一类振荡奇异积分及其局部化算子的HK_p（R~n）－有界性

记１＜Ｐ＜∞．本文考察了一类卷积型的振荡奇异积分及其局部化算子的ＨＫｐ（Ｒｎ）－有界性．     

Infinite Dimensional Oscillatory Integrals with Polynomial Phase and Applications to Higher-Order Heat-Type Equations

The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation for solutions of a general class of higher-order heat-type equations.     

Damped Oscillatory Integrals and Maximal Operators

In the present paper, we consider estimates of the Fourier transform of Borel measures concentrated on analytic hypersurfaces and containing a mitigating factor. The mitigating factors are expressed in terms of principal curvatures of the surface. The resulting estimates are applied to investigating the boundedness of the corresponding maximal operators.     

Estimates of Oscillatory Integrals with a Damping Factor

Mathematical Notes - Estimates of the Fourier transform of measures concentrated on analytic hypersurfaces containing a damping factor are considered. The solution of the Sogge and Stein problem on...     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

Uniform Estimates for Oscillatory Integrals and Volumes with the Varchenko Phase

We obtain uniform estimates for oscillatory integrals and volumes with a perturbed Varchenko phase, which follow from the statements of general theorems (the case of an arbitrary perturbation). The estimates turn out to be nearly exact. We also derive exact estimates by an argument similar to the proofs of general theorems (the case of a partial perturbation).     

Polynomial Carleson Operators Along Monomial Curves in the Plane

We prove \(L^p\) bounds for partial polynomial Carleson operators along monomial curves \((t,t^m)\) in the plane \(\mathbb {R}^2\) with a phase polynomial consisting of a single monomial. These operators are “partial” in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, \(L^2\) bounds for partial operators along curves imply the full strength of the \(L^2\) bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator.     

On Some Measure Convolution Operators in Neutron Transport Theory

We show how some fundamental spectral properties of neutron transport semigroups in L ^p spaces (1<p<+∞), such as stability of essential spectra or critical spectra and related results, can be inferred from the study of two measure convolution operators on  \(\mathbb{R}^{n}\) .     

带多项式相位的高维振荡积分算子的有界性

考虑如下的振荡积分算子:T_(m,k,n)f(x):=∫_(R~n)e~(i(x_1~2+…+x_n~2))~m(y_1~2+…+y_n~2)~kf(y)dy,其中函数f为定义在R~n上的Schwartz函数,并且满足m,k0.本文给出算子T_(m,k,n).从L~p(R~n)(1≤p∞)到L~q(R~n)有界的一个充分必要条件.此外,我们还证明了算子T_(m,k,n)把L~1(R~n)映到C_0(R~n).     

Projective Generation of Curves in Polynomial Extensions of an Affine Domain (II)

Let Abe an affine domain of dimension nover an algebraically closed field kof characteristic 0. Let I A[T]be a local complete intersection ideal of height nsuch that I/I2 is generated by n elements. It is proved that there exists a projective A[T]module Pof rank nsuch that Iis a surjective image of P.     

Volterra Convolution Operators with Values in Rearrangement Invariant Spaces

The Volterra convolution operator Vf(x) = ^x₀(x–y)f(y)dy,where (·) is a non-negative non-decreasing integrablekernel on [0, 1], is considered. Under certain conditions onthe kernel , the maximal Banach function space on [0, 1] onwhich the Volterra operator is a continuous linear operatorwith values in a given rearrangement invariant function spaceon [0, 1] is identified in terms of interpolation spaces. Thecompactness of the operator on this space is studied.     

Contour Integrals and Vector Calculus on Fractal Curves and Interfaces

This article develops the definition of contour integrals over fractal curves in the plane by introducing the notion of oriented Iterated Function Systems and directional pseudo-measures. An expression for the contour integral of continuous functions over fractal interfaces is obtained through renormalization. As a result, a vector calculus on fractal interfaces which are boundaries of regular two-dimensional domains is developed by extending Greens theorem in the plane, also to fractal curves.The use of moment analysis makes it possible to obtain recursive relations and closed-form expressions for contour integrals of algebraic functions. Several physical applications are analyzed, including the properties of double-layer potentials and connections with the solution of the Dirichlet problem on bounded two-dimensional domains possessing fractal boundaries.     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

Convolution Operators on Expanding Polyhedra: Limits of the Norms of Inverse Operators and Pseudospectra

We consider matrix convolution operators with integrable kernels on expanding polyhedra. We study their connection with convolution operators on the cones at the vertices of polyhedra. We prove that the norm of the inverse operator on a polyhedron tends to the maximum of the norms of the inverse operators on the cones, and the pseudospectrum tends to the union of the corresponding pseudospectra. The study bases on the local method adapted to this kind of problems.     

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.     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.