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• W1513563721 abstract "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 0 α [ μ ( {x ∈M : |f(x)| > α} )] 1 p . Let us now consider the case where all the R ’s are given by actions on points. That is, for all 1 ≤ j ≤ k and for all u ∈ G, there exist maps U j u : M → M such that the representations R u have the special form (0.7) (R uf)(x) = f(U j u−1x). In this case, we replace condition (0.3) by (0.8) U j uvf = U j uU 0 v f for all j = 1, . . . , k, all u, v ∈ G, and all f ∈ D. We now have the following Theorem 2. Assume that the R u’s satisfy (0.1), (0.2), (0.7), and (0.8). Assume that T given by (0.5) extends to a bounded operator from L1(G)× · · · ×Lk(G)→ Lp,∞(G) with norm N . Then T can be extended to a bounded operator L1(M)×· · ·×Lk(M)→ Lp,∞(M) with a bound no larger than NC0C1 . . . Ck. Finally, observe that an immediate consequence of (0.3) is (0.9) R vR 0 v−1R j u = R j u 3 for all u, v ∈ G and 1 ≤ j ≤ k. 1. The proof of Theorem 1 We first assume that L = support(K) is compact in all variables and that K is bounded in absolute value by some constant CK on L. Once the required estimate is proved for such kernels K, with bounds independent of their support and their size, a density argument will give the conclusion for all kernels K. The amenability of G is equivalent to Leptin’s condition: given > 0 and B a compact subset of G, there exists an open subset V of G, such that B is compact and (1.1) λ(B−1V ) ≤ (1 + )λ(V ). For a given > 0 and L = support(K), fix such a V . Also fix f1, . . . , fk ∈ D. The multiplicative property of R v and (0.9) imply (1.2) T (f1, . . . , fk)(x) = ∫ Gk K(u1, . . . , uk)R 0 v [ k ∏ j=1 (R v−1ujfj) ] (x)dλ(u1) . . . dλ(uk) for all v in G. By the continuity of R v, we can “move” R 0 v outside the k-fold integral in (1.2). Since T (f1, . . . , fk) is in L (M), (with bounds that depend on K) and R v is bounded on L(M) uniformly in v ∈ G, the following estimate holds ∫ M |T (f1, . . . , fk)(x)| dμ(x) ≤ C 0 ∫ M ∣∣∣∣ ∫ Gk K(u1, . . . , uk) k ∏ j=1 ( R v−1ujfj ) (x) dλ(u1) . . . dλ(uk) ∣∣∣∣p dμ(x), (1.3) for all v in G. Next, we average inequality (1.3) over V and we interchange the order of 4 integration to the right hand side of the averaged inequality. We obtain ∫ M |T (f1, . . . , fk)(x)| dμ(x) ≤ C 0 λ(V ) ∫" @default.
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• W1513563721 date "1996-06-01" @default.
• W1513563721 modified "2023-09-26" @default.
• W1513563721 title "Transference of multilinear Operators" @default.
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• W1513563721 doi "https://doi.org/10.1215/ijm/1255986110" @default.
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