The type set for homogeneous singular measures on R^3 of polynomial type

FERREYRA, ELIDA; GODOY, TOMÁS; URCIUOLO, MARTA

COLLOQUIUM MATHEMATICUM

Año: 2006 vol. 106 p. 161 - 175

0010-1354

L^{p} improving measures, Convolution operators<abstract/>Let ϕ:R²→R be a homogeneous polynomial function of degree m≥2, let μ the Borel measure on R³ defined by μ(E)=∫_{D}χ_{E}(x,ϕ(x))dx where D ={x∈R²:|x|≤1} and let T_{μ} be the convolution operator with the measure μ. Let ϕ=ϕ₁^{e₁}...ϕ_{n}^{e_{n}} be the decomposition of ϕ in irreducible factors. In this paper we show that if e_{i}≠(m/2) for each ϕ_{i} of degree 1, then the type set E_{μ}:={((1/p),(1/q))∈[0,1]×[0,1]:‖T_{μ}‖_{p,q}<∞} can be explicitely described as a closed polygonal region.