Operator θ$\theta$‐Hölder functions with respect to ∥·∥p$\Vert \cdot \Vert _p$, 0<p⩽∞$0< p\leqslant \infty$

Abstract

Let θ ∈ ( 0 , 1 ) $\theta \in (0,1)$ and ( M , τ ) $({\mathcal {M}},\tau )$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (J. London Math. Soc. (2) 95 (2017), 157–176; Geom. Funct. Anal. 27 (2017), 676–725) (denoted by S d , θ $S_{d,\theta }$ ), showing that there exists a constant d > 0 $d>0$ depending on p $p$ , 0 < p ⩽ ∞ $0<p\leqslant \infty$ , only such that every function f : R → C ∈ S d , θ $f:{\mathbb {R}}\rightarrow {\mathbb {C}}\in S_{d,\theta }$ is operator θ $\theta$ -Hölder with respect to ∥ · ∥ p $\Vert \cdot \Vert _p$ , that is, there exists a constant C p , f $C_{p,f}$ depending on p $p$ and f $f$ only such that the estimate f ( A ) − f ( B ) p ⩽ C p , f A − B θ p \begin{equation*} \hspace*{7pc}{\left\Vert f(A) -f(B)\right\Vert} _p \leqslant C_{p,f}{\left\Vert {\left| A-B \right|}^\theta \right\Vert} _p\hspace*{-7pc} \end{equation*} holds for arbitrary self-adjoint τ $\tau$ -measurable operators A $A$ and B $ B$ . In particular, we obtain a sharp condition such that a function f $f$ is operator θ $\theta$ -Hölder with respect to all quasi-norms ∥ · ∥ p $\Vert \cdot \Vert _p$ , 0 < p ⩽ ∞ $0<p\leqslant \infty$ , which complements the results on the case for 1 θ < p < ∞ $ \frac{1}{\theta }< p<\infty$ by Aleksandrov and Peller (J. Funct. Anal. 258 (2010), 3675–3724), and the case when p = ∞ $p=\infty$ treated by Aleksandrov and Peller (Adv. Math. 224 (2010), 910–966), and by Nikol ′ $^\prime$ skaya and Farforovskaya (Algebra i Analiz 22 (2010), 198–213 (Russian)). As an application, we show that this class of functions is operator θ $\theta$ -Hölder with respect to a wide class of symmetrically quasi-normed operator spaces affiliated with M ${\mathcal {M}}$ , which unifies the results on specific functions due to Birman, Koplienko and Solomjak (Izv. Vysš. Učebn. Zaved. Matematika 3 (1975), no. 154, 3–10; Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989)), Bhatia (Comm. Math. Phys. 111 (1987), 33–39), Ando (Math. Z. 197 (1988), 403–409) and Ricard (Arch. Math. (Basel) 104 (2015), 37–45; Adv. Math. 333 (2018), 194–211) with significant extension. In addition, when θ > 1 $\theta >1$ , we obtain a reverse of the Birman–Koplienko–Solomjak inequality, which extends a couple of existing results on fractional powers t ↦ t θ $t\mapsto t^\theta$ by Ando et al.