In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded...In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.展开更多
The iterated spherical average∆(A1)^(N)is an important operator in harmonic analysis,and has very important applications in approximation theory and probability theory,where∆is the Laplacian,A_(1)is the unit spherical...The iterated spherical average∆(A1)^(N)is an important operator in harmonic analysis,and has very important applications in approximation theory and probability theory,where∆is the Laplacian,A_(1)is the unit spherical average and(A1)^(N)is its iteration.In this paper,we mainly study the sufficient and necessary conditions for the boundedness of this operator in Besov-Lipschitz space,and prove the boundedness of the operator in Triebel-Lizorkin space.Moreover,we use above conclusions to improve the existing results of the boundedness of this operator in L^(p)space.展开更多
Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<s...Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>） such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>（Ω）={f∈B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>）: f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.展开更多
In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the c...In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.展开更多
基金Supported in part by the Natural Science Foundation of China under the Grant 10771221Natural Science Foundation of Beijing under the Grant 1092004
文摘In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by H^ab, is bounded from L^Pxy （R＋） to L^qxδ （R＋） with the bound explicitly worked out.
基金Project supported by the NSFC(No.10171111)and the Foundation of Advanced Research Center,zhongshan University.The second author is partially supported by a grant from Australia Research Council and NSF of Guangdong Province
文摘The iterated spherical average∆(A1)^(N)is an important operator in harmonic analysis,and has very important applications in approximation theory and probability theory,where∆is the Laplacian,A_(1)is the unit spherical average and(A1)^(N)is its iteration.In this paper,we mainly study the sufficient and necessary conditions for the boundedness of this operator in Besov-Lipschitz space,and prove the boundedness of the operator in Triebel-Lizorkin space.Moreover,we use above conclusions to improve the existing results of the boundedness of this operator in L^(p)space.
基金This research was partially supported by the SEDFthe NNSF of China.
文摘Let Ω be a bounded Lipschitz domain. Define B<sub>1,r</sub><sup>0.1</sup>（Ω）={f∈L<sup>1</sup>（Ω）: there is an F∈ B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>） such that F|Ω=f| and B<sub>1,z</sub><sup>0.1</sup>（Ω）={f∈B<sub>1</sub><sup>0.1</sup>（R<sup>n</sup>）: f=0 on R<sup>n</sup>\}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R<sub>+</sub><sup>n</sup>.
基金supported by Natural Science Foundation of Inner Mongolia(Grant No.2011MS0103)supported by National Natural Science Foundation of China(Grant No.10671019)
文摘In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.
