Estimates with global range for oscillatory integrals with concave phase
Volume 91 / 2002
Colloquium Mathematicum 91 (2002), 157-165
MSC: 42A45, 42B08, 42B25.
DOI: 10.4064/cm91-2-1
Abstract
We consider the maximal function $\|(S^af)[x]\|_{L^\infty[-1,1]}$ where $(S^af) (t)^\wedge (\xi) = e ^ {i t |\xi| ^ a} \widehat f(\xi)$ and $0 < a < 1$. We prove the global estimate $$ \| {S ^ a f}\|_ {L ^ 2 (\mathbb R , L ^ \infty [ -1 , 1 ])} \leq C \| f \| _{H^ s(\mathbb R)}, \quad\ s > a/4, $$ with $C$ independent of $f$. This is known to be almost sharp with respect to the Sobolev regularity $s$.
