Abstract:
The classical Littlewood-Paley inequality says that, for $1<p<\infty$, the norm of an arbitrary function in $L^p$ admits a two-sided estimate in terms of the $L^p$-norm of the quadratic expression composed of the Fourier sums for this function over nonoverlapping intervals whose endpoints are consecutive powers of $2$. If should be noted that this inequality is an indispensable ingredient of the proof of the Marcinkiewicz multiplier theorem, and this theorem finds fairly wide application outside Fourier analysis.
The year of 1984 was marked by an unexpected and important turn in this classical theme: Rubio de Francia discovered that a one-sided estimate (upper or lower, depending on the position of the exponent p relative to the number $2$) in the Littlewood-Paley inequality survives for quadratic expressions produced by an arbitrary partition of the line into disjoint intervals. Multidimensional extensions of that result emerged soon after that, as well as some applications (new multiplier theorems), but the subject seemed to be exhausted by the beginning of 1990s.
However, in the recent years several new substantial questions arose, which finally where answered by my students and myself. Now the entire plot seems to reach its endpoint (once again?).