Local smoothing estimates related to the circular maximal theorem (Q1357637)

For \(f\) in the Schwartz space \(S({\mathbb{R}}^2)\), \(t\) in \([1,2]\), and \(\alpha\) in \({\mathbb{R}}^+\), define \(\widehat{f}\) to be the Fourier transform of \(f\), and \(A_\alpha f(t, \cdot)\) to be the inverse Fourier transform of the function \(\xi\mapsto\widehat{f}(\xi) (1+|\xi|)^{-\alpha} e^{it|\xi|}\). The main result of this paper is that \(|A_\alpha f|_{L^q([1,2]\times {\mathbb{R}}^2)}\) may be estimated in terms of \(|f|_{L^p({\mathbb{R}}^2)}\), when \(1\leq p\leq s/2\) and \(q=3p/(p-1)\). This sheds more light on Stein's circular maximal function on \({\mathbb{R}}^2\). The result is extended to more general integral operators.