Convolution operators with anisotropically homogeneous measures on ${ℝ}^{2n}$ with n-dimensional support

E. Ferreyra, T. Godoy, and M. Urciuolo. "Convolution operators with anisotropically homogeneous measures on $ℝ^{2n}$ with n-dimensional support." Colloquium Mathematicae 93.2 (2002): 285-293. <http://eudml.org/doc/285302>.

@article{E2002,
abstract = {Let $α_i,β_i > 0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t • x = (t^\{α₁\}x₁,..., t^\{αₙ\}xₙ)$, $t ∘ x = (t^\{β₁\}x₁,..., t^\{βₙ\}xₙ)$ and $||x|| = ∑_\{i = 1\}^\{n\} |x_i|^\{1/α_i\}$. Let φ₁,...,φₙ be real functions in $C^∞(ℝⁿ-\{0\}) $ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on $ℝ^\{2n\}$ given by $μ(E) = ∫_\{ℝⁿ\} χ_E(x,φ(x)) ||x||^\{γ-α\} dx$, where $α = ∑_\{i=1\}^\{n\} α_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_μf = μ ∗ f$ and let $||T_μ||_\{p,q\}$ be the operator norm of $T_μ$ from $L^\{p\}(ℝ^\{2n\})$ into $L^q(ℝ^\{2n\})$, where the $L^\{p\}$ spaces are taken with respect to the Lebesgue measure. The type set $E_μ$ is defined by $E_μ = \{(1/p,1/q): ||T_μ||_\{p,q\} < ∞, 1 ≤ p,q ≤ ∞\}$. In the case $α_i ≠ β_k$ for 1 ≤ i,k ≤ n we characterize the type set under certain additional hypotheses on φ.},
author = {E. Ferreyra, T. Godoy, M. Urciuolo},
journal = {Colloquium Mathematicae},
keywords = { improving measures; convolution operators},
language = {eng},
number = {2},
pages = {285-293},
title = {Convolution operators with anisotropically homogeneous measures on $ℝ^\{2n\}$ with n-dimensional support},
url = {http://eudml.org/doc/285302},
volume = {93},
year = {2002},
}

TY - JOUR
AU - E. Ferreyra
AU - T. Godoy
AU - M. Urciuolo
TI - Convolution operators with anisotropically homogeneous measures on $ℝ^{2n}$ with n-dimensional support
JO - Colloquium Mathematicae
PY - 2002
VL - 93
IS - 2
SP - 285
EP - 293
AB - Let $α_i,β_i > 0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t • x = (t^{α₁}x₁,..., t^{αₙ}xₙ)$, $t ∘ x = (t^{β₁}x₁,..., t^{βₙ}xₙ)$ and $||x|| = ∑_{i = 1}^{n} |x_i|^{1/α_i}$. Let φ₁,...,φₙ be real functions in $C^∞(ℝⁿ-{0}) $ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on $ℝ^{2n}$ given by $μ(E) = ∫_{ℝⁿ} χ_E(x,φ(x)) ||x||^{γ-α} dx$, where $α = ∑_{i=1}^{n} α_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_μf = μ ∗ f$ and let $||T_μ||_{p,q}$ be the operator norm of $T_μ$ from $L^{p}(ℝ^{2n})$ into $L^q(ℝ^{2n})$, where the $L^{p}$ spaces are taken with respect to the Lebesgue measure. The type set $E_μ$ is defined by $E_μ = {(1/p,1/q): ||T_μ||_{p,q} < ∞, 1 ≤ p,q ≤ ∞}$. In the case $α_i ≠ β_k$ for 1 ≤ i,k ≤ n we characterize the type set under certain additional hypotheses on φ.
LA - eng
KW - improving measures; convolution operators
UR - http://eudml.org/doc/285302
ER -