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On Marcinkiewicz means of double Fourier integrals in \(H_ p(p\leq 1)\). (Russian) Zbl 0531.42007

Let \(H_ p=H_ p({\mathbb{R}}^ 2)\) (\(p\leq 1)\) be real Hardy spaces with the quasinorm \[ \| f\|_{H_ p({\mathbb{R}}^ 2)}=\| u^+\|_{L^ p({\mathbb{R}}^ 2)}, \] where \(u^+(x)=\sup_{t>0}| u(x,t)| \in L_ p({\mathbb{R}}^ 2),\) u(x,t) is a harmonic extension of f from \({\mathbb{R}}^ 2\) to \({\mathbb{R}}^ 3_+=\{(x,t);x\in {\mathbb{R}}^ 2,t>0\). For \(x=(x_ 1,x_ 2)\in {\mathbb{R}}^ 2\), set \(| x|_{\infty}=\max(| x_ 1|,| x_ 2|), | x|\) denotes the ordinary Euclidean norm of x. Define the Marcinkiewicz-Abel means of \(f\in H_ p({\mathbb{R}}^ 2)\) by \[ (1)\quad M_{\epsilon}f(x)=(2\pi)^{-2}\int_{{\mathbb{R}}^ 2}\exp(-\epsilon | y|_{\infty})\hat f(y)e^{ix\cdot y}dy, \] (\^f is the Fourier transform of f). For \(\delta>0\), \(\epsilon>0\) define the Marcinkiewicz- Riesz means of \(f\in H_ p({\mathbb{R}}^ 2)\) by \[ (2)\quad M^{\delta}_{\epsilon}f(x)=(2\pi)^{-2}\int_{{\mathbb{R}}^ 2}(1- (\epsilon | y|_{\infty})^ 2)_+^{\delta}\hat f(y)e^{ix\cdot y}dy. \] The main results of the paper deal with the almost everywhere and quasinorm convergence of the means (1) and (2). The following result is typical for this paper.
Theorem: Let \(f\in H_ p({\mathbb{R}}^ 2)\), 1/2\(\leq p\leq 1\). Define the maximal functions \(M_*(f(x))=\sup_{\epsilon>0}| M_{\epsilon}f(x)|.\) Then, \(M_*f(x)<\infty\) a.e. More precisely, \(mes\{x\in {\mathbb{R}}^ 2:M_*f(x)>t\}\leq Ct^{-1/2}(\| f\|_{H_{1/2}({\mathbb{R}}^ 2)})^{1/2}, t>0\) for \(p=1/2\) and \(\| M_*f\|_{L_ p({\mathbb{R}}^ 2)}\leq C_ p\| f\|_{H_ p({\mathbb{R}}^ 2)}\) for \(1/2<p\leq 1.\)
As a corollary, one obtains that if \(f\in H_ p({\mathbb{R}}^ 2)\), \(1/2<p\leq 1\), then \(\lim_{\epsilon \to 0}\| f- M_{\epsilon}f\|_{H_ p}=0\). It is also shown that later results fail for \(p\leq 1/2\).
Reviewer: D.Khavinson

MSC:

42B30 \(H^p\)-spaces