Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ&l...Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ</sup>（f） the Cesàro means of order δ of the Fourier-Laplace series of f.The special value λ∶=（n-2）/2 of δ is known as the critical index.In the case when n is even,this paper proves the existence of the‘rare’sequence {n<sub>k</sub>} such that the summability 1/N sum from k=1 to N σ<sub>n<sub>k</sub></sub><sup>λ</sup>（f）（x）→f（x）,N→∞ takes place at each Lebesgue point satisfying some antipole conditions.展开更多
We show that in a Q-doubling space （X, d, μ）, Q 〉 1, which satisfies a chain condition, if we have a Q-Poincare inequality for a pair of functions （u, g） where g ∈ LQ（X）, then u has Lebesgue points 7-th-a.e. ...We show that in a Q-doubling space （X, d, μ）, Q 〉 1, which satisfies a chain condition, if we have a Q-Poincare inequality for a pair of functions （u, g） where g ∈ LQ（X）, then u has Lebesgue points 7-th-a.e. for h（t） = log1-Q-c（1/t）. We also discuss how the existence of Lebesgue points follows for u ∈ W1,Q（x） where （X, d, μ） is a complete Q-doubling space supporting a Q-Poincar; inequality for Q 〉 1.展开更多
In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t）...In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]展开更多
In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a^b ∫_a^b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,...In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a^b ∫_a^b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ ∈ Λ [0,∞),(0.1)are given. Here f belongs to the function space L_1( <a,b >~2), where <a,b> is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].展开更多
We will give a survey on results concerning Girsanov transforma- tions, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will b...We will give a survey on results concerning Girsanov transforma- tions, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will be arisen.展开更多
基金Project supported by the Natural Science Foundation of China under Grant ≠ 19771009
文摘Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ</sup>（f） the Cesàro means of order δ of the Fourier-Laplace series of f.The special value λ∶=（n-2）/2 of δ is known as the critical index.In the case when n is even,this paper proves the existence of the‘rare’sequence {n<sub>k</sub>} such that the summability 1/N sum from k=1 to N σ<sub>n<sub>k</sub></sub><sup>λ</sup>（f）（x）→f（x）,N→∞ takes place at each Lebesgue point satisfying some antipole conditions.
基金supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research(Grant No.271983)
文摘We show that in a Q-doubling space （X, d, μ）, Q 〉 1, which satisfies a chain condition, if we have a Q-Poincare inequality for a pair of functions （u, g） where g ∈ LQ（X）, then u has Lebesgue points 7-th-a.e. for h（t） = log1-Q-c（1/t）. We also discuss how the existence of Lebesgue points follows for u ∈ W1,Q（x） where （X, d, μ） is a complete Q-doubling space supporting a Q-Poincar; inequality for Q 〉 1.
文摘In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]
文摘In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a^b ∫_a^b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ ∈ Λ [0,∞),(0.1)are given. Here f belongs to the function space L_1( <a,b >~2), where <a,b> is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].
文摘We will give a survey on results concerning Girsanov transforma- tions, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will be arisen.
