A restriction theorem for the quaternion Heisenberg group

We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L p to L p if 1 ≤ p ≤ 4 3 . This is different from the Heisenberg group, on which the restriction operator is not bounded from L p to L p unless p = 1.     

Strong Unique Continuation Properties of Generalized Baouendi–Grushin Operators

This paper gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi–Grushin operator, which generalizes the result due to Aronszaijn, Krzywicki, and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients.     

一类Grushin亚椭圆算子的特征集

研究了与亚椭圆算子2x x 2β2y相对应的G rush in平面上余维数为1的子流形S的特征集C(S)在对应的G rush in度量下的H ausdorff维数和H ausdorff测度.     

带势加权散度形式的 Grushin 型退化椭圆算子的 Dirichlet 特征值的上界*

本文考虑了带有位势的散度形式的 Grushin 型退化椭圆算子的 Dirichlet 加权特征值的估计.利 用傅里叶变换的方法得到了特征值的精确下界估计.然后通过试验函数的方法得到了特征值上界的杨型 不等式.     

Blowup solutions of Grushin’s operator

In this note, we consider the blowup phenomenon of Grushin’s operator. By using the knowledge of probability, we first get an expression of heat kernel of Grushin’s operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solution will blow up in finite time.     

Morse指数与含Grushin算子的Liouville型定理

本文研究退化椭圆型方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Rm×Rk和方程-Δxu-(α+1)2|x|~(2α)Δyu=|u|~(p-1)u,(x,y)∈Π的Liouville型定理,其中-Δx-(α+1)2|x|~(2α)Δy是Grushin算子,Π={(x,y)∈Rm×Rk:x10}或{(x,y)∈Rm×Rk:y10}.本文将证明,当1p(Q+2)/(Q-2)时,上述方程Morse指数有限的有界解只有零解,其中Q=m+(α+1)k为齐次空间的维数,因此,本文将Laplace方程的结果推广到含Grushin算子的方程.     

加权的退化椭圆系统稳定解的Liouville定理

该文研究了加权的退化椭圆系统{-△Gu =ω(x)v, RN=RN1×RN2,-△Gv =ω(x)uq,其中△Gu=△xu+(a+1)2|x|2α△yu是Grushin算子, a,β≥0,q＞1,ω(x)=(1+||x||2(a+1))β/2(α+1).超临界指数正稳定解的Liouville定理被建立.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

ON MIN＇S ZETA-FUNCTION AND HERMITE ELLIPTIC OPERATOR

In this note, the authors study some fundamental properties on a Min＇s zeta- function and explore its connection with Hermite elliptic operator.     

A Weighted Trudinger–Moser Inequality on R^N and Its Application to Grushin Operator

Let x=(x',x')with x'∈Rk and x'∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.     

Liouville-type theorem for nonlinear elliptic equations involving p-Laplace–type Grushin operators

In this article, we prove the Liouville-type theorem for stable solutions of weighted p-Laplace–type Grushin equations (1) and (2) where p ≥ 2, q>0 and are nonnegative functions satisfying and as ‖z‖_G ≥ R₀ with p−N_γ<b<θ+p, R₀,C_i(i=1,2) are some positive constants. ∇_G=(∇_x,(1+γ)|x|^γ∇_y),γ ≥ 0, and The results hold true for N_γ<μ₀(p,b,θ) in 1 and q>q_c(p,N_γ,b,θ) in 2 . Here, μ₀ and q_c are new exponents, which are always larger than the classical critical ones and depend on the parameters p,b and θ. N_γ=N₁+(1+γ)N₂ is the homogeneous dimension of      

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.