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Almost-everywhere summability of Fourier integrals. (English) Zbl 0631.42004

Let

(T λ f) ^(x)=(1-|x| 2 /R 2 ) + λ f ^(x)

denote the Bochner-Riesz operator of order λ 0. We investigate the almost everywhere convergence of T R λ f(x) as R·

Theorem 1. Let λ>0 and n2. For all fL p ( n ) with 2p<2n/(n-1-2λ), lim R T R λ f(x)=f(x) almost everywhere.

Theorem 2. For all fL p ( n ) with 2p<2n/(n-1), lim k T R k 0 f(x)=f(x) a.e. if {R k } is a lacunary sequence.

The proofs proceed via domination of the maximal function by square functions; however instead of seeking L p bounds for the square function we examine weighted L 2 inequalities, which, by duality, correspond to studying pointwise multipliers of Sobolev spaces.


MSC:
42A45Multipliers, one variable
42B25Maximal functions, Littlewood-Paley theory
42A55Lacunary series of trigonometric and other functions; Riesz products