Global range estimates for maximal oscillatory integrals with radial test functions

Abstract

We consider the maximal function of oscillatory integrals $S^{a}f$ where $(S^{a}f)(t)\widehat{\ }(\xi )=\exp(it|\xi|^{a})\widehat{f}(\xi)$ and $a\in \ ]0,1[$. For a fixed $n\geq2$ we prove the global estimate

\[\bigl\|S^{a}f\bigr\|_{L^{2}(\mathbf{R}^{n},L^{\infty}(-1,1))}\leq C\|f\|_{H^{s}(\mathbf{R}^{n})},\quad s>a/4\]

with $C$ independent of the radial function $f$. We also prove that this result is almost sharp with respect to the Sobolev regularity $s$. This extends work of Sjölin who proved these result for $a>1$.