This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
In this paper, we prove an $L^2-L^2-L^2$ decay estimate for a trilinear oscillatory integral of convolution type in $mathbb{R}^d,$ which recovers the earlier result of Li (2013) when $d=1.$ We discuss the sharpness of our result in the $d=2$ case. Ou
r main hypothesis has close connections to the property of simple nondegeneracy studied by Christ, Li, Tao and Thiele (2005).
Using Guths polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrodinger equation in $mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $lim_{t
to 0} e^{itDelta}f(x)=f(x)$ almost everywhere for all $f in H^s (mathbb{R}^2)$ provided that $s>3/8$.
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 (2016).
Let $ E subset mathbb{R}^2 $ be a finite set, and let $ f : E to [0,infty) $. In this paper, we address the algorithmic aspects of nonnegative $C^2$ interpolation in the plane. Specifically, we provide an efficient algorithm to compute a nonnegative
$C^2(mathbb{R}^2)$ extension of $ f $ with norm within a universal constant factor of the least possible. We also provide an efficient algorithm to approximate the trace norm.
Yangkendi Deng
,Zuoshunhua Shi
,Dunyan Yan
.
(2021)
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"$L^2$ estimates of trilinear oscillatory integrals of convolution type on $mathbb{R}^2$"
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Zuoshunhua Shi
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