On projections in \(L_ 1\) and \(L_{\infty}\) (Q1081048)

Let \(X_ n\) be a nested family of n-dimensional subspaces of \(L_{\infty}\). Let \(P_ n\) be a sequence of projections from either \(L_ 1\) or \(L_{\infty}\) onto \(X_ n\). Suppose the projections are written in their integral form \((P_ nf)(t)=\int K_ n(s,t)f(s)ds.\) It is proved that if \(\| K_ n-K_{n-1}\|_{\infty}<o(\log (n)),\) then \(\sup \| P_ n\| =\infty\). The essence of the paper is a complicated construction proving a lower bound for the average of the integrals (double) of the kernels \(K_ n\). This inequality is then applied to prove three theorems including the one above. This application extends known results.