Affine Restriction estimates imply Affine Isoperimetric inequalities

One thing I absolutely love about harmonic analysis is that it really has something interesting to say about nearly every other field of Analysis. Today’s example is exactly of this kind: I will show how a Fourier Restriction estimate can say something about Affine Geometry. This was first noted by Carbery and Ziesler (see below for references).

1. Affine Isoperimetric Inequality

Recall the Affine Invariant Surface Measure that we have defined in a previous post. Given a hypersurface \Sigma \subset \mathbb{R}^d sufficiently smooth to have a well-defined Gaussian curvature \kappa_{\Sigma}(\xi) (where \xi ranges over \Sigma ) and with surface measure denoted by d\sigma_{\Sigma} , we can define the Affine Invariant Surface measure as the weighted surface measure

\displaystyle d\Omega_{\Sigma}(\xi) := |\kappa_{\Sigma}(\xi)|^{1/(d+1)} \, d\sigma_{\Sigma}(\xi);

this measure has the property of being invariant under the action of SL(\mathbb{R}^d) – hence the name. Here invariant means that if \varphi is an equi-affine map (thus volume preserving) then

\displaystyle \Omega_{\varphi(\Sigma)}(\varphi(E)) = \Omega_{\Sigma}(E)

for any measurable E \subseteq \Sigma .
The Affine Invariant Surface measure can be used to formulate a very interesting result in Affine Differential Geometry – an inequality of isoperimetric type. Let K \subset \mathbb{R}^d be a convex body – say, centred in the origin and symmetric with respect to it, i.e. K = - K . We denote by \partial K the boundary of the convex body K and we can assume for the sake of the argument that \partial K is sufficiently smooth – for example, piecewise C^2-regular, so that the Gaussian curvature is defined at every point except maybe a \mathcal{H}^{d-1} -null set. Then the Affine Isoperimetric Inequality says that (with \Omega = \Omega_{\partial K} )

\displaystyle \boxed{ \Omega(\partial K)^{d+1} \lesssim |K|^{d-1}.  } \ \ \ \ \ \ \ (\dagger)


Notice that the inequality is invariant with respect to the action of SL(\mathbb{R}^d) indeed – thanks to the fact that d\Omega is. Observe also the curious fact that this inequality goes in the opposite direction with respect to the better known Isoperimetric Inequality of Geometric Measure Theory! Indeed, the latter says (let’s say in the usual \mathbb{R}^d ) that (a power of) the volume of a measurable set is controlled by (a power of) the perimeter of the set; more precisely, for any measurable E \subset \mathbb{R}^d

\displaystyle |E|^{d-1} \lesssim P(E)^d,

where P(E) denotes the perimeter1 of E – in case E = K a symmetric convex body as above we would have P(K) = \sigma(\partial K) . But in the affine context the “affine perimeter” is \Omega(\partial K) and is controlled by the volume instead of viceversa. This makes perfect sense: if K is taken to be a cube Q then \kappa_{\partial Q} = 0 and so the “affine perimeter” cannot control anything. Notice also that the power of the perimeter is d for the standard isoperimetric inequality and it is instead d+1 for the affine isoperimetric inequality. Informally speaking, this is related to the fact that the affine perimeter is measuring curvature too instead of just area.
So, the inequality should actually be called something like “Affine anti-Isoperimetric inequality” to better reflect this, but I don’t get to choose the names.

The inequality above is formulated for convex bodies since those are the most relevant objects for Affine Geometry. However, below we will see that Harmonic Analysis provides a sweeping generalisation of the inequality to arbitrary hypersurfaces that are not necessarily boundaries of convex bodies. Before showing this generalisation, we need to introduce Affine Fourier restriction estimates, which we do in the next section.

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Affine Invariant Surface Measure

In this short post I want to introduce an instance of certain objects that will be the subject of a few more posts. This particular object arises naturally in Affine Differential Geometry and turned out to have a relevant rôle in Harmonic Analysis too (in both Fourier restriction and in the theory of Radon transforms).

1. Affine Invariant measures

Affine Differential Geometry is the study of (differential-)geometric properties that are invariant with respect to SL(\mathbb{R}^d) . A very interesting object arising in Affine Geometry is the notion of an Affine Invariant Measure. Sticking to examples rather than theory (since the theory is still quite underdeveloped!), consider a hypersurface \Sigma \subset \mathbb{R}^{d} sufficiently smooth to have well-defined Gaussian curvature, which we denote by \kappa (a function on \Sigma ). If we let d\sigma denote the surface measure on \Sigma (induced from the Lebesgue measure on the ambient space \mathbb{R}^d for example, or by taking directly d\sigma = d\mathcal{H}^{d-1}\big|_{\Sigma} , the restriction of the (d-1) -dimensional Hausdorff measure to the hypersurface) then this crafty little object is called Affine Invariant Surface Measure and is given by

\displaystyle d\Omega(\xi) = |\kappa(\xi)|^{1/(d+1)} \,d\sigma(\xi).

It was first introduced by Blaschke for d=3 (finding the reference seems impossible; it’s [B] in this paper, if you feel luckier) and by Leichtweiss for general d . The reason this measure is so interesting is that it is (equi)affine invariant in the sense that if \varphi(\xi) = A \xi + \eta is an equi-affine transformation (thus with A \in SL(\mathbb{R}^d) and so volume-preserving since \det A = \pm 1) then, using subscripts to distinguish the two surfaces, we have

\displaystyle \boxed{ \Omega_{\varphi(\Sigma)}(\varphi(E)) = \Omega_{\Sigma}(E) } \ \ \ \ \ \ \ (1)


for any measurable E \subseteq \Sigma . We remark the following fact: that seemingly mysterious power \frac{1}{d+1} in the definition of d\Omega is the only exponent for which the resulting measure is (equi)affine-invariant.

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Proof of the square-function characterisation of L(log L)^{1/2}: part II

This is the 3rd post in a series that started with the post on the Chang-Wilson-Wolff inequality:

1. The Chang-Wilson-Wolff inequality using a lemma of Tao-Wright
2. Proof of the square-function characterisation of L(log L)^{1/2}: part I

In today’s post we will finally complete the proof of the Tao-Wright lemma. Recall that in the 2nd post of the series we proved that the Tao-Wright lemma follows from its discrete version for Haar/dyadic-martingale-differences, which is as follows:


Lemma 2 – Square-function characterisation of L(\log L)^{1/2} for martingale-differences:
For any function f : [0,1] \to \mathbb{R} in L(\log L)^{1/2}([0,1]) there exists a collection (F_j)_{j \in \mathbb{Z}} of non-negative functions such that:

  1. for any j \in \mathbb{N} and any I \in \mathcal{D}_j

    \displaystyle  |\langle f, h_I \rangle|\lesssim \frac{1}{|I|^{1/2}} \int_{I} F_j \,dx;

  2. they satisfy the square-function estimate

    \displaystyle  \Big\|\Big(\sum_{j \in \mathbb{N}} |F_j|^2\Big)^{1/2}\Big\|_{L^1} \lesssim \|f\|_{L(\log L)^{1/2}([0,1])}.

Today we will prove this lemma.

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Interlude: Atomic decomposition of L(log L)^r

This is going to be a shorter post about a technical fact that will be used in concluding the proof of the Tao-Wright lemma.
What we are going to see today is an atomic decomposition of the Orlicz spaces of L (\log L)^r type. Surprisingly, I could find no classical references that explicitely state this useful little fact – some attribute it to Titchmarsh, Zygmund and Yano; indeed, something resembling the decomposition can be found for example in Zygmund’s book (Volume II, page 120). However, I could only find a proper statement together with a proof in a paper of Tao titled “A Converse Extrapolation Theorem for Translation-Invariant Operators“, where he claims it is a well-known fact and proves it in an appendix (the paper is about reversing the implication in an old extrapolation theorem of Yano [1951], a theorem that tells you that if the operator norms \|T\|_{L^p \to L^p} blow up only to finite order as p \to 1^{+} , then you can “extrapolate” this into an endpoint inequality of the type \|Tf\|_{L^1} \lesssim \|f\|_{L(\log L)^r} ).

Briefly stated, the result is as follows. We will consider only L(\log L)^r ([0,1]), that is the Orlicz space of functions on [0,1] with Orlicz/Luxemburg norm

\displaystyle \|f\|_{L(\log L)^r ([0,1])} = \inf \Big\{\mu > 0 \text{ s.t. } \int_{0}^{1} \frac{|f(x)|}{\mu} \Big(\log \Big(2 + \frac{|f(x)|}{\mu}\Big)\Big)^{r} \,dx \leq 1 \Big\}.

Our atoms will be quite simply normalised characteristic functions: that is, for any measurable set E \subset [0,1] we let a_E denote the atom associated to E , given by

\displaystyle a_E := \frac{\mathbf{1}_E}{\|\mathbf{1}_E\|_{L(\log L)^r}};

obviously \|a_E\|_{L(\log L)^r} = 1 .
The statement is then the following.

Atomic decomposition of L(\log L)^r :
Let f \in L(\log L)^{r}([0,1]) . Then there exist measurable sets (E_j)_j and coefficients (\alpha_j)_j such that

\displaystyle f = \sum_{j} \alpha_j a_{E_j}

and

\displaystyle \sum_{j} |\alpha_j| \lesssim \|f\|_{L(\log L)^r}.

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Proof of the square-function characterisation of L(log L)^{1/2}: part I

This is a follow-up on the post on the Chang-Wilson-Wolff inequality and how it can be proven using a lemma of Tao-Wright. The latter consists of a square-function characterisation of the Orlicz space L(\log L)^{1/2} analogous in spirit to the better known one for the Hardy spaces.
In this post we will commence the proof of the Tao-Wright lemma, as promised. We will start by showing how the lemma, which is stated for smooth frequency projections, can be deduced from its discrete variant stated in terms of Haar coefficients (or equivalently, martingale differences with respect to the standard dyadic filtration). This is a minor part of the overall argument but it is slightly tricky so I thought I would spell it out.

Recall that the Tao-Wright lemma is as follows. We write \widetilde{\Delta}_j f for the smooth frequency projection defined by \widehat{\widetilde{\Delta}_j f}(\xi) = \widehat{\psi}(2^{-j}\xi) \widehat{f}(\xi) , where \widehat{\psi} is a smooth function compactly supported in 1/2 \leq |\xi| \leq 4 and identically equal to 1 on 1 \leq |\xi| \leq 2 .


Lemma 1 – Square-function characterisation of L(\log L)^{1/2} [Tao-Wright, 2001]:
Let for any j \in \mathbb{Z}

\displaystyle  \phi_j(x) := \frac{2^j}{(1 + 2^j |x|)^{3/2}}

(notice \phi_j is concentrated in [-2^{-j},2^{-j}] and \|\phi_j\|_{L^1} \sim 1).
If the function {f} is in L(\log L)^{1/2}([-R,R]) and such that \int f(x) \,dx = 0 , then there exists a collection (F_j)_{j \in \mathbb{Z}} of non-negative functions such that:

  1. pointwise for any j \in \mathbb{Z}

    \displaystyle \big|\widetilde{\Delta}_j f\big| \lesssim F_j \ast \phi_j ;

  2. they satisfy the square-function estimate

    \displaystyle  \Big\|\Big(\sum_{j \in \mathbb{Z}} |F_j|^2\Big)^{1/2}\Big\|_{L^1} \lesssim \|f\|_{L(\log L)^{1/2}}.

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The Chang-Wilson-Wolff inequality using a lemma of Tao-Wright

Today I would like to introduce an important inequality from the theory of martingales that will be the subject of a few more posts. This inequality will further provide the opportunity to introduce a very interesting and powerful result of Tao and Wright – a sort of square-function characterisation for the Orlicz space L(\log L)^{1/2} .

1. The Chang-Wilson-Wolff inequality

Consider the collection \mathcal{D} of standard dyadic intervals that are contained in [0,1] . We let \mathcal{D}_j for each j \in \mathbb{N} denote the subcollection of intervals I \in \mathcal{D} such that |I|= 2^{-j} . Notice that these subcollections generate a filtration of \mathcal{D}, that is (\sigma(\mathcal{D}_j))_{j \in \mathbb{N}}, where \sigma(\mathcal{D}_j) denotes the sigma-algebra generated by the collection \mathcal{D}_j . We can associate to this filtration the conditional expectation operators

\displaystyle  \mathbf{E}_j f := \mathbf{E}[f \,|\, \sigma(\mathcal{D}_j)],

and therefore define the martingale differences

\displaystyle  \mathbf{D}_j f:= \mathbf{E}_{j+1} f - \mathbf{E}_{j}f.

With this notation, we have the formal telescopic identity

\displaystyle  f = \mathbf{E}_0 f + \sum_{j \in \mathbb{N}} \mathbf{D}_j f.

Demystification: the expectation \mathbf{E}_j f(x) is simply \frac{1}{|I|} \int_I f(y) \,dy, where I is the unique dyadic interval in \mathcal{D}_j such that x \in I .

Letting f_j := \mathbf{E}_j f for brevity, the sequence of functions (f_j)_{j \in \mathbb{N}} is called a martingale (hence the name “martingale differences” above) because it satisfies the martingale property that the conditional expectation of “future values” at the present time is the present value, that is

\displaystyle  \mathbf{E}_{j} f_{j+1} = f_j.

In the following we will only be interested in functions with zero average, that is functions such that \mathbf{E}_0 f = 0. Given such a function f : [0,1] \to \mathbb{R} then, we can define its martingale square function S_{\mathcal{D}}f to be

\displaystyle  S_{\mathcal{D}} f := \Big(\sum_{j \in \mathbb{N}} |\mathbf{D}_j f|^2 \Big)^{1/2}.

With these definitions in place we can state the Chang-Wilson-Wolff inequality as follows.

C-W-W inequality: Let {f : [0,1] \to \mathbb{R}} be such that \mathbf{E}_0 f = 0. For any {2\leq p < \infty} it holds that

\displaystyle  \boxed{\|f\|_{L^p([0,1])} \lesssim p^{1/2}\, \|S_{\mathcal{D}}f\|_{L^p([0,1])}.} \ \ \ \ \ \ (\text{CWW}_1)

An important point about the above inequality is the behaviour of the constant in the Lebesgue exponent {p} , which is sharp. This can be seen by taking a “lacunary” function {f} (essentially one where \mathbf{D}_jf = a_j \in \mathbb{C} , a constant) and randomising the signs using Khintchine’s inequality (indeed, {p^{1/2}} is precisely the asymptotic behaviour of the constant in Khintchine’s inequality; see Exercise 5 in the 2nd post on Littlewood-Paley theory).
It should be remarked that the inequality extends very naturally and with no additional effort to higher dimensions, in which [0,1] is replaced by the unit cube [0,1]^d and the dyadic intervals are replaced by the dyadic cubes. We will only be interested in the one-dimensional case here though.

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Carbery's proof of the Stein-Tomas theorem

Writing the article on Bourgain’s proof of the spherical maximal function theorem I suddenly recalled another interesting proof that uses a trick very similar to that of Bourgain – and apparently directly inspired from it. Recall that the “trick” consists of the following fact: if we consider only characteristic functions as our inputs, then we can split the operator in two, estimate these parts each in a different Lebesgue space, and at the end we can combine the estimates into an estimate in a single L^p space by optimising in some parameter. The end result looks as if we had done “interpolation”, except that we are “interpolating” between distinct estimates for distinct operators!

The proof I am going to talk about today is a very simple proof given by Tony Carbery of the well-known Stein-Tomas restriction theorem. The reason I want to present it is that I think it is nice to see different incarnations of a single idea, especially if applied to very distinct situations. I will not spend much time discussing restriction because there is plenty of material available on the subject and I want to concentrate on the idea alone. If you are already familiar with the Stein-Tomas theorem you will certainly appreciate Carbery’s proof.

As you might recall, the Stein-Tomas theorem says that if R denotes the Fourier restriction operator of the sphere \mathbb{S}^{d-1} (but of course everything that follows extends trivially to arbitrary positively-curved compact hypersurfaces), that is

\displaystyle Rf = \widehat{f} \,\big|_{\mathbb{S}^{d-1}}

(defined initially on Schwartz functions), then

Stein-Tomas theorem: R satisfies the a-priori inequality

\displaystyle \|Rf\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim_p \|f\|_{L^p(\mathbb{R}^d)} \ \ \ \ \ \ (1)


for all exponents {p} such that 1 \leq p \leq \frac{2(d+1)}{d+3} (and this is sharp, by the Knapp example).

There are a number of proofs of such statement; originally it was proven by Tomas for every exponent except the endpoint, and then Stein combined the proof of Tomas with his complex interpolation method to obtain the endpoint too (and this is still one of the finest examples of the power of the method around).
Carbery’s proof obtains the restricted endpoint inequality directly, and therefore obtains inequality (1) for all exponents 1 \leq p < \frac{2(d+1)}{d+3} by interpolation of Lorentz spaces with the p=1 case (which is a trivial consequence of the Hausdorff-Young inequality).

In other words, Carbery proves that for any (Borel) measurable set {E} one has

\displaystyle \|R \mathbf{1}_{E}\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim |E|^{\frac{d+3}{2(d+1)}}, \ \ \ \ \ \ (2)

where the LHS is clearly the L^{2(d+1)/(d+3)} norm of the characteristic function \mathbf{1}_E . Notice that we could write the inequality equivalently as \|\widehat{\mathbf{1}_{E}}\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim |E|^{\frac{d+3}{2(d+1)}} .

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Bourgain's proof of the spherical maximal function theorem

Recently I have presented Stein’s proof of the boundedness of the spherical maximal function: it was in part III of a set of notes on basic Littlewood-Paley theory. Recall that the spherical maximal function is the operator

\displaystyle \mathscr{M}_{\mathbb{S}^{d-1}} f(x) := \sup_{t > 0} |A_t f(x)|,

where A_t denotes the spherical average at radius {t} , that is

\displaystyle A_t f(x) := \int_{\mathbb{S}^{d-1}} f(x - t\omega) d\sigma_{d-1}(\omega),

where d\sigma_{d-1} denotes the spherical measure on the (d-1) -dimensional sphere (we will omit the subscript from now on and just write d\sigma since the dimension will not change throughout the arguments). We state Stein’s theorem for convenience:

Spherical maximal function theorem [Stein]: The maximal operator \mathcal{M}_{\mathbb{S}^{d-1}} is L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d) bounded for any \frac{d}{d-1} < p \leq \infty .

There is however an alternative proof of the theorem due to Bourgain which is very nice and conceptually a bit simpler, in that instead of splitting the function into countably many dyadic frequency pieces it splits the spherical measure into two frequency pieces only. The other ingredients in the two proofs are otherwise pretty much the same: domination by the Hardy-Littlewood maximal function, Sobolev-type inequalities to control suprema by derivatives and oscillatory integral estimates for the Fourier transform of the spherical measure (and its derivative). However, Bourgain’s proof has an added bonus: remember that Stein’s argument essentially shows L^p \to L^p boundedness of the operator for every 2 \geq p > \frac{d}{d-1} quite directly; Bourgain’s argument, on the other hand, proves the restricted weak-type endpoint estimate for \mathcal{M}_{\mathbb{S}^{d-1}} ! The latter means that for any measurable E of finite (Lebesgue) measure we have

\displaystyle |\{x \in \mathbb{R}^d \; : \; \mathcal{M}_{\mathbb{S}^{d-1}}\mathbf{1}_E(x) > \alpha \}| \lesssim \frac{|E|}{\alpha^{d/(d-1)}}, \ \ \ \ \ \ (1)


which is exactly the L^{d/(d-1)} \to L^{d/(d-1),\infty} inequality but restricted to characteristic functions of sets (in the language of Lorentz spaces, it is the L^{d/(d-1),1} \to L^{d/(d-1),\infty} inequality). The downside of Bourgain’s argument is that it only works in dimension d \geq 4 , and thus misses the dimension d=3 that is instead covered by Stein’s theorem.

It seems to me that, while Stein’s proof is well-known and has a number of presentations around, Bourgain’s proof is less well-known – it does not help that the original paper is impossible to find. As a consequence, I think it would be nice to share it here. This post is thus another tribute to Jean Bourgain, much in the same spirit as the posts (III) on his positional-notation trick for sets.

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Oscillatory integrals II: several-variables phases

This is the second part of a two-part post on the theory of oscillatory integrals. In the first part we studied the theory of oscillatory integrals whose phases are functions of a single variable. In this post we will instead study the case in which the phase is a function of several variables (and we integrate in all of them). Here the theory becomes weaker because these objects can indeed have a worse behaviour. We will proceed by analogy following the same footsteps as in the single-variable case.
Part I

3. Oscillatory integrals in several variables

In the previous section we have analysed the situation for single variable phases, that is for integrals over (intervals of) {\mathbb{R}}. In this section, we will instead study the higher dimensional situation, when the phase is a function of several variables. Things are unfortunately generally not as nice as in the single variable case, as you will see.

In order to avoid having to worry about connected open sets of {\mathbb{R}^d} (see Exercise 18 for the sort of issues that arise in trying to deal with general open sets of {\mathbb{R}^d}), in this section we will study mainly objects of the form

\displaystyle  \mathcal{I}_{\psi}(\lambda) := \int_{\mathbb{R}^d} e^{i \lambda u(x)} \psi(x) dx,

where {\psi} has compact support. We have switched to {u} for the phase to remind the reader of the fact that it is a function of several variables now.

3.1. Principle of non-stationary phase – several variables

The principle of non-stationary phase we saw in Section 2 of part I continues to hold in the several variables case.
Given a phase {u}, we say that {x_0} is a critical point of {u} if

\displaystyle  \nabla u(x_0) = (0,\ldots,0).

Proposition 8 (Principle of non-stationary phase – several variables) Let {\psi \in C^\infty_c(\mathbb{R}^d)} (that is, smooth and compactly supported) and let the phase {u\in C^\infty} be such that {u} does not have critical points in the support of {\psi}. Then for any {N >0} we have

\displaystyle  |\mathcal{I}_\psi(\lambda)|\lesssim_{N,\psi,u} |\lambda|^{-N}.

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Oscillatory integrals I: single-variable phases

You might remember that I contributed some lecture notes on Littlewood-Paley theory to a masterclass, which I then turned into a series of three posts (IIIIII). I have also contributed a lecture note on some basic theory of oscillatory integrals, and therefore I am going to do the same and share them here as a blog post in two parts. The presentation largely follows the one in Stein’s “Harmonic Analysis: Real-variable methods, orthogonality, and oscillatory integrals“, with inputs from Stein and Shakarchi’s “Functional Analysis: Introduction to Further Topics in Analysis“, from some lecture notes by Terry Tao for his 247B course, from a very interesting paper by Carbery, Christ and Wright and from a number of other places that I would now have trouble tracking down.
In this first part we will discuss the theory of oscillatory integrals when the phase is a function of a single variable. There are extensive exercises included that are to be considered part of the lecture notes; indeed, in order to keep the actual notes short and engage the reader, I have turned many things into exercises. If you are interested in learning about oscillatory integrals, you should not ignore them.
In the next post, we will study instead the case where the phases are functions of several variables.

0. Introduction

A large part of modern harmonic analysis is concerned with understanding cancellation phenomena happening between different contributions to a sum or integral. Loosely speaking, we want to know how much better we can do than if we had taken absolute values everywhere. A prototypical example of this is the oscillatory integral of the form

\displaystyle  \int e^{i \phi_\lambda (x)} \psi(x) dx.

Here {\psi}, called the amplitude, is usually understood to be “slowly varying” with respect to the real-valued {\phi_\lambda}, called the phase, where {\lambda} denotes a parameter or list of parameters and \phi'_\lambda gets larger as \lambda grows; for example {\phi_\lambda(x) = \lambda \phi(x)}. Thus the oscillatory behaviour is given mainly by the complex exponential {e^{i \phi_\lambda(x)}}.
Expressions of this form arise quite naturally in several problems, as we will see in Section 1, and typically one seeks to provide an upperbound on the absolute value of the integral above in terms of the parameters {\lambda}. Intuitively, as {\lambda} gets larger the phase {\phi_\lambda} changes faster and therefore {e^{i \phi_\lambda}} oscillates faster, producing more cancellation between the contributions of different intervals to the integral. We expect then the integral to decay as {\lambda} grows larger, and usually seek upperbounds of the form {|\lambda|^{-\alpha}}. Notice that if you take absolute values inside the integral above you just obtain {\|\psi\|_{L^1}}, a bound that does not decay in {\lambda} at all.
The main tool we will use is simply integration by parts. In the exercises you will also use a little basic complex analysis to obtain more precise information on certain special oscillatory integrals.

1. Motivation

In this section we shall showcase the appearance of oscillatory integrals in analysis with a couple of examples. The reader can find other interesting examples in the exercises.

1.1. Fourier transform of radial functions

Let {f : \mathbb{R}^d \rightarrow \mathbb{C}} be a radially symmetric function, that is there exists a function {f_0: \mathbb{R}^{+} \rightarrow \mathbb{C}} such that {f(x) = f_0(|x|)} for every {x \in \mathbb{R}^d}. Let’s suppose for simplicity that {f\in L^1(\mathbb{R}^d)} (equivalently, that {f_0 \in L^1(\mathbb{R}, r^{d-1} dr)}), so that it has a well-defined Fourier transform. It is easy to see (by composing {f} with a rotation and using a change of variable in the integral defining {\widehat{f}}) that {\widehat{f}} must also be radially symmetric, that is there must exist {g: \mathbb{R}^{+} \rightarrow \mathbb{C}} such that {\widehat{f}(\xi) = g(|\xi|)}; we want to understand its relationship with {f_0}. Therefore we write using polar coordinates

\displaystyle \begin{aligned} \widehat{f}(\xi) = & \int_{\mathbb{R}^d} f(x) e^{-2\pi i x \cdot \xi} dx  \\  = & \int_{0}^{\infty}\int_{\mathbb{S}^{d-1}} f_0(r) e^{-2\pi i r \omega\cdot \xi} r^{d-1} d\sigma_{d-1}(\omega) dr, \\  = & \int_{0}^{\infty} f_0(r) r^{d-1} \Big(\int_{\mathbb{S}^{d-1}} e^{-2\pi i r \omega\cdot \xi} d\sigma_{d-1}(\omega)\Big) dr  \end{aligned}

where {d\sigma_{d-1}} denotes the surface measure on the unit {(d-1)}-dimensional sphere {\mathbb{S}^{d-1}} induced by the Lebesgue measure on the ambient space {\mathbb{R}^{d}}. By inspection, we see that the integral in brackets above is radially symmetric in {\xi}, and so if we define

\displaystyle  J(t) := \int_{\mathbb{S}^{d-1}} e^{-2\pi i t \omega\cdot \mathbf{e}_1} d\sigma_{d-1}(\omega),

with {\mathbf{e}_1 = (1, 0, \ldots, 0)}, we have

\displaystyle  \widehat{f}(\xi) = g(|\xi|) = \int_{0}^{\infty} f_0(r) r^{d-1} J(r|\xi|) dr. \ \ \ \ \ (1)

This is the relationship we were looking for: it allows one to calculate the Fourier transform of {f} directly from the radial information {f_0}.

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