Abstract. We examine the dependence on the Ap norm of w of the operator norms of singular integrals, maximal functions, and other operators in LP(w). We also examine connections between some fairly general reverse Jensen inequalities and the Ap and RHP weight conditions.
Abstract. We establish necessary conditions for the validity of Sobolev-Poincaré type inequalities. We give a geometric characterisation for the validity of this inequality for simply connected plane domains.
We prove the equivalence of three different geometric properties of metric-measure spaces with controlled geometry. The first property is the Gromov hyperbolicity of the quasihyperbolic metric. The second is a slice condition and the third is a combination of the Gehring-Hayman property and a separation condition.
Abstract. We investigate geometric conditions related to H older imbeddings, and show, among other things, that the only bounded Euclidean domains of the form U V that are quasiconformally equivalent to inner uniform domains are inner uniform domains. Inner uniform domains, as de ned by V ais al a V5], satisfy a uniformity condition with respect to the inner Euclidean metric. These domains form a class intermediate between uniform and John domains and, in particular, they include all Lipschitz domains see Section 1 for de nitions. We prove the following theorem which indicates that this class is well suited to the study of quasiconformal equivalence.
Physical interactions between proteins are essential for most biological processes governing life1. However, the molecular determinants of such interactions have been challenging to understand, even as genomic, proteomic and structural data increase. This knowledge gap has been a major obstacle for the comprehensive understanding of cellular protein–protein interaction networks and for the de novo design of protein binders that are crucial for synthetic biology and translational applications2–9. Here we use a geometric deep-learning framework operating on protein surfaces that generates fingerprints to describe geometric and chemical features that are critical to drive protein–protein interactions10. We hypothesized that these fingerprints capture the key aspects of molecular recognition that represent a new paradigm in the computational design of novel protein interactions. As a proof of principle, we computationally designed several de novo protein binders to engage four protein targets: SARS-CoV-2 spike, PD-1, PD-L1 and CTLA-4. Several designs were experimentally optimized, whereas others were generated purely in silico, reaching nanomolar affinity with structural and mutational characterization showing highly accurate predictions. Overall, our surface-centric approach captures the physical and chemical determinants of molecular recognition, enabling an approach for the de novo design of protein interactions and, more broadly, of artificial proteins with function.
Abstract. We define a notion of inversion valid in the general metric space setting. We establish several basic facts concerning inversions; e.g., they are quasimöbius homeomorphisms and quasihyperbolically bilipschitz. In a certain sense, inversion is dual to sphericalization. We demonstrate that both inversion and sphericalization preserve local quasiconvexity and annular quasiconvexity as well as uniformity.
Abstract. The concept of an H-chain set in a doubling space X, which generalizes that of a Hölder domain in Euclidean space, is defined and investigated. We show that every H-chain set is mean porous, and that its outer layer has measure bounded by a power of its thickness. As a consequence, we show that a John-Nirenberg type inequality holds on an open subset Ω of X if, and often only if, Ω is an H-chain set.
Introductionif either of the following equivalent conditions is satisfied:where Q is an arbitrary cube and 2Q is its concentric double dilate. We write u ∈ EI(Ω) if it satisfies the integrability conditionand we define u EI(Ω) to be the smallest C for which this condition holds; the peculiar "16" on the right is merely for later convenience (a comparable norm is obtained by replacing it by any fixed factor larger than 1).
We study the action of fractional differentiation and integration on weighted Bergman spaces and also the Taylor coefficients of functions in certain subclasses of these spaces. We then derive several criteria for the multipliers between such spaces, complementing and extending various recent results. Univalent Bergman functions are also considered.
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