$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy; P. Rocha

L^{p} - L^{q}

estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy ; P. Rocha

Colloquium Mathematicae, Tome 131 (2013), p. 101-111 / Harvested from The Polish Digital Mathematics Library

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Résumé

We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $ν (E) = \int_{ℂ ⁿ} χ_{E} (w, φ (w)) η (w) d w$ , where $φ (w) = \sum_{j = 1}^{n} a_{j} | w_{j} | ²$ , w = (w₁,...,wₙ) ∈ ℂⁿ, $a_{j} \in ℝ$ , and η(w) = η₀(|w|²) with $η ₀ \in C_{c}^{\infty} (ℝ)$ . We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^{p} (ℍ ⁿ) - L^{q} (ℍ ⁿ)$ bounded. We also obtain $L^{p}$ -improving properties of measures supported on the graph of the function $φ (w) = {| w |}^{2 m}$ .

Publié le : 2013-01-01

Zbl 1277.43013

EUDML-ID : urn:eudml:doc:283862

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     author = {T. Godoy and P. Rocha},
     title = {$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group},
     journal = {Colloquium Mathematicae},
     volume = {131},
     year = {2013},
     pages = {101-111},
     zbl = {1277.43013},
     language = {en},
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T. Godoy; P. Rocha. $L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group. Colloquium Mathematicae, Tome 131 (2013) pp. 101-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-8/