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Remarks on singular convolution operators. (English) Zbl 0711.42024

Summary: We prove some endpoint estimates for singular convolution operators. For example, let m be a bounded function such that for some suitable \(C^{\infty}_ 0({\mathbb{R}}^ n\setminus \{0\})\)-function \(\phi\), \(\phi\) m(t\(\cdot)\) is in the Besov space \(B^ q_{\alpha 1}\), uniformly in \(t>0\). Then m is a Fourier multiplier on \(L^ p({\mathbb{R}}^ n)\) if \(\alpha =n(1/p-1/2)>n/q\), \(1<p\leq 2\), and on \(H^ 1\) if \(\alpha =n/2\), \(2<q\leq \infty\). If m is radial we may replace \(B^ q_{n/2,1}\) by \(B^ 2_{n/2,1}\).

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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