Ergodic properties of pseudo-differential operators on compact Lie groups

Document Type : Research Paper

Authors

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Abstract

Let $ \mathbb{G} $ be a compact Lie group. This  article shows that a contraction pseudo-differential operator  $ A_{\tau} $ on $ L^{p}(\mathbb{G}) $ has a Dominated Ergodic Estimate (DEE), and is trigonometrically well-bounded. Then we express ergodic generalization of the Vector-Valued M. Riesz theorem for invertible contraction pseudo-differential operator  $ A_{\tau} $ on $ L^{p}(\mathbb{G}) $. For this purpose, we show that $ A_{\tau} $ is a Lamperti operator. Then we find a formula for its symbols $ \tau$. According to this formula, a representation for the symbol of adjoint and products is given.

Keywords

[1] M.A. Akcoglu, A pointwise ergodic theorem in Lp-spaces, Canad. J. Math. 27 (1975), no. 5, 1075–1082.
[2] S. Banach, Th´eorie des op´erations lin´eaires, Z. Subwencji funduszu kultury Narodowej, 1979.
[3] E. Berkson and T.A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1169–1189.
[4] R.V. Chacon and U. Krengel, Linear modulus of a linear operator, Proc. Amer. Math. Soc. 15 (1964), no. 4, 553–559.
[5] Z. Faghih and M.B. Ghaemi, Characterizations of pseudo-differential operators on S1 based on separationpreserving operators, J. Pseudo-Diff. Oper. Appl. 12 (2021), no. 1, 1–14.
[6] M.B. Ghaemi and M. Jamalpour Birgani, Lp-boundedness, compactness of pseudo-differential operators on compact Lie groups, J. Pseudo-Differ. Oper. Appl. 8 (2017), no. 1, 1–11.
[7] M.B. Ghaemi, M. Jamalpour Birgani and M.W. Wong, Characterizations of nuclear pseudo-differential operators on S1 with applications to adjoints and products, J. Pseudo-Differ. Oper. Appl. 8 (2017), no. 2, 191–201.
[8] M.B. Ghaemi, E. Nabizadeh, M. Jamalpour Birgani and M.K. Kalleji, A study on the adjoint of pseudo-differential operators on compact lie groups, Complex Var. Elliptic Equ. 63 (2018), no. 10, 1408–1420.
[9] C.H. Kan, Ergodic properties of Lamperti operators, Canad. J. Math. 30 (1978), no. 6, 1206–1214.
[10] J. Lamperti, On the isometries of certain function-spaces, Pacific J. Math. 8 (1958), no. 3, 459–466.
[11] W. Rudin, Real and complex analysis, McGraw-Hill Book Company, 1987.
[12] M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries: Background analysis and advanced topics, Vol. 2. Birkhauser and Boston, 2009.
[13] A. Tulcea, Ergodic properties of isometries in Lpspaces 1 < p < ∞, Bull. Amer. Math. Soc. 70 (1964), no. 3,366–371.
[14] A. Wolfgang, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), no. 2, 199–215.
Volume 13, Issue 2
July 2022
Pages 1703-1711
  • Receive Date: 02 January 2022
  • Accept Date: 13 March 2022