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.

Fourier neural operator approach to large eddy simulation of three-dimensional turbulence

Fourier neural operator(FNO)model is developed for large eddy simulation(LES)of three-dimensional(3D)turbulence.Velocity fields of isotropic turbulence generated by direct numerical simulation(DNS)are used for training the FNO model to predict the filtered velocity field at a given time.The input of the FNO model is the filtered velocity fields at the previous several time-nodes with large time lag.In the a posteriori study of LES,the FNO model performs better than the dynamic Smagorinsky model(DSM)and the dynamic mixed model(DMM)in the prediction of the velocity spectrum,probability density functions(PDFs)of vorticity and velocity increments,and the instantaneous flow structures.Moreover,the proposed model can significantly reduce the computational cost,and can be well generalized to LES of turbulence at higher Taylor-Reynolds numbers.

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)).

Fourier neural operator with boundary conditions for efficient prediction of steady airfoil flows

An efficient data-driven approach for predicting steady airfoil flows is proposed based on the Fourier neural operator(FNO),which is a new framework of neural networks.Theoretical reasons and experimental results are provided to support the necessity and effectiveness of the improvements made to the FNO,which involve using an additional branch neural operator to approximate the contribution of boundary conditions to steady solutions.The proposed approach runs several orders of magnitude faster than the traditional numerical methods.The predictions for flows around airfoils and ellipses demonstrate the superior accuracy and impressive speed of this novel approach.Furthermore,the property of zero-shot super-resolution enables the proposed approach to overcome the limitations of predicting airfoil flows with Cartesian grids,thereby improving the accuracy in the near-wall region.There is no doubt that the unprecedented speed and accuracy in forecasting steady airfoil flows have massive benefits for airfoil design and optimization.

Multi-function terahertz wave manipulation utilizing Fourier convolution operation metasurface

We propose a novel metasurface based on a combined pattern of outer C-shaped ring and inner rectangular ring.By Fourier convolution operation to generating different predesigned sequences of metasurfaces,we realize various functionalities to flexible manipulate terahertz waves including vortex terahertz beam splitting,anomalous vortex terahertz wave deflection,vortex terahertz wave splitting and deflection simultaneously.The incident terahertz wave can be flexibly controlled in a single metasurface.The designed metasurface has an extensive application prospect in the field of future terahertz communication and sensing.

On the bilinear square Fourier multiplier operators and related multilinear square functions

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).

Data-driven parametric soliton-rogon state transitions for nonlinear wave equations using deep learning with Fourier neural operator

In this paper,we develop the deep learning-based Fourier neural operator(FNO)approach to find parametric mappings,which are used to approximately display abundant wave structures in the nonlinear Schr?dinger(NLS)equation,Hirota equation,and NLS equation with the generalized PT-symmetric Scarf-II potentials.Specifically,we analyze the state transitions of different types of solitons(e.g.bright solitons,breathers,peakons,rogons,and periodic waves)appearing in these complex nonlinear wave equations.By checking the absolute errors between the predicted solutions and exact solutions,we can find that the FNO with the Ge Lu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures.Moreover,we find that the approximation errors via the physics-informed neural networks(PINNs)are similar in magnitude to those of the FNO.However,the FNO can learn the entire family of solutions under a given distribution every time,while the PINNs can only learn some specific solution each time.The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.

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^(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.

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.

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.

On a Linear Partial Differential Equation of the Higher Order in Two Variables with Initial Condition Whose Coefficients are Real-valued Simple Step Functions

By using the method developed in the paper [Georg. Inter. J. Sci. Tech., Volume 3, Issue 1 （2011）, 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions.

Ellipticity on spaces with higher singularities

We study corner-degenerate pseudo-differential operators of any singularity order and develop ellipticity based on the principal symbolic hierarchy, associated with the stratification of the underlying space. We construct parametrices within the calculus and discuss the aspect of additional trace and potential conditions along lower-dimensional strata.