Spectral multipliers on Lie groups of polynomial growth

Abstract

Abstract. Let L be a left invariant sub-Laplacian on a connected Lie group G of polynomial volume growth, and let {Ex, A > 0} be the spectral resolution of L and m a bounded Borel measurable function on [0, oo). In this article we give a sufficient condition on m for the operator m{L) = /0°° m{k) dE^ to extend to an operator bounded on LP{G), 1 < p < co , and also from L1{G) to weak-L'(G).

Joint Spectral Multipliers for Mixed Systems of Operators

“…with d = 1) Marcinkiewicz condition (3.1) of order ρ, then the multiplier operator 3 In the single operator case it might seem better to use the term 'Hörmander functional calculus', cf. [32,Theorem 2].…”

“…Moreover, there are many results in the literature, see e.g. [3,4,11,13,21,27,40], which imply that a single operator has a Marcinkiewicz functional calculus. Consequently, using the corollary we obtain a joint Marcinkiewicz functional calculus for a vast class of systems of operators acting on separate variables.…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…The argument we give to verify Theorem 1.1 is strongly influenced by [2,9,23]. The argument only requires bounds on a concrete number of derivates of the multiplier (see Theorem 4.1); however, in the interests of clarity and simplicity, we have made no effort to optimize this number.…”

$$L^p$$ L p Fourier multipliers on compact Lie groups

“…There is extensive literature providing criteria for central multipliers, see e.g. Weiss [23], Coifman and Weiss [5], Stein [22], Cowling [8], Alexopoulos [2], to mention only very few. There are also results for functions of the sub-Laplacian, for example on SU(2), see Cowling and Sikora [9].…”

“…For the case of the group SU(2) a characterisation for operators leading to Calderon-Zygmund kernels in terms of certain symbols was given by Coifman and Weiss in [5] based on a criterion for CalderonZygmund operators from [4] (see also [6]). The proofs and formulations, however, rely on explicit formulae for representations and for the Clebsch-Gordan coefficients available on SU (2) and are not extendable to other groups. In general, in the case when we do not deal with functions of a fixed operator, it is even unclear in which terms to formulate criteria for the L p -boundedness.…”