The type set for homogeneous singular measures on $\mathbb{R}^{3}$ of polynomial type

E. Ferreyra¹ ; T. Godoy¹

¹ FaMAF Universidad Nacional de Córdoba and CIEM (UNC – CONICET) Ciudad Universitaria 5000 Córdoba, Argentina

Colloquium Mathematicum, Tome 106 (2006) no. 2, pp. 161-175

Résumé

Let $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a homogeneous polynomial function of degree ${m\geq2}$, let $\mu$ be the Borel measure on $\mathbb{R}^{3}$ defined by $\mu( E) =\int_{D}\chi_{E} (x,\varphi(x)) \,dx$ with $D=\{ x\in\mathbb{R} ^{2}:| x | \leq 1\} $ and let $T_{\mu}$ be the convolution operator with the measure $\mu$. Let $\varphi= \varphi_{1}^{e_{1}}\cdots \varphi_{n}^{e_{n}}$ be the decomposition of $\varphi$ into irreducible factors. We show that if $e_{i}\neq{m}/{2}$ for each $% \varphi_{i}$ of degree $1$, then the type set $E_{\mu}:=\{( {1}/{p},{1}/{q})\in[ 0,1] \times [ 0,1] :\| T_{\mu}\| _{p,q}\infty\} $ can be explicitly described as a closed polygonal region.

Détail

DOI : 10.4064/cm106-2-1

Keywords: varphi mathbb rightarrow mathbb homogeneous polynomial function degree geq borel measure mathbb defined int chi varphi mathbb leq convolution operator measure varphi varphi cdots varphi decomposition varphi irreducible factors neq each varphi degree type set times infty explicitly described closed polygonal region