Triebel-Lizorkin空间和Besov空间上关于一类带Hardy核的振荡奇异积分

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.     

Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces

In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.     

Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón's identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting.     

Oscillatory integrals for phase functions having certain degenerate critical points

The paper is concerned with oscillatory integrals for phase functions having certain de- generate critical points. Under a finite type condition of phase functions we show the estimate of oscillatory integrals of the first kind. The decay of the oscillatory integral depends on indices of the finite type, the spatial dimension and the symbol.     

ROUGH OPERATORS AND COMMUTATORS ON HOMOGENEOUS WEIGHTED HERZ SPACES

The authors establish the boundedness on homogeneous weighted Herz spaces for a large class of rough operators and their commutators with BMO functions. In particular, the Calderon-Zygmund singular integrals and the rough R. Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.     

ROUGH OPERATORS AND COMMUTATORS ON HOMOGENEOUS WEIGHTED HERZ SPACES

The authors esstablish the boundedness on homogeneous weighted Herz spaces for a large class of rough operators and their commutators with BMO functions.In particular ,the Cadderon-Zygmund singular integrals and the rough R.Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.     

Rough operators and commutators on homogeneous weighted Herz spaces

The authors establish the boundedness on homogeneous weighted Herz spaces for a large class of rough grals and the rough R. Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.     

Boundedness of Multilinear Oscillatory Singular Integral on Weighted Weak Hardy Spaces

In this paper, by using the atomic decomposition of the weighted weak Hardy space WH_ω~1(R~n), the authors discuss a class of multilinear oscillatory singular integrals and obtain their boundedness from the weighted weak Hardy space WH_ω~1(R~n) to the weighted weak Lebesgue space WL_ω~1(R~n) for ω∈A_1(R~n).     

APPLICATIONS OF HERZ-TYPE TRIEBEL-LIZORKIN SPACES

In this paper, the authors first establish the connections between the Herz-type Triebel-Lizorkin spaces and the well-known Herz-type spaces; the authors then studythe pointwise multipliers for the Herz-type Triebel-Lizorkin spaces and show that pseudo-differential operators are bounded on these spaces by using pointwise multipliers.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

Boundedness of an oscillating multiplier on Triebel-Lizorkin spaces

Abstract In this paper, we study Triebel-Lizorkin space estimates for an oscillating multiplier mΩ,α,β. This operator was initially studied by Wainger and by Fefferman-Stein in the Lebesgue spaces. We obtain the boundedness results on the Triebel-Lizorkin space Fpα,q（R^n） for different p, q.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy’s Theorem for Jacobi transform by using the sharp estimate for the heat kernel.     

Commutators generated by multilinear Calderón-Zygmund type singular integral and Lipschitz functions

In this paper, we establish the boundedness of commutators generated by the multilinear Calderon- Zygmud type singular integrals and Lipschitz functions on the Triebel-Lizorkin space and Lipschitz spaces.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Certain averaging operators on Triebel-Lizorkin spaces

Abstract. In this article, we study the boundedness properties of the averaging operator S_t^γon Triebel-Lizorkin spaces■ for various p, q. As an application, we obtain the norm convergence rate for S_t^γ(f ) on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S_t^γis given.     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

本刊英文版Vol.29(2013),No.11论文摘要(英文)

<正>Vitali Type Convergence Theorems for Banach Space Valued Integrals Marek BALCERZAK Kazimierz MUSIAL Abstract Let(Ω,∑,μ)be a complete probability space and let X be a Banach space.We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem.We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.On Universally Left-stability ofε-Isometry     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

数值积分的可选择误差界(英文)

Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.     

Development of the Berezin Number Inequalities

We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.