Estimates with global range for oscillatory integrals with concave phase (Q2773344)

The author considers the maximal function \(\|(S^a f)[x]\|_{L^\infty[-1,1]}\), where NEWLINE\[NEWLINE(S^a f)(t)^\land(\xi)=e^{it|\xi|^a}\hat f(\xi)NEWLINE\]NEWLINE for \(\xi\in\mathbb R\) and \(0<a<1\). He proves the global estimate NEWLINE\[NEWLINE\|S^a f\|_{L^2(\mathbb R,L^\infty[-1,1])}\leq C\|f\|_{H^s(\mathbb R)}NEWLINE\]NEWLINE for \(s>a/4\), where \(C\) is independent of \(f\), which is known to be almost sharp with respect to the Sobolev regularity \(s\).