A note on unconditionally converging series in 𝐿_{𝑝}

Abstract

In an effort to find a principle underlying such generalizations, we were led to the theorem stated in the abstract which reduces certain questions of a.e. convergence for u.c. series in L to the same questions for orthogonal series.Recall that a series 27; in X is unconditionally convergent if the map T(otj) = 2 ajfj is continuous as an operator from lx to X. Further, since the natural injection of L into L, is continuous when/? > 1, an unconditionally convergent series in L with p > 1 may as well be considered to be an unconditionally convergent series in L,. We now are in a position to prove the theorem stated in the abstract.

The Bilinear Maximal Functions Map into L p for 2/3 < p ≤ 1

“…Such sequences f j are called "unconditionally convergent," and are wellknown to enjoy many of the properties of orthogonal series. The most elegant result in this direction is one due to Ørno [8] which dilates the f j to an orthogonal series, but we cannot easily deduce our quantitative inequality from his. Our lemma is a close cousin of one due to G. Bennett, Theorem 2.5 of [1]; we do not include a proof because the lemma is easily obtained from the classical method of dyadic decomposition.…”