Abstract
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 \({\mathcal {L}}_p\).
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This research was funded by Australian Research Council, Grant no [FL170100052].
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Appendix A: Automatic complete boundedness of \({\mathcal {L}}_p\)-bounded Schur multipliers
Appendix A: Automatic complete boundedness of \({\mathcal {L}}_p\)-bounded Schur multipliers
The following is a recently published result of Aleksandrov and Peller [5, Theorem 3.1].
Theorem A.0.1
Let \(A \in M_n({\mathbb {C}})\) be a matrix, where \(1\le n\le \infty \). Let \(N\ge 1\) and denote by \(\mathrm {id}_{M_N({\mathbb {C}})}\) the \(N\times N\) matrix with all entries equal to 1. Then for all \(0< p < 1\) we have
In other words, bounded Schur multipliers of \({\mathcal {L}}_p\) are automatically completely bounded. The analogous statement for \(p=1\) is well-known, see [39, Theorem 5.1]. For the sake of completeness we include a proof of Theorem A.0.1. The proof is different from that of [5], and is instead closely modelled on a proof for the \(p=1\) case due to Smith [48, Theorem 2.1].
Recall that we denote by \(\ell _2^n\) the n-dimensional Hilbert space.
Proof of Theorem A.0.1
Let \(1\le n\le \infty \) and \(N\ge 1\). Let \(u \in \ell _2^n\otimes \ell _2^N\) be a unit vector. Write the components of u as \(u =\sum _{j,l} u_{j,l}e_j\otimes e_l\). Consider the mapping:
given by
The adjoint of \(Q_u\) is easily computed. We have
or \(Q_u^*(e_j\otimes e_l) = 0\) if \(\sum _{r=1}^N |u_{j,r}|^2 = 0\).
Then we compute \(Q_u^*Q_ue_j\). If \(\sum _{l=1}^N |u_{j,l}|^2 = 0\) then \(Q_u^*Q_ue_j = 0\) and so assuming otherwise we have for every \(1\le j\le n\),
So \(Q_u\) is indeed a contraction.
Given \(u \in \ell _2^n\otimes \ell _2^N\), define \({\widetilde{u}} \in \ell _{2}^n\) as,
We have that \(\Vert {\widetilde{u}}\Vert _{\ell _2^n} = \Vert u\Vert _{\ell _2^n\otimes \ell _2^N}\).
We now assert that
It suffices to check (A.1) entrywise. On the left hand side, we have
and on the right
This verifies (A.1).
Since \(Q_u\) and \(Q_v\) are contractions, (A.1) implies that
Taking the supremum over \(u,v \in \ell _2^n\otimes \ell _2^N\) with norm at most 1 and using Lemma 2.2.1 yields the conclusion
The reverse inequality is clear. \(\square \)
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McDonald, E., Sukochev, F. Lipschitz estimates in quasi-Banach Schatten ideals. Math. Ann. 383, 571–619 (2022). https://doi.org/10.1007/s00208-021-02247-x
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DOI: https://doi.org/10.1007/s00208-021-02247-x