ISSN 1439-8516 CN 11-2039/O1
The main purpose of this paper is to establish the Hörmander-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:
Tmf(x1, x2, x3)=(1)/((2π)n1+n2+n3)∫Rn1×Rn2×Rn3 m(ξ)f(ξ)e2πix·ξdξ
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×Rn2×Rn3 satisfying
.jpg){kind=small}
||mj,k,l||W (s1,s2,s3) < ∞
with si > ni(1/p -1/2) for 1 ≤ i ≤ 3. Then Tm is bounded from Hp(Rn1×Rn2×Rn3) to Hp(Rn1×Rn2×Rn3) for all 0 < p ≤ 1 and
||Tm Hp→Hp ≤ ||mj,k,l||W (s1,s2,s3)
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 for 1 ≤ i ≤ 3, and W(s1,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.
Let{fn} be a sequence of functions meromorphic in a domain D, let {hn} be a sequence of holomorphic functions in D, such that hn(z)
h(z), where h(z)≠ 0 is holomorphic in D, and let k be a positive integer. If for each n ∈ N+, fn(z)≠0 and fn(k)(z)-hn(z) has at most k distinct zeros (ignoring multiplicity) in D, then {fn} is normal in D.
