Hormander Fourier multiplier theorems with optimal regularity in bi-parameter Besov spaces

The main aim of this paper to establish a bi-parameter version of a theorem of Baernstein and Sawyer [1] on boundedness of Fourier multipliers on one-parameter Hardy spaces H-p(R-n) which improves an earlier result of Calderon and Torchinsky [2]. More precisely, we prove the boundedness of the bi-parameter Fourier multiplier operators on the Lebesgue spaces L-p(R-n1 x R-n2) (1 < p < infinity) and bi-parameter Hardy spaces H-p(R-n1 x R-n2) (0 < p <= 1) with optimal regularity for the multiplier being in the bi-parameter Besov spaces B-2,1((n1/2 , n2/2)) (R-n1 x R-n2) and B-2,q((s1 , s2)) (R-n1 x R-n2). The Besov regularity assumption is clearly weaker than the assumption of the Sobolev regularity. Thus our results sharpen the known Hormander multiplier theorem for the bi-parameter Fourier multipliers using the Sobolev regularity in the same spirit as Baernstein and Sawyer improved the result of Calderon and Torchinsky. Our method is differential from the one used by Baernstein and Sawyer in the one-parameter setting. We employ the bi-parameter Littlewood-Paley-Stein theory and atomic decomposition for the bi-parameter Hardy spaces H-p(R-n1 x R-n2) (0 < p <= 1) to establish our main result (Theorem 1.6). Moreover, the bi-parameter nature involves much more subtlety in our situation where atoms are supported on arbitrary open sets instead of rectangles.