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On multipliers on compact Lie groups. (English. Russian original) Zbl 1276.43001
Funct. Anal. Appl. 47, No. 1, 72-75 (2013); translation from Funkts. Anal. Prilozh. 47, No. 1, 87-91 (2013).
From the text: “In this note we announce $$L^p$$ multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hörmander-Mikhlin theorem on $$\mathbb R^n$$ and its variants on tori $$\mathbb T^n$$. Applications are given to the mapping properties of pseudo-differential operators on $$L^p$$-spaces and to a-priori estimates for non-hypoelliptic operators.”
“The following condition (6) below is a natural relaxation from the $$L^p$$-boundedness of zero order pseudo-differential operators to a multiplier theorem and generalises the Hörmander–Mikhlin theorem [S. G. Michlin, Dokl. Akad. Nauk SSSR, 109, 701–703 (1956; Zbl 0073.08402), Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr., 12, 143–155 (1957; Zbl 0092.31701)], [L. Hörmander, Acta Math. 104, 93–140 (1960; Zbl 0093.11402)] to arbitrary [compact Lie] groups.
Theorem 1. Denote by $$\kappa$$ the smallest even integer larger than $$\frac 12\dim G$$. Let $$A: C^\infty(G) \to \mathcal D'(G)$$ be left-invariant. Assume that its symbol $$\sigma_A$$ satisfies the inequalities $\|\triangle\!\!\!\!\!\ast\,^{\kappa/2} \sigma_A(\xi)\|_{op} \leq C \langle \xi\rangle^{-\kappa} \qquad\text{and}\qquad \|{\mathbb D}^{\alpha} \sigma_A(\xi) \|_{op} \leq C_\alpha \langle\xi\rangle^{-|\alpha|}\quad\quad\quad(6)$ for all multi-indices $$\alpha$$ with $$|\alpha|\leq \kappa-1$$, and for all $$[\xi]\in \widehat G$$. Then the operator $$A$$ is of weak type $$(1,1)$$ and $$L^p$$-bounded for all $$1<p<\infty$$.”
The authors then give some applications of their main theorem above. Full proofs can be found in [M. Ruzhansky and J. Wirth, “$$L^p$$-multipliers on compact Lie groups”, preprint arXiv:1102.3988].

MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Full Text:
References:
 [1] R. Coifman and M. de Guzman, Rev. Un. Mat. Argentina, 25 (1970/71), 137–143. [2] R. Coifman and G. Weiss, Rev. Un. Mat. Argentina, 25 (1970/71), 145–166. [3] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Springer-Verlag, 1971. · Zbl 0224.43006 [4] L. Hörmander, Acta Math., 104 (1960), 93–140. · Zbl 0093.11402 · doi:10.1007/BF02547187 [5] S. G. Mihlin, Dokl. Akad. Nauk SSSR, 109 (1956), 701–703. [6] S. G. Mihlin, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr., 12: 7 (1957), 143–155. [7] M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkhäuser, Basel, 2010. · Zbl 1193.35261 [8] M. Ruzhansky, V. Turunen, and J. Wirth, http://arxiv.org/abs/1004.4396 . [9] M. Ruzhansky and J. Wirth, http://arxiv.org/abs/1102.3988 .
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