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

Abstract

Let θ∈false(0,1false)$\theta \in (0,1)$ and false(scriptM,τfalse)$({\mathcal {M}},\tau )$ 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 Sd,θ$S_{d,\theta }$), showing that there exists a constant d>0$d&gt;0$ depending on p$p$, 0<p⩽∞$0&lt;p\leqslant \infty$, only such that every function f:double-struckR→double-struckC∈Sd,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 Cp,f$C_{p,f}$ depending on p$p$ and f$f$ only such that the estimate ffalse(Afalse)−ffalse(Bfalse)p⩽Cp,fA−Bθp\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⩽∞$0&lt;p\leqslant \infty$, which complements the results on the case for 1θ<p<∞$ \frac{1}{\theta }&lt; p&lt;\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$\theta &gt;1$, we obtain a reverse of the Birman–Koplienko–Solomjak inequality, which extends a couple of existing results on fractional powers t↦tθ$t\mapsto t^\theta$ by Ando et al.