Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators

Abstract

Abstract To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journé class, Fefferman discovered a boundedness criterion on bi-parameter Hardy spaces H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} . In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to L ω p ⁢ ( ℝ n 1 + n 2 ) {L_{\omega}^{p}(\mathbb{R}^{n_{1}+n_{2}})} and h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for all 0 < p ≤ 1 {0<p\leq 1} , which sharpens our earlier result even in the unweighted case requiring max ⁡ { n 1 n 1 + 1 , n 2 n 2 + 1 } < p ≤ 1 . \max\Bigl{\{}\frac{n_{1}}{n_{1}+1},\frac{n_{2}}{n_{2}+1}\Bigr{\}}<p\leq 1.