The type set for homogeneous singular measures on $\mathbb{R}^{3}$ of polynomial type
Volume 106 / 2006
Colloquium Mathematicum 106 (2006), 161-175
MSC: Primary 42B20; Secondary 42B25.
DOI: 10.4064/cm106-2-1
Abstract
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.
