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$L^p$-improving properties of measures supported on curves on the Heisenberg group

100%

Secco S.

Studia Mathematica

1999

tom 132

nr 2

179-201

$L^p$-$L^q$ boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.

$L^{p}-L^{q}$ boundedness of analytic families of fractional integrals

63%

Casarino V., Secco S.

Studia Mathematica

2008

tom 184

nr 2

153-174

We consider a double analytic family of fractional integrals $S^{γ,α}_{z}$ along the curve $t ↦ |t|^{α}$, introduced for α = 2 by L. Grafakos in 1993 and defined by $(S^{γ,α}_{z}f)(x₁,x₂): = 1/Γ(z+1/2) ∫∫ |u-1|^{z}ψ(u-1)f(x₁-t,x₂-u|t|^{α})du|t|^{γ}dt/t$, where ψ is a bump function on ℝ supported near the origin, $f ∈ 𝓒^{∞}_{c}(ℝ²)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S^{γ,α}_{z}$ maps $L^{p}(ℝ²)$ to $L^{q}(ℝ²)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K^{iϱ,α}_{-1+iθ}$ is a product kernel on ℝ², adapted to the curve $t ↦ |t|^{α}$; as a consequence, we show that the operator $S^{iϱ,α}_{-1+iθ}$, θ, ϱ ∈ ℝ, is bounded on $L^{p}(ℝ²)$ for 1 < p < ∞.

Ograniczanie wyników

2 Studia Mathematica

2 Secco S.

1 Casarino V.

1 2008

1 1999

Wyniki wyszukiwania

$L^p$-improving properties of measures supported on curves on the Heisenberg group

$L^{p}-L^{q}$ boundedness of analytic families of fractional integrals