Weak type operator Lipschitz and commutator estimates for commuting tuples

Caspers, Martijn; Sukochev, Fedor; Zanin, Dmitriy

doi:10.5802/aif.3195

Cited by 9 publications

(4 citation statements)

References 32 publications

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“…MC is supported by the NWO Vidi grant 'Noncommutative harmonic analysis and rigidity of operator algebras', VI. Vidi.192.018. interpolation [CPSZ19] (see also [CSZ18]) it implies the main results of [PS11] and [CMPS14] as well as many previous results on perturbation of commutators and non-commutative Lipschitz properties. In this sense the so-called weak type (1, 1) estimate (1.3) is the optimal one.…”

Section: Introductionsupporting

confidence: 78%

Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions

Caspers¹,

Sukochev²,

Zanin³

2020

Preprint

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Consider the generalized absolute value function defined byFurther, consider the n-th order divided difference function a [n] : R n+1 → C and let 1 < p1, . . . , pn < ∞ be such that n l=1 p −1 l = 1. Let Sp l denote the Schatten-von Neumann ideals and let S1,∞ denote the weak trace class ideal. We show that for any (n + 1)-tuple A of bounded self-adjoint operators the multiple operator integral T A a [n] maps Sp 1 × . . . × Sp n to S1,∞ boundedly with uniform bound in A. The same is true for the class of C n+1 -functions that outside the interval [−1, 1] equal a. In [CLPST16] it was proved that for a function f in this class such boundedness of T A f [n] from Sp 1 × . . . × Sp n to S1 may fail, resolving a problem by V.Peller. This shows that the estimates in the current paper are optimal. The proof is based on a new reduction method for arbitrary multiple operator integrals of divided differences.

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Section: Introductionsupporting

confidence: 78%

Weak $(1,1)$ estimates for multiple operator integrals and generalized absolute value functions

Caspers¹,

Sukochev²,

Zanin³

2020

Preprint

Self Cite

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“…It was pointed out in [21] that the noncommutative Calderón-Zygmund decomposition and the related method should open a door to work for more general class of operators. For the subsequent works related to weak type (1,1) problem and noncommutative Calderón-Zygmund decomposition, we refer to see [20], [1], [5], [6], [15] and the references therein.…”

Section: Introduction and State Of Main Resultsmentioning

confidence: 99%

Noncommutative maximal operators with rough kernels

Lai¹

2019

Preprint

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This paper is devoted to the study of noncommutative maximal operators with rough kernels. More precisely, we prove the weak type (1, 1) boundedness for noncommutative maximal operator with a rough kernel. Then, via the noncommutative Marcinkiewicz type interpolation theorem, together with a trivial (∞, ∞) estimate, we obtain its Lp boundedness for all 1 < p < ∞. The proof of weak type (1,1) estimate is based on the noncommutative Calderón-Zygmund decomposition. To deal with the rough kernel, we use the microlocal decomposition in the proofs of both bad and good functions.

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“…As in many problems of a semiclassical character, the spectral estimates of Cwikel, Kato-Seiler-Simon and Solomyak play an fundamental role in its proof. More specific to the problem at hand and of crucial importance both here and in [35,31] are techniques from double operator integrals; see also [9,10,11,20]. The abovementioned second and third steps are the topic of Sections 5, 6 and 7.…”

Section: Thus Limmentioning

confidence: 99%

“…Proof. The proof of this lemma requires some tools from the theory of Double Operator Integrals, for which we refer the reader to [10,11,9,20]. The symbol T A φ denotes the double operator integral, based on the operator A, with the symbol φ as defined in these papers.…”

Section: 2mentioning

confidence: 99%