# Basic Littlewood-Paley theory III: applications

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 $L^2$ 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

In this section we will present two applications of the Littlewood-Paley theory developed so far. You can find further applications in the exercises (see particularly Exercise 22 and Exercise 23).

7.1. Marcinkiewicz multipliers

Given an ${L^\infty (\mathbb{R}^d)}$ function ${m}$, one can define the operator ${T_m}$ given by

$\displaystyle \widehat{T_m f}(\xi) := m(\xi) \widehat{f}(\xi)$

for all ${f \in L^2(\mathbb{R}^d)}$. The operator ${T_m}$ is called a multiplier and the function ${m}$ is called the symbol of the multiplier1. Since ${m \in L^\infty}$, Plancherel’s theorem shows that ${T_m}$ is a linear operator bounded in ${L^2}$; its definition can then be extended to ${L^2 \cap L^p}$ functions (which are dense in ${L^p}$). A natural question to ask is: for which values of ${p}$ in ${1 \leq p \leq \infty}$ is the operator ${T_m}$ an ${L^p \rightarrow L^p}$ bounded operator? When ${T_m}$ is bounded in a certain ${L^p}$ space, we say that it is an ${L^p}$multiplier.

The operator ${T_m}$ introduced in Section 1 of the first post in this series is an example of a multiplier, with symbol ${m(\xi,\tau) = \tau / (\tau - 2\pi i |\xi|^2)}$. It is the linear operator that satisfies the formal identity $T \circ (\partial_t - \Delta) = \partial_t$. 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 ${|\partial^{\alpha}m(\xi,\tau)| \lesssim_\alpha |(\xi,\tau)|^{-|\alpha|} = (|\xi|^2 + \tau^2)^{-|\alpha|/2}}$ is clearly incompatible with the rescaling invariance of the symbol ${m}$, which satisfies ${m(\lambda \xi, \lambda^2 \tau) = m(\xi,\tau)}$ for any ${\lambda \neq 0}$. However, the derivatives of ${m}$ actually satisfy some other superficially similar conditions that are of interest to us. Indeed, letting ${(\xi,\tau) \in \mathbb{R}^2}$ for simplicity, we can see for example that ${\partial_\xi \partial_\tau m(\xi, \tau) = \lambda^3 \partial_\xi \partial_\tau m(\lambda\xi, \lambda^2\tau)}$. When ${|\tau|\lesssim |\xi|^2}$ we can therefore argue that ${|\partial_\xi \partial_\tau m(\xi, \tau)| = |\xi|^{-3} |\partial_\xi \partial_\tau m(1, \tau |\xi|^{-2})| \lesssim |\xi|^{-1} |\tau|^{-1} \sup_{|\eta|\lesssim 1} |\partial_\xi \partial_\tau m(1, \eta)|}$, and similarly when ${|\tau|\gtrsim |\xi|^2}$; this shows that for any ${(\xi, \tau)}$ with ${\xi,\tau \neq 0}$ one has

$\displaystyle |\partial_\xi \partial_\tau m(\xi, \tau)| \lesssim |\xi|^{-1} |\tau|^{-1}.$

This condition is comparable with the corresponding Hörmander-Mikhlin condition only when ${|\xi| \sim |\tau|}$, and is vastly different otherwise, being of product type (also notice that the inequality above is compatible with the rescaling invariance of ${m}$, as it should be).

The Littlewood-Paley theory developed in these notes allows us to treat multipliers with symbols that satisfy product-type conditions like the above, and which typically are beyond the reach of Calderón-Zygmund theory. Indeed, we have the following general result, where the conditions assumed of the multiplier are a generalisation of the pointwise ones above.

Theorem 8 (Marcinkiewicz multiplier theorem) Let ${m}$ be a function of ${\mathbb{R}^d}$ that is of class ${C^d}$ away from the hyperplanes where one of the coordinates is zero. Suppose that ${m}$ satisfies the following conditions:

1. ${m \in L^\infty}$;
2. there is a constant ${C>0}$ such that for every ${0 < \ell \leq d}$ and for every permutation ${j_1, \ldots,j_\ell,j_{\ell+1}, \ldots, j_d}$ of the set ${\{1,\ldots, d\}}$ we have

$\displaystyle \sup_{\boldsymbol{k} \in \mathbb{Z}^\ell} \sup_{\xi_{j_{\ell+1}}, \ldots, \xi_{j_d} \neq 0} \int_{R_{\boldsymbol{k}}} \big|\partial_{\xi_{j_1}}\cdots \partial_{\xi_{j_\ell}} m(\xi)\big| d\xi_{j_1} \ldots d\xi_{j_\ell} \leq C,$

where the ${R_{\boldsymbol{k}}}$ are the dyadic Littlewood-Paley rectangles as in § 6 of the second post in this series.

Then the multiplier ${T_m}$ associated to the symbol ${m}$ satisfies for any ${1 the inequality

$\displaystyle \|T_m f\|_{L^p(\mathbb{R}^d)} \lesssim_p \|f\|_{L^p(\mathbb{R}^d)}.$

To be more specific, if $\boldsymbol{k}= (k_1, \ldots,k_d)$ and $I_k := [2^k, 2^{k+1}]\cup [-2^{k+1},-2^k]$, the rectangle $R_{\boldsymbol{k}}$ is given by $I_{k_1} \times \ldots \times I_{k_d}$.

A multiplier whose symbol satisfies the conditions of the above theorem is called a Marcinkiewicz multiplier. Condition 2., spelled out in words, says that a certain subset of the partial derivatives ${\partial^\alpha m}$ of ${m}$ (precisely, the derivatives ${\partial^\alpha}$ where the components of the multi-index ${\alpha}$ have values in ${\{0,1\}}$ and ${0<|\alpha|\leq d}$) has the property that their integral over any ${|\alpha|}$-dimensional dyadic Littlewood-Paley rectangle is uniformly bounded, where the integration is happening in those variables ${\xi_j}$ such that ${\alpha_j \neq 0}$.

Remark 1 The conditions above are very general but also very cumbersome to spell out, as the above attempt demonstrates. The pointwise conditions, analogous to the one seen before, that imply condition 2. of the theorem are usually easier to check and are as follows:

1. ${m \in L^\infty}$ exactly as before;
2. (pointwise) for any multi-index ${\alpha \in \{0,1\}^d}$ such that ${0< |\alpha| \leq d}$ we have

$\displaystyle |\partial^\alpha m(\xi)| \lesssim_\alpha |\xi_1|^{-\alpha_1}\cdot \ldots \cdot |\xi_d|^{-\alpha_d}.$

See Exercise 14.

When the dimension ${d}$ is equal to ${1}$, the statement of the theorem can be superficially generalised and reformulated in the following way, that perhaps clarifies the moral meaning of condition 2.

Theorem 9 (Marcinkiewicz multiplier theorem for ${d=1}$) Let ${m}$ be a function of ${\mathbb{R}}$ that is of bounded variation on any interval not containing the origin. Suppose that ${m}$ satisfies the following conditions:

1. ${m \in L^\infty}$;
2. there is a constant ${C>0}$ such that, with ${\int_I |dm|}$ the total variation2 of ${m}$ over the interval ${I}$,

$\displaystyle \sup_{k \in \mathbb{Z}} \int_{[2^{k}, 2^{k + 1}] \cup [-2^{k + 1}, -2^{k}]} |dm|(\xi) \leq C.$

Then the multiplier ${T_m}$ with symbol ${m}$ satisfies for any ${1 the inequality

$\displaystyle \|T_m f\|_{L^p(\mathbb{R})} \lesssim_p \|f\|_{L^p(\mathbb{R})}.$

Condition 2. is now saying that the total variation of ${m}$ on any dyadic Littlewood-Paley interval is bounded. Observe that it is implied by the pointwise condition ${|m'(\xi)| \lesssim |\xi|^{-1}}$, but since we are in dimension ${d=1}$ now this stronger condition would just coincide with the Hörmander-Mikhlin one. There is in general a certain degree of overlap between the Marcinkiewicz multiplier theorem and the Hörmander-Mikhlin theorem in any dimension, but neither implies the other.

We will prove Theorem 9 here, and the proof we will give will already contain all the main ingredients needed for the proof of the full Theorem 8. We leave it to you to extend the proof to higher dimensions in Exercise 15.

Proof:
Let ${T = T_m}$ be the multiplier with symbol ${m}$. Observe that by Theorem 5 of the second part of these notes we have

$\displaystyle \|Tf\|_{L^p} \lesssim_p \Big\| \Big(\sum_{j \in \mathbb{Z}} |\Delta_j Tf|^2 \Big)^{1/2} \Big\|_{L^p},$

and thus it will suffice to prove that

$\displaystyle \Big\| \Big(\sum_{j \in \mathbb{Z}} |\Delta_j Tf|^2 \Big)^{1/2} \Big\|_{L^p} \lesssim_p \Big\| \Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2} \Big\|_{L^p}, \ \ \ \ \ (1)$

by a second application of the boundedness of the Littlewood-Paley square function. Now, ${\Delta_j T}$ is a multiplier with symbol ${m(\xi) \mathbf{1}_{[2^j,2^{j+1}]}(\xi)}$ (we discard the negative frequencies for ease of notation), and by condition 2) we see that there exists a complex Borel measure3 ${dm}$ such that for ${\xi \in [2^j,2^{j+1}]}$

$\displaystyle m(\xi) = m(2^j) + \int_{2^j}^{\xi} dm(\eta).$

By Fubini we see therefore that

$\displaystyle \Delta_j T f(x) = m(2^j) \Delta_j f(x) + \int_{2^j}^{2^{j+1}} \Delta_{[\eta,2^{j+1}]}f(x) \, dm(\eta),$

where we recall that ${\Delta_{[\eta,2^{j+1}]}}$ denotes frequency projection onto the interval ${[\eta,2^{j+1}]}$. By condition 1), the first term in the right-hand side contributes at most ${\|m\|_{L^\infty} Sf}$ to the left-hand side of (1) overall, and therefore we can safely discard it; let ${T_j f(x)}$ be the second term. We have by Cauchy-Schwarz and condition 2) that

$\displaystyle |T_j f(x)|^2 \leq C \int_{2^j}^{2^{j+1}} |\Delta_{[\eta,2^{j+1}]}f(x)|^2 \, |dm|(\eta),$

and therefore we have reduced to control the square function

$\displaystyle \Big( \sum_{j\in\mathbb{Z}}\int_{2^j}^{2^{j+1}} |\Delta_{[\eta,2^{j+1}]}f(x)|^2 \, |dm|(\eta) \Big)^{1/2}. \ \ \ \ \ (2)$

Now comes the tricky part: we want to realise the above expression as the norm in an abstract Hilbert space ${\mathscr{H}}$ of a vector-valued frequency projection, as in Proposition 1 of the first part of this series. There are many ways of doing so, and the following is the one we choose. Let ${\Gamma := \bigcup_{j \in \mathbb{Z}} (2^j, 2^{j+1}] \times \{j\}}$, that is the set of elements ${(\eta,j)}$ such that ${2^j < \eta \leq 2^{j+1}}$; observe that a set ${E \subseteq \Gamma}$ has necessarily the form ${E = \bigcup_{j \in A} E_j \times \{j\}}$ for a certain ${A \subseteq \mathbb{Z}}$ and sets ${E_j \subseteq (2^j, 2^{j+1}]}$. We define the measure ${d\mu}$ on ${\Gamma}$ to be given by

$\displaystyle \mu(E) := \sum_{j \in A} \int_{E_j} |dm|(\eta),$

provided the ${E_j}$‘s are ${|dm|}$-measurable. Thus the Hilbert space ${\mathscr{H} = L^2(\Gamma, d\mu)}$ consists of those functions ${G(\eta,j)}$ such that

$\displaystyle \sum_{j\in\mathbb{Z}} \int_{2^j}^{2^{j+1}} |G(\eta,j)|^2 |dm|(\eta)$

is finite. Finally, to any ${(\eta,j) \in \Gamma}$ we assign the interval ${I_{\eta,j} := [\eta,2^{j+1}]}$. With these definitions, we see that (2) corresponds to

$\displaystyle \|(\Delta_{I_{\eta,j}} g_{\eta,j}(x))_{(\eta,j) \in \Gamma}\|_{\mathscr{H}},$

where in our particular case we can take ${g_{\eta,j}(x) = \Delta_j f(x)}$ because ${\Delta_{I_{\eta,j}} \Delta_j = \Delta_{I_{\eta,j}}}$. By Proposition 1 of the first part of this series we have for generic vector-valued functions ${(g_{\eta,j})_{(\eta,j) \in \Gamma}}$ that ${\|(\Delta_{I_{\eta,j}} g_{\eta,j})_{(\eta,j) \in \Gamma}\|_{L^p(\mathbb{R};\mathscr{H})} \lesssim_{p} \|(g_{\eta,j})_{(\eta,j) \in \Gamma}\|_{L^p(\mathbb{R};\mathscr{H})}}$; applying this to ${g_{\eta,j} = \Delta_j f}$ we see that we have bounded ${\|(\sum_{j}|T_j f|^2)^{1/2}\|_{L^p}}$ by the ${L^p(\mathbb{R})}$ norm of

$\displaystyle \Big( \sum_{j \in \mathbb{Z}} \int_{2^j}^{2^{j+1}} |\Delta_j f(x)|^2 |dm|(\eta) \Big)^{1/2},$

but the integrand does not depend on ${\eta}$ and thus condition 2) shows that the above is bounded pointwise by ${C \, Sf(x)}$. This concludes the proof of (1) and thus the proof of the theorem. $\Box$

7.2. Boundedness of the spherical maximal function

Recall that the boundedness of the spherical maximal function ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ implies, through the method of rotations, that the Hardy-Littlewood maximal function ${M}$ is ${L^q(\mathbb{R}^d) \rightarrow L^q(\mathbb{R}^d)}$ bounded for any ${1 < q < \infty}$ with constant independent of the dimension. In the final part of this lecture we will finally prove the boundedness of ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ (when ${d\geq 3}$).

Theorem 10 (Stein, ’76) Let ${d \geq 3}$. Then for any ${\frac{d}{d-1} < p \leq \infty}$ the inequality

$\displaystyle \| \mathscr{M}_{\mathbb{S}^{d-1}} f\|_{L^p(\mathbb{R}^d)} \lesssim_{p} \|f\|_{L^p(\mathbb{R}^d)}$

holds for any ${f \in L^p(\mathbb{R}^d)}$.

The range of ${p}$‘s stated in the above theorem is sharp (prove it in Exercise 17).
A small caveat: the proof we will give below will yield a constant ${C_{d,p}}$ for the above inequality that depends on the dimension ${d}$. It is only later – through the method of rotations – that one can remove this dependence on the dimension, showing that if one has a bound ${\| \mathscr{M}_{\mathbb{S}^{d-1}}\|_{L^p(\mathbb{R}^{d}) \rightarrow L^p(\mathbb{R}^{d})} \leq A}$ for dimension ${d}$ then one also has the bound ${\| \mathscr{M}_{\mathbb{S}^{d}}\|_{L^p(\mathbb{R}^{d+1}) \rightarrow L^p(\mathbb{R}^{d+1})} \leq A}$ for dimension ${d+1}$.
Let me stress once again that the following is merely a slight re-elaboration of Tao’s excellent post on the subject.

Proof: It will suffice to prove the theorem for ${\frac{d}{d-1}< p<2}$, since the rest of the exponents can be obtained by Marcinkiewicz interpolation with the trivial ${L^\infty}$ estimate. It will also suffice to assume that ${f}$ is a Schwartz function for convenience, as they are dense in ${L^p}$.
We will prove the theorem first for a local version of ${\mathscr{M}_{\mathbb{S}^{d-1}}}$, namely the operator

$\displaystyle \mathscr{M}_{\mathbb{S}^{d-1}}^{\mathrm{loc}} f(x) := \sup_{1 \leq r \leq 2} |A_r f(x)|,$

where ${A_r}$ denotes the average over the sphere of radius ${r}$, that is

$\displaystyle A_r f(x) := \int_{\mathbb{S}^{d-1}} f(x - r y) d\sigma(y),$

where ${d\sigma}$ is the normalised surface measure on the sphere. Then in Exercise 21 you will conclude the proof for the full case of ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ with little extra effort.
We will use an annular frequency decomposition as the one in Section 6 of the second post in this series, but slightly modified to suit our purposes. We let ${\psi}$ be a smooth radial function compactly supported in the annulus ${\{ \xi \in \mathbb{R}^d : 1/2 < |\xi| < 2\}}$ and such that4 for any ${\xi\neq 0}$ we have ${\sum_{j \in \mathbb{Z}} \psi(2^{-j}\xi) = 1}$. We let ${\psi_j (\xi) := \psi(2^{-j}\xi)}$ for convenience. Thus, with ${P_j}$ being given by ${\widehat{P_j f}(\xi) = \psi_j(\xi)\widehat{f}(\xi)}$ we have that for all functions ${f \in L^2}$ we can decompose

$\displaystyle f = \sum_{j \in \mathbb{Z}} P_j f.$

Since the radius ${r}$ will be ${\sim 1}$, we will not need to consider the projections to extremely low frequencies separately (see below for an explanation); therefore we define ${P_{low}f := \sum_{j<0} P_j f}$, so that ${f = P_{low}f + \sum_{j \in \mathbb{N}} P_j f}$. In view of the above decomposition, to conclude the ${L^p \rightarrow L^p}$ boundedness of ${\mathscr{M}_{\mathbb{S}^{d-1}}^{\mathrm{loc}}}$ it will suffice by triangle inequality to prove

$\displaystyle \Big\| \sup_{1 \leq r \leq 2} |A_r P_{low} f| \Big\|_{L^p} \lesssim_p \, \|f\|_{L^p}, \ \ \ \ \ (3)$

$\displaystyle \Big\| \sup_{1 \leq r \leq 2} |A_r P_j f| \Big\|_{L^p} \lesssim_p \, 2^{-\delta j}\|f\|_{L^p}, \ \ \ \ \ (4)$

for some ${\delta=\delta_{p,d} >0}$.
Now, the heuristic motivation behind such a decomposition is that ${|P_j f|}$ is now roughly constant at scale ${2^{-j}}$. Indeed, one way to appreciate this informal principle (which is a manifestation of the Uncertainty Principle) is to show for example that ${|P_j f(x)|}$ is pointwise dominated by its average at scale ${2^{-j}}$: since ${\widehat{P_j f}}$ is supported in the ball ${B(0,2^{j+1})}$, we can take a smooth bump function ${\phi}$ such that ${\phi(2^{-j}\xi)}$ is identically ${1}$ on this ball and vanishes outside ${B(0,2^{j+2})}$ and see that therefore ${\widehat{P_j f}(\xi) = \phi(2^{-j}\xi)\widehat{P_j f}(\xi)}$. The latter means that ${P_j f = P_j f \ast 2^{jd} \widehat{\phi}(2^{j}\cdot)}$ and the smoothness of ${\phi}$ means that we have ${|\widehat{\phi}(x)| \lesssim (1 + |x|)^{-10d}}$ (say); therefore we have by expanding the convolution that

$\displaystyle |P_j f(x)| \lesssim \int_{\mathbb{R}^d} |P_j f(x- 2^{-j}y)| \frac{dy}{(1 + |y|)^{10d}}.$

This observation suggests that ${A_r P_{low} f}$ should just be an average of ${f}$ at unit scale, since ${r \sim 1}$ and ${P_{low} f}$ is roughly constant at scales ${\lesssim 1}$. Indeed, one can realise that ${P_{low}}$ is given by convolution with a Schwartz function ${\varphi}$ and thus by Fubini ${A_r P_{low} f = f \ast A_r \varphi}$; it is not hard to show that ${\varphi}$ being Schwartz and ${r}$ being ${\sim 1}$ implies bounds such as ${|A_r \varphi(x)| \lesssim (1+|x|)^{-10d}}$ (or any other exponent really), and therefore we have as seen before

$\displaystyle |A_r P_{low} f(x)| \lesssim Mf(x)$

uniformly in ${r}$, so that the pointwise domination continues to hold after we take the supremum. This immediately implies (3) for any ${1 (so even outside our desired range).
Repeating the argument for ${P_j f}$ does not allow us to conclude. Indeed, you can see that in this case ${|A_r \widehat{\psi_j}|}$ is essentially a function of magnitude ${\sim 2^j}$ concentrated in a shell of width ${\sim 2^{-j}}$ and radius ${r}$ (see Exercise 20). A greedy argument, disregarding the width information, shows the rough bound ${|A_r \widehat{\psi_j}(x)| \lesssim 2^j (1 + |x|)^{-10 d}}$, which in turn implies ${|A_r P_j f| \lesssim 2^j Mf}$ uniformly in ${r \sim 1}$. Therefore this argument gives at best the bound

$\displaystyle \Big\| \sup_{1 \leq r \leq 2} |A_r P_j f| \Big\|_{L^p} \lesssim_p \, 2^{j}\|f\|_{L^p}, \ \ \ \ \ (5)$

which blows up as ${j \rightarrow \infty}$. However, one advantage of this bound is that it holds even for ${1, which would allow for some interpolation if we had a decaying bound for some other exponent to compensate with. A particularly nice exponent is ${p=2}$ of course, because Plancherel's theorem is available, so we should explore what happens in this case. Consider ${r}$ fixed and observe that

$\displaystyle \widehat{A_r f}(\xi) = \widehat{d\sigma}(r \xi) \widehat{f}(\xi),$

where ${\widehat{d\sigma}}$ denotes the Fourier transform of ${d\sigma}$, that is ${\widehat{d\sigma}(\xi) = \int_{\mathbb{S}^{d-1}} e^{-2\pi i y \cdot \xi} d\sigma(y)}$ (verify the formula). It turns out that, due to the curvature of the sphere, ${\widehat{d\sigma}}$ has a certain amount of decay as ${\xi}$ gets large – in particular, one has the pointwise bound

$\displaystyle |\widehat{d\sigma}(\xi)| \lesssim (1 + |\xi|)^{-(d-1)/2}.$

We will take this for granted here (it is a well known fact) but if you want to see a proof, prove the decay yourself in Exercise 19. Now, combining the decay information above with the frequency localisation of ${P_j f}$, we see by Plancherel that, since ${r\sim 1}$,

$\displaystyle \|A_r P_j f\|_{L^2} \lesssim 2^{-j(d-1)/2} \|f\|_{L^2}, \ \ \ \ \ (6)$

a bound nicely decaying in ${j}$. However, this bound is not directly useful to us as we still have to take the supremum in ${r}$! To deal with this issue we will do something a bit unconventional: we will replace the supremum by the integral of the derivative in the parameter! Indeed, observe that the fact that $|P_j f|$ is approximately constant at scale $2^{-j}$ suggests that $A_r P_j f$ cannot change by much if we change ${r}$ by $\ll 2^{-j}$; that is, we expect $A_r P_j f$ to be somewhat “smooth” in the parameter ${r}$. We could take advantage of this heuristic as follows: since for a generic function ${\phi(r)}$ we have by the Fundamental Theorem of Calculus that

$\displaystyle \sup_{1 \leq r \leq 2} |\phi(r)| \leq |\phi(1)| + \int_{1}^{2} |\phi'(r)|dr,$

we could replace the supremum by the right-hand side. However, if we proceeded with this approach we would get bad bounds in the end – the second term at the right-hand side would contribute much more than the other one. We need therefore some more sophisticated inequality which allows us to optimally balance the two contributions, and the following will do. Observe that, by the Fundamental Theorem of Calculus and Cauchy-Schwarz, if ${\phi(1)=0}$ we have

\displaystyle \begin{aligned} \frac{1}{2}\sup_{1 \leq r \leq 2} |\phi(r)|^2 \leq & \int_{1}^{2} |\phi(r)||\phi'(r)|dr \\ \leq & \Big(\int_{1}^{2} |\phi(r)|^2 dr\Big)^{1/2} \Big(\int_{1}^{2} |\phi'(r)|^2 dr\Big)^{1/2}, \end{aligned}

and by the trivial inequality ${xy \lesssim B x^2 + B^{-1} y^2}$ we have therefore

$\displaystyle \sup_{1 \leq r \leq 2} |\phi(r)| \lesssim B \Big(\int_{1}^{2} |\phi(r)|^2 dr\Big)^{1/2} + B^{-1} \Big(\int_{1}^{2} |\phi'(r)|^2 dr\Big)^{1/2} \ \ \ \ \ (7)$

for any ${B>0}$, which crucially we are free to choose, and which will therefore allow us to balance two distinct contributions equally. We take ${\phi(r) = A_r P_j f - A_1 P_j f}$ and estimate the terms on the right-hand side separately. The ${L^2}$ bound (6) shows that, since the ${L^2}$ expressions commute,

$\displaystyle \Big\|\Big(\int_{1}^{2} |A_r P_j f(x)|^2 dr\Big)^{1/2} \Big\|_{L^2} \lesssim 2^{-j(d-1)/2} \|f\|_{L^2}$

(and similarly for ${A_1 P_j f}$). For the other term, we need to evaluate ${\frac{d}{dr} A_r P_j f}$ (the ${A_1 P_j f}$ term disappears), and this is easy on the Fourier side, giving

$\displaystyle \widehat{\frac{d}{dr} A_r P_j f }(\xi) = \xi \cdot \nabla \widehat{d\sigma}(r\xi) \widehat{P_j f}(\xi).$

The gradient ${ \nabla \widehat{d\sigma}}$ has the same decay as ${\widehat{d\sigma}}$, namely ${| \nabla \widehat{d\sigma}(\xi)| \lesssim (1 + |\xi|)^{-(d-1)/2}}$ (and this can be proven in the same way). Combining this decay information with the frequency localisation of ${P_j f}$ we obtain by Plancherel’s theorem that

$\displaystyle \Big\| \Big(\int_{1}^{2}\Big|\frac{d}{dr} A_r P_j f\Big|^2 dr\Big)^{1/2} \Big\|_{L^2} \lesssim 2^{j} 2^{-j(d-1)/2}\|f\|_{L^2} = 2^{-j(d-3)/2}\|f\|_{L^2}.$

Putting all this information together, we see that inequality (7) gives us a bound of ${(B 2^{-j(d-1)/2} + B^{-1} 2^{-j(d-3)/2)}) \|f\|_{L^2}}$ for ${\| \sup_{1 \leq r \leq 2} |A_r P_j f| \|_{L^2}}$, which is optimised if we choose ${B = 2^{j/2}}$, resulting in

$\displaystyle \| \sup_{1 \leq r \leq 2} |A_r P_j f| \|_{L^2} \lesssim 2^{-j(d-2)/2} \|f\|_{L^2} \ \ \ \ \ (8)$

(observe that when ${d = 2}$ this expression gives no decay at all, thus explaining why the proof does not work in that case). For a given ${1 we can then interpolate between estimates (5) and (8) to obtain a constant for the ${L^p \rightarrow L^p}$ norm of ${\sup_{1 \leq r \leq 2} |A_r P_j f|}$ that is at most ${\lesssim_\epsilon 2^{-j(d-1 -d/p) + \epsilon j}}$ for any small ${\epsilon >0}$ (do the calculation; you will need to interpolate between the ${p=2}$ exponent and an exponent extremely close to ${1}$ but not ${1}$ – hence the ${\epsilon}$). The expression ${d-1 -d/p}$ is only positive when ${p > d/(d-1)}$, and therefore in this regime we have that (4) holds for some ${\delta = \delta_{p,d}>0}$, allowing us to sum in ${j \in \mathbb{N}}$ and thus to conclude that ${\mathscr{M}_{\mathbb{S}^{d-1}}^{\mathrm{loc}}}$ is bounded. $\Box$

Exercises:

Exercise 13 Assume that function ${m(\xi)}$ satisfies ${\|m\|_{L^\infty} < \infty}$ and that it has bounded variation over all of ${\mathbb{R}}$ (that is ${\int_{\mathbb{R}} |dm|(\xi) < \infty}$). Show, without appealing to the Marcinkiewicz multiplier theorem, that ${m}$ defines a multiplier which is ${L^p(\mathbb{R}) \rightarrow L^p(\mathbb{R})}$ bounded for all ${1 < p < \infty}$.
[hint: simplify the proof of Theorem 9 as much as you can.]

Exercise 14 Show that the pointwise condition 2 in Remark 1 implies condition 2 in Theorem 8.

Exercise 15 In this exercise you will prove Theorem 8 in dimensions ${d>1}$. Actually, the notation required becomes nightmarish pretty quickly, and thus we will content ourselves with proving the case ${d=2}$, since it already contains the full generality of the argument. Let ${T = T_m}$ and ${m}$ satisfy the conditions stated in the theorem. The proof is essentially a repetition of the argument given for Theorem 9.

1. Show that, by Theorem 7 of the second part of these notes, it will suffice to show

$\displaystyle \Big\|\Big( \sum_{\boldsymbol{k} \in \mathbb{Z}^2} |\Delta_{R_{\boldsymbol{k}}} Tf|^2 \Big)^{1/2} \Big\|_{L^p} \lesssim_p \Big\|\Big( \sum_{\boldsymbol{k} \in \mathbb{Z}^2} |\Delta_{R_{\boldsymbol{k}}} f|^2 \Big)^{1/2} \Big\|_{L^p}$

under the assumption that ${\widehat{f}}$ is supported in the first quadrant ${[0,\infty) \times [0,\infty)}$.

2. Show that in each rectangle ${R_{k_1,k_2}}$ we can write

\displaystyle \begin{aligned} m(\xi,\eta) = & m(2^{k_1},2^{k_2}) + \int_{2^{k_1}}^{\xi} \partial_\xi m(\zeta_1,2^{k_2}) d\zeta_1 \\ & + \int_{2^{k_2}}^{\eta} \partial_\eta m(2^{k_1},\zeta_2) d\zeta_2 + \int_{2^{k_2}}^{\eta} \int_{2^{k_1}}^{\xi} \partial_\xi \partial_\eta m(\zeta_1, \zeta_2) d\zeta_1 d\zeta_2. \end{aligned}

3. Show that by ii) and Fubini we have

\displaystyle \begin{aligned} \Delta_{R_{k_1,k_2}} Tf = & \,m(2^{k_1},2^{k_2}) \Delta_{R_{k_1,k_2}}f + \int_{2^{k_1}}^{2^{k_1 + 1}} \partial_\xi m(\zeta_1,2^{k_2}) \Delta_{[\zeta_1, 2^{k_1 + 1}]}^{(1)}f d\zeta_1 \\ & + \int_{2^{k_2}}^{2^{k_2 + 1}} \partial_\eta m(2^{k_1},\zeta_2) \Delta_{[\zeta_2, 2^{k_2 + 1}]}^{(2)}f d\zeta_2 \\ & +\int_{2^{k_2}}^{2^{k_2 + 1}} \int_{2^{k_1}}^{2^{k_1 + 1}} \partial_\xi \partial_\eta m(\zeta_1, \zeta_2) \Delta_{[\zeta_1, 2^{k_1 + 1}] \times [\zeta_2, 2^{k_2 + 1}]}f d\zeta_1 d\zeta_2. \end{aligned}

4. Use condition 1. of the theorem to dispense with the first term at the right-hand side of the last expression.
5. Use condition 2. and Cauchy-Schwarz to show that the (square of) the remaining terms is bounded by

\displaystyle \begin{aligned} C & \Big( \int_{2^{k_1}}^{2^{k_1 + 1}} \big|\Delta_{[\zeta_1, 2^{k_1 + 1}]}^{(1)}f\big|^2 \,|\partial_\xi m(\zeta_1,2^{k_2})| d\zeta_1 \\ & + \int_{2^{k_2}}^{2^{k_2 + 1}} \big|\Delta_{[\zeta_2, 2^{k_2 + 1}]}^{(2)}f|^2 \,|\partial_\eta m(2^{k_1},\zeta_2)\big| d\zeta_2 \\ & +\int_{2^{k_2}}^{2^{k_2 + 1}} \int_{2^{k_1}}^{2^{k_1 + 1}} \big|\Delta_{[\zeta_1, 2^{k_1 + 1}] \times [\zeta_2, 2^{k_2 + 1}]}f\big|^2 \,|\partial_\xi \partial_\eta m(\zeta_1, \zeta_2)| d\zeta_1 d\zeta_2 \Big). \end{aligned}

6. Find measure spaces ${(\Gamma_j, d\mu_j)}$ and collections of rectangles or intervals for ${j=1,2,3}$ such that each of the ${3}$ terms above can be treated by Proposition 1 of the first part of this series, as in the proof of the ${d=1}$ case.
7. Conclude using condition 2. one last time.

Exercise 16 Show that the multiplier given by symbol

$\displaystyle m(\xi_1, \ldots, \xi_d) := \frac{|\xi_1|^{\alpha_1} \cdot \ldots \cdot |\xi_d|^{\alpha_d}}{(\xi_1^2 + \ldots + \xi_d^2)^{|\alpha|/2}},$

where ${\alpha_j > 0}$ and ${|\alpha|:= \alpha_1 + \ldots + \alpha_d}$, is a Marcinkiewicz multiplier.

Exercise 17 Show that the range ${\frac{d}{d-1} < p \leq \infty}$ for the boundedness of the spherical maximal function ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ is sharp, in the sense that ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ is not bounded on ${L^p(\mathbb{R}^d)}$ for any ${1 \leq p \leq \frac{d}{d-1}}$ (find a counterexample).

Exercise 18 Let the dimension be ${d=3}$ and let ${A_r}$ denote the spherical average ${A_r f(x) := \int_{\mathbb{S}^{2}} f(x - ty) d\sigma(y)}$, where ${d\sigma}$ is the normalised surface measure on ${\mathbb{S}^{2}}$. Show that ${u(x,t):= t A_t g(x)}$ solves the wave equation

$\displaystyle \begin{cases} \partial_t^2 u - \Delta u = 0, \\ u(x,0) = 0, \\ \partial_t u(x,0) = g(x), \end{cases}$

with ${x \in \mathbb{R}^3}$, ${t \geq 0}$; here we assume ${g \in C^3(\mathbb{R}^3) \cap L^2 (\mathbb{R}^3)}$. This is an instance of Huygens’ principle, and more in general if the initial data becomes ${u(x,0) = f(x)}$ with ${f \in C^2}$, the complete solution is ${t A_t g + \frac{d}{dt}(t A_t f)}$. Similar formulas hold in higher dimensions, but only when ${d}$ is odd they involve only the spherical averages ${A_t}$. When ${d}$ is even, one needs to average over balls instead (that is, the classical Huygens’ principle fails in even dimensions).

1. Using Stokes theorem, show that

$\displaystyle \frac{d}{dt} A_t g(x) = \frac{1}{t^2} \Delta\Big( \int_{|y|\leq t} g(x-y) dy\Big).$

2. Using polar coordinates, show that

$\displaystyle \int_{|y|\leq t} g(x-y) dy = \int_{0}^{t} s^2 A_s g(x) ds.$

3. Use i)-ii) to show that

$\displaystyle \frac{d}{dt} \Big( t^2 \frac{d}{dt} A_t g(x)\Big) = t^2 \Delta A_t g(x).$

4. Show that for any ${F}$ twice differentiable it holds that ${\frac{1}{t} \frac{d}{dt} \Big( t^2 \frac{d}{dt} F(t)\Big) = \Big(\frac{d}{dt}\Big)^2 (t F(t))}$, and combine this with iii) to show that the wave equation is satisfied.
5. It remains to check the initial conditions. Using Theorem 10 argue that ${\lim_{t\rightarrow 0} A_t g(x) = g(x)}$ for every ${x}$, and then use this fact together with i) to argue that the initial conditions are indeed satisfied in the limit ${t \rightarrow 0}$ (and ${u, \partial_t u}$ are continuous in ${t\geq 0}$).

Exercise 19 Let ${\psi}$ be a smooth function compactly supported in the ball ${B(0,1)}$ of ${\mathbb{R}^{d-1}}$. In this exercise you will show that

$\displaystyle \Big|\int_{\mathbb{R}^{d-1}} e^{i (x',x_d)\cdot(\xi,|\xi|^2)} \psi(\xi) d\xi\Big| \lesssim (1 + |x'| + |x_d|)^{-(d-1)/2} \ \ \ \ \ (9)$

for any ${(x',x_d) \in \mathbb{R}^{d-1}\times \mathbb{R} = \mathbb{R}^d}$. It is a matter of doing a smooth partition of unity on ${\mathbb{S}^{d-1}}$ and a change of variables with a suitable diffeomorphism to show that the above implies ${|\widehat{d\sigma}(\xi)|, |\nabla \widehat{d\sigma}(\xi)| \lesssim (1 + |\xi|)^{-(d-1)/2}}$; however, we will content ourselves with the object above (you can see it as the Fourier transform of a measure supported on the elliptic paraboloid parametrised by ${(\xi, |\xi|^2)}$, instead of the sphere; but the two surfaces are locally the same).

1. Square the left-hand side of (9) and show by a change of variables that the result equals

$\displaystyle I = \iint_{\mathbb{R}^{d-1} \times \mathbb{R}^{d-1}} e^{i (x' \cdot \eta_1 + x_d \eta_1 \cdot \eta_2)} \Psi(\eta_1,\eta_2) d\eta_1 d\eta_2$

for some smooth function ${\Psi}$ compactly supported in ${B(0,2)\times B(0,2)}$.

2. You will show (9) in two complementary cases. First let ${|x'|< 200 |x_d|}$. Show that

$\displaystyle |I| \leq \int_{|\eta_1| \leq 2} |\mathcal{F}_2 \Psi(\eta_1, x_d \eta_1)| d\eta_1,$

where ${\mathcal{F}_2}$ denotes the Fourier transform in the second variable. Argue that ${|\mathcal{F}_2 \Psi(\eta_1, x_d \eta_1)| \lesssim_N (1 + |\eta_1|+ |x_d||\eta_1|)^{-N}}$ for any arbitrarily large ${N>0}$ and conclude that this shows ${|I| \lesssim (1 + |x_d|)^{-(d-1)} \sim (1 + |x'| + |x_d|)^{-(d-1)}}$, thus proving (9) in this regime.

3. Let now ${|x'|\geq 200 |x_d|}$. Show that

$\displaystyle |I| \leq \int_{|\eta_2|\leq 2} |\mathcal{F}_1 \Psi(x' + x_d \eta_2, \eta_2)| d\eta_2;$

next argue that ${|\mathcal{F}_1 \Psi(x' + x_d \eta_2, \eta_2)| \lesssim_{N} (1 + |x' + x_d \eta_2| + |\eta_2|)^{-N} \lesssim_N (1 + |x'|)^{-N}}$ for any ${N>0}$ and show that this proves (9) in this case as well (even with arbitrarily large exponents).

Exercise 20 Let ${\psi_j}$ be as in the proof of Theorem 10. Show that, thanks to the fact that ${|\widehat{\psi}(x)| \lesssim_N (1 + |x|)^{-N}}$ for any ${N>0}$, one has the rough bound

$\displaystyle |A_r \widehat{\psi_j}(x)| \lesssim 2^j (1 + |x|)^{-10 d}$

for any ${r \sim 1}$ and any ${x \in \mathbb{R}^d}$.

Exercise 21 In this exercise you will finish the proof of Theorem 10. Here you will need to use a smooth square function at some point, but other than that the proof is just a repetition of the one given for ${\mathscr{M}_{\mathbb{S}^{d-1}}^{\mathrm{loc}}}$.

1. We see ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ as

$\displaystyle \sup_{k \in \mathbb{Z}} \sup_{2^k \leq r \leq 2^{k+1}} |A_r f|.$

For a fixed ${k \in \mathbb{Z}}$ let ${P_{\leq k} = \sum_{j \leq k} P_j}$ be the operator that will take on the rôle of ${P_{low}}$, so that

$\displaystyle f = P_{\leq k} f + \sum_{j \in \mathbb{N}} P_{j+k} f.$

Show that it suffices to prove for any ${d/(d-1) < p < 2}$

$\displaystyle \| \sup_{k \in \mathbb{Z}} \sup_{2^{-k} \leq r \leq 2^{-k+1}} |A_r P_{\leq k}f| \|_{L^p} \lesssim_p \, \|f\|_{L^p}, \ \ \ \ \ (10)$

$\displaystyle \| \sup_{k \in \mathbb{Z}} \sup_{2^{-k} \leq r \leq 2^{-k+1}} |A_r P_{j+k}f| \|_{L^p} \lesssim_p \, 2^{-\delta j} \|f\|_{L^p}, \ \ \ \ \ (11)$

for some ${\delta = \delta_{p,d} >0}$.

2. Show that for ${r \sim 2^{-k}}$ one has

$\displaystyle |A_r P_{\leq k} f| \lesssim Mf$

uniformly in ${r,k}$ and conclude (10). You can simply rescale the argument given for the analogous part of ${\mathscr{M}_{\mathbb{S}^{d-1}}^{\mathrm{loc}}}$, that is for ${k=0}$.

3. Show that for ${r \sim 2^{-k}}$ and ${j>0}$ one has

$\displaystyle |A_r P_{j+k} f| \lesssim 2^j \, Mf$

uniformly in ${r}$ and ${k}$ (again, just rescale the argument used when ${k=0}$). Conclude that

$\displaystyle \| \sup_{k \in \mathbb{Z}} \sup_{2^{-k} \leq r \leq 2^{-k+1}} |A_r P_{j+k}f| \|_{L^q} \lesssim_p \, 2^{j} \|f\|_{L^q} \ \ \ \ \ (12)$

for any ${q > 1}$.

4. Show, using the decay of ${\widehat{d\sigma}}$, that when ${r \sim 2^{-k}}$

$\displaystyle \| A_r P_{j+k}f \|_{L^2} \lesssim \, 2^{-j(d-1)/2} \|f\|_{L^2}.$

5. Show, using the decay of ${\nabla\widehat{d\sigma}}$, that when ${r \sim 2^{-k}}$

$\displaystyle \Big\| \frac{d}{dr} A_r P_{j+k}f \Big\|_{L^2} \lesssim \, 2^{k} 2^{-j(d-3)/2} \|f\|_{L^2}.$

6. Show, using a suitably rescaled inequality (7), that for any ${k \in \mathbb{Z}}$

$\displaystyle \| \sup_{2^{-k} \leq r \leq 2^{-k+1}} |A_r P_{j+k}f| \|_{L^2} \lesssim 2^{-j(d/2 - 1)} \|f\|_{L^2}. \ \ \ \ \ (13)$

7. Let ${\widetilde{\psi}}$ be a smooth function compactly supported in the annulus ${\{ \xi \in \mathbb{R}^d \,:\, 1/4 \leq |\xi| \leq 4\}}$ and identically equal to ${1}$ in the annulus ${\{ \xi \in \mathbb{R}^d \,:\, 1/2 \leq |\xi| \leq 2\}}$; finally, let ${\widetilde{P}_j}$ be the annular projection given by ${\widehat{\widetilde{P}_j f}(\xi) = \widetilde{\psi}(2^{-j}\xi) \widehat{f}(\xi)}$ (thus they are just a slight modification of the projections in Section 6 of the second post). Show that in (13) we can replace ${\|f\|_{L^2}}$ in the right-hand side by ${\|\widetilde{P}_{j+k} f\|_{L^2}}$.
8. Replace the supremum in ${k \in \mathbb{Z}}$ by the ${\ell^2(\mathbb{Z})}$ sum and show that, by Proposition 6 of the second part in this series, we have

$\displaystyle \| \sup_{k \in \mathbb{Z}} \sup_{2^{-k} \leq r \leq 2^{-k+1}} |A_r P_{j+k}f| \|_{L^2} \lesssim \, 2^{-j (d/2 - 1)} \|f\|_{L^2}. \ \ \ \ \ (14)$

9. Interpolate between (12) and (14) to show that (11) holds. This concludes the proof.

Exercise 22 The Carleson operator for ${\mathbb{R}}$ is given by

$\displaystyle \mathcal{C}f(x):= \sup_{N >0} \Big|\int \mathbf{1}_{[-N,N]}(\xi) \widehat{f}(\xi) e^{2\pi i \xi x} d\xi\Big|;$

recall that its ${L^p \rightarrow L^{p,\infty}}$ boundedness for a certain ${p}$ implies the a.e. convergence of the integral above to ${f(x)}$ as ${N\rightarrow \infty}$ when ${f \in L^p}$. The methods developed in these notes are not powerful enough to show the boundedness of ${\mathcal{C}}$, but they are enough for a simplified version of it. Indeed, in this exercise you will prove the ${L^p}$ boundedness of the lacunary Carleson operator

$\displaystyle \mathcal{C}_{\mathrm{lac}}f(x):= \sup_{k \in \mathbb{Z}} \Big|\int \mathbf{1}_{[-2^k, 2^k]}(\xi) \widehat{f}(\xi) e^{2\pi i \xi x} d\xi\Big|.$

Fix ${p}$ such that ${1 < p < \infty}$.

1. First, consider the additional simplification given by replacing the characteristic function ${\mathbf{1}_{[-N,N]}(\xi)}$ by the smooth function ${\varphi(\xi / N)}$ where ${\varphi}$ is a non-negative smooth function compactly supported in ${[-1,1]}$ and identically equal to ${1}$ on ${[-1/2,1/2]}$. Show that the associated maximal operator

$\displaystyle \widetilde{\mathcal{C}}f(x) := \sup_{N >0} \Big|\int \varphi(\xi/N) \widehat{f}(\xi) e^{2\pi i \xi x} d\xi\Big|$

is bounded pointwise by a constant multiple of the Hardy-Littlewood maximal function ${Mf(x)}$ and conclude ${L^p \rightarrow L^p}$ boundedness of ${\widetilde{\mathcal{C}}}$.

2. Show that it suffices to prove the ${L^p}$ boundedness of the operator

$\displaystyle \mathscr{C}_{\mathrm{lac}}f(x):= \sup_{k \in \mathbb{Z}} \Big|\int (\mathbf{1}_{[-2^k, 2^k]}(\xi) - \varphi(2^{-k}\xi)) \widehat{f}(\xi) e^{2\pi i \xi x} d\xi\Big|$

to conclude that of ${\mathcal{C}_{\mathrm{lac}}}$.

3. Show that there is a smooth non-negative function ${\theta}$ compactly supported in ${[1/2,2] \cup [-2,-1/2]}$ and such that

$\displaystyle \mathbf{1}_{[-2^k, 2^k]}(\xi) - \varphi(2^{-k}\xi) = \mathbf{1}_{[2^{k-1},2^k]\cup[-2^k,-2^{k-1}]}(\xi) \cdot \theta(2^{-k}\xi).$

4. Let ${\Theta_k}$ denote the frequency projections associated to ${\theta}$, that is ${\widehat{\Theta_k f}(\xi) = \theta(2^{-k}\xi) \widehat{f}(\xi)}$. Show that it suffices to show that the square function

$\displaystyle \mathfrak{S}f := \Big(\sum_{k\in\mathbb{Z}} |\Delta_k \Theta_k f|^2 \Big)^{1/2}$

is ${L^p \rightarrow L^p}$ bounded to conclude the same for ${\mathscr{C}_{\mathrm{lac}}}$.

5. Show that ${\mathfrak{S}}$ is indeed ${L^p \rightarrow L^p}$ bounded.
6. Argue that the argument generalises to treat the operators

$\displaystyle \mathcal{C}'_{\mathrm{lac}}f(x):= \sup_{k \in \mathbb{N}} \Big|\int \mathbf{1}_{[-N_k, N_k]}(\xi) \widehat{f}(\xi) e^{2\pi i \xi x} d\xi\Big|$

where ${(N_k)_{k \in \mathbb{N}}}$ is any fixed lacunary sequence – that is, a sequence that satisfies ${\liminf_{k \rightarrow \infty} N_{k+1} / N_k =: \rho > 1}$. However, the constant produced by the proof will end up depending on ${\rho}$.

Exercise 23 There is a known application of Riesz transforms that provides important motivation for singular integrals. The application is the following: the Riesz transforms are defined by ${\widehat{R_j f}(\xi) = (\xi_j/|\xi|) \widehat{f}(\xi)}$, and working on the Fourier transform side one can verify that

$\displaystyle R_i R_j \Delta f = \partial_{x_i} \partial_{x_j} f$

(modulus some constants); thanks to the above identity and the boundedness of the Riesz transforms, one can therefore control the ${L^p}$ norms of all the partial derivatives of order ${2}$ by the Laplacian alone: for any ${i,j \in \{1,\ldots,d\}}$

$\displaystyle \|\partial_{x_i} \partial_{x_j} u\|_{L^p} \lesssim_p \|\Delta u\|_{L^p} \qquad \text{ for all } 1< p < \infty.$

With a little spherical harmonics theory, this can be generalised to generic homogeneous elliptic differential operators, always within the framework of Calderón-Zygmund theory. More precisely, let ${Q(X) \in \mathbb{C}[X_1,\ldots,X_d]}$ be a homogeneous elliptic polynomial of degree ${k}$, that is

1. ${Q(\lambda X) = \lambda^k Q(X)}$ for all ${\lambda \neq 0}$;
2. the zero set ${Z(Q)}$ of ${Q(X)}$ consists only of the origin.

If ${Q(X) = \sum_{\alpha \in \mathbb{N}^d : |\alpha|=k} c_\alpha X^\alpha}$, the homogeneous elliptic differential operator ${Q(\partial)}$ is defined as

$\displaystyle Q(\partial) := \sum_{\alpha \in \mathbb{N}^d : |\alpha|=k} c_\alpha \partial^\alpha.$

Then one can show that for all multi-indices ${\alpha \in \mathbb{N}^d}$ such that ${|\alpha| = k}$ and (at least) for all the functions ${u}$ of class ${C^k}$ and compact support, one has

$\displaystyle \|\partial^\alpha u\|_{L^p} \lesssim_{p,Q} \|Q(\partial) u\|_{L^p} \qquad \text{ for all } 1< p < \infty.$

The point is that the operator ${T}$ such that ${\partial^\alpha = T \circ Q(\partial)}$ exists and is a bounded Calderón-Zygmund singular integral operator with homogeneous kernel, that is of the form $\Omega\Big(\frac{x-y}{|x-y|}\Big)/|x-y|^d$ with $\int_{\mathbb{S}^{d-1}} \Omega(\omega) d\sigma(\omega) = 0$ (or a composition of such operators).
Now, as a further application of Littlewood-Paley theory, you will show that the result above generalises even further. Indeed, you will show that if ${P(X) \in \mathbb{C}[X_1,\ldots,X_d]}$ is a polynomial of degree ${k}$ whose top-degree component is elliptic, then for all multi-indices ${\alpha \in \mathbb{N}^d}$ such that ${|\alpha| \leq k}$ and (at least) for all the functions ${u}$ of class ${C^k}$ and compact support, one has

$\displaystyle \|\partial^\alpha u\|_{L^p} \lesssim_{p,P} \|P(\partial) u\|_{L^p} + \|u\|_{L^p} \qquad \text{ for all } 1< p < \infty.$

1. Let ${Z(P)}$ denote the zero set of ${P}$. Show that ${Z(P)}$ is contained in a ball ${B(0,R)}$ for some large ${R}$ depending on ${P}$.
2. Let ${\varphi}$ be a smooth function compactly supported in a neighbourhood of ${Z(P)}$ and that vanishes outside ${B(0,2R)}$.
3. With ${\xi^\alpha = \xi_1^{\alpha_1} \cdot \ldots \cdot \xi_d^{\alpha_d}}$, show that ${m_1(\xi) := \xi^\alpha \varphi(\xi)}$ is a Schwartz function and deduce that ${T_{m_1}}$ is ${L^p \rightarrow L^p}$ bounded for ${1 < p < \infty}$.
4. Show that ${m_2 (\xi) := (1 - \varphi(\xi))\frac{\xi^\alpha}{P(\xi)}}$ is a Marcinkiewicz multiplier and therefore ${T_{m_2}}$ is ${L^p \rightarrow L^p}$ bounded for ${1 < p < \infty}$.
5. Decompose ${\xi^\alpha \widehat{f}(\xi) = m_1(\xi)\widehat{f}(\xi) + m_2(\xi) P(\xi)\widehat{f}(\xi)}$ and use this to conclude the inequality.

Footnotes:
1: The etymology of the term “multiplier” should be clear from the definition.
2: Recall that the total variation of a function ${f}$ on the interval ${[a,b]}$ is defined as ${\sup_{N} \sup_{a \leq \xi_1 < \ldots < \xi_N \leq b} \sum_{j=1}^{N-1} |f(\xi_{j+1}) - f(\xi_j)|}$; if ${f}$ has finite total variation on ${[a,b]}$ then there exists a complex Borel measure ${\mu}$ such that ${f(x) = f(a) + \int_{a}^{x} d\mu}$.
3: If ${m}$ is absolutely continuous we simply have ${dm(\eta) = m'(\eta) d\eta}$.
4: Recall that such a function exists, and can be realised for example by taking ${\psi(\xi) = \varphi(\xi) - \varphi(\xi/2)}$ with ${\varphi}$ smooth compactly supported in ${[-2,2]}$ and identically ${1}$ on ${[-1,1]}$.