## The type set for some measures on $$\mathbb R^{2n}$$ with $$n$$-dimensional support.(English)Zbl 1012.42012

Summary: Let $$\varphi _1,\dots ,\varphi _n$$ be real homogeneous functions in $$C^\infty (\mathbb R^n-\{0\})$$ of degree $$k \geq 2$$, let $$\varphi (x) = (\varphi _1 (x), \dots , \varphi _n(x))$$ and let $$\mu$$ be the Borel measure on $$\mathbb R^{2n}$$ given by $\mu (E) = \int _{\mathbb R^n} \chi _E (x,\varphi (x)) |x|^{\gamma - n} dx$ where $$dx$$ denotes the Lebesgue measure on $$\mathbb R^n$$ and $$\gamma >0$$. Let $$T_{\mu }$$ be the convolution operator $$T_{\mu } f(x)=(\mu * f)(x)$$ and let $$E_{\mu } = \{(1/p, 1/q): \|T_{\mu }\|_{p,q} < \infty$$, $$1\leq p,q \leq \infty \}.$$
Assume that, for $$x \neq 0$$, the following two conditions hold: $$\operatorname {det}(d^2 \varphi (x) h)$$ vanishes only at $$h=0$$ and $$\operatorname {det}(d\varphi (x)) \neq 0$$. In this paper we show that if $$\gamma >n(k+1)/3$$ then $$E_\mu$$ is the empty set and if $$\gamma \leq n(k+1)/3$$ then $$E_\mu$$ is the closed segment with endpoints $$D=(1-\frac \gamma {n(k+1)},1-\frac {2\gamma } {n(k+1)})$$ and $$D'=(\frac {2\gamma } {n(1+k)},\frac \gamma {n(1+k)})$$. Also, we give some examples.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B15 Multipliers for harmonic analysis in several variables 47B38 Linear operators on function spaces (general)

### Keywords:

singular measures; convolution operators
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### References:

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