Singular integrals on product spaces with variable coefficients (Q1100382)

Let \(K^{(1)}(x)=\pi_ 1(x)/\| x\|\) n and \(K^{(2)}(y)=\pi_ 2(y)/\| y\|\) m denote the kernels that define the classical singular integrals that commute with dilations in \(R^ n\) and \(R^ m\), respectively. Then define the following singular integral with variable coefficients in \(R^ n\times R^ m\) \[ Tf(x,y)=\int \sum_{k}K^{(1)}_{(x')}\phi_ k(x')\int \sum_{h\geq \delta (k,y)}K^{(2)}(y')\psi_ h(y')f(x-x',y-y')dx'dy' \] where \(\phi\) and \(\psi\) are \(C^{\infty}\), radial functions supported respectively, on \(\{\leq \| x'\| \leq 2\}\), \((\leq \| y'\| \leq 2\}\) and \(\delta(k,y)\) is an arbitrary function. It is proved that T is a bounded operator from \(L^ p\) to itself, \(1<p<\infty\), with norm depending only on n,m,p and also from \(H^ p\) to \(L^ p\), \(0<p\leq 1\) and from \(LlgL^{M(n,m)}\) to weak-L 1. Moreover it is considered the maximal operator \[ \tilde Tf(x,y)=_{k_ 0>0}| \sum_{k\leq k_ 0}K^{(1)}\phi_ k*\sum_{h\geq \delta (k,y\quad)}K^{(2)}\psi_ n*f(x,y)| \] and it is proved the following pointwise estimate from above \[ Tf(x,y)\leq c\{M_ 1T^{(2)}f(x,y)+M_ 1(Tf)(x,y)\} \] where \(M_ 1\) denotes the maximal function in the x' variable and \[ \tilde T^{(2)}g(y)=_{h_ 0>0}| \sum_{h\leq h_ 0}K^{(2)}\psi_ k*g(y)|. \]