Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describin...Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].展开更多
A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
In this paper, the author establishes a discrete characterization of the Herz-type Triebel-Lizorkin spaces which is used to prove the boundedness of pseudo-differential operators on these function spaces.
In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class A q and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.
We give a Clifford algebra characterization of classical Hardy spaces Hp(R+n+1),0<p≤1,The man new feature is the role played by the matrix valued Clifford algebra.
The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.
A multidirectional discrete space consists of numerous hypercubic lattices each of which contains one of the spatial directions. In such a space, several groups of lattices can be distinguished with a certain property...A multidirectional discrete space consists of numerous hypercubic lattices each of which contains one of the spatial directions. In such a space, several groups of lattices can be distinguished with a certain property. Each group is determined by the number of lattices it comprises, forming the characterizing numbers of the space. Using the specific properties of a multidirectional discrete space, it is shown that some of the characterizing numbers can be associated with a physical constant. The fine structure constant appears to be equal to the ratio of two of these numbers, which offers the possibility of calculating the series of smallest numerical values of these numbers. With these values, a reasoned estimate can be made of the upper limit of the smallest distance of the discrete space of approximately the Planck length.展开更多
This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from...This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.展开更多
We consider Hardy spaces with variable exponents defined by grand maximal function on the Heisenberg group. Then we introduce some equivalent characterizations of variable Hardy spaces. By using atomic decomposition a...We consider Hardy spaces with variable exponents defined by grand maximal function on the Heisenberg group. Then we introduce some equivalent characterizations of variable Hardy spaces. By using atomic decomposition and molecular decomposition we get the boundedness of singular integral operators on variable Hardy spaces. We investigate the Littlewood-Paley characterization by virtue of the boundedness of singular integral operators.展开更多
文摘Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].
文摘A new maximal function is introduced in the dual spaces of test function spaces on spaces of homogeneous type. Using this maximal function, we get new characterization of atomic H^p spaces.
文摘In this paper, the author establishes a discrete characterization of the Herz-type Triebel-Lizorkin spaces which is used to prove the boundedness of pseudo-differential operators on these function spaces.
基金The projectsupported by NSF of China and the Foundation of Advanced Research Center of Zhongshan Universi-ty
文摘In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class A q and our weighted Triebel-Lizorkin spaces are weighted atomic Triebel-Lizorkin spaces.
文摘We give a Clifford algebra characterization of classical Hardy spaces Hp(R+n+1),0<p≤1,The man new feature is the role played by the matrix valued Clifford algebra.
基金This work was supported by KBN grant 2 P301 019 06
文摘The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.
文摘A multidirectional discrete space consists of numerous hypercubic lattices each of which contains one of the spatial directions. In such a space, several groups of lattices can be distinguished with a certain property. Each group is determined by the number of lattices it comprises, forming the characterizing numbers of the space. Using the specific properties of a multidirectional discrete space, it is shown that some of the characterizing numbers can be associated with a physical constant. The fine structure constant appears to be equal to the ratio of two of these numbers, which offers the possibility of calculating the series of smallest numerical values of these numbers. With these values, a reasoned estimate can be made of the upper limit of the smallest distance of the discrete space of approximately the Planck length.
基金supported by National Natural Science Foundation of China(Grant No.11661075)
文摘This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.
文摘We consider Hardy spaces with variable exponents defined by grand maximal function on the Heisenberg group. Then we introduce some equivalent characterizations of variable Hardy spaces. By using atomic decomposition and molecular decomposition we get the boundedness of singular integral operators on variable Hardy spaces. We investigate the Littlewood-Paley characterization by virtue of the boundedness of singular integral operators.
