We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey sp...We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.展开更多
The main purpose of this paper is to establish the HSrmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy Spaces for k≥ 3 using the multi- parameter Littlewood-Paley theory...The main purpose of this paper is to establish the HSrmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy Spaces for k≥ 3 using the multi- parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k 3, and the method works for all the cases k ≥ 3： Tmf（x1,x2,x3） =1/（（2π）＋n1＋n2＋n3） ∫ R n1×R n2×R n3 m（ξ）f（ξ）e 2π ix.ξ dξ. where x = （x1,x2,x3） ∈ Rn1 × Rn2 × R n3 and ξ = （ξ1,ξ2,ξ3） ∈ R n1 × Rn2 ×R n3. One of our main results is the following： Assume that m（ξ） is a function on Rn1＋n2＋n3 satisfying sup j,k,l ∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3）〈∞ with si 〉 ni（1/p-1/2） for 1 ≤ i ≤ 3. Then Tm is bounded from HP（R n1 × R n2 ×R n3） to HP（R n1 ×R n2 × R n3） for all 0 〈 p ≤ 1 and ｜｜Tm｜｜ Hp→Hp≤ sup j,k,l∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3） Moreover, the smoothness assumption on sl for 1 ≤ i ≤ 3 is optimal. Here we have used the notations mj,k,l （ξ）= m（2 j ξ1,2 k ξ2, 2 l ξ3） ψ（ξ1） ψ（ξ2） ψ（ξ3） and ψ（ξi） is a suitable cut-off function on R ni for 1 ≤ i ≤ 3, and W（s1,s2,s3） is a three-parameter Sobolev space on R n × R n2 × Rn 3. Because the Fefferman criterion breaks down in three parameters or more, we consider the Lp boundedness of the Littlewood-Paley square function of T mf to establish its boundedness on the multi-parameter Hardy spaces.展开更多
The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As...The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.展开更多
In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the varia...In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.展开更多
文摘We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.
文摘The main purpose of this paper is to establish the HSrmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy Spaces for k≥ 3 using the multi- parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k 3, and the method works for all the cases k ≥ 3： Tmf（x1,x2,x3） =1/（（2π）＋n1＋n2＋n3） ∫ R n1×R n2×R n3 m（ξ）f（ξ）e 2π ix.ξ dξ. where x = （x1,x2,x3） ∈ Rn1 × Rn2 × R n3 and ξ = （ξ1,ξ2,ξ3） ∈ R n1 × Rn2 ×R n3. One of our main results is the following： Assume that m（ξ） is a function on Rn1＋n2＋n3 satisfying sup j,k,l ∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3）〈∞ with si 〉 ni（1/p-1/2） for 1 ≤ i ≤ 3. Then Tm is bounded from HP（R n1 × R n2 ×R n3） to HP（R n1 ×R n2 × R n3） for all 0 〈 p ≤ 1 and ｜｜Tm｜｜ Hp→Hp≤ sup j,k,l∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3） Moreover, the smoothness assumption on sl for 1 ≤ i ≤ 3 is optimal. Here we have used the notations mj,k,l （ξ）= m（2 j ξ1,2 k ξ2, 2 l ξ3） ψ（ξ1） ψ（ξ2） ψ（ξ3） and ψ（ξi） is a suitable cut-off function on R ni for 1 ≤ i ≤ 3, and W（s1,s2,s3） is a three-parameter Sobolev space on R n × R n2 × Rn 3. Because the Fefferman criterion breaks down in three parameters or more, we consider the Lp boundedness of the Littlewood-Paley square function of T mf to establish its boundedness on the multi-parameter Hardy spaces.
基金Acknowledgements The authors would like to thank Professor Yonghua Mao for his helpful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11131003）, the Specialized Research Fund for the Doctoral Program of Higher Education （Grant No. 20100003110005）, the ＇985＇ project from the Ministry of Education in China, and the Fundamental Research Funds for the Central Universities.
文摘The mixed principal eigenvalue of p-Laplacian （equivalently, the optimal constant of weighted Hardy inequality in Lp space） is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.
基金National Natural Science Foundation of China (No.10671022)
文摘In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.
