Hardy's inequality and the boundary size

We establish a self-improving property of the Hardy inequality and an estimate on the size of the boundary of a domain supporting a Hardy inequality.

     

A Radial Uniqueness Theorem for Sobolev Functions

We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rⁿ and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .     

Dimension gap under Sobolev mappings

We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Hölder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.     

Kinetics and mechanistic details of bulk ZnO dissolution using a thiol–imidazole system

Oxide dissolution is important for metal extraction from ores and has become an attractive route for the preparation of inks for thin film solution deposition; however, oxide dissolution is often kinetically challenging. While binary “alkahest” systems comprised of thiols and N-donor species, such as amines, are known to dissolve a wide range of oxides, the mechanism of dissolution and identity of the resulting solute(s) remain unstudied. Here, we demonstrate facile dissolution of both bulk synthetic and natural mineral ZnO samples using an “alkahest” that operates via reaction with thiophenol and 1-methylimidazole (MeIm) to give a single, pseudotetrahedral Zn(SPh)₂(MeIm)₂ molecular solute identified by X-ray crystallography. The kinetics of ZnO dissolution were measured using solution ¹H NMR, and the reaction was found to be zero-order in the presence of excess ligands, with more electron withdrawing para-substituted thiophenols resulting in faster dissolution. A negative entropy of activation was measured by Eyring analysis, indicating associative ligand binding in, or prior to, the rate determining step. Combined experimental and computational surface binding studies on ZnO reveal stronger, irreversible thiophenol binding compared to MeIm, leading to a proposed dissolution mechanism initiated by thiol binding to the ZnO surface with the liberation of water, followed by alternating MeIm and thiolate ligand additions, and ultimately cleavage of the ligated zinc complex from the ZnO surface. Design rules garnered from the mechanistic insight provided by this study should inform the dissolution of other bulk oxides into inks for solution processed thin films.

Oxide dissolution is important for metal extraction from ores and has become an attractive route for the preparation of inks for thin film solution deposition; however, oxide dissolution is often kinetically challenging.     

LR-CAHSQC: an application of a Carr-Purcell-Meiboom-Gill-type sequence to heteronuclear multiple bond correlation spectroscopy

A new pulse sequence, long-range CPMG-adjusted heteronuclear single quantum coherence (LR-CAHSQC), is proposed for the determination of long-range JCH coupling constants from a long-range 1H-13C correlation experiment. The long-range heteronuclear coupling constants can be directly extracted from COSY-type antiphase peak patterns. The current approach utilizes CPMG-sequences for polarization transfer, and thus avoids the evolution of homonuclear JHH couplings, which normally may introduce abnormalities into the cross peak pattern. The differences between LR-CAHSQC and normal LR-HSQC are discussed.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established.