Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther¹

¹ Royal Institute of Technology SE-100 44 Stockholm, Sweden and Brown University Providence, RI 02912-1917 U.S.A.

Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 157-165.

Résumé

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 $s$.

Détail

DOI : 10.4064/cm91-2-1

Keywords: consider maximal function infty where wedge widehat prove global estimate mathbb infty leq mathbb quad independent known almost sharp respect sobolev regularity