Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38].The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (?,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.     

Cohomology,extensions and automorphisms of skew braces

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial brace is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.     

A Triptycene-Based 2D MOF with Vertically Extended Structure for Improving the Electrocatalytic Performance of CO2 to Methane

Two-dimensional conductive metal-organic frameworks (2D-c-MOFs) have attracted extensive attention owing to their unique structures and physical-chemical properties. However, the planarly extended structure of 2D-c-MOFs usually limited the accessibility of the active sites. Herein, we designed a triptycene-based 2D vertically conductive MOF (2D-vc-MOF) by coordinating 2,3,6,7,14,15-hexahydroxyltriptycene (HHTC) with Cu²⁺. The vertically extended 2D-vc-MOF(Cu) possesses a weak interlayer interaction, which leads to a facile exfoliation to the nanosheet. Compared with the classical 2D-c-MOFs with planarly extended 2D structures, 2D-vc-MOF(Cu) exhibits a 100 % increased catalytic activity in terms of turnover number and a two-fold increased selectivity. Density functional theory (DFT) calculations further revealed that higher activity originated from the lower energy barriers of the vertically extended 2D structures during the CO₂ reduction reaction process.     

Asymmetric C1 Extension of Aldehydes through Biocatalytic Cascades for Stereodivergent Synthesis of Mandelic Acids

The development of mild, efficient, and enantioselective methods for preparing chiral building blocks from simple, renewable carbon units has been a long-term goal of the sustainable chemical industry. Mandelate derivatives are valuable pharmaceutical intermediates and chiral resolving agents, but their manufacture relies heavily on highly toxic cyanide. Herein, we report (S)-4-hydroxymandelate synthase (HmaS)-centered biocatalytic cascades for the synthesis of mandelates from benzaldehydes and glycine. We show that HmaS can be engineered to perform R-selective hydroxylation by single-point mutation, empowering the stereodivergent synthesis of both (S)- and (R)-mandelate derivatives. These biocatalytic cascades enabled the production of various mandelate derivatives with high atom economy as well as excellent yields (up to 98 %) and ee values (up to >99 %). This methodology offers an effective cyanide-free technology for greener and sustainable production of mandelate derivatives.