I consider, for $0<s<1$, the Bessel function space $$ L^{s,2}(\mathbb{R}^d) = \left\{f\in L^2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L^2(\mathbb{R}^n)\right\}. $$ The question I cannot understand is the following: denoting by $*$ the usual symmetric-decreasing rearrangement, does the inequality $$ \left\| (I-\Delta)^{s/2}f^* \right\|_2 \leq \left\| (I-\Delta)^{s/2}f \right\|_2 $$ hold in $L^{s,2}(\mathbb{R}^n)$?
I learnt from a famous paper by Almgren and Lieb (Theorem 9.2 in [1]) that this is true for the more familiar norm $\| (-\Delta)^{s/2}f\|_2$, but I believe that their proof does not carry over to Bessel norms. Does anyone has any reference or hint?
[1] F. J. Almgren, Jr.; E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. J. Amer. Math. Soc. 2 (1989), no. 4, 683–773.
