We study the class of functions f on \({\mathbb {R}}\) satisfying a Lipschitz estimate in the Schatten ideal \({\mathcal {L}}_p\) for \(0 < p \le 1\). The corresponding problem with \(p\ge 1\) has been extensively studied, but the quasi-Banach range \(0< p < 1\) is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class \({\dot{B}}^{\frac{1}{p}}_{\frac{p}{1-p},p}({\mathbb {R}})\) obey the estimate

for all bounded self-adjoint operators A and B with \(A-B\in {\mathcal {L}}_p\). In the case \(p=1\), our methods recover and provide a new perspective on a result of Peller that \(f \in {\dot{B}}^1_{\infty ,1}\) is sufficient for a function to be Lipschitz in \({\mathcal {L}}_1\). We also provide related Hölder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on \({\mathbb {R}}\) are not Lipschitz in \({\mathcal {L}}_p\) for any \(0< p < 1\). This gives counterexamples to a 1991 conjecture of Peller that \(f \in {\dot{B}}^{1/p}_{\infty ,p}({\mathbb {R}})\) is sufficient for f to be Lipschitz in \({\mathcal {L}}_p\).

我们研究\({\mathbb {R}}\)上的函数f类满足 Schatten 理想中的 Lipschitz 估计\({\mathcal {L}}_p\) for \(0 < p \le 1\) . \(p\ge 1\)的相应问题已被广泛研究，但相比之下，准巴拿赫范围\(0< p < 1\)却知之甚少。使用小波分析技术，我们证明了属于齐次 Besov 类的 Lipschitz 函数\({\dot{B}}^{\frac{1}{p}}_{\frac{p}{1-p}, p}({\mathbb {R}})\)服从估计

对于所有有界自伴随算子A和B与\(AB\in {\mathcal {L}}_p\)。在\(p=1\)的情况下，我们的方法恢复并提供了对 Peller 结果的新视角，即\(f \in {\dot{B}}^1_{\infty ,1}\)足以满足在\({\mathcal {L}}_1\) 中成为 Lipschitz 的函数。我们还提供了相关的 Hölder 类型估计，扩展了 Aleksandrov 和 Peller 的结果。此外，我们证明了令人惊讶的事实上非恒定的周期函数\（{\ mathbb {R}} \）不李氏在\（{\ mathcal {L}} _ p \）对于任何\（0 <P < 1\)。这给出了 1991 年佩勒猜想的反例：\(f \in {\dot{B}}^{1/p}_{\infty ,p}({\mathbb {R}})\)足以让f在\({\mathcal { L}}_p\)。