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

• Autorzy: E. Ferreyra , T. Godoy , M. Urciuolo
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 42B15: Multipliers
• 42B20: Singular and oscillatory integrals (Calder\'on-Zygmund, etc.)
• 44A35: Convolution

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2002
• Tom: 93
• Numer: 2
• Strony: 285-293

Daty

wydano

2002

Twórcy

autor

E. Ferreyra

• FaMAF, Universidad Nacional de Córdoba, CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina

autor

T. Godoy

• FaMAF, Universidad Nacional de Córdoba, CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina

autor

M. Urciuolo

• FaMAF, Universidad Nacional de Córdoba, CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina

Identyfikatory

DOI

10.4064/cm93-2-8