Hörmander 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 Calder´on and Torchinsky [2]. More pre-cisely, we prove the boundedness of the bi-parameter Fourier multiplier operators on the Lebesgue spaces L p ( R n 1 × R n 2 ) (1 < p < ∞ ) and bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) (0 < p ≤ 1) with optimal regularity for the multiplier being in the bi-parameter Besov spaces B ( n 12 , n 22 ) 2 , 1 ( R n 1 × R n 2 ) and B ( s 1 ,s 2 ) 2 ,q ( R n 1 × R n 2 ). The Besov regularity assumption is clearly weaker than the assumption of the Sobolev regularity. Thus our results sharpen the known H¨ormander multiplier theorem for the bi-parameter Fourier multipliers using the Sobolev regularity in the same spirit as Baernstein and Sawyer improved the result of Calder´on 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 n 1 × R n 2 ) (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.