Related papers: Sharp $L^2$ decay rate for (1+2)-dimensional oscil…

We consider an oscillatory integral operator with Loomis-Whitney multilinear form. The phase is real analytic in a neighborhood of the origin in $\mathbb{R}^d$ and satisfies a nondegeneracy condition related to its Newton polyhedron.…

We prove the sharp L^1-L^{\infty} time-decay estimate for the 2D-Schroedinger equation with a general family of scaling critical electromagnetic potentials.

In this paper we derive sharp $L^p-L^q$ estimates, $1\leq p\leq q\leq \infty$ (including endpoint estimates as $L^1-L^1$ and $L^1-L^\infty$) for dissipative wave-type equations, under the assumption that the dissipation dampen the…

We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+i\alpha x$ for non-negative values of the parameter $\alpha$, by getting the exponential growth rate of $\|\Pi_n(\alpha)\|$, where $\Pi_n(\alpha)$ is…

We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as…

For any integer $n \geq 2$, we establish $L^p(\R^n)$ inequalities for the $r$-variations of Stein-Wainger type oscillatory integral operators with general phase functions. These inequalities closely related to Carleson's theorem are sharp,…

We establish an almost sharp L^r to L^p estimate for oscillatory integral operators satisfying the cinematic curvature condition. The proof combines Wolff's two-ends reduction with refined decoupling inequalities.

In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove…

We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation…

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the…

We propose, analyze, and implement a quadrature method for evaluating integrals of the form $\int_0^2 f(s)\exp(zs)\, {\rm d}s$, where $z$ is a complex number with a possibly large negative real part. The integrand may exhibit exponential…

In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier…

In this paper, we propose a numerical method of computing an integral whose integrand is a slowly decaying oscillatory function. In the proposed method, we consider a complex analytic function in the upper-half complex plane, which is…

This paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin--Voigt damping $$ u_{tt}(x,t)-[u_{x}(x,t)+b(x)u_{x,t}(x,t)]_{x}=0,\; x\in(-1,1),\; t>0,$$ where $b(x)=0$ on $x\in (-1,0]$, and…

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the…

The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of…

We consider a class of wave equations of the type $\partial_{tt} u + Lu + B\partial_{t} u = 0$, with a self-adjoint operator $L$, and various types of local damping represented by $B$. By establishing appropriate and raher precise estimates…

In this paper, we establish an improved variable coefficient version of square function inequality, by which the local smoothing estimate $L^p_\alpha\rightarrow L^p$ for the Fourier integral operators satisfying cinematic curvature…

In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schr\"{o}dinger equations. Our aim of this research is to specify the asymptotic profile of the solution in $L^{\infty}$ as $t \to \infty$. It is…

In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the…