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1. Zygmund dilations: bilinear analysis and commutator estimates

   https://doi.org/10.1090/tran/9366  

   Airta, Emil; Li, Kangwei; Martikainen, Henri (April 2025, Transactions of the American Mathematical Society)

   We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $$(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$$ in $$\mathbb{R}^3$$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates. 

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2. On the equivalence between weak BMO and the space of derivatives of the Zygmund class

   https://doi.org/10.1515/dema-2021-0013  

   Kwessi, Eddy (January 2021, Demonstratio Mathematica)

   Abstract In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy space H 1 {H}^{1} strictly contains the special atom space. 

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3. Multilinear paraproducts on Sobolev spaces

   https://doi.org/10.1007/s40574-024-00444-5  

   Di_Plinio, Francesco; Green, A Walton; Wick, Brett D (March 2025, Bollettino dell'Unione Matematica Italiana)

   Paraproducts are a special subclass of the multilinear Calderón-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space T(1)-type theorems for multilinear Calderón-Zygmund operators. 

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4. Mappings of generalized finite distortion and continuity

   https://doi.org/10.1112/jlms.12835  

   Doležalová, Anna; Kangasniemi, Ilmari; Onninen, Jani (November 2023, Journal of the London Mathematical Society)

   Abstract We study continuity properties of Sobolev mappings , , that satisfy the following generalized finite distortion inequalityfor almost every . Here and are measurable functions. Note that when , we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion , where a sharp condition for continuity is that is in the Zygmund space for some . We also show that one can slightly relax the boundedness assumption on to an exponential class with , and still obtain continuous solutions when with . On the other hand, for all with , we construct a discontinuous solution with and , including an example with and . 

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5. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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