Global estimates of maximal operators generated by dispersive equations
The general dispersive equation is defined by \[ \begin{aligned} &i u_t (x) = - \phi (D) u(x), \quad (x,t) \in \mathbb{R}^n \times \mathbb{R}, \\ &u(x,0) = f(x), \end{aligned} \] where \(D = -i \nabla\) and \(\phi\) is a smooth phase function which satisfies the following: \(\phi\) is a radial and \(\phi \in C^2(\mathbb{R}^n \setminus \{ 0 \})\), and there exist \(a \in \mathbb{R} ( \neq 0,1)\) and positive constants \(c_1, c_2\) such that \[ c_1 | \xi |^{a-k} \leq | \phi^{(k)} (\xi ) | \leq c_2 | \xi |^{a-k} \quad (k=0,1,2), \quad \text{in} \quad \mathbb{R}^n \setminus \{ 0 \}. \] The formal solution of this equation is \[ u(x,t) = Tf(x,t) = \frac{1}{(2\pi)^n} \int e^{i ( x \cdot \xi + t \phi (\xi ))} \hat{f}( \xi )d\xi. \] Let us define maximal operator \[ T^{**}f(x) \equiv \sup_{t \in R} | Tf(x,t) |. \] This maximal operator is motivated from the well-known pointwise convergence problem: \[ \lim_{t \to 0} u(x,t) = f(x) \quad \text{a.e.}\;x, \quad \text{for}\quad f\in H^{1/4}(\mathbb{R}^n). \] One of the authors' results is the following. Assume that \(1/4 < s < 1/2, m > (3n-5)/6 - 2/p, 4n/(2n-1) \leq p < 2n/(n-2s)\) and \(0 < a \neq 1\). Then \[ \| T^{**} f \|_{L^p} \leq C \| f \|_{H^s ( H^m_{\omega})}, \] where \( \| f \|_{H^s ( H^m_{\omega})} = \| ( 1 - \Delta)^{s/2} ( 1 - \Delta_{\omega})^{m/2} f \|_{L^2} \) and \(\Delta_{\omega}\) is the Laplace-Beltrami operator defined on the unit sphere \(S^{n-1}\).
scientific article; zbMATH DE number 5548685