Notes on interpolation of Hardy spaces

Quanhua Xu

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 4, page 875-889
  • ISSN: 0373-0956

Abstract

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

How to cite

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Xu, Quanhua. "Notes on interpolation of Hardy spaces." Annales de l'institut Fourier 42.4 (1992): 875-889. <http://eudml.org/doc/74977>.

@article{Xu1992,
abstract = {Let $H_ p$ denote the usual Hardy space of analytic functions on the unit disc $(0&lt; p\le \infty )$. We prove that for every function $f\in H_ 1$ there exists a linear operator $T$ defined on $L_ 1(\{\bf T\})$ which is simultaneously bounded from $L_ 1(\{\bf T\})$ to $H_ 1$ and from $L_ \infty (\{\bf T\})$ to $H_ \infty $ such that $T(f)=f$. Consequently, we get the following results $(1\le p_ 0,p_ 1\le \infty )$:1) $(H_\{p_ 0\},H_\{p_ 1\})$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F(H_\{p_ 0\},H_\{p_ 1\})=H(F(L_\{p_ 0\}(\{\bf T\}),L_\{p_ 1\}(\{\bf T\})))$, where$H(F(L_\{p_ 0\}(\{\bf T\}),L_\{p_ 1\}(\{\bf T\})))$ denotes the closed subspace of $F(L_\{p_ 0\}(\{\bf T\}),L_\{p_ 1\}(\{\bf T\}))$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy spaces associated to general rearrangement invariant spaces on the unit circle.},
author = {Xu, Quanhua},
journal = {Annales de l'institut Fourier},
keywords = {Hardy space of analytic functions on the unit disc; Calderón-Mitjagin couple; interpolation functor; rearrangement invariant spaces on the unit circle},
language = {eng},
number = {4},
pages = {875-889},
publisher = {Association des Annales de l'Institut Fourier},
title = {Notes on interpolation of Hardy spaces},
url = {http://eudml.org/doc/74977},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Xu, Quanhua
TI - Notes on interpolation of Hardy spaces
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 4
SP - 875
EP - 889
AB - Let $H_ p$ denote the usual Hardy space of analytic functions on the unit disc $(0&lt; p\le \infty )$. We prove that for every function $f\in H_ 1$ there exists a linear operator $T$ defined on $L_ 1({\bf T})$ which is simultaneously bounded from $L_ 1({\bf T})$ to $H_ 1$ and from $L_ \infty ({\bf T})$ to $H_ \infty $ such that $T(f)=f$. Consequently, we get the following results $(1\le p_ 0,p_ 1\le \infty )$:1) $(H_{p_ 0},H_{p_ 1})$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F(H_{p_ 0},H_{p_ 1})=H(F(L_{p_ 0}({\bf T}),L_{p_ 1}({\bf T})))$, where$H(F(L_{p_ 0}({\bf T}),L_{p_ 1}({\bf T})))$ denotes the closed subspace of $F(L_{p_ 0}({\bf T}),L_{p_ 1}({\bf T}))$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy spaces associated to general rearrangement invariant spaces on the unit circle.
LA - eng
KW - Hardy space of analytic functions on the unit disc; Calderón-Mitjagin couple; interpolation functor; rearrangement invariant spaces on the unit circle
UR - http://eudml.org/doc/74977
ER -

References

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  1. [1] C. BENNETT, R. SHARPLEY, Interpolation of operators, Pure and applied Mathematics, 129, Academic Press, 1988. Zbl0647.46057MR89e:46001
  2. [2] J. BERGH, J. LÖFFSTRÖM, Interpolation spaces, An introduction, Berlin-Heidelberg-New York, Springer-Verlag, 1976. Zbl0344.46071
  3. [3] J. BOURGAIN, Bilinear forms on H∞ and bounded bianalytic functions, Trans. Amer. Math. Soc., 286 (1984), 313-338. Zbl0572.46048MR86c:46060
  4. [4] A.P. CALDERÓN, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math., 26 (1966), 273-299. Zbl0149.09203MR34 #3295
  5. [5] M. CWIKEL, Monotonicity properties of interpolation spaces, Ark. Mat., 14 (1976), 213-236. Zbl0339.46024MR56 #1095
  6. [6] J.B. GARNETT, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, 1981. Zbl0469.30024MR83g:30037
  7. [7] G.H. HARDY, J.E. LITTLEWOOD, G. PÓLYA, Inequalities, Cambridge University Press, Cambridge, 1934 (2nd ed., 1952). Zbl0010.10703JFM60.0169.01
  8. [8] P.W. JONES, L∞-estimates for the ∂-problem in the half-plane, Acta Math., 150 (1983), 137-152. Zbl0516.35060MR84g:35135
  9. [9] P.W. JONES, On interpolation between H1 and H∞, Lect. Notes in Math. Springer, 1070 (1984), 143-151. Zbl0573.46044MR86c:46021
  10. [10] S.V. KISLIAKOV, Extension of (q,p)-summing operators defined on the disc-algebra with an appendix on Bourgain's analytic projections, preprint, 1990. 
  11. [11] S.V. KISLIAKOV, Truncating functions in weighted Hp and two theorems of J. Bourgain, preprint, 1989. 
  12. [12] S.V. KISLIAKOV, (q,p)-summing operators on the disc algebra and a weighted estimate for certain outer functions, LOMI, preprint E-11-89, Leningrad, 1989. 
  13. [13] J. LINDENSTRAUSS, L. TZAFRIRI, Classical Banach spaces II, Berlin-New York, Springer-Verlag, 1979. Zbl0403.46022MR81c:46001
  14. [14] G.G. LORENTZ, T. SHIMOGAKI, Interpolation theorems for the pairs of spaces (Lp, L∞) and (L1, Lq), Trans. Amer. Math. Soc., 59 (1971), 207-221. Zbl0244.46044MR52 #1347
  15. [15] P.F.X. MÜLLER, Holomorphic martingales and interpolation between Hardy spaces, to appear in J. d'Analyse Math.. Zbl0796.60051
  16. [16] G. PISIER, Interpolation between Hp spaces and non-commutative generalizations, preprint, 1991. 
  17. [17] G. SPARR, Interpolation of weighted Lp-spaces, Studia Math., 62 (1973), 229-236. Zbl0393.46029MR80d:46055

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