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Transferring $L^{p}$ eigenfunction bounds from $S^{2n+1}$ to hⁿ

100%

Casarino V., Ciatti P.

Studia Mathematica

2009

tom 194

nr 1

23-42

By using the notion of contraction of Lie groups, we transfer $L^{p} - L²$ estimates for joint spectral projectors from the unit complex sphere $S^{2n+1}$ in $ℂ^{n+1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge's work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

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

100%

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 Casarino V.

1 Ciatti P.

1 Secco S.

1 2009

1 2008

Wyniki wyszukiwania

Transferring $L^{p}$ eigenfunction bounds from $S^{2n+1}$ to hⁿ

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