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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)
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References:

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