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

T. Godoy, and P. Rocha. "$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group." Colloquium Mathematicae 132.1 (2013): 101-111. <http://eudml.org/doc/283862>.

@article{T2013,
abstract = {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 $φ(w) = |w|^\{2m\}$.},
author = {T. Godoy, P. Rocha},
journal = {Colloquium Mathematicae},
keywords = {singular measures; group Fourier transform; Heisenberg group; convolution operators; spherical transform},
language = {eng},
number = {1},
pages = {101-111},
title = {$L^\{p\} - L^\{q\}$ estimates for some convolution operators with singular measures on the Heisenberg group},
url = {http://eudml.org/doc/283862},
volume = {132},
year = {2013},
}

TY - JOUR
AU - T. Godoy
AU - P. Rocha
TI - $L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group
JO - Colloquium Mathematicae
PY - 2013
VL - 132
IS - 1
SP - 101
EP - 111
AB - 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 $φ(w) = |w|^{2m}$.
LA - eng
KW - singular measures; group Fourier transform; Heisenberg group; convolution operators; spherical transform
UR - http://eudml.org/doc/283862
ER -