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Zbl 0760.46060
Xu, Quanhua
Notes on interpolation of Hardy spaces. (English)
[J] Ann. Inst. Fourier 42, No.4, 875-889 (1992). ISSN 0373-0956; ISSN 1777-5310/e

Let $H\sb p$ denote the usual Hardy space of analytic functions on the unit disc $(0<p\leq\infty)$. We prove that for every function $f\in H\sb 1$ there exists a linear operator $T$ defined on $L\sb 1({\bold T})$ which is simultaneously bounded from $L\sb 1({\bold T})$ to $H\sb 1$ and from $L\sb \infty({\bold T})$ to $H\sb \infty$ such that $T(f)=f$. Consequently, we get the following results $(1\leq p\sb 0,p\sb 1\leq\infty)$:\par 1) $(H\sb{p\sb 0},H\sb{p\sb 1})$ is a Calderón-Mitjagin couple;\par 2) for any interpolation functor $F$, we have $F(H\sb{p\sb 0},H\sb{p\sb 1})=H(F(L\sb{p\sb 0}({\bold T}),L\sb{p\sb 1}({\bold T})))$, where\par $H(F(L\sb{p\sb 0}({\bold T}),L\sb{p\sb 1}({\bold T})))$ denotes the closed subspace of $F(L\sb{p\sb 0}({\bold T}),L\sb{p\sb 1}({\bold T}))$ of all functions whose Fourier coefficients vanish on negative integers.\par These results also extend to Hardy spaces associated to general rearrangement invariant spaces on the unit circle.
[Q.Xu (Paris)]

MSC 2000:: *46M35 Abstract interpolation of topological linear spaces
46E15 Banach spaces of functions defined by smoothness properties
46B70 Interpolation between normed linear spaces
46E30 Spaces of measurable functions
42B30 Hp-spaces (Fourier analysis)

Keywords: Hardy space of analytic functions on the unit disc; Calderón-Mitjagin couple; interpolation functor; rearrangement invariant spaces on the unit circle

Cited in: Zbl 0783.46037

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