Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and ...Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.展开更多
The main purpose of this paper is to establish, using the Littlewood–Paley–Stein theory(in particular, the Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fouri...The main purpose of this paper is to establish, using the Littlewood–Paley–Stein theory(in particular, the Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp(Rn;A) associated with expensive dilation A:■Our main Theorem is the following: Assume that m(ξ) is a function on Rn satisfying ■with s > ζ--1(1/p-1/2). Then Tm is bounded from Hp(Rn;A) to Hp(Rn;A) for all 0 < p ≤ 1 and ■where A* denotes the transpose of A. Here we have used the notations mj(ξ) = m(A*jξ)φ(ξ) and φ(ξ) is a suitable cut-off function on Rn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on Rn.展开更多
In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))sati...In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))satisfying the Hormander's condition,f(x,u)∈C(Ω×R),g(x,u)is a Carathéodory function onΩ×R,andΩis an open bounded domain in R~n with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.展开更多
LET T:D→D′be a continuous linear operatorl, and K denote the distribution kernel of T. Assume that restriction of K to the set{( x, y ) ∈R^n x R^n, x≠y}satisfies the following "size" and "smoothness...LET T:D→D′be a continuous linear operatorl, and K denote the distribution kernel of T. Assume that restriction of K to the set{( x, y ) ∈R^n x R^n, x≠y}satisfies the following "size" and "smoothness" conditions:展开更多
基金Supported by the National Natural Science Foundation of China(11871436,12071437)。
文摘Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.
基金supported partly by NNSF of China(Grant No.11371056)supported by NNSF of China(Grant No.11801049)Technology Pro ject of Chongqing Education Committee(Grant No.KJQN201800514)
文摘The main purpose of this paper is to establish, using the Littlewood–Paley–Stein theory(in particular, the Littlewood–Paley–Stein square functions), a Calderón–Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces Hp(Rn;A) associated with expensive dilation A:■Our main Theorem is the following: Assume that m(ξ) is a function on Rn satisfying ■with s > ζ--1(1/p-1/2). Then Tm is bounded from Hp(Rn;A) to Hp(Rn;A) for all 0 < p ≤ 1 and ■where A* denotes the transpose of A. Here we have used the notations mj(ξ) = m(A*jξ)φ(ξ) and φ(ξ) is a suitable cut-off function on Rn, and Ws(A*) is an anisotropic Sobolev space associated with expansive dilation A* on Rn.
基金supported by National Natural Science Foundation of China(Grant No.12131017)supported by National Natural Science Foundation of China(Grant No.12201607)+3 种基金National Key R&D Program of China(Grant No.2022YFA1005602)Knowledge Innovation Program of Wuhan-Shuguang Project(Grant No.2023010201020286)China Postdoctoral Science Foundation(Grant No.2023T160655)supported by China National Postdoctoral Program for Innovative Talents(Grant No.BX20230270)。
文摘In this paper,we study the semilinear subelliptic equation{−△Xu=f(x,u)+g(x,u)inΩ,u=0 on∂Ω,where△X=−∑m i=1 Xi∗Xi is the self-adjoint Hormander operator associated with the vector fields X=(X_(1),X_(2),...,X_(m))satisfying the Hormander's condition,f(x,u)∈C(Ω×R),g(x,u)is a Carathéodory function onΩ×R,andΩis an open bounded domain in R~n with smooth boundary.Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values,we obtain two kinds of existence results for multiple weak solutions to the problem above.Furthermore,we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations.Compared with the classical elliptic cases,both approaches here have their own strengths in the degenerate cases.This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases.
文摘LET T:D→D′be a continuous linear operatorl, and K denote the distribution kernel of T. Assume that restriction of K to the set{( x, y ) ∈R^n x R^n, x≠y}satisfies the following "size" and "smoothness" conditions:
