This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from t...This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from the known Poincare estimates. The main ingredients of the proofs rely on the fractional maximal functions. These results evidently have applications to the regularity of subelliptic PDE.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
A boundedness criterion is set up for some convolution operators on a compact Lie group.By this criterion a Hormander multiplier theorem is proved in the Hardy spaces on SU(2).
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
Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（...Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（R^n） into the weak L^1 （R^n） space and from certain atomic Hardy space Hb^1 （R^n） into the Lebesgue space L^1 （R^n）, when the multiplier m satisfies the conditions of Hoermander type.展开更多
基金Research supported in part by he National Sience Foundation Grant # DMS93-15963
文摘This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from the known Poincare estimates. The main ingredients of the proofs rely on the fractional maximal functions. These results evidently have applications to the regularity of subelliptic PDE.
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.
文摘A boundedness criterion is set up for some convolution operators on a compact Lie group.By this criterion a Hormander multiplier theorem is proved in the Hardy spaces on SU(2).
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 by the Research Funds of Zhejiaug Sci-Tech University (No. 0313055-Y).
文摘Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（R^n） into the weak L^1 （R^n） space and from certain atomic Hardy space Hb^1 （R^n） into the Lebesgue space L^1 （R^n）, when the multiplier m satisfies the conditions of Hoermander type.
