LECTURE 1: CLASSICAL METHODS IN RESTRICTION

LECTURE 1: CLASSICAL METHODS IN RESTRICTION

THEORY

JONATHAN HICKMAN AND MARCO VITTURI

1. Basic restriction theory

The purpose of these notes is to describe some exciting recent work of Bourgainand Demeter [4] in Fourier restriction theory. Here we will focus on restriction toa compact piece of the parabola, defining

Pn´1 :“ tpx, |x|2q : x P r´1, 1sn´1u

and letting σ denote the surface measure supported on Pn´1 given by1

ż

fdσ :“

ż

r´1,1sn´1

fpx, |x|2qdx.

We begin by recalling the following fundamental conjecture.

Conjecture 1 (Fourier Restriction Conjecture). The inequality

f |Pn´1LqpPn´1,σq À fLppRnq (1)

holds whenever p1 ą 2npn´ 1q and q ď pn´ 1qp1pn` 1q.

Throughout these notes, the dependence of constants on the dimension n andany Lebesgue exponents will be suppressed in the À notation.

As always, the conjecture requires (1) to hold whenever f is assumed to belong

to some suitable dense class of functions, f denotes the Fourier transform of f and

f |Pn´1 its restriction to Pn´1. Thus, the problem is to determine the mapping

properties of the restriction operator R : f ÞÑ f |Pn´1 . This appears to be an ex-tremely difficult question and despite a vast amount of work by many pre-eminentmathematicians, only the n “ 2 case is known in full (due to Fefferman and Zyg-mund). There is a complicated array of partial results in higher dimensions (i.e.with restricted ranges of p and q) which we will not survey here.

It is often convenient to work with the adjoint extension operator leading to anequivalent formulation of Conjecture 1.

Conjecture 2 (Fourier Restriction Conjecture (Adjoint Form)). The inequality

pgdσqqLp1 pRnq À gLq1 pPn´1,σq (2)

holds whenever p1 ą 2npn´ 1q and q ď pn´ 1qp1pn` 1q.

The restriction conjecture has a rich history and is intimately connected withnumerous other important questions in harmonic analysis and beyond. However,here such details, fascinating as they are, are not our main concern and we arecontent to consider Conjectures 1 and 2 without further motivation.2

1To be precise, the surface measure is a weighted version of σ; the weight is „ 1 everywhere onPn´1 and therefore innocuous as far as we are concerned and so we choose to ignore it. Further,

we remark that σ, though not precisely the surface measure, has a geometric interpretation as the

affine surface measure on Pn´1. This choice of weight has some special significance in restrictiontheory but we will not focus at all on the “affine perspective” here.

2The interested reader is directed to the excellent and accessible survey article [12] and coursenotes [10] for background information.

1

2 JONATHAN HICKMAN AND MARCO VITTURI

There is always a trivial range of estimates for which (1). Indeed, if f P L1pRnq,then by Riemann-Lebesgue f is a continuous function satisfying

pfL8ppRq ď fL1pRnq.

From these facts one immediately deduces the conjectures always hold in the re-stricted range given by p “ 1, 1 ď q ď 8.

Remark 3. One geometric property of the parabola is fundamental to the problem:Pn´1 has everywhere non-vanishing Gaussian curvature.3 It is natural to formulatea general conjecture with the parabola replaced by any (compact) hypersurface Ssatisfying this property (an important alternative example of such a hypersurfacebeing the sphere). However, for simplicity of exposition, we will adhere to theconcrete situation presented above. If the curvature of the surface is allowed tovanish, then the full range of estimates given in the statement of the conjecturesis not possible. In the extreme case, one can show relatively easily that the trivialrange of estimates discussed in the previous remark is best possible if and only ifS contains a point at which the Gaussian curvature vanishes to infinite order.

The range of exponents in Conjecture 1 is suggested by testing the inequality onsome simple examples:

‚ The condition p1 ą 2npn´ 1q is due to the well-known fact that pdσqqpξqhas a decay rate of the order of |ξ|´pn´1q2 as |ξ| Ñ 8 (which itself is aconsequence of the curvature properties discussed above).

‚ We mention in particular that the condition q ď pn ´ 1qp1pn ` 1q arisesfrom the Knapp example. Let R " 1 and consider the small rectangle

D :“ r´R´12, R´12sn´1 ˆ r´R´1, R´1s.

The intersection of D with Pn´1 is a R´12ˆ¨ ¨ ¨ˆR´12 cap on the surface,we let g denote its characteristic function so that

gLq1 pPn´1,σq „ R´pn´1q2q1 . (3)

By the uncertainty principle one expects pgdσqq to be concentrated on thecone generated by the normals to the surface along the cap. In particular,it is reasonable to suppose that this function should be large on the dualrectangle

D˚ :“ r´R12, R12sn´1 ˆ r´R,Rs.

Indeed, it is easy to see |pgdσqq pxq| Á R´pn´1q2χcD˚pxq leading to theestimate

pgdσqqLp1 pRnq Á R´pn´1q2|D˚|1p1

“ R´pn´1q2`pn`1q2p1 . (4)

Comparing (3) and (4) gives the desired condition on the exponents.Notice that one could take D to be any R´12 ˆ ¨ ¨ ¨ ˆ R´12 ˆ R´1

rectangle which is centred “tangentially” at some point on the parabola(and therefore intersects Pn´1 on a R´12 ˆ ¨ ¨ ¨ ˆR´12 cap). In this case,the dual rectangle D˚ is centred at the origin and has the same dimensionsas before but is now orientated in the normal direction to the centre of thecap.

One important partial result on the restriction conjecture is the Stein-Tomastheorem, which provides a sharp L2-based estimate.

3The fact Pn´1 is compact or, moreover, that σ is a finite measure is also significant, but onemay develop a meaningful restriction theory for non-compact surfaces provided a certain amountof symmetry is present in the problem.

`2 DECOUPLING INEQUALITY 3

Theorem 4 (Stein-Tomas restriction theorem). For 1 ď p ď 2pn ` 1qpn ` 3q wehave

f |Pn´1L2pPn´1,σq À fLppRnq (5)

Here we present a simple argument due to Carbery which gives the full range ofinequalities except for the p “ 2pn ` 1qpn ` 3q endpoint, where a restricted-typeestimate is obtained. This weakened version of Theorem 4 is a consequence of thefollowing lemma and interpolation (with the trivial estimates).

Lemma 5 (Almost sharp Stein-Tomas). For any Borel set E Ă Rn, we haveż

Pn´1

|pχEpξq|2 dσpξq À χEL2pn`1qpn`3qpRnq. (6)

Proof. Let T denote the convolution operator Tf :“ f ˚ qµ and fix a Borel setE Ă Rn. Observe, by duality,

ż

Pn´1

|pχEpξq|2 dσpξq “

ż

RnχEpxqTχEpxqdx

We proceed by estimating the convolution operator.4 Fix a radially decreasingSchwartz function ϕ satisfying ϕpxq “ 1 for x P Bp0, 1q and supppϕq Ă Bp0, 2q andfor r ą 0 define ϕrpxq :“ ϕpr´1xq. Decompose the measure σ by writing σ “ σ1`σ2

where pσ1p´xq “ qσpxqϕrpxq and pσ2p´xq “ qσpxqp1 ´ ϕrpxqq for some appropriatevalue of r ą 0 to be chosen later. Thus, T “ T1`T2 where Tjf :“ f ˚qσj for j “ 1, 2and it suffices to show

|E|12T1χEL2pRnq ` |E|T2χEL8pRnq À χE2L2pn`1qpn`3qpRnq.

Observe σ1pξq “ pϕr ˚ σpξq and so

σ1pξq “

ż

Bpξ,1rq

pϕrpξ ´ ηqdσpηq `8ÿ

k“1

ż

Bpξ,2krqzBpξ,2k´1rq

pϕrpξ ´ ηqdσpηq

“: I ` II

Using the simple estimate pϕrL8ppRnq ď |Bp0, 2rq|, one may deduce

|I| ď |Bp0, 2rq|σpBpξ, 1rq X Pn´1q À r,

where the latter inequality is due to the dimensionality of Pn´1. Furthermore, therapid decay of ϕ implies |pϕrpξ ´ ηq| À 2´pk´1qnrn whenever η R Bpξ, 2k´1rq andso

|II| À

ˆ 8ÿ

k“1

2´pk´1qnσ`

Bpξ, 2krq X Pn´1˘

˙

rn À

ˆ 8ÿ

k“1

2´k˙

r.

Combining these observations σ1L8ppRnq À r and so

T1χEL2pRnq “ σ1pχEL2ppRnq À r|E|12.

On the other hand, stationary phase calculations show the measure σ satisfiesthe Fourier decay estimate

|qσpξq| À p1` |ξ|q´pn´1q2

and, since suppp1´ ϕrq Ď RnzBp0, rq, it follows qσ2L8pRnq À r´pn´1q2 and hence

T2χEL8pRnq ď qσ2L8pRnqχEL1ppRnq À r´pn´1q2|E|.

4This trick of reducing the problem of estimating the restriction operator to estimating theconvolution operator is an example of the so-called TT˚ method. A similar argument can be

applied to obtain Strichartz estimates for solutions to the Schrodinger equation in Rn, as discussedlater in these notes.


Thus, combining these observations and choosing r so that r1`pn´1q2 „ |E|, thedesired estimate (6) follows.

Remarks. 1) Inequalities of the form (5) were first considered by Stein in unpub-lished work dating back to the late 1960s; sharp estimates were later establishedby Stein and Tomas. More precisely, Stein obtained the sharp result by usinganalytic interpolation techniques whilst Tomas provided a much simpler argu-ment which gave estimates only in the restricted range 1 ď p ă 2pn`1qpn`3q,missing the endpoint. It was observed by Carbery that Tomas’ proof can beadapted to give a restricted-type inequality when p “ 2pn ` 1qpn ´ 3q (this isthe argument presented above). More recently, Bak and Seeger developed thesemethods to obtain the full range of strong-type estimates and thereby gave analternative and, it transpires, more robust proof of Stein’s theorem.

2) The argument presented above is quite general and relies only on the dimension-ality and Fourier decay of the measure σ. In particular, Mitsis and Mockenhauptobserved one may use these methods to formulate and prove Stein-Tomas-type

theorems for general Borel measures µ on pRn which satisfy for some 0 ă a ă nand b ď a:5

‚ The dimensionality condition

µpBpξ, rqq À ra for all ξ P pR;

‚ The Fourier decay condition

|pµpxq| À r´b2 for all ´x R Bp0, rq.

One interesting consequence of this is that it allows one to develop a non-trivialFourier restriction theory for subsets of R.

3) Moreover, in unpublished work Wright observed that one may push these meth-ods further and apply them in a general locally compact abelian group G, pro-vided G admits a basic form of Littlewood-Paley theory. One particularly in-teresting example is given by G :“ rZpsZsn; in this setting one may develop adiscretised analogue of (Euclidean) restriction theory.

4) An argument of Bak and Seeger can be applied in these abstract settings tostrengthen all the results to strong-type estimates at the relevant endpoints.

2. Restriction and Kakeya

The Knapp example introduced in the previous section is central to much of thefollowing discussion. Note that, necessarily, the restriction estimate (2) “just fails”for the exponents p1 “ q1 “ 2npn ´ 1q (albeit the failure is rather dramatic [1]).In particular, one can show using Holder’s inequality that if Conjecture 1 is true,then for all ε ą 0, the estimate

pgdσqqL2npn´1qpBp0,CRqq Àε RεgL2npn´1qpPn´1,σq (7)

holds for all R " 1. Such “ε-loss” in R will be a regular feature of our analysis and itis useful to introduce the following notation: if X,Y are non-negative real numbersthen X Æ Y or Y Ç X is taken to mean that for all ε ą 0 we have X Àε R

εY forall R " 1. Hence, (8) can be concisely expressed as

pgdσqqL2npn´1qpBp0,CRqq Æ gL2npn´1qpPn´1,σq. (8)

Fix R " 1 and consider many Knapp examples placed around the parabola;that is, take g “

ř

κ χκ to be the sum of characteristic functions of many disjoint

5These restrictions on the parameters a, b are natural; see, for instance, [13].


R´12 ˆ ¨ ¨ ¨ ˆ R´12 caps on Pn´1. Letting Ω Ď Sn´1 denote the set normaldirections to these caps, we have

gL2npn´1qpPn´1q „ pR´pn´1q2#Ωqpn´1q2n. (9)

On the other hand, our heuristics tell us pχκdσqq is large on some „ R12 ˆ ¨ ¨ ¨ ˆ

R12ˆR rectangle Tω which is centred at the origin and orientated in the directionω. Thus, heuristically,

pgdσqqpxq „ R´pn´1q2ÿ

ωPΩ

e2πiξω.xχTω pxq (10)

for some collection of frequencies ξω. By modulating the summands of g one mayreplace each Tω in (10) with any translate of itself whilst maintaining (9). Westipulate that the tubes are contained in Bp0, CRq, but otherwise arrange them inan arbitrary fashion.

There is some cancellation arising from the exponentials in (10) and we manipu-late this using a standard randomisation argument. In very rough and non-rigorousterms, randomisation allows us to model the cancellation by the estimate

|pgdσqqpxq| „ R´pn´1q2

ˆ

ÿ

ωPΩ

χTω pxq

˙12

. (11)

If no cancellation was present, then |pgdσqqpxq| would roughly be the product ofR´pn´1q2 and an `1 sum of the χTω ; the presence of the modulations allows us toknock this down to a (smaller) `2 sum. In practice one works with the expectedvalues of a randomised version of g and shows (11) holds “on average”.

To make the above discussion precise, let εk “ ˘1 denote a sequence of indepen-dent, identically distributed random signs which take the values `1 and ´1 withequal probability 12. We assign to each cap κ a (unique) random sign εκ andredefine g by taking

gpξq “ÿ

κ

εκe2πixκ.ξχκpξq

for some choice of xκ P Rn, so that

pgdσqqpxq “ÿ

κ

εκpχκdσqqpx´ xκq.

The randomisation can be exploited by appealing to Khinchin’s inequality. Thisstates that for all 0 ă p ă 8, if εk are as above,

E”

ˇ

ˇ

mÿ

k“1

εkakˇ

ˇ

pı1p

„

´

mÿ

k“1

|ak|2¯12

for any sequence a1, . . . , am P C; details of the proof can be found in [9]. In thepresent situation Khinchin’s inequality implies

E“

|pgdσqqpxq|2npn´1q‰pn´1q2n

„

´

ÿ

κ

|pχκdσqqpx´ xκq|2¯12

Á R´pn´1q2´

ÿ

ωPΩ

χTω pxq¯12

for almost every x P Rn, which is the rigorous version (11). Thus, taking Lp-normsand applying Fubini’s theorem,

E“

pgdσqqLppRnq‰

Á R´pn´1q2›

›

›

ÿ

ωPΩ

χTω

›

›

›

12

Lnpn´1qpRnq(12)


On the other hand, the value of |g| is independent of the outcome of the εk andthus

E“

gL2npn´1qpPn´1q

‰

„ pR´pn´1q2#Ωqpn´1q2n. (13)

Combining (8), (12) and (13) it follows›

›

›

›

ÿ

ωPΩ

χTω

›

›

›

›

Lnpn´1qpRnqÆ Rn´1´pn´1q22n#Ωpn´1qn.

Thus, we conclude that the restriction conjecture, if true, would imply the followingresult concerning the size of a union of distinctly-orientated rectangles.

Conjecture 6 (Kakeya Maximal Conjecture). Let Ω Ď Sn´1 be a maximal setof R´12-separated directions and tTωuωPΩ a collection of R ˆ R12 ˆ ¨ ¨ ¨ ˆ R12

rectangles where Tω is orientated in the direction of ω. Then the inequality

›

›

›

ÿ

ωPΩ

χTω

›

›

›

Lnpn´1qpRnqÆ

ˆ

ÿ

ωPΩ

|Tω|

˙pn´1qn

(14)

holds.

Remarks. a) In much of the literature the above conjecture is stated in an equiv-alent, rescaled form involving 1 ˆ R´1 ˆ ¨ ¨ ¨ ˆ R´1 rectangles. The scale usedhere is suited to the geometry restriction problem.

b) If the rectangles Tω were mutually disjoint, then›

›

›

ÿ

ωPΩ

χTω

›

›

›

Lnpn´1qpRnq“

´

ÿ

ωPΩ

|Tω|¯pn´1qn

and so (14) can be interpreted as a quantifying the principle that rectangleswhich point in distinct directions must have small intersection (or, in otherwords, they must be “essentially disjoint”).

The Kakeya conjecture is a major open problem in geometric measure theorywhich is closely connected to many classical problems in Fourier analysis. A simpleand elegant argument of Cordoba can be used to establish the two dimensionalcase6 but in all higher dimensions only partial results are known.7

3. Local restriction theory

The restriction conjecture implies the Kakeya conjecture and one is tempted toask to what extent, if any, the converse holds. Since oscillation clearly plays asubstantial role in the former problem whilst the latter is a pure size estimate (i.e.there is no oscillation present), it seems infeasible that Conjecture 1 should be aconsequence of Kakeya alone. One method of capturing the cancellation presentin Conjecture 1 is to attempt to interpose a certain kind of square function;8 we’llsee that if one assumes the necessary square function estimate holds, then therestriction conjecture follows from the Kakeya estimate (14). Before introducingthese square functions, however, it will be useful to develop local restriction theory.

6Alternatively, Conjecture 6 can be proved for n “ 2 using the (known) n “ 2 case of the

restriction conjecture via the method outlined above.7For further information, the surveys of Wolff [13] and Katz and Tao [8] provide an excellent

introduction to the Kakeya conjecture.8This is similar in spirit to the estimate (11) in the proof of Restriction ùñ Kakeya, the key

difference being that before we were constructing a specific function g and therefore had the useful

technique of randomisation at our disposal. Since now we want to prove restriction estimates forgeneral g, it does not make sense to assume g is randomised.


Rather than considering the restriction conjecture per se, most of the time we willfocus on establishing ostensibly weaker local estimates. In particular, we considerinequalities of the form

pgdσqqLp1 pBRq Àα RαgLq1 pPn´1q (15)

for R " 1, where BR Ă Rn is an arbitrary ball of radius R (by the translationinvariance of the problem, the choice of centre of the ball is irrelevant). It is clearthat the local estimate (15) holds with α “ 0 if and only if the global restrictioninequality (2) is true. Furthermore, it is easy to see (15) always holds for α “ np1

and (15) becomes progressively harder to prove as α decreases: the problem becomesto try to push down the value of α.

We have already had a glimpse of the utility of such local estimates in the proofof Restriction ùñ Kakeya where it was noted that for any ε ą 0 the restrictionconjecture implies the local estimate

pgdσqqL2npn´1qpBRq Àε RεgL2npn´1qpPn´1q. (16)

Indeed, one of the many advantages of this localised set-up is that it now makessense to consider restriction estimates at the endpoint p1 “ q1 “ 2npn´ 1q. Moreimportantly, Tao [11] showed that (16) with its ε-loss in the R-exponent can beconverted into a global estimate with a loss in p (i.e. we move away from theendpoint); moreover, if the ε-loss can be made arbitrarily small, then one mayobtain global estimates arbitrarily close to the endpoint. From this one can thenobtain the full range of estimates for the restriction conjecture via interpolationwith the trivial range and Holder’s inequality. Consequently, Conjecture 1 is infact equivalent to showing (16) holds for arbitrarily small ε.

Thus, the restriction conjecture can be reformulated as:

Conjecture 7 (Restriction conjecture, local form). The inequality

pgdσqqL2npn´1qpBRq Æ gL2npn´1qpPn´1q

holds for a suitable class of functions g on Pn´1.

Henceforth we will concentrate on proving localised restriction estimates, takingadvantage of many phenomena which do not arise in the global setting. First of all,we observe that localising to scale R in the spatial variable should impact upon thesituation on the frequency side via the uncertainty principle and one expects g to be“blurred out” or “essentially constant” at scales R´1. This should allow us to safelyfatten up the parabola to NR´1pPn´1q, where NrpEq denotes the r-neighbourhoodof a set E Ď Rn. As these neighbourhoods are “approximating” the parabola, it isnatural that they should be endowed with a normalised measure; this prompts theintroduction of the following notation.

Definition (Lp-averages). Let 1 ď p ă 8.

i) If Λ is a finite set and f : Λ Ñ C is any function, then

f`pavgpΛq :“´ 1

#Λ

ÿ

xPΛ

|fpxq|p¯1p

.

ii) If Ω Ď Rn is measurable with finite Lebesgue measure and f P LppΩq, then

fLpavgpΩq :“´ 1

|Ω|

ż

Ω

|fpxq|p dx¯1p

.

The definition of these averages extends in the obvious manner to the p “ 8 case.


Lemma 8. To prove the local inequality (15) for some fixed R ě 1 it suffices toshow

GLp1 pBRq Àα Rα´1G

Lq1

avgpNR´1 pPn´1qq(17)

holds whenever G is a smooth function supported in the annular region NR´1pPn´1q.

Proof. Fix R " 1 and ψ P C8c ppRq with suppψ Ď Bp0, 1q and |ψpxq| Á 1 for all

x P Bpx0, 1q. Defining G :“ ψR´1 ˚ gdσ where ψR´1pξq :“ RnψpRξq, it follows

pgdσqqLp1 pBpx0,RqqÀ pgdσqqψR´1Lp1 pBpx0,Rqq

“ GLp1 pBpx0,Rqq.

Since G is supported in NR´1pPn´1q, we may apply (17) to deduce

pgdσqqLp1 pBpx0,RqqÀ Rα´1ψR´1 ˚ gdσ

Lq1

avgpNR´1 pPn´1qq

and it therefore remains to show

ψR´1 ˚ gdσLq1 ppRnq À R1q1gLq1 pPn´1q.

Observe if q1 replaced by 1, then the estimate

ψR´1 ˚ gdσL1ppRnq À gL1pPn´1q

is a simple consequence of Fubini’s theorem and so, by interpolation, the proof isconcluded if we can show the corresponding estimate when q1 is replaced with 8.That is, it suffices to show

ψR´1 ˚ gdσL8ppRnq À RgL8pPn´1q.

This in turn is reduced to showingż

Pn´1

|ψR´1pξ ´ ηq|dσpηq À R (18)

uniformly in ξ P pRn. It is clear heuristically why (18) is true: the support ofthe integrand intersects Pn´1 on at most a R´1 ˆ ¨ ¨ ¨ ˆR´1 cap and the functionhas height at most Rn leading to the base ˆ height bound R´pn´1q ˆ Rn “ R.Turing to the rigorous proof of the estimate (18), we’ll in fact prove the more

general statement that whenever ψ is a Schwartz function on pRn and S Ă pRn isany compact hypersurface (no curvature conditions are required here), it follows

ż

S

|ψR´1pξ ´ ηq|dσpηq À R

uniformly in ξ P pRn and R " 1. By rapid decay the left-hand side of this expressioncan be bounded by

Ipξq :“ Rnż

S

dσpηq

p1`R|ξ ´ η|qn.

We form a dyadic decomposition of the above integral based on the size of the

denominator appearing in the integrand. In particular, fix ξ P pRn and consider

partitioning pRn into the closed ball

A´1pξq :“ tη P pRn : R|ξ ´ η| ď 1u

together with the dyadic annuli

Akpξq “ tη P pRn : 2k ă R|ξ ´ η| ď 2k`1u

defined for all k P N. Let Skpξq “ Akpξq X S for all k ě ´1 and write

Ipξq À Rn8ÿ

k“´1

ż

Skpξq

dσpηq

p1`R|ξ ´ η|qn.


To estimate the terms of this sum we observe, due to the dimensionality of S, thatfor r ą 0 we have

σpBpξ, rq X Sq À rn´1 (19)

Indeed, for r large the result is trivial whilst at small scales (0 ă r ! 1) the surfaceis essentially flat and so, provided it is non-empty, Bpξ, rq X S is approximately adisc of radius r, leading to the estimate in this case. The inequality (19) impliesσpS´1pξqq À R´pn´1q and σpSkpξqq À pR

´12kqn´1 for all k ě 0 and so

Ipξq À RnS´1pξq `Rn8ÿ

k“0

σpSkpξqq

2knÀ R,

as required.

In fact, (15) and (17) are equivalent; the converse implication will not be usedin the present notes but will be referred to later.

Lemma 9. If the local extension estimate (15) holds, then the inequality (17) isvalid.

Proof. Without loss of generality (by the translation and rotation invariance of theproblem, together with the triangle inequality) one may assume G is supported inNR´1pPn´1q X Bp0, 12q. Consequently, suppG is contained in the disjoint unionof vertical translates Pn´1

ζ :“ Pn´1 ` p0, ζq of the paraboloid as ζ varies over

p´R´1, R´1q Ă R. By Fubini’s theorem and a simple change of variables,

Gpxq “

ż

|ζ|ăR´1

ż

ξ1Pr´1,1sn´1

Gpξ1, |ξ1|2 ` ζqe2πipx1¨ξ1`xnp|ξ1|2`ζqq dξ1 dζ

“

ż

|ζ|ăR´1

pG|Pn´1ζ

dσζqqpxqdζ,

where σζ is the obvious measure on Pn´1ζ .

Assuming the local extension estimate (15) holds, it immediately follows fromtranslation invariance that

pG|Pn´1ζ

dσζqqLp1 pBRq Àα RαG|Pn´1

ζLq1 pPn´1

ζ q

for all ζ. Combining the above estimate with Minkowski’s inequality one deducesthat

GLp1 pBRq ď

ż

|ζ|ăR´1

pG|Pn´1ζ

dσζqqLp1 pBRq dζ Àα Rα

ż

|ζ|ăR´1

G|Pn´1ζLq1 pPn´1

ζ qdζ

and Holder’s inequality bounds the latter by

RαR´p1´1q1q´

ż

|ζ|ăR´1

G|Pn´1ζq1

Lq1 pPn´1ζ q

dζ¯1q1

“ Rα´1GLq1

avgpNR´1 pPn´1qq,

as required.

4. Restriction estimates via Reverse Littlewood-Paley

Here we discuss a possible approach to proving inequalities of the form (16) viaa Littlewood-Paley theory for slabs on a neighbourhood of the paraboloid.

By our earlier discussion, to prove the restriction conjecture it suffices to showfor any f with smooth Fourier transform support in NR´1pPn´1q, the inequality

fL2npn´1qpBRq Æ R´1fL

2npn´1qavg pNR´1 pPn´1qq


holds. We’ll make the situation slightly simpler by fixing f to be a smooth functionwhose Fourier transform belongs to the unit ball of L8pNR´1pPn´1qq and aimingto prove

fL2npn´1qpBRq Æ R´1.

By Holder’s inequality, this is clearly a weaker statement; it turns out, however,that symmetry considerations show the two estimates are in fact equivalent, butthis is a technical point and we won’t discuss it here.

Arguing in a similar manner to our proof of Restriction ùñ Kakeya, we firstdecompose our neighbourhood of the parabola into a collection of essentially disjointcurved regions (which we will refer to as slabs) θ which are each contained withina „ R´12 ˆ ¨ ¨ ¨ ˆ R´12 ˆ R´1 rectangle. An explicit way to do this is to coverr´1, 1sn´1 with 2R´12 ˆ ¨ ¨ ¨ ˆ 2R´12 cubes tQu whose centres lie in the latticeR´12Zn´1 and define each θ by

θ “ tpξ1, η ` |ξ1|2q : ξ1 P Qθ, |η| À R´1u

for some choice of Qθ P tQu. It is important to note in our construction the slabsdo not have disjoint interiors: the overlap is included for technical reasons9 whichwill become manifest much later in the discussion. Furthermore, letting Ω denotesthe set of normals to these slabs (that is, the set of vectors which correspond to theunit normal to the paraboloid at the centre of a slab), an important consequenceof this construction is that these normals are „ R´12-separated.10

Henceforth, we will let fθ denote the Fourier restriction of f to θ; that is,

fθ :“ fχθ.

With this new notation observe

f „ÿ

θ:R´12´slab

fθ,

where the summation is over the entire collection of slabs. Indeed, if the slabs haddisjoint interiors then this identity would be valid with equality, but almost everypoint in NR´1pPn´1q lies in some fixed number of slabs Cn ą 1 which correspondsto the implied constant in the above equation. Thus, we wish to establish

›

›

›

ÿ

θ:R´12´slab

fθ

›

›

›

L2npn´1qpBRqÆ R´1.

In fact, we’ll aim to prove the (at least ostensibly) stronger inequality›

›

›

ÿ

θ:R´12´slab

fθ

›

›

›

L2npn´1qpRnqÆ R´1; (20)

indeed, fattening the paraboloid on the frequency side encapsulates the spatiallocalistation and so this global estimate is essentially no more difficult to provethan the local version.

9In particular, it allows one to construct a partition of unity for NR´1 pPn´1q adapted to the

family of slabs.10Indeed, if pξ1j , |ξ

1j |

2q P Pn´1 for j “ 1, 2 and νj denotes the unit normal at pξ1j , |ξ1j |

2q, then

ν1 ¨ ν2 “4ξ11 ¨ ξ

12 ` 1

p4|ξ11|2 ` 1q12p4|ξ12|

2 ` 1q12.

Suppose |ξ11´ξ12| Á R´12 and, without loss of generality, assume the above dot product is positive

(otherwise ν1 and ν2 are Op1q-separated). Then

ν1 ¨ ν2 ď´

1´4|ξ11 ´ ξ

12|

2

p4|ξ11|2 ` 1qp4|ξ12|

2 ` 1q

¯12ď p1´ CR´1q12

and it follows that |ν1 ´ ν2|2 “ 2p1´ ν1 ¨ ν2q Á R´1, by the mean value theorem.


We would like to estimate the left-hand sum in (20) in terms of individual contri-butions from the fθ; we’ll see later that these individual portions basically behavelike a superposition of many ‘parallel’ Knapp examples and can be relatively easilyunderstood.11 The main difficultly here is to understand the cancellation betweenthe fθ. One approach would be to try to replace |

ř

θ fθ| with an `2 expression, inthe manner of (11) from the Restriction ùñ Kakeya argument. This has the ef-fect of separating the contributions from the individual fθ whilst accounting for anydestructive interference between terms. In particular, we would like an inequalityof the following form:

Conjecture 10 (Reverse Littlewood-Paley inequality for slabs). Suppose f hasfrequency support in NR´1pPn´1q. With the above notation,

fLppRnq Æ›

›

›

`

ÿ

θ:R´12´slab

|fθ|2˘12

›

›

›

LppRnq(21)

holds whenever 2 ď p ď 2npn´ 1q.

Thus, the Littlewood-Paley (or square-function) estimate (21) would tells usthere is significant destructive interference present between the different frequencylocalised parts of f : enough to allow one to improve the trivial `1 inequality fp ďř

θ |fθ|p to an `2 bound.It is easy to see the p “ 2 case of the conjecture follows immediately from the

almost-orthogonality of the fθ.

Lemma 11. If f has frequency support in NR´1pPn´1q, then

fL2pRnq À›

›

›

`

ÿ

θ:R´12´slab

|fθ|2˘12

›

›

›

L2pRnq.

Proof. By Plancherel’s theorem,

fL2pRnq „›

›

›

ÿ

θ:R´12´slab

fθ

›

›

›

L2pRnq“

›

›

›

ÿ

θ:R´12´slab

fθ

›

›

›

L2ppRnq

and Cauchy-Schwarz bounds this by›

›

›

`

ÿ

θ:R´12´slab

|fθ|2˘12` ÿ

θ:R´12´slab

χθ˘12

›

›

›

L2ppRnq.

Since the slabs are finitely-overlapping the latter expression is dominated by (aconstant multiple of)

›

›

›

`

ÿ

θ:R´12´slab

|fθ|2˘12

›

›

›

L2ppRnq“

›

›

›

`

ÿ

θ:R´12´slab

|fθ|2˘12

›

›

›

L2pRnq,

where we have once again applied Plancherel’s theorem (after reordering the norms).

The `2-decoupling inequalities studied in these lectures are closely related to andpartially motivated by (21). At present, however, we will simply assume Conjecture10 holds and examine the consequences for the Fourier restriction problem.

We have now isolated the function f into frequency localised parts fθ and havecontrolled their oscillatory interactions via the above Littlewood-Paley estimate.This more-or-less takes care of the significant cancellation in the problem and whatremains are pure size considerations. Returning to the uncertainty principal, sincethe function fθ is frequency supported in essentially a „ R´12ˆ¨ ¨ ¨ˆR´12ˆR´1

11That is, provided we have at our disposal pretty heavy-weight tools such as the full Kakeyaconjecture! Here we really interested in understanding the cancellation in the problem and aretherefore rather cavalier when it comes to “pure size estimates”.


rectangle, it should be essentially constant on rectangles with reciprocal dimensions.In order to make this statement rigorous, we employ what is known as a wave packetdecomposition of fθ.

To introduce the wave packet decomposition, we first need to define wave packets.These objects are essentially smoothed out copies of the Knapp example introducedearlier. Fix a bump function φ whose Fourier transform is supported on r´12, 12sn

and equals 1 on r´14, 14sn. For any rectangle T let aT denote an affine trans-formation whose linear part has determinant „ |T | which maps r´14, 14sn to Tbijectively and define φT :“ φ ˝ a´1

T .We will consider rectangles orientated in directions determined by the slabs. In

particular, suppose ω is the normal direction to Pn´1 at the centre ξθ of the slab θ(in this situation we say θ has normal ω). We let Tpθq denote a finitely-overlappingcollection of „ R12 ˆ ¨ ¨ ¨ ˆR12 ˆR rectangles which cover Rn and are orientatedin the direction of ω.

Definition (Wave packet adapted to T ). The wave packet adapted to T P Tpθq isthe function given by

ψT pxq :“ |T |´1e2πiξθ.xφT pxq.

Notice that (provided the implicit constants are suitably chosen) ψT is supportedin a dilate of θ and has modulus 1 on θ. Indeed, a simple computation yields

|ψpξq| „ |φpa˚T pξ ´ ξθqq|

where a˚T is the adjoint of the linear part of aT and, consequently,

tξ P pRn : |ψpξq| “ 1u Ď pa˚T q´1pr´14, 14snq ` ξθ.

It is easy to see pa˚T q´1pr´14, 14snq is a rectangle centred at the origin which is

dual to T . Furthermore, if the sidelengths of T are chosen correctly (dependingonly on some innocuous constant), then θ is contained in the translate of this dualrectangle by ξθ, as required. A similar argument can be used to show the supportcondition holds.

Lemma 12 (Wave packet decomposition). Let f be a smooth function on Rn. Forany slab θ there exists a decomposition

fθpxq “ÿ

TPTpθqfTψT pxq

where the constants fT satisfy`

ÿ

TPTpθq|fT |

2˘12

À fθL2avgpθq

Proof. Let T0 denote the „ R12 ˆ ¨ ¨ ¨ ˆ R12 ˆ R rectangle which is orientated inthe direction of the normal to the slab θ and centred at 0. It follows that, providedthe implied constant defining the dimensions of T0 is chosen correctly,

gθpξq :“ fθppa˚T0q´1ξ ` ξθq

is supported in r´14, 14sn and so we can think of it as a function on the torusTn – r´12, 12sn. Expressing this function in terms of its Fourier series we deducethe formula

fθpξq “ÿ

kPZnuke

´2πik.a˚T0pξ´ξθq for ξ P pa˚T0

q´1pr´12, 12snq ` ξθ.

Here the tukukPZ are the Fourier coefficients of gθ which satisfy`

ÿ

kPZ|uk|

2˘12

“ gθL2pr´12,12snq À fθL2avgpθq

.


By our earlier observations, the function ξ ÞÑ φpa˚T0pξ´ξθqq equals 1 on the support

of fθ and is itself supported in pa˚T0q´1pr´12, 12snq ` ξθ so that

fθpξq “ÿ

kPZnuke

´2πik.a˚T0pξ´ξθqφpa˚T0

pξ ´ ξθqq,

which is valid on the whole of pRn. Taking the inverse Fourier transform,

fθpxq “ÿ

kPZnuk|det a´1

T0|e2πix.ξθφT0

px´ aT0kq

“ cÿ

kPZnukψT0`aT0k

pxq,

and the proof is concluded by defining the Tpθq to be the collection of rectangles ofthe form T0 ` aT0

k.

Applying the conjectured Littlewood-Paley estimate together with the wave-packet decomposition, we now wish to bound

R´pn`1q2›

›

›

´

ÿ

θ:R´12´slab

“

ÿ

TPTpθq|fT ||φT |

‰2¯12›

›

›

L2npn´1qpRnq. (22)

The Schwartz function φT rapidly decays away from T ; we would like to replacethis “smooth indicator function” with the sharp cut-off χT in order to place our-selves in position to apply the (hypothesised) Kakeya estimates. Of course, φT isnot compactly supported and a modicum of extra work is required to make such a“substitution” rigorous. Let χT,l denote the characteristic function of the rectangleAT pr´14, 14sn` l2q for each l P Zn so that the tχT,lulPZn form a rough partitionof unity of Rn. By the rapid decay of φ we have

|φT pxq| “ÿ

lPZn|φT pxq|χT,lpxq

Àÿ

lPZn

χT,lpxq`

1` |a´1T pxq|

˘n`1 Àÿ

lPZn

χT,lpxq`

1` |l|˘n`1

We combine the preceding observations together with a two-fold application ofMinkowski’s inequality to deduce (22) is bounded above by (a constant multipleof)

R´pn`1q2ÿ

lPZn

`

1` |l|˘´pn`1q

›

›

›

´

ÿ

θ:R´12´slab

“

ÿ

TPTpθq|fT |χT,l

‰2¯12›

›

›

L2npn´1qpRnq.

Since the supports of the χT,l are essentially disjoint as T varies over Tpθq, for a

fixed value of l the L2npn´1q-norm in the above expression can be bounded by›

›

›

ÿ

θ:R´12´slab

ÿ

TPTpθq|fT |

2χT,l

›

›

›

12

Lnpn´1qpRnq. (23)

To conclude the proof, it suffices to show that (23) is OpRpn´1q2q; one may thensum in l to obtain the desired restriction inequality.

Recall, the properties of the wave packet decomposition and our initial hypothesison f imply

ÿ

TPTpθq|fT |

2 À 1

for each θ. One may therefore find sequences pcT qTPTpθq of non-negative real num-bers such that

ÿ

TPTpθqcT “ 1


and such that (23) is dominated by›

›

ÿ

θ:R´12´slab

ÿ

TPTpθqcTχT,l

›

›

12

Lnpn´1qpRnq. (24)

Consider randomly selecting a sequence of rectangles, one for each direction θwhere each T is chosen from Tpθq with probability cT . This corresponds to endowingthe space

ś

θ Tpθq with the probability measure that assigns the probabilityś

θ cTθto each singleton tpTθqu. For fixed x P Rn consider the random variable

ř

θ χTθ,lpxqwhich counts the number of rectangles in a randomly selected sequence pTθq forwhich x P suppχTθ,l. It is easy to see, using the linearity of the expectation, that

E“

ÿ

θ:R´12´slab

χTθ,lpxq‰

“ÿ

θ:R´12´slab

ÿ

TPTpθqχTθ,lpxq

and so, by Minkowski’s inequality,›

›

ÿ

θ:R´12´slab

ÿ

TPTpθqcTχT,l

›

›

Lnpn´1qpRnq ď E”

›

›

ÿ

θ:R´12´slab

χTθ,l›

›

Lnpn´1qpRnq

ı

.

The argument is concluded by applying the hypothesised Kakeya estimate›

›

ÿ

θ:R´12´slab

χTθ,l›

›

Lnpn´1qpRnq Æ Rn´1,

which is valid for every choice of l P Zn and pTθq Pś

θ Tpθq.

5. Progress on the Littlewood-Paley inequality for slabs

We have already seen that a global version of the Littlewood-Paley inequalityfor slabs is true (and somewhat trivial) for p “ 2 in all dimensions. In this section aclassical argument is presented which proves (21) for n “ 2 and p “ 4. This can becombined with the above analysis to give a full proof of the restriction conjecture forn “ 2. In later notes we will present a simpler proof of two-dimensional restrictionrelying on bilinear techniques.

Proposition 13 (Cordoba and Fefferman). Conjecture 10 holds when n “ 2. Inparticular, with the above notation,

fL4pR2q À

›

›

›

`

ÿ

θ:R´12´slabs

|fθ|2˘12

›

›

›

L4pR2q.

whenever f has Fourier support in NR´1pP 1q. (So in this case the result holdswithout any ε-leakage in the R-exponent).

Proof. The Fourier support condition ensures

fL4pR2q „

›

›

›

ÿ

θ,θ1:R´12´slabs

fθfθ1›

›

›

2

L2pR2q(25)

and, due to the trivial Cauchy-Schwarz estimateˇ

ˇ

ˇ

ÿ

θ,θ1:R´12´slabs

distpθ,θ1qÀR´12

fθfθ1ˇ

ˇ

ˇÀ

ÿ

θ,θ1:R´12´slabs

|fθ|2,

it is enough to bound the right-hand side of (25) with the summation restricted topairs of well-separated slabs pθ, θ1q for which distpθ, θ1q Á R´12. In particular, itsuffices to show

›

›

›

ÿ

θ,θ1:R´12´slabs

distpθ,θ1qÁR´12

fθfθ1›

›

›

2

L2pR2qÀ

ÿ

θ,θ1:R´12´slabs

fθfθ12L2pR2q,


which can be interpreted as the statement that the fθfθ1 are almost orthogonal.Observe

supp fθ ˚ˆfθ1 Ď θ ´ θ1

and, by Plancherel, the problem now reduces to showing these Minkowski differenceshave bounded overlap as pθ, θ1q varies over well-separated pairs.

Suppose θ1 ´ θ11 X θ2 ´ θ

12 ‰ H for R´12-slabs θj , θ

1j which satisfy distpθj , θ

1jq Á

R´12 for j “ 1, 2. Thus, there exist yj P θj and y1j P θ1j for j “ 1, 2 such that

y1 ´ y11 ´ y2 ` y

12 “ 0

Since each slab belongs to NR´1pP 1q, there exist tj , t1j P r´1, 1s such that

|yj ´ ptj , t2j q|, |y

1j ´ pt

1j , pt

1jq

2q| ă R´1

for j “ 1, 2 and so

|pt1 ´ t11q ´ pt2 ´ t

12q| À R´1; (26)

|pt21 ´ pt11q

2q ´ pt22 ´ pt12q

2q| À R´1.

It follows that

|t1 ´ t11||pt1 ` t

11q ´ pt2 ` t

12q| À R´1

and the separation condition on the θj , θ1j then implies

|pt1 ` t11q ´ pt2 ` t

12q| À R´12. (27)

Comparing (26) and (27), one deduces |t1 ´ t2|, |t11 ´ t

12| À R´12 and therefore

|y1 ´ y2|, |y11 ´ y

12| À R´12.

Consequently, given θ1, θ11 there are at most Op1q choices of θ2, θ

12 for which θ1 ´

θ11 X θ2 ´ θ12 ‰ H and

#

pθ, θ1q : distpθ, θ1q Á R´12 and ξ P θ ´ θ1(

À 1

for all ξ P pR2, as required.

In higher dimensions partial results are known, which are typically non-optimalin the R exponent. We mention, for instance, the work of Bourgain [2] which shows(21) holds for the wider range12 2 ď p ď 2pn ` 1qpn ´ 1q provided the R-loss isstepped up from Æ 1 to À Rα where α :“ pn´ 1qp12´ 1pq4.

6. The `2 decoupling conjecture

We saw in the previous sections that, if true, a Littlewood-Paley inequality forslabs, (21), would provide an effective tool for understanding significant cancella-tion phenomena in the Fourier restriction problem. We remark that an argumentof Carbery [5] shows that (21) would also imply the Kakeya conjecture and, con-sequently, the preceding argument “reduces” proving the restriction problem toestablishing Conjecture 10.13

For the remainder of these lectures we will investigate a weaker variant of (21);namely, we are interested in estimates of the form

fLppRnq Æ`

ÿ

θ:R´12´slab

fθ2LppRnq

˘12. (28)

12The exponent 2pn ` 1qpn ´ 1q pervades restriction theory and, in particular, forms the

endpoint for the Stein-Tomas theorem discussed earlier.13Attempting to prove the whole restriction conjecture from this direction may be a somewhat

optimistic strategy: Conjecture 10 appears to be very powerful and in all likelihood considerablymore difficult than the restriction conjecture.


Using the terminology introduced by Bourgain and Demeter, we will refer to (28) asan `2-decoupling inequality. The idea is that, as with the square function estimate,this inequality separates (or “decouples”) the contributions to fp from the manyfrequency localised portions fθ. This is done in an efficient way, taking into accountthe cancellation between the fθ. In this regard, however, the decoupling inequalitiesare, a priori, not quite as effective as the Littlewood-Paley inequality and, indeed,(28) is clearly weaker than (21) for 2 ď p ď 2npn´ 1q by Minkowski’s inequality.

We remark that (28) does not act as a substitute for the square function estimatein the sense that it is not clear if it is possible to modify our earlier arguments toprove the restriction conjecture using decoupling inequalities. Decoupling theorydoes, however, have a plethora of applications and [4] uses (28) to not only studyrestriction theory, but problems in PDE, additive combinatoric and number theory.

Given the square-function conjecture, it makes sense to conjecture the following:

Conjecture 14 (`2-decoupling conjecture, preliminary version). With the abovenotation,

fLppRnq Æ`

ÿ

θ:R´12´slab

fθ2LppRnq

˘12.

holds whenever 2 ď p ď 2npn´ 1q.

In the next few lectures we’ll present a proof of this conjecture. The key toolwill be multilinear restriction theory, which is well understood thanks to the workof Bennett, Carbery, Tao and Guth, et al. We mention that the above conjecturewas known prior to Bourgain and Demeter’s paper, appearing in [3].

It turns out that the range of exponents 2 ď p ď 2npn ´ 1q is no longer opti-mal when considering the weaker decoupling inequalities and a more appropriateendpoint is the Stein-Tomas exponent 2pn` 1qpn´ 1q. For larger values of p onemay still obtain decoupling inequalities, but necessarily with a constant dependingpolynomially on R. The full version of the conjecture is the following.

Conjecture 15 (`2-decoupling conjecture, full version). With the above notation,

fLppRnq Æ Rαppq`

ÿ

θ:R´12´slab

fθ2LppRnq

˘12

holds with

αppq :“

"

0 if 2 ď p ď 2pn` 1qpn´ 1q;pn´ 1q4´ pn` 1q2p if 2pn` 1qpn´ 1q ă p.

The above corollary can be seen to be best possible, except possibly for the ε-lossin R.

We leave it as an exercise to show no decoupling estimates are possible for p ă 2.Recall that prior to Bourgain and Demeter’s work it was shown in [3] that the

conjecture holds for 2 ď p ď 2npn´ 1q. In addition, partial results in the “super-critical” regime p ě 2pn` 1qpn´ 1q were established in [7] and [6]. Bourgain andDemeter [4] were able to adapt and develop multilinear restriction arguments toestablish the full super-critical range of estimates from which one may establishConjecture 15 in full.14 In the applications it is often necessary to have the fullpower of Conjecture 15 and, after discussing the preliminary Conjecture 14, we willdetail how the complete range of estimates was proved in later lectures.

14The sub-critical estimates follow from the p “ 2pn` 1qpn´ 1q case together with the trivial

p “ 2 inequality - the details of this argument will be discussed in later lectures.


References

[1] W. Beckner, A. Carbery, S. Semmes, and F. Soria, A note on restriction of the Fourier trans-

form to spheres, Bull. London Math. Soc. 21 (1989), no. 4, 394–398. MR 998638 (90i:42023)

[2] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom.Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257 (92g:42010)

[3] , Moment inequalities for trigonometric polynomials with spectrum in curved hyper-

surfaces, Israel J. Math. 193 (2013), no. 1, 441–458. MR 3038558[4] J. Bourgain and C. Demeter, The proof of the `2 Decoupling Conjecture, To appear in Ann.

of Math. arXiv:1403.5335v2.

[5] A. Carbery, A remark on reverse Littlewood-Paley, Restriction and Kakeya, Private commu-nication.

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[7] G. Garrigos and A. Seeger, A mixed norm variant of Wolff’s inequality for paraboloids,Harmonic analysis and partial differential equations, Contemp. Math., vol. 505, Amer. Math.

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national Conference on Harmonic Analysis and Partial Differential Equations (El Escorial,

2000), no. Vol. Extra, 2002, pp. 161–179. MR 1964819 (2003m:42036)[9] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math-

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#7280)[10] T. Tao, Restriction theorems and applications (course notes), Available at:

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[11] , The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96(1999), no. 2, 363–375. MR 1666558 (2000a:42023)

[12] , Some recent progress on the restriction conjecture, Fourier analysis and convex-ity, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004, pp. 217–243.

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[13] T. H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, AmericanMathematical Society, Providence, RI, 2003, With a foreword by Charles Fefferman and

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Jonathan Hickman, Room 5409, James Clerk Maxwell Building, University of Edin-

burgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD.E-mail address: [email protected]

Marco Vitturi, Room 4606, James Clerk Maxwell Building, University of Edinburgh,Peter Guthrie Tait Road, Edinburgh, EH9 3FD.

E-mail address: [email protected]

LECTURE 1: CLASSICAL METHODS IN RESTRICTION

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