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Littlewood-Paley Theory and Its Applications

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Littlewood-Paley Theory and Its Applications LITTLEWOOD-PALEY THEORY AND APPLICATIONS Atelier d’Analyse Harmonique 2019 Harmonic analysis makes a fundamental use of divide-et-impera approaches. A particularly fruitful one is the decomposition of a function in terms of the frequen- cies that compose it, which is prominently incarnated in the theory of the Fourier transform and Fourier series. In many applications however it is not necessary or even useful to resolve the function f at the level of single frequencies and it suffices instead to consider how wildly different frequency components behave instead. One example of this is the (formal) decomposition of functions of R given by X f = ∆jf; j2Z where ∆jf denotes the operator Z 2πiξ·x ∆jf(x) := fb(ξ)e dξ; j j+1 fξ2R:2 ≤|ξj<2 g commonly referred to as a (dyadic) frequency projection (you have encountered a similar one in the study of Hörmander-Mikhlin multipliers). Thus ∆jf represents the portion of f with frequencies of magnitude ∼ 2j. The Fourier inversion for- mula can be used to justify the above decomposition if, for example, f 2 L2(R). Heuristically, since any two ∆jf; ∆kf oscillate at significantly different frequencies when jj − kj is large, we would expect that for most x’s the different contributions to the sum cancel out more or less randomly; a probabilistic argument typical of random walks (see Exercise 1) leads to the conjecture that jfj should behave “most 1=2 P 2 of the time” like j∆jfj (the last expression is an example of a square j2Z function). While this is not true in a pointwise sense, we will see in these notes that the two are indeed interchangeable from the point of view of Lp-norms: more precisely, we will show that for any 1 < p < 1 it holds that 1=2 X 2 kfkLp( ) ∼p j∆jfj : (y) R p L (R) j2Z This is a result historically due to Littlewood and Paley, which explains the name given to the related theory. It is easy to see that the p = 2 case is obvious thanks to Plancherel’s theorem, to which the statement is essentially equivalent. Therefore one could interpret the above as a substitute for Plancherel’s theorem in generic Lp spaces when p 6= 2. In developing a framework that allows to prove (y) we will encounter some variants of the square function above, including ones with smoother frequency projections that are useful in a variety of contexts. We will moreover show some applications of the above fact and its variants. One of these applications will be a proof of the boundedness of the spherical maximal function d−1 introduced in a previous MS lecture. Notation: We will use A . B to denote the estimate A ≤ CB where C > 0 is some absolute constant, and A ∼ B to denote the fact that A . B . A. If the constant C depends on a list of parameters L we will write A .L B. 1 2 LITTLEWOOD-PALEY THEORY 1. Motivation: estimates for the heat equation with sources In this section we present certain natural objects that cannot be dealt with by (euclidean) Calderón-Zygmund theory alone, as a way to motivate the study of Littlewood-Paley theory. Consider the heat equation on Rd ( @ u − ∆u = f; t (1.1) u(x; 0) = f0(x); with f = f(x; t) representing the instantaneous heat generated by the source at point x at instant t. If the total heat introduced by the sources is small, we expect to see that the solution u evolves slowly (that is, @tu is also small). One sense in which this conjecture can be made rigorous is to control the Lp(Rd × R)-norms of @tu by the corresponding norms of the data f. If we take a Fourier transform in both x and t we have @dtu(ξ; τ) = 2πiτub(ξ; τ); 2 2 ∆cu(ξ; τ) = −4π jξj ub(ξ; τ): Now, we are trying to relate @tu and f on the Fourier side (that is, we are trying to use spectral calculus). The simplest way to do so is to observe that, by taking the Fourier transform of both sides of (1.1), we have 2 2 @dtu − ∆cu = f;b that is (2πiτ + 4π jξj )ub(ξ; τ) = fb(ξ; τ); thus we can write with a little algebra 2πiτ + 4π2jξj2 τ @dtu(ξ; τ) = 2πiτu = 2πiτ u = f:b b 2πiτ + 4π2jξj2 b τ − 2πijξj2 τ Let m(ξ; τ) := τ−2πijξj2 , so that the result can be restated as @dtu = mfb. Observe that both the real and imaginary parts of m are discontinuous at (0; 0) and nowhere else. By Fourier inversion we can interpret this equality as defining a linear operator T , given by m Z i(ξ·x+tτ) Tmg(x; t) = m(ξ; τ)g(ξ; τ)e dξdτ; d b R ×R thus we have come to realise that the partial derivative @tu can be obtained (at least formally) from the function f by @ u = T f. To bound k@ uk p d is t m t L (R ×R) p therefore equivalent to bounding kT fk p d : if T is bounded from L into m L (R ×R) m Lp then we will have k@ uk p d = kT fk p d kfk p n ; t L (R ×R) m L (R ×R) . L (R ×R) which is the type of control we are looking for. Observe that by the properties of the Fourier transform our operator is actually a convolution operator: precisely,1 Z Z Tmf(x; t) = f ∗ m(x; t) = f(x − y; t − s)m(y; s)dyds: b d b R R If we knew that the convolution kernel mb satisfied kmb kL1 < 1 we would be done, since we could appeal to Young’s inequality. Unfortunately, this is not the case. We have already encountered this issue in dealing with singular integrals, so the natural thing to do is to check whether mb is a Calderón-Zygmund kernel. However, this 1 With a slight abuse of notation we have used mb to denote the inverse Fourier transform as d well, in place of the more usual mˇ , motivated by the fact that on R the two differ only by a reflection. We will continue to do so in the rest of these notes as it will always be clear from context which one is meant. LITTLEWOOD-PALEY THEORY 3 is also not the case! Indeed, if K is a (euclidean) Calderón-Zygmund convolution −d−1 kernel, you have seen that it must satisfy properties such as jrK(x)j . jxj . However, mb enjoys a certain (parabolic) scaling invariance that makes it incompat- ible with the last inequality: indeed, one can easily see that for any λ > 0 d+2 2 mb (y; s) = λ mb (λy; λ s) and deduce from this a similar anisotropic rescaling invariance for rmb . Since −d−1 j(y; s)j is not invariant with respect to this rescaling, the inequality jrmb (y; s)j . j(y; s)j−d−1 cannot possibly hold for all y; s. One solution to this issue is to develop a theory of parabolic singular integrals that extends the results of classical Calderón-Zygmund theory to those kernels that satisfy anisotropic rescaling invariance identities of the kind above. While this is a viable approach, we will take a different one in these notes. The approach we will take will not make use of said identities and will thus be more general in na- ture. With the methods of next section we will be able to show that Tm is indeed Lp ! Lp bounded for all 1 < p < 1. 2. Littlewood-Paley theory We begin by considering the frequency projections introduced in the beginning. Let I be an interval and define the frequency projection ∆I to be Z 2πiξx ∆I f(x) := 1I (ξ)fb(ξ)e dξ: An equivalent way of defining this is to work directly on the frequency side of things and stipulate that ∆I is the operator that satisfies ∆dI f(ξ) = 1I (ξ)fb(ξ); this is well-defined for f 2 L2 and can be therefore extended to functions f 2 2 p p p L \L . We claim that ∆I defines an L ! L bounded operator for any 1 < p < 1 and any (possibly semi-infinite) interval I and, importantly, that its Lp ! Lp norm is bounded independently of I. The point is that the operator ∆I is essentially a linear combination of two (modulated) Hilbert transforms, that are bounded in the same stated range. Indeed, observe that the Fourier transform of the Hilbert transform kernel p:v:1=x is −iπsign(ξ); it is easy to show that, if I = (a; b), then sign(ξ − a) − sign(ξ − b) 1 (ξ) = : (a;b) 2 Now, the Fourier transform exchanges translations with modulations, and specifi- cally fb(ξ + θ) = (f(·)e−2πiθ·)^(ξ); we see therefore that we can write, with H the Hilbert transform, Z −iπ fb(ξ)sign(ξ − a)e2πiξxdξ Z = − iπe2πiax fb(ξ + a)sign(ξ)e2πiξxdξ = e2πiaxH(e−2πia·f(·))(x): −2πiθx If we let Modθ denote the modulation operator Modθf(x) := e f(x), we have therefore shown that 1 −iπ∆ = Mod ◦ H ◦ Mod − Mod ◦ H ◦ Mod ; (a;b) 2 −a a −b b and from the Lp ! Lp boundedness of H it follows that k∆I fkLp .p kfkLp (2.1) p (since Modθ does not change the L norms at all).
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