Multipliers on modules over the Fourier algebra
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- by Charles F. Dunkl and Donald E. Ramirez PDF
- Trans. Amer. Math. Soc. 172 (1972), 357-364 Request permission
Abstract:
Let $G$ be an infinite compact group and $\hat G$ its dual. For $1 \leq p < \infty ,{\mathfrak {L}^p}(\hat G)$ is a module over ${\mathfrak {L}^1}(\hat G) \cong A(G)$, the Fourier algebra of $G$. For $1 \leq p,q < \infty$, let ${\mathfrak {M}_{p,q}} = {\operatorname {Hom} _{A(G)}}({\mathfrak {L}^p}(\hat G),{\mathfrak {L}^q}(\hat G))$. If $G$ is abelian, then ${\mathfrak {M}_{p,p}}$ is the space of ${L^P}(\hat G)$-multipliers. For $1 \leq p < 2$ and $p’$ the conjugate index of $p$, \[ A(G) \cong {\mathfrak {M}_{1,1}} \subset {\mathfrak {M}_{p,p}} = {\mathfrak {M}_{p’,p’}} \subsetneqq {\mathfrak {M}_{2,2}} \cong {L^\infty }(G).\] Further, the space ${\mathfrak {M}_{p,p}}$ is the dual of a space called $\mathcal {A}_p$, a subspace of ${\mathcal {C}_0}(\hat G)$. Using a method of J. F. Price we observe that \[ \cup \{ {\mathfrak {M}_{q,q}}:1 \leq q < p\} \subsetneqq {\mathfrak {M}_{p,p}} \subsetneqq \cap \{ {\mathfrak {M}_{q,q}}:p < q < 2\} \] (where $1 < p < 2$). Finally, ${\mathfrak {M}_{q,p}} = \{ 0\}$ for $1 \leq p < q < \infty$.References
- Charles F. Dunkl and Donald E. Ramirez, Topics in harmonic analysis, The Appleton-Century Mathematics Series, Appleton-Century-Crofts [Meredith Corporation], New York, 1971. MR 0454515
- Charles F. Dunkl and Donald E. Ramirez, Existence and nonuniqueness of invariant means on ${\scr L}^{\infty }(\hat G)$, Proc. Amer. Math. Soc. 32 (1972), 525–530. MR 296609, DOI 10.1090/S0002-9939-1972-0296609-1
- Charles F. Dunkl and Donald E. Ramirez, Helson sets in compact and locally compact groups, Michigan Math. J. 19 (1972), 65–69. MR 324315
- Alessandro Figà-Talamanca, Translation invariant operators in $L^{p}$, Duke Math. J. 32 (1965), 495–501. MR 181869
- Sigurđur Helgason, Lacunary Fourier series on noncommutative groups, Proc. Amer. Math. Soc. 9 (1958), 782–790. MR 100234, DOI 10.1090/S0002-9939-1958-0100234-5
- J. F. Price, Some strict inclusions between spaces of $L^{p}$-multipliers, Trans. Amer. Math. Soc. 152 (1970), 321–330. MR 282210, DOI 10.1090/S0002-9947-1970-0282210-5
- Marc A. Rieffel, Multipliers and tensor products of $L^{p}$-spaces of locally compact groups, Studia Math. 33 (1969), 71–82. MR 244764, DOI 10.4064/sm-33-1-71-82
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 357-364
- MSC: Primary 43A22
- DOI: https://doi.org/10.1090/S0002-9947-1972-0324318-3
- MathSciNet review: 0324318