Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =（x1,x2,x3）∈Rn1×Rn2×Rn3 and ξ =（ξ1,ξ2,ξ3）∈Rn1×Rn2×Rn3. One of our main results is the following:Assume that m（ξ） is a function on Rn1+n2+n3 satisfying ■ with si >ni（1/p-1/2） for 1≤i≤3. Then Tm is bounded from Hp(Rn1×Rn2×Rn3 to Hp(Rn1×Rn2×Rn3for all 0 < p≤1 and ■ Moreover, the smoothness assumption on si for 1≤i≤3 is optimal. Here we have used the notations mj,k,l（ξ）=m（2jξ1,2kξ2,2lξ3）Ψ（ξ1）Ψ（ξ2）Ψ（ξ3） and Ψ（ξi） is a suitable cut-off function on Rni for1≤i≤3, and Ws1,s2,s3 is a three-parameter Sobolev space on Rn1×Rn2× Rn3.Because the Fefferman criterion breaks down in three parameters or more, we consider the Lp boundedness of the Littlewood-Paley square function of Tmf to establish its boundedness on the multi-parameter Hardy spaces.