Wyniki wyszukiwania

1

Endpoint bounds for convolution operators with singular measures

100%

Ferreyra E., Godoy T., Urciuolo M.

Colloquium Mathematicum

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1998

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tom 76

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nr 1

35-47

EN

Let $S\subset \R^{n+1}$ be the graph of the function $\varphi:[ -1,1]^n\rightarrow \R $ defined by $\varphi ( x_1,\dots,x_n) =\sum_{j=1}^n| x_j|^{\beta_j},$ with 1<$\beta_1\leq \dots \leq \beta_n,$ and let $\mu $ the measure on $\R^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu $ is $L^p$-$L^q$ bounded.

2

On some singular integral operatorsclose to the Hilbert transform

100%

Godoy T., Saal L., Urciuolo M.

Colloquium Mathematicum

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1997

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tom 72

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nr 1

9-17

EN

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.

3

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

100%

Ferreyra E., Godoy T., Urciuolo M.

Colloquium Mathematicum

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2002

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tom 93

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nr 2

285-293

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.

4

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

100%

Ferreyra E., Godoy T., Urciuolo M.

Studia Mathematica

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2004

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tom 160

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nr 3

249-265

EN

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = {(x,φ(x)): |x| ≤ 1} and let σ be the Borel measure on Σ defined by $σ(A) = ∫_{B} χ_{A}(x,φ(x))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p}(ℝ³)$ to $L^{q}(Σ,dσ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Ograniczanie wyników

3 Colloquium Mathematicum

1 Studia Mathematica

4 Godoy T.

4 Urciuolo M.

3 Ferreyra E.

1 Saal L.

1 2004

1 2002

1 1998

1 1997

Endpoint bounds for convolution operators with singular measures

On some singular integral operatorsclose to the Hilbert transform

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

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³