Transference of bilinear multipliers on Lorentz spaces

Abstract

We study DeLeeuw type transference theorems for multi-linear multiplier operators on the Lorentz spaces. To be detail, we show that, under some mild conditions on m, a bilinear multiplier operator $T_{m,1}(f,g)$ is bounded on the Lorentz space in ${\mathbb {R}} ^{n}$ if and only if its periodic version ${\widetilde{T}}_{m,\varepsilon }({\widetilde{f}},{\widetilde{g}})$ is bounded on the Lorentz space in the n-torus $T^{n}\$uniformly on $\varepsilon >0.$ Most significantly, we prove that these two operators share the same operator norm. We also obtain the same results on their restriction versions and their maximal versions $T_{m}^{*}(f,g)$ and ${\widetilde{T}}_{m}^{*}({\widetilde{f}},{\widetilde{g}})$. The previous method by Kenig and Tomas to treat the sub-linear operator $T_{m}^{*}(f)$ is to linearize the operator and then invoke the duality argument. This approach seems complicated and difficult to be used when we study the sub-bilinear operator $T_{m}^{*}(f,g)$. Thus, we will use a simpler, but different method. Our results are substantial improvements and extensions of many known theorems.