Convergence of Solutions of General Dispersive Equations Along Curve

Abstract

In this paper, the authors give the local L ² estimate of the maximal operator$S_{\phi ,\gamma }^ * $ of the operator family {S _{t,ϕ, γ}} defined initially by${S_{t,\phi ,\gamma }}f(x): = {{\rm{e}}^{{\rm{i}}\,t\phi (\sqrt { - \Delta } )}}f(\gamma (x,t)) = {(2\pi )^{ - 1}}\int_\mathbb{R} {{{\rm{e}}^{{\rm{i}}\gamma (x,t) \cdot \xi + {\rm{i}}\,t\phi ({\rm{|}}\xi {\rm{|}})}}} \hat f(\xi ){\rm{d}}\xi ,\;\;\;\;\;\;\;\;f \in {\cal S}(\mathbb{R}),$ which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}: $\left\{ {\matrix{ {{\rm{i}}{\partial _t}u + \phi (\sqrt { - {\rm{\Delta }}} )u = 0,} \hfill \;\;\;\;\; {(x,t) \in \mathbb{R}{^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$ where {itϕ}: ℝ⁺ → ℝ satisfies some suitable conditions and$\phi (\sqrt { - {\rm{\Delta }}} )$ is a pseudo-differential operator with symbol {itϕ}(∣{itξ}∣). As a consequence of the above result, the authors give the pointwise convergence of the solution (when {itn} = 1) of the equation (*) along curve {itγ}. Moreover, a global {itL}² estimate of the maximal operator$S_{\phi ,\gamma }^ * $ is also given in this paper.

Keywords

L ₂ estimate / Global maximal operator / Dispersive equation / Curve

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