E. G. Reyes, Nonlinear equations with infinitely many derivatives, Complex Anal. Oper. Theory 5 (2011), no. 1, 313–323. 10.1007/s11785-009-0043-zSuche in Google Scholar
[10] P. Górka, H. Prado and E. G. Reyes, Generalized Euclidean bosonic string equations, Spectral Analysis of Quantum Hamiltonians, Oper. Theory Adv. Appl. 224, Birkhäuser/Springer, Basel (2012), 147–169. 10.1007/978-3-0348-0414-1_8Suche in Google Scholar
[11] P. Górka and T. Kostrzewa, Sobolev spaces on metrizable groups, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 837–849. 10.5186/aasfm.2015.4045Suche in Google Scholar
[12] P. Górka and T. Kostrzewa, Pego everywhere, J. Algebra Appl. 15 (2016), no. 4, Article ID 1650074. 10.1142/S0219498816500742Suche in Google Scholar
[13] P. Górka and T. Kostrzewa, A second look of Sobolev spaces on metrizable groups, Ann. Acad. Sci. Fenn. Math. 45 (2020), no. 1, 95–120. 10.5186/aasfm.2020.4507Suche in Google Scholar
[14] P. Górka, T. Kostrzewa and E. G. Reyes, The Rellich lemma on compact abelian groups and equations of infinite order, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 2, Article ID 1220030. 10.1142/S0219887812200307Suche in Google Scholar
[15] P. Górka and E. G. Reyes, Sobolev spaces on locally compact abelian groups and the bosonic string equation, J. Aust. Math. Soc. 98 (2015), no. 1, 39–53. 10.1017/S1446788714000433Suche in Google Scholar
[16] P. Hajł asz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no. 4, 403–415. Suche in Google Scholar
[17] E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Math. 1635, Springer, Berlin, 1996. 10.1007/BFb0092907Suche in Google Scholar
[18] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups, Grundlehren Math. Wiss. 152, Springer, New York, 1970. 10.1007/978-3-662-26755-4_3Suche in Google Scholar
[19] M. I. Ostrovskii, Sobolev spaces on graphs, Quaest. Math. 28 (2005), no. 4, 501–523. 10.2989/16073600509486144Suche in Google Scholar
[20] J. J. Rodríguez-Vega and W. A. Zúñiga Galindo, Elliptic pseudodifferential equations and Sobolev spaces over p-adic fields, Pacific J. Math. 246 (2010), no. 2, 407–420. 10.2140/pjm.2010.246.407Suche in Google Scholar
[21] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries. Background Analysis and Advanced Topics, Pseudo Diff. Oper. 2, Birkhäuser, Basel, 2010. 10.1007/978-3-7643-8514-9Suche in Google Scholar
[22] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), no. 2, 243–279. 10.4171/RMI/275Suche in Google Scholar
[23] M. E. Taylor, Partial Differential Equations. I, Appl. Math. Sci. 115, Springer, New York, 1996. Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Arithmetic of Châtelet surface bundles revisited
- Sobolev spaces on compact groups
- Left invariant Ricci flat metrics on Lie groups
- Structural theorems on the distance sets over finite fields
- On sums of hyper-Kloosterman sums
- Estimates for the Fourier coefficients of the Duke–Imamoglu–Ikeda lift
- The Pierce decomposition and Pierce embedding of endomorphism rings of abelian p-groups
- Joint distribution of the cokernels of random p-adic matrices
- A profinite approach to complete bifix decodings of recurrent languages
- Representation of relative sheaf cohomology
- Sublinearly Morse boundaries from the viewpoint of combinatorics
-
Explicit Sato–Tate type distribution for a family of
- Sharp constants for multilinear Hausdorff operators on local Morrey-type spaces
- Explicit solutions of the Dirichlet–Schrödinger problem via the new cylindrical Poisson–Schrödinger kernel method
Artikel in diesem Heft
- Frontmatter
- Arithmetic of Châtelet surface bundles revisited
- Sobolev spaces on compact groups
- Left invariant Ricci flat metrics on Lie groups
- Structural theorems on the distance sets over finite fields
- On sums of hyper-Kloosterman sums
- Estimates for the Fourier coefficients of the Duke–Imamoglu–Ikeda lift
- The Pierce decomposition and Pierce embedding of endomorphism rings of abelian p-groups
- Joint distribution of the cokernels of random p-adic matrices
- A profinite approach to complete bifix decodings of recurrent languages
- Representation of relative sheaf cohomology
- Sublinearly Morse boundaries from the viewpoint of combinatorics
-
Explicit Sato–Tate type distribution for a family of
- Sharp constants for multilinear Hausdorff operators on local Morrey-type spaces
- Explicit solutions of the Dirichlet–Schrödinger problem via the new cylindrical Poisson–Schrödinger kernel method
