Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces

Abstract

Abstract We will establish the boundedness of the Fourier multiplier operator T m ⁢ f T_{m}f on multi-parameter Hardy spaces H p ⁢ ( R n 1 × ⋯ × R n r ) H^{p}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}) ( 0 < p ≤ 1 0<p\leq 1 ) when the multiplier 𝑚 is of optimal smoothness in multi-parameter Besov spaces B 2 , q ( s 1 , … , s r ) ⁢ ( R n 1 × ⋯ × R n r ) B^{{(s_{1},\ldots,s_{r})}}_{2,q}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}) , where T m ⁢ f ⁢ ( x ) = ∫ R n 1 × ⋯ × R n r m ⁢ ( ξ ) ⁢ f ^ ⁢ ( ξ ) ⁢ e 2 ⁢ π ⁢ i ⁢ x ⋅ ξ ⁢ d ξ T_{m}f(x)=\int_{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}}m(\xi)\hat{f}(\xi)e^{2\pi ix\cdot\xi}\,d\xi for x ∈ R n 1 × ⋯ × R n r x\in{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}} . We will show ∥ T m ∥ H p → H p ≲ sup j 1 , … , j r ∈ Z ⁡ ∥ m j 1 , … , j r ∥ B 2 , q ( s 1 , … , s r ) , \lVert T_{m}\rVert_{H^{p}\to H^{p}}\lesssim\sup_{j_{1},\ldots,j_{r}\in\mathbb{Z}}\lVert m_{j_{1},\ldots,j_{r}}\rVert_{B^{{(s_{1},\ldots,s_{r})}}_{2,q}}, where 0 < q < ∞ 0<q<\infty and s i > n i ⁢ ( 1 p - 1 2 ) s_{i}>n_{i}\bigl{(}\frac{1}{p}-\frac{1}{2}\bigr{)} . Here we have used the notation m j 1 , … , j r ⁢ ( ξ ) = m ⁢ ( 2 j 1 ⁢ ξ 1 , … , 2 j r ⁢ ξ r ) ⁢ ψ ( 1 ) ⁢ ( ξ 1 ) ⁢ ⋯ ⁢ ψ ( r ) ⁢ ( ξ r ) , m_{j_{1},\ldots,j_{r}}(\xi)=m(2^{j_{1}}\xi_{1},\ldots,2^{j_{r}}\xi_{r})\psi^{(1)}(\xi_{1})\cdots\psi^{(r)}(\xi_{r}), and ψ ( i ) ⁢ ( ξ i ) \psi^{(i)}(\xi_{i}) is a suitable cut-off function on R n i \mathbb{R}^{n_{i}} for 1 ≤ i ≤ r 1\leq i\leq r . This multi-parameter Hörmander multiplier theorem is in the spirit of the earlier work of Baernstein and Sawyer in the one-parameter setting and sharpens our recent result of Hörmander multiplier theorem in the bi-parameter case which was established using R. Fefferman’s boundedness criterion. Because R. Fefferman’s boundedness criterion fails in the cases of three or more parameters, it is substantially more difficult to establish such Hörmander multiplier theorems in three or more parameters than in the bi-parameter case. To assume only the optimal smoothness on the multipliers, delicate and hard analysis on the sharp estimates of the square functions on arbitrary atoms are required. Our main theorems give the boundedness on the multi-parameter Hardy spaces under the smoothness assumption of the multipliers in multi-parameter Besov spaces and show the regularity conditions to be sharp.