Operator θ-Hölder functions with respect to (Formula presented.)

Let (Formula presented.) and (Formula presented.) 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 (Formula presented.)), showing that there exists a constant (Formula presented.) depending on (Formula presented.), (Formula presented.), only such that every function (Formula presented.) is operator (Formula presented.) -Hölder with respect to (Formula presented.), that is, there exists a constant (Formula presented.) depending on (Formula presented.) and (Formula presented.) only such that the estimate (Formula presented.) holds for arbitrary self-adjoint (Formula presented.) -measurable operators (Formula presented.) and (Formula presented.). In particular, we obtain a sharp condition such that a function (Formula presented.) is operator (Formula presented.) -Hölder with respect to all quasi-norms (Formula presented.), (Formula presented.), which complements the results on the case for (Formula presented.) by Aleksandrov and Peller (J. Funct. Anal. 258 (2010), 3675–3724), and the case when (Formula presented.) treated by Aleksandrov and Peller (Adv. Math. 224 (2010), 910–966), and by Nikol (Formula presented.) skaya and Farforovskaya (Algebra i Analiz 22 (2010), 198–213 (Russian)). As an application, we show that this class of functions is operator (Formula presented.) -Hölder with respect to a wide class of symmetrically quasi-normed operator spaces affiliated with (Formula presented.), 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 (Formula presented.), we obtain a reverse of the Birman–Koplienko–Solomjak inequality, which extends a couple of existing results on fractional powers (Formula presented.) by Ando et al.