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3 Colloquium Mathematicum

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

100%

Godoy T. , Rocha P.

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 $φ(w) = |w|^{2m}$.

On some singular integral operatorsclose to the Hilbert transform

80%

Godoy T. , Saal L. , Urciuolo M.

nr 1

9-17

Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p(ℝ)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_m f(x) = p.v. \int k(x-y) m(x+y) f(y)dy$ where $k(x) = \sum_{j ∈ ℤ} 2^j φ _j (2^j x)$ for a family of functions ${φ_j}_{j∈ℤ}$ satisfying conditions (1.1)-(1.3) given below.

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

80%

Ferreyra E. , Godoy T. , Urciuolo M.

nr 2

285-293

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.