Rearrangements, arrangements of sings and convergence of sequences of operators

A. B. Gulisashvili

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 46-70

Résumé

Let $(S,\Sigma,\mu)$ be a non-atomic measure space and $T_n$, $n\ge1$, be a sequence of integral operators $$ (T_nf)(x)=\int_S f(u)K_n(x,u)\,d\mu(u),\quad f\in L^1,\quad n\ge1, $$ with measurable and bounded kernels $K_n$. We prove that under some addtitional assumptions any function $f\in L^p$, $1\le p<\infty$, can be rearranged so that for the rearranged function $g$ the sequence $T_ng$ is convergent in the space $L^p$. As a corollary we obtain that any function $f\in L^p$, $1\le p<2$, can be rearranged so that the Fourier series with respect to any given complete orthonormal (in $L^2$) family of bounded functions is convergent in the space $L^p$. Similar questions are studied for arrangements of signs and in the case of the a.e. convergence and integrability of the maximal operator $T^*f=\sup_n|T_nf|$.