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Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

We consider the maximal function | | ( S a f ) [ x ] | | L [ - 1 , 1 ] where ( S a f ) ( t ) ( ξ ) = e i t | ξ | a f ̂ ( ξ ) and 0 < a < 1. We prove the global estimate | | S a f | | L ² ( , L [ - 1 , 1 ] ) C | | f | | H s ( ) , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

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