Maximal operators of pseudo-differential operators with rough symbols

Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.

Boundedness for Multilinear Operators of Pseudo-Differential Operators

In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the Lp (1 p norm inequalities for the multilinear operators.

ENDPOINT ESTIMATES FOR THE COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS

It is well known that the commutator Tb of the Calderbn-Zygmund singular integral operator is bounded on LP（Rn） for 1 〈 p 〈 ＋∞ if and only if b E BMO [1]. On the other hand, the commutator Tb is bounded from H1（Rn） into L1（Rn） only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H1. Let Tσ be the operators that its symbol is Sσ1,δ with 0 ≤δ〈 1, if b ∈ LMO∞, then, the commutator [b, Tσ] is bounded from H1（Rn） into L1（Rn） and from L∞（Rn） into BMO（Rn）; If [b,Tσ] is bounded from H1（Rn） into L1（Rn） or L1（Rn） into BMO（Rn）, then, b ∈ LMOtoc.

General Formulae for Two Pseudo-differential Operator

Some general formulas in the Sato theory related to the nonisospectral KP and mKP hierarchies are derived for simplifying calculations.

Pseudo-differential Operators and Generalized Lax Equations in Symbolic Computation

In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.

Pseudo-Differential Operators of Homogeneous Symbol Class Associated with the Weinstein Transform

In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced.The boundedness of pseudo-differential operators and commutator between two pseudo-differential operators on H_(α,2)^(r) are proven with the help of the Weinstein transform technique.

A Pair of Pseudo-differential Operators Involving Index Whittaker Transform in L_(2)^(a) (R_(+);m_(a)(x)dx)

Pseudo-differential operators(PDO)Q(x,L_(a,x))and Q(x,L_(a,x))involving the index Whittaker transform are defined.Estimates for these operators in Hilbert space L_(2)^(a)(R+;m_(a)(x)dx)are obtained.A symbol classΩis introduced.Later product and commutators for the PDO are investigated and their boundedness results are discussed.

GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION

The Landau equation is studied for hard potential with-2≤γ≤1.Under a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_(v)^(2)(d>3/2),which extends the results of[11]in the torus domain to the whole space R_(x)^(3).Here we utilize the pseudo-differential calculus to derive our desired result.

Bilinear Pseudo-Differential Operator and Its Commutator on Generalized FractionalWeighted Morrey Spaces

The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator Ts and its commutator[b1,b2,Ts]generated by Ts and b1,b22 BMO(Rn)on generalized fractional weighted Morrey spaces Lp,h,j(w).Under assumption that a weight satisfies a certain condition,the authors prove that Ts is bounded from products of spaces Lp1,h1,j(w1)Lp2,h2,j(w2)into spaces Lp,h,j(~w),where~w=(w1,w2)2 A~P,~P=(p1,p2),h=h1+h2 and 1 p=1 p1+1 p2 with p1,p22(1,¥).Furthermore,the authors show that the[b1,b2,Ts]is bounded from products of generalized fractional Morrey spaces Lp1,h1,j(Rn)Lp2,h2,j(Rn)into Lp,h,j(Rn).As corollaries,the boundedness of the Ts and[b1,b2,Ts]on generalized weighted Morrey spaces Lp,j(w)and on generalized Morrey spaces Lp,j(Rn)is also obtained.

Investigation on the Natural Modes of A Semi-Closed Floating Tank

Vessels with semi-closed tanks(i.e.,well docks)are widely applied in the military operation and maritime engineer-ing.The water is bound by the semi-closed floating tank and forced by both the incident waves and ship’s motions.The free surface oscillations inside the flooded well dock is thus distinctive and very complicated.So far,the natural modes of semi-closed floating tanks have not yet been studied.This paper investigates the characteristics of natural modes of a floating semi-closed tank by combining a mode-resolving model based on mild-slope equations and a hydrodynamic model based on computational fluid dynamics.Results show that the first three natural periods(i.e.,74,23.6,and 14 s)of the tank fall into the band of swell and infragravity waves and they could be triggered under certain circumstance.Multi-period free surface oscillations are observed inside the tank,including the longest natural period(i.e.,74 s),though the incident waves are monochromatic.A possible generation mechanism for the long-period mode is explained on the basis of liquid sloshing and harbor oscillations.Moreover,a long-period component with a period close to the natural mode of well dock is observed in the ship motions,which is generated by the interaction between the waves and ship.

A DISCRETE CHARACTERIZATION OF HERZ-TYPE TRIEBEL-LIZORKIN SPACES AND ITS APPLICATIONS

In this paper, the author establishes a discrete characterization of the Herz-type Triebel-Lizorkin spaces which is used to prove the boundedness of pseudo-differential operators on these function spaces.

PSEUDO-DIFFERENTIAL OPERATORS AND DERIVATIVES ON LOCALLY COMPACT VILENKIN GROUPS

Many mathematicians pay their great attention to the study of the concept of derivatives for functions defined on locally compact Vilenkin groups. In this paper, this topic is investigated by virtue of so-called pseudo-differential operators. We give the definitions of derivatives and integrals of functions defined on locally compact Vilenkingroups and study their properties, then give an application example.

Getting pre-stack time migration travel times from the single square root operator

Improving the focusing capability of pre-stack time migration allows the imaged section to reflect structural characteristics, depth, and interface shape and it is a key step for the preparation of the initial depth migration velocity model. The traditional symmetrical travel time equation is derived based on the assumption of a layered model. It is difficult to achieve the desired effect of focusing in media with strong lateral variation. The nonsymmetrical travel time equation based on Lie algebra and a pseudo-differential operator contains a lateral velocity derivative which can improve the focusing capability even in strongly lateral variable media and also the computation precision of the weight coefficients for relative amplitude preservation. Compared with the symmetrical methods, the nonsymmetrical method is more effective. In this paper, we describe several key steps of nonsymmetric pre-stack travel time calculation and present some test results using synthetic and real data.

The stability problem of reverse time migration for viscoacoustic VTI media

In real strata anisotropy and viscosity extensively exists. They degraded waveforms in amplitude, resulting in which reducing of image resolution. To obtain high-precision imaging of deep reservoirs, we extended the separated viscous and anisotropic reverse time migration （RTM） to a stable viscoacoustic anisotropic RTM for vertical transverse isotropic （VTI） media, based on single generalized standard and linear solid （GSLS） media theory.. We used a pseudo-spectral method to develop the numerical simulation. By introducing a regularization operator to eliminate the high-frequency instability problem, we built a stable inverse propagator and achieved viscoacoustic VTI media RTM. High-resolution imaging results were obtained after correcting for the effects of anisotropy and viscosity. Synthetic tests verify the validity and accuracy of algorithm.

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.

Coiflet solution of strongly nonlinear p-Laplacian equations

A new boundary extension technique based on the Lagrange interpolat- ing polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coifiet-based solution procedure is established for general two-dimensional p^Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.

MEROMORPHIC SOLUTIONS OF SOME TYPES OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some properties of meromorphic solutions, and we ob- tain some results, which are the improvements and extensions of some results in references. Examples show that our results are precise.

Pseudo-Differential Operators on Sobolev and Lipschitz Spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.

Bilinear Pseudo-differential Operators on Local Hardy Spaces

In this paper, the authors consider a class of bilinear pseudo-differential operators with symbols of order 0 and type （1, 0） in the sense of HSrmander and use the atomic decompositions of local Hardy spaces to establish the boundedness of the bilinear pseudo-differential operators and the bilinear singular integral operators on the product of local Hardy spaces.