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Hilbertian Hardy-Sobolev spaces on a half-plane

José E. Galé, Valentin Matache, Pedro J. Miana, Luis Sánchez--Lajusticia

Abstract

In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel $K_{z,n}$ of $H^{(n)}_2$, from which we can find the estimate $\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}$ for $z\in\mathbb{C}^+$. Then composition operators $C_\varphi :H_2^{(n)} \to H_2^{(n)}$, $C_\varphi f=f\circ \varphi $, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps $\varphi: \mathbb{C}^+\to \mathbb{C}^+$ to induce bounded composition operators.

Hilbertian Hardy-Sobolev spaces on a half-plane

Abstract

In this paper we deal with a scale of reproducing kernel Hilbert spaces , , which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane . They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel of , from which we can find the estimate for . Then composition operators , , on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps to induce bounded composition operators.
Paper Structure (8 sections, 20 theorems, 126 equations)

This paper contains 8 sections, 20 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. The Sobolev-Lebesgue space ${\mathcal{T}}^{(n)}_2$
  3. The Hardy-Sobolev spaces $H^{(n)}_2$
  4. The Hardy-Sobolev spaces on the disc
  5. Reproducing kernel of $H^{(n)}_2$
  6. Analytic selfmaps of $\mathbb{C}^+$
  7. Boundedness of composition operators on $H^{(n)}_2$
  8. Concluding remarks

Key Result

Proposition 2.1

Take $n\in \mathbb{N}$ and $\varphi\in L^2(t^n)$. Then, for $1\le j\le n-1$, the mapping is well defined and $W^{-j}\varphi\in L^1(a,\infty)$ for every $a>0$. Moreover, for $1\le k\le n$, we have $W^{-k}\varphi$ is $(k-1)$-times differentiable with $(W^{-k}\varphi)^{(l)}(t)=(-1)^lW^{-(k-l)}\varphi$, that is, for every $1\le l\le k-1$, and $(W^{-k}\varphi)^{(k-1)}$ is absolutely continuous such

Theorems & Definitions (53)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 43 more