In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit posit...In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit positive constant C(n,p,λ),depending only on n,p,λ,such that,if∫MSn/2dM<∞,∫M(S−λ)n/2+dM<C(n,p,λ),then Mn is a totally geodetic sphere,where S denotes the square of the second fundamental form of the submanifold and∫+=max{0,f}.Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.展开更多
We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner...We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.展开更多
The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic ...The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szeg?, kemel and the Cauchy singular integral operator.展开更多
Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite diff...Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.展开更多
基金supported by the National Natural Science Foundation of China(11531012,12071424,12171423)the Scientific Research Project of Shaoxing University(2021LG016)。
文摘In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit positive constant C(n,p,λ),depending only on n,p,λ,such that,if∫MSn/2dM<∞,∫M(S−λ)n/2+dM<C(n,p,λ),then Mn is a totally geodetic sphere,where S denotes the square of the second fundamental form of the submanifold and∫+=max{0,f}.Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.
文摘We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.
文摘The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1 <p < ∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalise the work of L. K. Hua, A. Korányli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szeg?, kemel and the Cauchy singular integral operator.
基金Supported by King Saud University, Deanship of Scientific Research, College of Science Research Center
文摘Dual Toeplitz operators on the Hardy space of the unit circle are anti-unitarily equivalent to Toeplitz operators. In higher dimensions, for instance on the unit sphere, dual Toeplitz operators might behave quite differently and, therefore, seem to be a worth studying new class of Toeplitz-type operators. The purpose of this paper is to introduce and start a systematic investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space of the unit sphere in Cn . In particular, we establish a corresponding spectral inclusion theorem and a Brown-Halmos type theorem. On the other hand, we characterize commuting dual Toeplitz operators as well as normal and quasinormal ones.
