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1. 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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2. Logarithmic Sobolev Inequalities on Homogeneous Spaces

   https://doi.org/10.1093/imrn/rnae205  

   Gordina, Maria; Luo, Liangbing (September 2024, International Mathematics Research Notices)

   Abstract We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a sub-Riemannian structure induced by a transitive action by a Lie group. Then the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we show that the constant is independent of the dimension of the underlying spaces in several examples. 

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3. Sobolev spaces revisited

   https://doi.org/10.4171/RLM/976  

   Brezis, Haïm; Seeger, Andreas; Van Schaftingen, Jean; Yung, Po-Lam (January 2022, Atti della Accademia nazionale dei Lincei Rendiconti Lincei Matematica e applicazioni)
4. Fractional Sobolev metrics on spaces of immersions

   https://doi.org/10.1007/s00526-020-1719-5  

   Bauer, Martin; Harms, Philipp; Michor, Peter W. (April 2020, Calculus of Variations and Partial Differential Equations)
5. Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

   https://doi.org/10.3390/math10030522  

   Behzadan, Ali; Holst, Michael (February 2022, Mathematics)

   In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<1, and 1<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here. 

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