A Sobolev estimate for radial lp-multipliers on a class of semi-simple lie groups

Let G be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup K. Let ΩK be minus the radial Casimir operator. Let 1 4 dim(G/K) < SG < 1 2 dim(G/K), s ∈ (0, SG] and p ∈ (1,∞) be such that(1 p - 1 2 )< s 2SG . Then, there exists a constant CG,s,p > 0 such that for every m ∈ L∞(G) ∩ L2(G) bi-K-invariant with m ∈ Dom(Ωs K) and Ωs K(m) ∈ L2SG/s(G) we have, (0.1) ∥Tm : Lp(G) → Lp( G)∥ ≤ CG,s,p∥Ωs K(m)∥ L2SG/s(G), where Tm is the Fourier multiplier with symbol m acting on the noncommutative Lp-space of the group von Neumann algebra of G. This gives new examples of Lp-Fourier multipliers with decay rates becoming slower when p approximates 2.