论文标题
Lipschitz估计值，准巴纳赫·沙顿理想
Lipschitz estimates in quasi-Banach Schatten ideals
论文作者
论文摘要
我们研究了$ \ mthbb {r}上的功能类$ f $，以满足lipschitz估计的schatten precional $ \ m varycal {l} _p $ for $ 0 <p \ p \ leq 1 $。 $ p \ geq 1 $的相应问题已经进行了广泛的研究，但是相比之下，准巴纳赫范围$ 0 <p <1 $。使用小波分析的技术，我们证明Lipschitz属于同质besov类$ \ dot {b}^{\ frac {\ frac {1} {p}} {p}} _ {\ frac {p} {p} \ | f（a）-f（b）\ | _ {p} \ leq c_ {p}（\ | f'\ | _ {l _ {\ infty}（\ Mathbb {r}）}}+\ | f \ | _ {\ dot {\ dot {b}^{ \ frac {1} {p}} _ {\ frac {p} {1-p}，p}，p}（\ mathbb {r}）}）\ | a-b \ | _ {p} $$对于所有有界的自动接合运算符$ a $ a $ a $ a $ a $ a $ a-b \ in \ mathcal {l} _p $。在$ p = 1 $的情况下，我们的方法恢复并提供了佩勒的结果的新观点，即$ f \ in \ dot {b}^1 _ {\ infty，1} $就足以使函数成为$ \ nathcal {l} _1 _1 $的函数。我们还提供相关的Hölder型估计值，扩展了Aleksandrov和Peller的结果。此外，我们证明了一个令人惊讶的事实，即$ \ mathbb {r} $上的非恒定周期性函数不是$ \ mathcal {l} _p $ in $ 0 <p <1 $的lipschitz。这给了1991年的peller的反描述：
We study the class of functions $f$ on $\mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $\mathcal{L}_p$ for $0 < p \leq 1$. The corresponding problem with $p\geq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})$ obey the estimate $$ \|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 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 $\mathcal{L}_p$.
