Singular integrals on product domains (Q5955999)

Suppose that the singular integral is defined by \[ Tf(x)= \int_{\mathbb{R}^d} K(x,t) f(t) dt\qquad (x\not\in\text{supp} f). \] If \(T\) is bounded on \(L^2(\mathbb{R}^d)\) and the kernel \(K\) satisfies \[ |K(x,t)- K(x,u)|\leq|t-u|^\delta|x-u|^{-d+\delta} \] then it is well-known that \(T\) is bounded from \(H^p(\mathbb{R}^d)\) to \(L^p(\mathbb{R}^d)\) for \(d/(d+\delta)< p\leq 2\) and for \(d/(d+N)< p\leq 2\) if the additional smoothness condition \[ |\partial^\alpha_t K(x,t)|\leq C|x-t|^{-d- |\alpha|}\quad (|\alpha|\leq N), \] is assumed on \(K\). To prove this boundedness, one only needs to prove that for any \(p\)-atom \(a(x)\), \(\|Ta\|_{L^p(\mathbb{R}^d)}\leq C\), where \(C\) is a constant independent of atoms. This method is not true on the product spaces, but a modified criterion was developed by Journé and by Pipher. In this reviewed paper, the author generalizes above results for the multi-parameter setting. He considers the Hardy spaces \(H^p(\mathbb{R}^{d_1}\times\cdots\times \mathbb{R}^{d_n})= H^p\) and supposes that \(T\) is an operator with the kernel \(K= K_1\cdots K_n\), where each \(K_i\) is a \(d_i\times d_i\)-dimensional kernel function. Then he establishes a criterion to verify that \(T\) is bounded from \(H^p\) to \(L^p\). He also proves that the multi-parameter Riesz transforms are bounded on \(H^p\) and they can be defined by a limit of non-singular integrals for \(f\in L(\text{Log }L)^{n-1}\).