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

E. Ferreyra; T. Godoy

E. Ferreyra ; T. Godoy

Colloquium Mathematicae, Tome 106 (2006), p. 161-175 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $μ (E) = \int_{D} χ_{E} (x, φ (x)) d x$ with D = x ∈ ℝ ²:|x| ≤ 1 and let $T_{μ}$ be the convolution operator with the measure μ. Let $φ = φ ₁^{e ₁} \dots φ ₙ^{e ₙ}$ be the decomposition of φ into irreducible factors. We show that if $e_{i} \neq m / 2$ for each $φ_{i}$ of degree 1, then the type set $E_{μ} : = (1 / p, 1 / q) \in [0, 1] \times [0, 1] : | | T_{μ} {| |}_{p, q} < \infty$ can be explicitly described as a closed polygonal region.

Publié le : 2006-01-01

Zbl 1146.42002

EUDML-ID : urn:eudml:doc:284155

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     author = {E. Ferreyra and T. Godoy},
     title = {The type set for homogeneous singular measures on $\mathbb{R}$ $^3$ of polynomial type},
     journal = {Colloquium Mathematicae},
     volume = {106},
     year = {2006},
     pages = {161-175},
     zbl = {1146.42002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-2-1}
}

E. Ferreyra; T. Godoy. The type set for homogeneous singular measures on ℝ ³ of polynomial type. Colloquium Mathematicae, Tome 106 (2006) pp. 161-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-2-1/