  We study the class of functions $f$ on $\mathbb{R}$ satisfying a Lipschitz
estimate in the Schatten ideal $\mathcal{L}_p$ for $0 < p \leq 1$. The
corresponding problem with $p\geq 1$ 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 $$
  \|f(A)-f(B)\|_{p} \leq
C_{p}(\|f'\|_{L_{\infty}(\mathbb{R})}+\|f\|_{\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})})\|A-B\|_{p}
$$ 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 H\"older-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
$\mathcal{L}_p$.
