Man, do I feel stupid for taking this long to figure it out.

Since Qian asked me to look over her paper on the equivalence of Besov norms defined relative to Klainerman-Rodnianski's Geometric-Littlewood-Paley projection and Besov norms relative to the classic LP projection (a paper I am still working on reading: she asked me to check the math carefully and so that's what I am doing), I've been thinking about Littlewood-Paley theory. One of the books that I am working through is Stein's Topics in Harmonic Analysis related to Littlewood-Paley Theory in which he introduced the notion of defining LP projections using heat-flow or Poisson flow on compact Lie groups. Sergiu and Igor's work is based on the heat-flow method on manifolds. The problem is that for the longest time I couldn't see how it relates to the classical Littlewood-Paley theory.

But now I think I understand.

The heart of Littlewood-Paley theory is probably the Littlewood-Paley Inequality, which gives a family of operators P^t that takes L^p functions to L^p functions for 1<p<∞, such that L^p(P^tf)≅L^p(f).

The philosophy behind Littlewood-Paley theory, however, is a different matter. The goal is to decompose a function in such a way that it is useful. For a decomposition to be useful in the study of partial differential equations, it helps to have the decomposition be frequency-localized in some way: that is because the size of a derivative is comparable to the size of the function multiplied by the frequency (think the function sin(kx) and its derivatives). A frequency decomposition would allow us to separate different types of interactions. The question then, is how to do the separation.

The simplest way is to take the Fourier transform of a function, multiply it by a localizing function in frequency space, and take the inverse Fourier transform. This pretty much defines the classical Littlewood-Paley theory. The problem is that such a method only works when the Fourier transform is well defined (which limits us to R^d and T^d, the Euclidean and Tori spaces). This simplest way also allows many very good estimates on the LP projections.

Let us look at the Fourier transform more carefully: rather than considering the full transform, let us look at the operator T_F defined by the following:

T_F[f](x,ξ) = e^ixξ∫f(x)e^-ixξdx

or, in other words, the integrand of the inverse Fourier transform. (All these is done formally of course.) Then the classical Littlewood-Paley decomposition is through a weight function m(ξ) = m(|ξ|):

E_k[f] = ∫ m(2^-2k|ξ|) T_F[f](x,ξ) dξ

The key is that the transform T_F produced a parametrized family of representation of the function f such that different frequency components of f are weighted differently by different parameters. In the classical Littlewood-Paley decomposition, the parameter is precisely the frequency.

So, in essence, we can define a Littlewood-Paley theory for any operator T taking f(x) to T[f](x,s) where s is the parameter such that for each s we have a projection unto the different frequency components according to a varying weight. Heuristically speaking, s parametrizes a family of measures μ(s) on frequency space. If we let g(ξ) denote the Fourier transform of f(x), then T[f] can be expressed as ∫g(ξ)e^ixξμ(s).

Of course, the requirement of the Littlewood-Paley inequality gives some limits to what families of measures we can choose. In general, Schwartz measures with the entire family covering the frequency space seems to be a good choice.

What remains is to find families of good measures that allow computations to be done.

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Back to the Geometric LP projections defined via heat-flow. Now we give an heuristic argument on how it (and the closely related cousin defined via Poisson-flow) defines a LP theory.

The heat-operator U take f(x) to U[f](t,x) such that it satisfies

∂_tU[f] = ΔU[f]

whereas the Poisson-operator P takes f(x) to P[f](t,x) with

∂_t²P[f] = -ΔP[f]

where Δ of course denote the Laplace-Beltrami operator. If we take the spacial Fourier transform of the RHS of both the equations, we get

-ξ²U[g](t,ξ)

where U[g] denotes the Fourier transform of U[f], and a corresponding equation for P[g].

It is then simple to see that both of the equations can be solved by a simple exponential, with the solutions

U[g](t,ξ) = e^-ξ²tg(ξ)
P[g](t,ξ) = e^-ξtg(ξ)

Notice how different frequencies dissipate with different speeds!

This "dispersive" property in which different waves travel at different speeds is what allows the definition of an LP theory from the operators.