Inverse theorems in \(L^ p\) for some multidimensional positive linear operators (Q1192403)

The paper is concerned with inverse theorems for bidimensional positive linear operators on \(L_ p(I\times I)\), \(I=[0,1]\) or \([0,\infty)\) of the form \[ L_{n,m}(F;x,y)=L_ n(L_ m(F(u,v)y);x)=L_ m(L_ n(F(u,v);x);y), \] for \(n,m\in N\), \(F\in L_ p(I\times I)\), \(1\leq p<\infty\), \((x,y)\in I\times I\) and \(\{L_ n\}\) a sequence of positive linear operators on \(L_ p(I)\). A corollary to the main result of the paper asserts that \[ \| L_{m,n} F-F\|_{L_ p(I\times I)}=O(n^{-\alpha}-m^{-\alpha}), \] if and only if \[ \|\sup_{0\leq h\leq t}\|\Delta^ 2_{h\varphi} F^ y\left\|_{L_ p[h^*,H^{**}]}\right\|_ p=O(t^{2\alpha}), \] where, for \(0\leq h\leq 1\), \(h^*=h^ 2/(1+h^ 2)\) or \(h^ 2\) and \(H^{**}=1/(1+h^ 2)\) or \(\infty\), according to \(I=[0,1]\) or \([0,\infty)\), and \(\varphi(x)=\sqrt{x(1-x)}\) or \(\sqrt x\). The considered class of operators contains as particular cases the Kantorovich, Szás-Mirakjan- Kantorovich, Meyer-König and Zeller, Baskakov-Kantorovich, Bernstein- Durrmeyer and Szás-Mirakjan clases of positive linear operators.