Transference of Multilinear Operators

We introduce the notion of transference (k + 1)-tuples of strongly continuous mappings defined on an amenable group G. We use these tuples to transfer boundedness properties of multilinear operators from products of Lebesgue spaces into Lp and weak Lp. 0. Introduction and statement of results Fix an integer k ≥ 2. Let G be an amenable group and (M,dμ) a measure space. For 0 ≤ j ≤ k, let 0 < pj ≤ ∞, and assume that p0 = p is given by 1 p0 = 1 p1 + · · · + 1 pk . Assume that for any 0 ≤ j ≤ k and any u ∈ G, R u is a bounded map from the Banach space Lj (M) into itself. We denote by ‖Rj u‖op the operator norm of R u : Lj (M) → Lj (M). We say that R u is strongly continuous if for any sequence un → u in the topology of G, we have ‖Rj unf−R uf‖Lpj (M) → 0 for all f ∈ Lj (M). We call the family (R u, R u, . . . , R u)u∈G a transference (k + 1)-tuple if the following are true: (0.1) for 0 ≤ j ≤ k, the maps u → R u are strongly continuous. (0.2) sup{‖Rj u‖op, u ∈ G} = Cj < ∞, for 0 ≤ j ≤ k. ∗Research partially supported by the National Science Foundation 1991 Mathematics Subject Classification. Primary 42. Typeset by AMS-TEX 1 (0.3) R vR j uf = R j vuf , for all u, v ∈ G, 1 ≤ j ≤ k, and all f ∈ D, where D is some dense subclass of all the spaces Lj (M) and we are implicitly assuming that the domain of any R u includes the ranges of each of the R j v. [BPW] used transference couples, (k = 1), to transfer boundedness properties of convolution and maximal operators. In this paper, we will use (k + 1)-tuples to transfer boundedness properties of multilinear operators from amenable groups into measure spaces. The general maximal transference presented in [BPW] can be extended to the multilinear setting, but this will not concern us in this paper. We need to make the additional assumption that each R u is multiplicative. More precisely, this means that R u(fg) = (R 0 uf)(R 0 ug) whenever f , g, and fg belong to D. This property is clearly satisfied if the R u’s are given by actions on the points of M , i.e. for all u ∈ G there exist maps Uu : M → M , such that (0.4) (R uf)(x) = f(U 0 u−1x). In this paper we shall, in fact, assume that (0.4) holds. In many settings (0.4) is a consequence of being multiplicative. Moreover, the restriction given by (0.4) is used explicitly for all the families R in the proof of the weak-type transference announced in Theorem 2. Let λ be left Haar measure on G. It is well known that if G is amenable with respect to left Haar measure λ, it is also amenable with respect to right Haar measure ρ. The spaces Lj (G) are defined with respect to left Haar measure λ. Consider the multilinear operator T on the group G defined by (0.5) T (g1, . . . , gk)(v) = ∫ Gk K(u1, . . . , uk)g1(u 1 v) . . . gk(u −1 k v) dλ(u1) . . . dλ(uk), for gj in some dense subspace of Lj (G), where K is a kernel on G which may not be integrable. For k = 1, T is a usual convolution operator but for k ≥ 2 it isn’t. We transfer the operator T to an operator T̃ defined by : (0.6) T̃ (f1, . . . , fk)(x) = ∫ Gk K(u1, . . . , uk)(R u1f1)(x) . . . (R k uk fk)(x) dλ(u1) . . . dλ(uk),