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

E. Ferreyra, and T. Godoy. "The type set for homogeneous singular measures on ℝ ³ of polynomial type." Colloquium Mathematicae 106.2 (2006): 161-175. <http://eudml.org/doc/284155>.

@article{E2006,
abstract = {Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $μ(E) = ∫_\{D\} χ_\{E\}(x,φ(x))dx$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_\{μ\}$ be the convolution operator with the measure μ. Let $φ = φ₁^\{e₁\} ⋯ φₙ^\{eₙ\}$ be the decomposition of φ into irreducible factors. 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 explicitly described as a closed polygonal region.},
author = {E. Ferreyra, T. Godoy},
journal = {Colloquium Mathematicae},
keywords = { improving measures; convolution operators},
language = {eng},
number = {2},
pages = {161-175},
title = {The type set for homogeneous singular measures on ℝ ³ of polynomial type},
url = {http://eudml.org/doc/284155},
volume = {106},
year = {2006},
}

TY - JOUR
AU - E. Ferreyra
AU - T. Godoy
TI - The type set for homogeneous singular measures on ℝ ³ of polynomial type
JO - Colloquium Mathematicae
PY - 2006
VL - 106
IS - 2
SP - 161
EP - 175
AB - Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by $μ(E) = ∫_{D} χ_{E}(x,φ(x))dx$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{μ}$ be the convolution operator with the measure μ. Let $φ = φ₁^{e₁} ⋯ φₙ^{eₙ}$ be the decomposition of φ into irreducible factors. 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 explicitly described as a closed polygonal region.
LA - eng
KW - improving measures; convolution operators
UR - http://eudml.org/doc/284155
ER -