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Almost-everywhere summability of Fourier integrals. (English) Zbl 0631.42004
Let $(T^{\lambda}_{{\mathbb{R}}}f){\hat{\;}}(x)=(1-| x| ^ 2/R^ 2)_ +^{\lambda}\hat f(x)$ denote the Bochner-Riesz operator of order $$\lambda$$ $$\geq 0$$. We investigate the almost everywhere convergence of $$T_ R^{\lambda}f(x)$$ as $$R\to \infty.$$
Theorem 1. Let $$\lambda >0$$ and $$n\geq 2$$. For all $$f\in L^ p({\mathbb{R}}^ n)$$ with $$2\leq p<2n/(n-1-2\lambda)$$, $$\lim _{R\to \infty}T_ R^{\lambda}f(x)=f(x)$$ almost everywhere.
Theorem 2. For all $$f\in L^ p({\mathbb{R}}^ n)$$ with $$2\leq p<2n/(n-1)$$, $$\lim _{k\to \infty}T^ 0_{R_ k}f(x)=f(x)$$ 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:
 42A45 Multipliers in one variable harmonic analysis 42B25 Maximal functions, Littlewood-Paley theory 42A55 Lacunary series of trigonometric and other functions; Riesz products
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