Higher order Poincaré inequalities associated with linear operators on stratified groups and applications

 This paper considers the dual of anisotropic Sobolev spaces on any stratified groups 𝔾. For 0≤k<m and every linear bounded functional T on anisotropic Sobolev space W ^m−k,p (Ω) on Ω⊂𝔾, we derive a projection operator L from W ^m,p (Ω) to the collection 𝒫 _k+1 of polynomials of degree less than k+1 such that T(X ^I (Lu))=T(X ^I u) for all uW ^m,p (Ω) and multi-index I with d(I)≤k. We then prove a general Poincaré inequality involving this operator L and the linear functional T. As applications, we often choose a linear functional T such that the associated L is zero and consequently we can prove Poincaré inequalities of special interests. In particular, we obtain Poincaré inequalities for functions vanishing on tiny sets of positive Bessel capacity on stratified groups. Finally, we derive a Hedberg-Wolff type characterization of measures belonging to the dual of the fractional anisotropic Sobolev spaces W ^α,p 𝔾. Received: 25 March 2002; in final form: 10 September 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 46E35, 41A10, 22E25 The second author was supported partly by U.S NSF grant DMS99-70352 and the third author was supported partly by NNSF grant of China.     

Non-Quadratic Transverse Magnetoresistance in the Nodal Line Dirac Semimetal InBi

JETP Letters - The transverse magnetoresistance of the nodal line Dirac semimetal InBi has been studied. It is found that the magnetoresistance is not quadratic. In the region of small magnetic...     

Sub-elliptic global high order Poincaré inequalities in stratified Lie groups and applications

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot-Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincaré inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space , where 1?p<Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of f is globally Lp integrable; i.e., is finite (but assume that lower order sub-elliptic derivatives are only locally Lp integrable). We denote the space of such functions as Bm,p(G). We prove a high order Poincaré inequality for f minus a polynomial of order m−1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in Bm,p(G) modulus a finite-dimensional subspace.     

Über einige Derivate des Phenylindoxazens

Ohne ZusammenfassungHerr Prof. Gärtner wird die besondere Freundlichkeit haben, die Hydrobase auf ihre physiologischen Wirkungen zu untersuchen.     

Zur Bestimmung von Dampfdichten

Ohne Zusammenfassung     

Selbstth?tige Auswaschvorrichtungen

Ohne Zusammenfassung