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

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Godoy T., Rocha P.

Colloquium Mathematicum

2013

tom 132

nr 1

101-111

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

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1 Colloquium Mathematicum

1 Godoy T.

1 Rocha P.

1 2013

Wyniki wyszukiwania

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