A Sobolev estimate for radial ^{ }-multipliers on a class of semi-simple Lie groups

• Published in: Transactions of the American Mathematical Society 2023-09
• Main Author: Caspers, Martijn
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/9052

Let G G be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup K K . Let Ω K \Omega _K be minus the radial Casimir operator. Let 1 4 dim ⁡ ( G / K ) > S G > 1 2 dim ⁡ ( G / K ) , s ∈ ( 0 , S G ] \frac {1}{4} \dim (G/K) > S_G > \frac {1}{2} \dim (G/K) , s \in (0, S_G] and p ∈ ( 1 , ∞ ) p \in (1,\infty ) be such that \[ | 1 p − 1 2 | > s 2 S G . \left | \frac {1}{p} - \frac {1}{2} \right | > \frac {s}{2 S_G}. \] Then, there exists a constant C G , s , p > 0 C_{G,s,p} >0 such that for every m ∈ L ∞ ( G ) ∩ L 2 ( G ) m \in L^\infty (G) \cap L^2(G) bi- K K -invariant with m ∈ D o m ( Ω K s ) m \in Dom(\Omega _K^s) and Ω K s ( m ) ∈ L 2 S G / s ( G ) \Omega _K^s(m) \in L^{2S_G/s}(G) we have, ‖ T m : L p ( G ^ ) → L p ( G ^ ) ‖ ≤ C G , s , p ‖ Ω K s ( m ) ‖ L 2 S G / s ( G ) , \begin{equation} \Vert T_m: L^p(\widehat {G}) \rightarrow L^p(\widehat {G}) \Vert \leq C_{G, s,p} \Vert \Omega _K^s(m) \Vert _{L^{2S_G/s}(G)}, \end{equation} where T m T_m is the Fourier multiplier with symbol m m acting on the non- commutative L p L^p -space of the group von Neumann algebra of G G . This gives new examples of L p L^p -Fourier multipliers with decay rates becoming slower when p p approximates 2 2 .