Related papers: A bilinear oscillatory integral along parabolas

The QED radiative corrections are calculated in the leading log approximation up to ${\cal O}(\alpha^2)$ for different definitions of the kinematical variables using jet measurement, the 'mixed' variables, the double angle method, and a…

We prove that the bilinear Hilbert transforms and maximal functions along certain general plane curves are bounded from $L^2(\mathbb{R})\times L^2(\mathbb{R})$ to $L^1(\mathbb{R})$.

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…

We introduce an analogue of Calder\'on's first commutator along a parabola, and establish its $L^2$ boundedness under essentially sharp hypotheses.

Working with a general class of linear Hamiltonian systems with at least one singular boundary condition, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated…

Let $\Omega \subseteq \mathbb{R}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^\infty$ coefficients, and $V$ a locally integrable function on $\Omega$ whose negative part is…

Based on geometric considerations, longitudinal and transverse Lagrangian velocity increments are introduced as components along, and perpendicular to, the displacement of fluid particles during a time scale {\tau}. It is argued that these…

This paper addresses a doubly nonlinear parabolic inclusion of the form $A(u_t)+B(u)\ni f$. Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $A$ and $B$, which in…

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad…

A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two finite-energy solutions to the wave equation are…

In this paper, we prove $L^p$ decay estimates for multilinear oscillatory integrals in $\mathbb{R}^2$, establishing sharpness through a scaling argument. The result in this paper is a generalization of the previous work by Gressman and Xiao…

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the…

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be H\"older or Dini continuous in the time variable and all but one spatial variables. This…

We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm…

We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the…

We derive a short wave length approximation of a boundary integral operator for two-dimensional isotropic and homogeneous elastic bodies of arbitrary shape. Trace formulae for elastodynamics can be deduced in this way from first principles…

We obtain global $W^{2,\delta}$ estimates for a type of singular fully nonlinear elliptic equations where the right hand side term belongs to $L^\infty$. The main idea of the proof is to slide paraboloids from below and above to touch the…

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$…

We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing perturbations of the Couette flow. We prove estimates on the L_2 norm of the resolvent of this operator…

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients has the form $t^{-\alpha}a(t)$,~$\alpha>0$, where $a(t)$ is trigonometric polynomial with an arbitrary set of frequencies.…