Transference of bilinear multipliers on Lorentz spaces

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 Tm,1( f, g) is bounded on the Lorentz space in Rn if and only if its periodic version (T) over tilde (m,e)( (f) over tilde, (g) over tilde) is bounded on the Lorentz space in the n-torus T-n uniformly on epsilon > 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 (T) over tilde *m((f) over tilde, (g) over tilde). 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.