In this paper, we establish the L^p （R-n＋1） boundedness for the commutators of singular integrals associated to surfaces of revolution, { （t,Ф （ ｜ t｜ ） ）： t ∈R^n }, with rough kernels Ω∈ L（IogL）^2（sn...In this paper, we establish the L^p （R-n＋1） boundedness for the commutators of singular integrals associated to surfaces of revolution, { （t,Ф （ ｜ t｜ ） ）： t ∈R^n }, with rough kernels Ω∈ L（IogL）^2（sn^-1）, if Ф（｜t｜） = ｜t｜.展开更多
In this article, certain Marcinkiewicz integral operators associated to surfaces of revolution on product domains were studied. The Lp boundedness for these operators are established under some rather weak conditions ...In this article, certain Marcinkiewicz integral operators associated to surfaces of revolution on product domains were studied. The Lp boundedness for these operators are established under some rather weak conditions on kernels. The main results essentially improve and extend some known results.展开更多
Here we consider the following strongly singular integral TΩ,γ,α,βf（x,t）=∫R^ne^i｜y｜^-βΩ（y/｜y｜）/｜y｜^n＋af（x-y,t-γ（｜y｜））dy, where Ω∈L^p（S^n-1）,p〉1,n〉1,α〉0 and γis convex on （0,∞）.We p...Here we consider the following strongly singular integral TΩ,γ,α,βf（x,t）=∫R^ne^i｜y｜^-βΩ（y/｜y｜）/｜y｜^n＋af（x-y,t-γ（｜y｜））dy, where Ω∈L^p（S^n-1）,p〉1,n〉1,α〉0 and γis convex on （0,∞）.We prove that there exists A（p,n） 〉 0 such that if β 〉 A（p,n） （1 ＋α）, then TΩ,γ,α,β is bounded from L^2 （R^n＋1） to itself and the constant is independent of γ Furthermore,when Ω∈ C^∞ （S^n-1 ）, we will show that TΩ,γ,α,β is bounded from L^2 （R^n＋l） to itself only if β〉 2α and the constant is independent of γ.展开更多
We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among th...We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.展开更多
基金supported by NSF of China(Grant No.11471033)NCET of China(Grant No.NCET-11-0574)the Fundamental Research Funds for the Central Universities(FRF-TP-12-006B)
文摘In this paper, we establish the L^p （R-n＋1） boundedness for the commutators of singular integrals associated to surfaces of revolution, { （t,Ф （ ｜ t｜ ） ）： t ∈R^n }, with rough kernels Ω∈ L（IogL）^2（sn^-1）, if Ф（｜t｜） = ｜t｜.
基金Supported by the NSF of China (G10571122) the NFS of Fujian Province of China (Z0511004)
文摘In this article, certain Marcinkiewicz integral operators associated to surfaces of revolution on product domains were studied. The Lp boundedness for these operators are established under some rather weak conditions on kernels. The main results essentially improve and extend some known results.
基金supported by NSFC(Nos.11471288,11371136 and 11671363)NSFZJ(LY14A010015)China Scholarship Council
文摘Here we consider the following strongly singular integral TΩ,γ,α,βf（x,t）=∫R^ne^i｜y｜^-βΩ（y/｜y｜）/｜y｜^n＋af（x-y,t-γ（｜y｜））dy, where Ω∈L^p（S^n-1）,p〉1,n〉1,α〉0 and γis convex on （0,∞）.We prove that there exists A（p,n） 〉 0 such that if β 〉 A（p,n） （1 ＋α）, then TΩ,γ,α,β is bounded from L^2 （R^n＋1） to itself and the constant is independent of γ Furthermore,when Ω∈ C^∞ （S^n-1 ）, we will show that TΩ,γ,α,β is bounded from L^2 （R^n＋l） to itself only if β〉 2α and the constant is independent of γ.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11771237).
文摘We study a functional modelling the progressive lens design,which is a combination of Willmore functional and total Gauss curvature.First,we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y=f(x)about the x-axis.Then,choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional,we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals.Our results not only provide a strictly mathematical proof for numerical methods,but also give a more reasonable and more extensive choice for the background surfaces.
