A+ CATEGORY SCIENTIFIC UNIT

Estimates with global range for oscillatory integrals with concave phase

Volume 91 / 2002

Bjorn Gabriel Walther 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 $s$.

Authors

  • Bjorn Gabriel WaltherRoyal Institute of Technology
    SE-100 44 Stockholm, Sweden
    and
    Brown University
    Providence, RI 02912-1917
    U.S.A.
    e-mail

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