## 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

### Keywords:

singular convolution operators; Fourier multiplier
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