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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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