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A class of approximate identities. (Chinese. English summary) Zbl 1147.42003

A sequence of functions \(\{\varphi_k\}\) is called an approximate identity if \(\varphi_k\geq 0\), \(\int_{{\mathbb{R}}^n}\varphi_k(x)\,dx=1\) and for every \(f\in{L_1({\mathbb{R}}^n)}\), \(\lim_{k\to\infty}\| \varphi_k*f-f\| _1=0\). For \(1{\leq}p \leq\infty\), an approximate identity \(\{\varphi_k\}\) is called \(L_p\)-good if \(\varphi_k*f{\to}f\) a.e. for all \(f \in L^p({\mathbb{R}}^n)\). Let \(\{a_k\}\) be a sequence of real numbers with \(\alpha_k>0\) and \(\lim_{k\to\infty}\alpha_k=0\). For a measurable function \(\varphi\geq 0\) with \(\int_{{\mathbb{R}}^n}\varphi(x)\,dx=1\), denote \(\varphi_k(x)= \frac{1}{\alpha_k^n} \varphi(\frac{x}{\alpha_k})\). Let \(1\leq p<\infty\), we say \(\varphi\) is of \(p-\)type, if the maximal operator \(M\) is of weak type \((p,p)\), where \(Mf(x)=\sup_{k\in\mathbb{N}}| \varphi_k*f(x)| \), \(\forall x\in {\mathbb{R}}^n\). The main results of the paper are the following two theorems:
Theorem 1: A sequence of functions \(\{\varphi_k\}\) with \(\varphi_k\geq 0\) and \(\int_{{\mathbb{R}}^n}\varphi_k(x)\,dx=1\) is an approximate identity if and only if for every \(\varepsilon>0\) there exists \(k_0\in\mathbb{N}\) such that for all \(k\geq k_0\) there has \(\int_{B(0,\varepsilon)}\varphi_k(x)\,dx > 1-\varepsilon.\)
Theorem 3: Let \(1\leq p<\infty\), if \(\varphi\) is \(p-\)type, then \(\{\varphi_k\}\) is an \(L_p\)-good approximate identity. The one-dimensional case of Theorem 1 was established by P. Avramidou [Proc. Am. Math. Soc. 133, No. 1, 175–184 (2005; Zbl 1068.47033)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
28A15 Abstract differentiation theory, differentiation of set functions
42A85 Convolution, factorization for one variable harmonic analysis
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)

Citations:

Zbl 1068.47033
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