Radial functions and maximal estimates for radial solutions to the Schrödinger equation (Q1277168)

The author establishes the following estimate for radial functions \(f\): If \(2\leq q\leq 8/3\), \(\alpha= q(2n-1)/4-n\), then \[ \Biggl(\int_{\mathbb{R}^n}| S^*f(x)|^q| x|^\alpha dx\Biggr)^{1/q}\leq C\| f\|_{H_{1/4}}, \] where the maximal functions \(S^*f\) is defined by \[ S^*f(x)= \sup_{0<t<1}| S_tf(x)|, \] \[ S_tf(x)= (2\pi)^{-n} \int_{\mathbb{R}^n} e^{ix\xi}e^{it|\xi|^\alpha} \widehat f(\xi)d\xi. \] As a special case the inequality \[ \Biggl(\int_{\mathbb{R}^n}| S^* f(x)|^qdx\Biggr)^{1/q}\leq C\| f\|_{H_{1/4}} \] holds if \(q= 4n/(2n- 1)\). The result is used to give a solution to Carleson's problem on the pointwise convergence of a solution of the Schrödinger equation to its initial value as \(t\to 0\).