THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e^(2)-n/2,thenT_(α,φ)is a bounded operator from L^(∞()R^(n))to BMO(R^(n)).

Evaluation of Some Fourier Integrals in N-Dimensional Space for the Theory of Dislocations in Quasicrystals

A method to evaluate some Fourier integrals is extended from two-dimensional (2-D) and three dimensional (3-D) spaces ton-dimensional (n-D) space, which are often used in the elasticity theory of dislocations in quasicrystals. Some key formulae have been given.

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

ON THE LOCALIZATION AND CONVERGENCE OF MULTIPLE FOURIER INTEGRAL BY BOCHNER-RIESZ MEANS

In this paper we consider lim _(R-) B_R^(f,x_0), in one case that f_x_0 (t) is a ABMV function on [0, ∞], and in another case that f∈L_(m-1)~1(R~) and x^k/~kf∈BV(R) when |k| = m-1 and f(x) = 0 when |x -x_0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).

FOURIER TRANSFORMATION AND SINGULAR INTEGRALS ON SELF-SIMILAR MEASURE

This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this op- erator is of type (p,p)(1<p<∞).

ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.

On the global L^(1)-boundedness of Fourier integral operators with rough amplitude and phase functions

Let T_(ϕ,a)be a Fourier integral operator with amplitude a and phase functions ϕ.In this paper,we study the boundedness of Fourier integral operator of rough amplitude a∈L^(∞)S_(ρ)^(m)and rough phase functionsϕ∈L^(m)ϕ^(2)with some measure condition.We prove the global L^(1)boundedness for T_(ϕ,a),when 1/<ρ≤1 and m<ρ-n+1/2.Our theorem improves some known results.

MECHANICAL ANALYSIS OF SHAFT LINING WHEN THE STRATUM SETTLEMENT RESULTING FROM WATER DRAINAGE

Based on the stratum settlement resulting from water drainage, this paper establishes the calculating method of stresses and displacements of shaft lining and stratum by using Fourier integration, obtains the calculating formulas of tangiential load which shaft lining is subjected to, and provides theoretical basis for design of shaft lining.

Analysis of mode Ⅲ crack perpendicular to the interface between two dissimilar strips

The present work is concerned with the problem of mode Ⅲ crack perpendicular to the interface of a bi-strip composite. One of these strips is made of a functionally graded material and the other of an isotropic material, which contains an edge crack perpendicular to and terminating at the interface. Fourier transforms and asymptotic analysis are employed to reduce the problem to a singular integral equation which is numerically solved using Gauss-Chebyshev quadrature formulae. Furthermore, a parametric study is carried out to investigate the effects of elastic and geometric characteristics of the composite on the values of stress intensity factor.

Exact analysis of the orientation-adjusted adhesive full stick contact of layered structures with the asymmetric bipolar coordinates

The adhesion failure has become one dominant factor in determining the reliability and service life of miniaturized devices subject to loadings with arbitrary orientations.This article establishes an adhesive full stick contact model between an elastic half-space and a rigid cylinder loaded in any direction.Using the Papkovich-Neuber functions,the Fourier integral transform,and the asymmetric bipolar coordinates,the exact solution is obtained.Unlike the Johnson-Kendall-Roberts(JKR)model,the present adhesive contact model takes into account the effects of the load direction as well as the coupling of the normal and tangential contact stresses.Besides,it considers the full stick contact which has large values of the friction coefficient between contacting surfaces,contrary to the frictionless contact supposed in the JKR model.The result shows that suitable angles can be found,which makes the contact surfaces difficult to be peeled off or easy to be pressed into.

DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

The dynamic behavior of two collinear cracks in a piezoelectric layer bonded to two half spaces under harmonic anti-plane shear waves was investigated by means of Schmidt method. The cracks are vertically to The boundary conditions of the electrical field the interfaces of the piezoelectric layer. were assumed to be the permeable crack surface. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. Numerical examples were presented to show the effect of the geometry of the interacting cracks, the piezoelectric constants of the materials and the frequency of the incident waves upon the stress intensity factors. The results show that the dynamic field will impede or enhance the propagation of the crack in a piezoelectric material at different stages of the frequency of the incident waves. It is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than that for the impermeable crack surface conditions.

Nano-Contact Problem with Surface Effects on Triangle Distribution Loading

This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.

L^(1) Boundedness of a class of rough Fourier integral operators

In this note,we consider a class of Fourier integral operators with rough amplitudes and rough phases.When the index of symbols in some range,we prove that they are bounded on L^(1) and construct an example to show that this result is sharp in some cases.This result is a generalization of the corresponding theorems of Kenig-Staubach and Dos Santos Ferreira-Staubach.

L^(p)Boundedness of Fourier Integral Operators in the Class S_(1,0)

We prove the following properties:(1)Let a∈Λ_(1,0,k,k’)^(m0)(R^(n)×R^(n))with m0=-1|1/p-1/2|(n-1),n≥2(1 n/p,k’>0;2≤p≤∞,k>n/2,k’>0 respectively).Suppose the phase function S is positively homogeneous inξ-variables,non-degenerate and satisfies certain conditions.Then the Fourier integral operator T is L^(p)-bounded.Applying the method of(1),we can obtain the L^(p)-boundedness of the Fourier integral operator if(2)the symbol a∈Λ_(1,δ,k,k’)^(m0),0≤δ≤1,with m0,k,k’and S given as in(1).Also,the method of(1)gives:(3)a∈Λ_(1,δ,k,k’),0≤δ<1 and k,k’given as in(1),then the L^(p)-boundedness of the pseudo-differential operators holds,1<p<∞.

L^(2) Boundedness of the Fourier Integral Operator with Inhomogeneous Phase Functions

In this paper,we investigate the L^(2) boundedness of the Fourier integral operator Tφ,a with smooth and rough symbols and phase functions which satisfy certain non-degeneracy conditions.In particular,if the symbol a∈L∞Smρ,the phase functionφsatisfies some measure conditions and ∇kξφ(·,ξ)L∞≤C|ξ|−k for all k≥2,ξ≠0,and some∈>0,we obtain that Tφ,a is bounded on L^(2) if m<n2 min{ρ−1,−2}.This result is a generalization of a result of Kenig and Staubach on pseudo-differential operators and it improves a result of Dos Santos Ferreira and Staubach on Fourier integral operators.Moreover,the Fourier integral operator with rough symbols and inhomogeneous phase functions we study in this paper can be used to obtain the almost everywhere convergence of the fractional Schr odinger operator.

L^(p) Boundedness of Fourier Integral Operators in the Class S_(0,0)

We first prove the L~2-boundedness of a Fourier integral operator where it’s symbol a ∈S_(1/2,1/2)~0(R~n× R~n) and the phase function S is non-degenerate,satisfies certain conditions and may not be positively homogeneous in ξ-variables.Then we use the above property,Paley’s inequality,covering lemma of Calderon and Zygmund etc.,and obtain the L~p-boundedness of Fourier integral operators if(1) the symbol a ∈ Λ_(k)^(m_(0)) and Supp a = E×R~n,with E a compact set of R~n(m_(0) =-|1/p-1/2|n,1<p≤2,k>n/2;2<p<∞,k>n/p),(2) the symbol a ∈ Λ_(0,k,k’)^(m_(0))(m_(0) =-|1/p-1/2|n,1<p ≤2,k>n/2,k’>n/p;2<p<∞,k>n/p,k’>n/2) with the phase function S(x,ξ) = xξ + h(x,ξ),x,ξ ∈ R~n non-degenerate,satisfying certain conditions and ?ξ h ∈ S_(1,0)~0(R~n× R~n),or(3) the symbol a ∈ Λ_(0,k,k’)^(m_(0)),the requirements for m_(0),k,k’ are the same as in(2),and ?_(ξ)h is not in S_(1,0)~0(R~n× R~n) but the phase function S is non-degenerate,satisfies certain conditions and is positively homogeneous in ξ-variables.

Conjectures and problems on Bochner-Riesz means

The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.

Improved Hrmander’s Theorem and New Methods for Oscillatory Integral Operators

This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.

An Explicit Formula for Szeg? Kernels on the Heisenberg Group

In this paper, we give an explicit formula for the Szego kernel for （0, q） forms on the Heisenberg group Hn＋1.

Stationary phase and practical numerical evaluation of ship waves in shallow water

A simple and highly-efficient method for numerically evaluating the waves created by a ship that travels at a constant speed in calm water, of large depth or of uniform depth, is given. The method, inspired by Kelvin＇s classical stationary-phase analysis, is suited for evaluating far-field as well as near-field waves. More generally, the method can be applied to a broad class of integrals with integrands that contain a rapidly oscillatory trigonometric function with a phase function whose first derivative（and possibly also higher derivatives） vanishes at one or several points, commonly called points of stationary phase, with the range of integration.