Boundedness of a class of bi-parameter square functions in the upper half-space (Q461704)

In this paper, the author introduces the bi-parameter square functions defined by NEWLINE\[NEWLINE Sf(y_1,y_2)=\left(\iint_{\Gamma(y_2)}\iint_{\Gamma(y_1)}\left|\theta_{t_1,t_2}f(x_1,x_2)\right|^2\frac{dx_1 dt_1}{t_1^{n+1}} \frac{dx_2 dt_2}{t_2^{m+1}}\right)^{1/2}, NEWLINE\]NEWLINEwhere \(\Gamma(y_1)=\{(x_1, t_1)\in \mathbb{R}_{+}^{n+1}: |x_1-y_1|<t_1\}\) and NEWLINE\[NEWLINE\theta_{t_1,t_2}f(x_1,x_2)=\iint_{\mathbb{R}^{n+m}} s_{t_1,t_2}(x_1,x_2,z_1,z_2) f(z_1,z_2) dz_1 dz_2.NEWLINE\]NEWLINENEWLINENEWLINEThe author together with \textit{M. Mourgoglou} [Proc. Am. Math. Soc. 142, No. 11, 3923--3931 (2014; Zbl 1325.42015)] proved a boundedness criterion for one-parameter square functions with general measures. In this paper, the author extends this efficient proof strategy to the case of two parameters. Precisely, under certain assumptions the author shows that NEWLINE\[NEWLINE\|Sf\|^2_{L^2(\mathbb{R}^{n+m})}\simeq \iint_{\mathbb{R}_{+}^{m+1}}\iint_{\mathbb{R}_{+}^{n+1}}\left|\theta_{t_1,t_2}f(x_1,x_2)\right|^2\frac{dx_1 dt_1}{t_1} \frac{dx_2 dt_2}{t_2}\lesssim \|f\|^2_{L^2(\mathbb{R}^{n+m})}.NEWLINE\]NEWLINENEWLINENEWLINEThe technique is based on a simple new averaging identity over good double Whitney regions.