This is the last part of a 3 part series on the basics of Littlewood-Paley theory. Today we discuss a couple of applications, that is Marcinkiewicz multiplier theorem and the boundedness of the spherical maximal function (the latter being an application of frequency decompositions in general, and not so much of square functions – though one appears, but only for estimates where one does not need the sophistication of Littlewood-Paley theory).
Part I: frequency projections
Part II: square functions
7. Applications of Littlewood-Paley theory
7.1. Marcinkiewicz multipliers
for all . The operator is called a multiplier and the function is called the symbol of the multiplier1. Since , Plancherel’s theorem shows that is a linear operator bounded in ; its definition can then be extended to functions (which are dense in ). A natural question to ask is: for which values of in is the operator an bounded operator? When is bounded in a certain space, we say that it is an –multiplier.
The operator introduced in Section 1 of the first post in this series is an example of a multiplier, with symbol . It is the linear operator that satisfies the formal identity . We have seen that it cannot be a (euclidean) Calderón-Zygmund operator, and thus in particular it cannot be a Hörmander-Mikhlin multiplier. This can be seen more directly by the fact that any Hörmander-Mikhlin condition of the form is clearly incompatible with the rescaling invariance of the symbol , which satisfies for any . However, the derivatives of actually satisfy some other superficially similar conditions that are of interest to us. Indeed, letting for simplicity, we can see for example that . When we can therefore argue that , and similarly when ; this shows that for any with one has
This condition is comparable with the corresponding Hörmander-Mikhlin condition only when , and is vastly different otherwise, being of product type (also notice that the inequality above is compatible with the rescaling invariance of , as it should be).