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1. Counterexamples for multi-parameter weighted paraproducts

   https://doi.org/10.5802/crmath.52  

   Mozolyako, Pavel; Psaromiligkos, Georgios; Volberg, Alexander (February 2020, Comptes rendus mathematiques de lAcademie des sciences)

   We build the plethora of counterexamples to bi-parameter two weight embedding theorems. Two weight one parameter embedding results (which is the same as results of boundedness of two weight classical paraproducts, or two weight Carleson embedding theorems) are well known since the works of Sawyer in the 80’s. Bi-parameter case was considered by S. Y. A. Chang and R. Fefferman but only when underlying measure is Lebesgue measure. The embedding of holomorphic functions on bi-disc requires general input measure. In [9] we classified such embeddings if the output measure has tensor structure. In this note we give examples that without tensor structure requirement all results break down. 

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2. Bi-parameter embedding and measures with restricted energy conditions

   Arcozzi, Nicola; Holmes, Irina; Volberg, Alexander (April 2020, Mathematische Annalen)

   Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree. 

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3. A modulation invariant Carleson embedding theorem outside local L^2

   Di Plinio, Francesco; Ou, Yumeng (April 2016, Journal d’analyse mathématique)

   The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer Lp spaces for the wave packet transform of functions in L^p, in the 2≤p≤∞ range referred to as local L^2. In this article, we formulate a suitable extension of this theory to exponents 1<2, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation. 

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4. Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

   https://doi.org/http://dx.doi.org/10.1090/tran/7536  

   Cavero, J.; Hofmann, S.; Martell, J.M. (January 2019, Transactions of the American Mathematical Society)

   Let $$\Omega\subset\re^{n+1}$$, $$n\ge 2$$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $$\partial\Omega$$ is $$n$$-dimensional Ahlfors regular. Consider $$L_0$$ and $$L$$ two real symmetric divergence form elliptic operators and let $$\omega_{L_0}$$, $$\omega_L$$ be the associated elliptic measures. We show that if $$\omega_{L_0}\in A_\infty(\sigma)$$, where $$\sigma=H^n\lfloor_{\partial\Omega}$$, and $$L$$ is a perturbation of $$L_0$$ (in the sense that the discrepancy between $$L_0$$ and $$L$$ satisfies certain Carleson measure condition), then $$\omega_L\in A_\infty(\sigma)$$. Moreover, if $$L$$ is a sufficiently small perturbation of $$L_0$$, then one can preserve the reverse H√∂lder classes, that is, if for some $$1<\infty$$, one has $$\omega_{L_0}\in RH_p(\sigma)$$ then $$\omega_{L}\in RH_p(\sigma)$$. Equivalently, if the Dirichlet problem with data in $$L^{p'}(\sigma)$$ is solvable for $$L_0$$ then so it is for $$L$$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $$A_\infty(\sigma)$$ then necessarily $$\Omega$$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable. 

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5. Bi-Lipschitz arcs in metric spaces with controlled geometry

   https://doi.org/10.4171/RMI/1484  

   Honeycutt, Jacob; Vellis, Vyron; Zimmerman, Scott (May 2024, Revista Matemática Iberoamericana)

   In this paper, we generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by bi-Lipschitz curves. 

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