Some strict inclusions between spaces of $L^{p}$-multipliers
HTML articles powered by AMS MathViewer
- by J. F. Price PDF
- Trans. Amer. Math. Soc. 152 (1970), 321-330 Request permission
Abstract:
Suppose that the Hausdorff topological group $G$ is either compact or locally compact abelian and that ${C_c}$ denotes the set of continuous complex-valued functions on $G$ with compact supports. Let $L_p^q$ denote the restrictions to ${C_c}$ of the continuous linear operators from ${L^p}(G)$ into ${L^q}(G)$ which commute with all the right translation operators. When $1 \leqq p < q \leqq 2$ or $2 \leqq q < p \leqq \infty$ it is known that \[ (1)\quad L_p^p \subset L_q^q.\] The main result of this paper is that the inclusion in (1) is strict unless $G$ is finite. In fact it will be shown, using a partly constructive proof, that when $G$ is infinite \[ \bigcup \limits _{1 \leqq q < p} {L_q^q \subsetneqq } L_p^p \subsetneqq \bigcap \limits _{p < q \leqq 2} {L_q^q} \] for $1 < p < 2$, with the first inclusion remaining strict when $p = 2$ and the second inclusion remaining strict when $p = 1$. (Similar results also hold for $2 \leqq p \leqq \infty$.) When $G$ is compact, simple relations will also be developed between idempotent operators in $L_p^q$ and lacunary subsets of the dual of $G$ which will enable us to find necessary conditions so that inclusion (1) is strict even if, for example, $L_p^p$ and $L_q^q$ are replaced by the sets of idempotent operators in $L_p^p$ and $L_q^q$ respectively.References
- B. Brainerd and R. E. Edwards, Linear operators which commute with translations. I. Representation theorems, J. Austral. Math. Soc. 6 (1966), 289–327. MR 0206725, DOI 10.1017/S1446788700004286
- R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
- R. E. Edwards, Changing signs of Fourier coefficients, Pacific J. Math. 15 (1965), 463–475. MR 200676, DOI 10.2140/pjm.1965.15.463
- R. E. Edwards, Fourier series: a modern introduction. Vol. II, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1967. MR 0222538
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
- Alessandro Figà-Talamanca, Translation invariant operators in $L^{p}$, Duke Math. J. 32 (1965), 495–501. MR 181869
- Alessandro Figà-Talamanca and G. I. Gaudry, Density and representation theorems for multipliers of type $(p,\,q)$, J. Austral. Math. Soc. 7 (1967), 1–6. MR 0209770, DOI 10.1017/S1446788700005012 —, Multipliers and sets of uniqueness of ${L^p}$, Michigan Math. J. (to appear).
- Alessandro Figà-Talamanca and Daniel Rider, A theorem of Littlewood and lacunary series for compact groups, Pacific J. Math. 16 (1966), 505–514. MR 206626, DOI 10.2140/pjm.1966.16.505
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. MR 0116177, DOI 10.1512/iumj.1960.9.59013
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 152 (1970), 321-330
- MSC: Primary 46.35; Secondary 42.00
- DOI: https://doi.org/10.1090/S0002-9947-1970-0282210-5
- MathSciNet review: 0282210