Some embedding theorems for W 1,p (Ω,H n )

Abstract

For the maps on the Heisenberg group target, we prove a Poincaré type inequality.Applying this Poincaré type inequality, we obtain the corresponding versions of Sobolev and Rellich embedding theorems. §1 IntroductionThe Heisenberg group H n (see [3]) is the Lie group whose underlying manifold is C n × R with n ∈ N, endowed with the group law algebra g = R 2n+1 and a nonablian group law (z, t) · (z , t ) = (z + z , t + t + 2 Im (z · z )), ∀z, z ∈ C n , t,t ∈ R,(1.

Construction of Defect‐Rich Ni‐Fe‐Doped K0.23MnO2 Cubic Nanoflowers via Etching Prussian Blue Analogue for Efficient Overall Water Splitting

“…27 In the O 1s spectrum (Figure 2e), the peaks at 530.0 and 531.3 eV were attributed to O 2− and –OH in the oxidized manganese species 28. The increased OH peak intensity in Ni‐Fe‐K 0.23 MnO 2 CNFs‐300 was related to the formation and increase of active (oxy)hydroxides and oxygen vacancies, possibly due to the coaction of the annealing process and doping effect 13,29. Interestingly, the O 1s main peak of Ni‐Fe‐K 0.23 MnO 2 CNFs‐300 also significantly shifted toward a more negative level, suggesting the alteration of the bonding configuration between O and metal ions 30.…”

Influence of hematite morphology on the CO oxidation performance of Au/α-Fe2O3

“…Many efforts have been made in this direction. For example, the influence of the shape/morphology of CeO2 has been extensively studied in the context of gold-catalyzed CO oxidation [27][28][29][30][31]. Huang and co-authors reported that ceria nanorods showed a much higher gold dispersion than gold on conventional ceria, explaining the higher CO oxidation activity [28].…”

Cavity Enhanced Laser Cooling of Solids

“…Seletskiy used the multi-pass scheme to do some meaningful experiments and obtain a milestone progress, that is, they cool a high pure Yb 3+ -doped YLF to cryogenic temperature 12 in this year. To obtain an efficient absorption, another scheme was proposed to insert a Brewster-cut sample into a resonant cavity and make the cavity to be resonated with the pump laser, and then the maximum absorption of the pump laser can be achieved 2,[13][14][15][16][17][18] . Seletskiy demonstrated this scheme in the experiment 13 .…”

“…Fortunately, the intra-cavity scheme proposed by Heeg can solve this problem in the case of low absorption 2 . He also suggested that inserting a cooling sample into the resonant cavity of a laser with a gain (so-called intra-cavity enhancement scheme) to realize the maximum absorption of the pump laser 2,14,19 and carried out it in the experiment 3 . Besides of enhancement laser cooling of bulk solids using resonant cavity, we propose another new scheme to optically cool some film materials (i.e., filmier solids) by using a high fineness planar waveguide, which is similar to the cavity enhanced cooling scheme.…”