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Register a new user$L^p$-estimates of maximal function related to Schr{o}dinger Equation in $mathbb{R}^2$
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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 $s>3/8$.
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In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr{o}dinger operator $P=-Delta+V(x)$ on $mathbb{R}^n, ngeq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q geq n/2$. Let $T_1=(-Delta+V)^{-1}V, T_2=(-Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-Delta+V)^{-1/2} abla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(mathbb{R}^n)$ when $p$ ranges in a interval, where $b in mathbf{BMO}(mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness.
In the first part of the paper we continue the study of solutions to Schrodinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrodinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.
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, for general plane curves $gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(mathbb{R}^2)$-boundedness of the Hilbert transforms $H^infty_{U,gamma}$ along the variable plane curves $(t,U(x_1, x_2)gamma(t))$ and the $L^p(mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^infty_{U,gamma}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<pleq 2$, under the additional conditions that $U$ is Lipschitz and making a $varepsilon_0$-truncation with $gamma(2 varepsilon_0)leq 1/4|U|_{textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{varepsilon_0}_{U,gamma}$ and $M^{varepsilon_0}_{U,gamma}$.
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
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