\(\ell_{p,q}\)-bounded sequences in function spaces (Q1885168)

Summary: A sequence of measurable functions \(\{f_n\}\) is called \(\ell_{p,q}\)-bounded sequence, \(0<p<\infty\), \(0<q\leq\infty\), if for any sequence of real numbers \(a=\{\alpha_n\}\in\ell_{p,q}\) we have \(\sup_n |\alpha_nF_n(\omega)| <\infty\) \(\omega\)-a.e., for any sequence \(\{F_n\}\) such that for every \(n\) the functions \(f_n\) and \(F_n\) are equimeasurable. The main result gives necessary and sufficient conditions for the sequence to be \(\ell_{p,q}\)-bounded.