A strong type $(2, 2)$ estimate for a maximal operator associated to the Schrödinger equation

Abstract:

Let ${T^{\ast } }f(x) = \sup _{t > 0}|{T_t}f(x)|$, where $({T_t}f)^{\hat {}}(\xi ) = {e^{it|\xi |^2}}\hat f(\xi )/|\xi {|^{1/4}}$. We show that, given any finite interval $I$, $\int _I {|{T^{\ast } }f{|^2}\;dx \leqslant {C_I}\int _{\mathbf {R}} {|f(x){|^2}\;dx} }$, and that the above inequality is false with $2$ replaced by any $p < 2$. This maximal operator is related to solutions of the Schrödinger equation.