A Sobolev estimate for radial $L^p$-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 ${\mathrm{\Omega }}_K$ be minus the radial Casimir operator. Let $\frac{1}{4}\dim (G\mathrm{/}K)<S_G<\frac{1}{2}\dim (G\mathrm{/}K),s\in (0,S_G]$ and $p\in (1,\mathrm{\infty })$ be such that $$\mid \frac{1}{p}-\frac{1}{2}\mid <\frac{s}{2S_G}\mathrm{.}$$ Then, there exists a constant $C_{G,s,p}>0$ such that for every $m\in L^{\mathrm{\infty }}(G)\cap L^2(G)$ bi-$K$-invariant with $m\in \mathrm{D}\mathrm{o}\mathrm{m}({\mathrm{\Omega }}_K^s)$ and ${\mathrm{\Omega }}_K^s(m)\in L^{2S_G\mathrm{/}s}(G)$ we have, $$\mathrm{\parallel }{T}_m:L^p(\widehat{G})\to L^p(\widehat{G})\mathrm{\parallel }\le C_{G,s,p}\mathrm{\parallel }{\mathrm{\Omega }}_K^s(m){\mathrm{\parallel }}_{L^{2S_G\mathrm{/}s}(G)},$$ where $T_m$ is the Fourier multiplier with symbol $m$ acting on the non-commutative $L^p$-space of the group von Neumann algebra of $G$. This gives new examples of $L^p$-Fourier multipliers with decay rates becoming slower when $p$ approximates $2$.