A pair of indices for function spaces on the circle
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- by Colin Bennett
- Trans. Amer. Math. Soc. 174 (1972), 289-304
- DOI: https://doi.org/10.1090/S0002-9947-1972-0333699-6
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Abstract:
We give here some of the basic properties of the classes $\{ {\Phi _r}\}$, $\{ {\Psi _r}\} , - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces $\mathfrak {X}$ on the circle It is shown that to each space $\mathfrak {X}$ there correspond two numbers $\xi ,\eta$, called indices, which satisfy $0 \leq \eta \leq \xi \leq 1$; these numbers represent the rate of growth or decay of $\left \| {{\Psi _r}} \right \|$ as $r \to \pm 1$. By using the operators ${\Psi _r}$ to obtain estimates for certain averaging operators ${A_\gamma }$, we are able to show that the indices $(\xi ,\eta )$ coincide with the Boyd indices $(\alpha ,\beta )$. As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.References
- C. Bennett, On the harmonic analysis of rearrangement-invariant Banach function spaces, Thesis, University of Newcastle, 1971.
- Colin Bennett and John E. Gilbert, Homogeneous algebras on the circle. I. Ideals of analytic functions, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 1–19 (English, with French summary). MR 338782
- Colin Bennett and John E. Gilbert, Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 21–50 (English, with French summary). MR 338783
- D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, DOI 10.4153/CJM-1967-053-7
- David W. Boyd, The spectral radius of averaging operators, Pacific J. Math. 24 (1968), 19–28. MR 221308
- David W. Boyd, Indices of function spaces and their relationship to interpolation, Canadian J. Math. 21 (1969), 1245–1254. MR 412788, DOI 10.4153/CJM-1969-137-x
- A.-P. Calderón, Spaces between $L^{1}$ and $L^{\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. MR 203444, DOI 10.4064/sm-26-3-301-304
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- G. G. Lorentz and T. Shimogaki, Interpolation theorems for operators in function spaces, J. Functional Analysis 2 (1968), 31–51. MR 0257775, DOI 10.1016/0022-1236(68)90024-4
- George G. Lorentz and Tetsuya Shimogaki, Interpolation theorems for the pairs of spaces $(L^{p},\,L^{\infty })$ and $(L^{1},\,L^{q})$, Trans. Amer. Math. Soc. 159 (1971), 207–221. MR 380447, DOI 10.1090/S0002-9947-1971-0380447-9
- Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Technische Hogeschool te Delft, Delft, 1955. Thesis. MR 0072440 —, Rearrangement-invariant Banach function spaces, Queen’s Papers in Pure and Appl. Math., no. 10, Queen’s University, Kingston, Ont., 1967, pp. 83-144.
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
- Tetsuya Shimogaki, An interpolation theorem on Banach function spaces, Studia Math. 31 (1968), 233–240. MR 234301, DOI 10.4064/sm-31-3-233-240
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 174 (1972), 289-304
- MSC: Primary 46E30
- DOI: https://doi.org/10.1090/S0002-9947-1972-0333699-6
- MathSciNet review: 0333699