The type set for some measures on $\mathbb{R}^{2n}$ with n-dimensional support

Abstract

Let $\phi _1 ,...,\phi _n$ be realhomogeneous functions in $C^\infty (\mathbb{R}^n - \;\{ 0\} )$ ofdegree $k \geqslant 2,{\text{ let }}\varphi {\text{(}}x{\text{)}}\; = (\varphi _1 (x),\;.\,.\;.\;,\;\varphi _n (x))$ and let μ bethe Borel measure on $\mathbb{R}^{2n}$ given by

$$\mu (E) = \int_{\mathbb{R}^n } {\chi _E } (x,\alpha (x))\;\left| x \right|^{\gamma - n} {\text{d}}x$$

where dx denotes theLebesgue measure on $\mathbb{R}^n$ and γ > 0. Let T _μ be the convolution operator $T_\mu f(x) = (\mu * f)(x)$ and let

$$E_\mu = \{ ({1 \mathord{\left/ {\vphantom {1 {p,\;{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}}}} \right. \kern-\nulldelimiterspace} {p,\;{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}}}):\;\;\parallel {\kern 1pt} T_\mu {\kern 1pt} \parallel _{p,q} \; < \infty ,\;\;1 \leqslant p,q \leqslant \infty \} .$$

Assume that, for x ≠ 0, the followingtwo conditions hold: $\det ({\text{d}}^{\text{2}} \varphi (x)h)$vanishes only at h = 0 and $\det ({\text{d}}\varphi (x)) \ne 0$. In this paper we show that if $\gamma >n(k + 1)/3$then E _μ is the empty set and if $\gamma \leqslant n(k + 1)/3$ then E _μ is the closed segment withendpoints $D = (1 - \tfrac{\gamma }{{n(k + 1)}},\;1 - \tfrac{{2\gamma }}{{n(k + 1)}})$ and $D' = (\tfrac{{2\gamma }}{{n(1 + k)}},\;\tfrac{\gamma }{{n(1 + k)}})$. Also, we give some examples.