Let \(\Omega _i\) be finite sets, \(w_i(a)\ge 0\) weights, and \(k_i(a,b)\ge 0\) kernels for \(i\in [n]\). For \(p,r\ge 1\), suppose \(\sum _{a,b}k_i(a,b)u(a)v(b)\le \| u\| _{p,w_i}\| v\| _{r,w_i}\) for every coordinate \(i\) and real \(u,v\). Then \(\sum _{x,y}(\prod _i k_i(x_i,y_i))f(x)g(y)\le \| f\| _{p,w}\| g\| _{r,w}\) for all real \(f,g\) on \(\prod _i\Omega _i\), where \(w(x)=\prod _iw_i(x_i)\) and \(\| h\| _{s,w}=(\sum _xw(x)|h(x)|^s)^{1/s}\).

Let \(\Omega _i\) be finite sets, \(w_i(a)\ge 0\) weights, and \(k_i(y,x)\ge 0\) kernels for \(i\in [n]\). For \(p\ge 1\) and \(q\gt 1\), suppose \(\| \sum _xk_i(\, \cdot \, ,x)h(x)\| _{q,w_i}\le \| h\| _{p,w_i}\) for every coordinate \(i\) and real \(h\). Then the product operator \(Tf(y)=\sum _x\prod _i k_i(y_i,x_i)f(x)\) satisfies \(\| Tf\| _{q,w}\le \| f\| _{p,w}\) for every real \(f\) on \(\prod _i\Omega _i\), with \(w(x)=\prod _iw_i(x_i)\) and \(\| h\| _{s,w}=(\sum _xw(x)|h(x)|^s)^{1/s}\).