In this paper, the authors give the local L2 estimate of the maximal operator
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\begin{document}$$S_{\phi ,\gamma }^ * $$\end{document} of the operator family {St,ϕ, γ} defined initially by
St,ϕ,γf(x):=eitϕ(−Δ)f(γ(x,t))=(2π)−1∫ℝeiγ(x,t)⋅ξ+itϕ(|ξ|)f^(ξ)dξ,f∈S(ℝ),\documentclass[12pt]{minimal}
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\begin{document}$${S_{t,\phi ,\gamma }}f(x): = {{\rm{e}}^{{\rm{i}}\,t\phi (\sqrt { - \Delta } )}}f(\gamma (x,t)) = {(2\pi )^{ - 1}}\int_\mathbb{R} {{{\rm{e}}^{{\rm{i}}\gamma (x,t) \cdot \xi + {\rm{i}}\,t\phi ({\rm{|}}\xi {\rm{|}})}}} \hat f(\xi ){\rm{d}}\xi ,\;\;\;\;\;\;\;\;f \in {\cal S}(\mathbb{R}),$$\end{document} which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}:
{i∂tu+ϕ(−Δ)u=0,(x,t)∈ℝn×ℝ,u(x,0)=f(x),f∈S(ℝn),\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ {\matrix{ {{\rm{i}}{\partial _t}u + \phi (\sqrt { - {\rm{\Delta }}} )u = 0,} \hfill \;\;\;\;\; {(x,t) \in \mathbb{R}{^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$$\end{document} where {itϕ}: ℝ+ → ℝ satisfies some suitable conditions and
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\begin{document}$$\phi (\sqrt { - {\rm{\Delta }}} )$$\end{document} 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
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\begin{document}$$S_{\phi ,\gamma }^ * $$\end{document} is also given in this paper.
