arXiv:2110.11185v1 [math.CA] 21 Oct 2021

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CIRCULAR AVERAGE RELATIVE TO

FRACTAL MEASURES

SEHEON HAM, HYERIM KO, AND SANGHYUK LEE

Abstract. We prove new Lp–Lq estimates for averages over dilates of the cir-cle with respect to α-dimensional fractal measure, which unify different typesof maximal estimates for the circular average. Our results are consequences ofLp–Lq smoothing estimates for the wave operator relative to fractal measures.We also discuss similar results concerning the spherical averages.

1. introduction

Let d ≥ 2. We consider the average

Af(x, t) =∫

Sd−1

f(x− ty)dσ(y), x ∈ Rd.

Here dσ denotes the normalized measure on the unit sphere Sd−1. We study esti-mates for A relative to fractal measures. To motivate our study, we briefly reviewsome of the previous results.

Stein [34] (d ≥ 3) and Bourgain [5] (d = 2) proved that the maximal operatorf 7→ supt>0 |Af(·, t)| is bounded on Lp if and only if d

d−1 < p ≤ ∞. If the supremum

is taken over a compact interval I contained in (0,∞), then the consequent maximalfunction has Lp improving property. That is to say, the estimate

(1.1) ‖Af‖LqxL

∞

t (Rd×I) ≤ Cd‖f‖Lp(Rd)

holds for some p < q. Except for a few endpoint cases, we now have a completecharacterization of p, q for which (1.1) holds. Let Pd ⊂ [0, 1]× [0, 1] be the convex

hull of the set consisting of the points O := (0, 0), Q1 :=(d(d−1)d2+1 ,

d−1d2+1

), and

Q2 :=(d−1d , 1d

), Q3 :=

(d−1d , d−1

d

). Schlag [31] (d = 2), and Schlag and Sogge

[32] (d ≥ 3) showed that the estimate (1.1) holds for (p−1, q−1) ∈ Pd \ ((O,Q1) ∪[Q1, Q2) ∪ [Q2, Q3]) and it fails if (p−1, q−1) 6∈ (Pd \ Q3). The third namedauthor [20] proved the borderline case (p−1, q−1) ∈ (O,Q1) ∪ (Q1, Q2) ∪ (Q2, Q3),however the cases (p−1, q−1) = Q1, Q2 remain open. See also [1, 29, 2] for recentdevelopments regarding the circular and spherical maximal functions.

On the other hand, different forms of maximal estimate have also been of interest.In particular, the estimate

(1.2) ‖Af‖LqtL

∞x (Rd×I) ≤ Cd‖f‖Lp

ǫ(Rd)

was used to study packing problems for the circle and sphere [36, 17, 24, 26]. Here‖f‖Lp

s(Rd) := ‖(1−∆)s2 f‖Lp(Rd). When d ≥ 3, Kolasa and Wolff [17] showed (1.2)

with p = q = 2 for any ǫ > 0. The corresponding estimate in R2 turned out to be

2020 Mathematics Subject Classification. 42B25.Key words and phrases. circular average, general measures.

1

http://arxiv.org/abs/2110.11185v1

2 SEHEON HAM, HYERIM KO, AND SANGHYUK LEE

more difficult. Wolff [36] obtained (1.2) with p = q = 3 for ǫ > 0. The results in[36, 17] are given in a different form in which averages over the sphere (circle) werereplaced by the average over annuli of thickness δ. However, it is elementary to seethat those estimates are equivalent to the abovementioned estimates in the form of(1.2). It is known that ǫ > 0 can not be removed (see [4, 16] or [17, Proposition 2.2]for example). Wolff’s result was extended to variable coefficient settings by Zahl[38, 39].

Definition 1.1. For α ∈ (0, d+1], we say a non-negative Borel measure ν on Rd+1

is α-dimensional if there is a constant Cν such that

(1.3) ν(Bd+1(z, ρ)) ≤ Cνρα, ∀(z, ρ) ∈ Rd+1 × R+,

where Bd+1(z, ρ) = z′ ∈ Rd+1 : |z − z′| < ρ. By Cd+1(α) we denote the class ofα-dimensional measure ν. For ν ∈ Cd+1(α), we also define

〈ν〉α := supz∈Rd+1, ρ>0

ρ−αν(Bd+1(z, ρ)).

For the rest of the paper, we set I = (1, 2). We consider the estimate

(1.4) ‖Af‖Lq(Rd×I;dν) ≤ C〈ν〉1qα‖f‖Lp(Rd).

By finite speed of propagation and translation invariance the estimate (1.4) is equiv-alent to the local estimate, i.e., (4.1) below when p ≤ q. Remarkably, the estimate(1.4) unifies the maximal estimates (1.1) and (1.2) in a single framework. Indeed, itis not difficult to see that (1.4) implies the seemingly unrelated estimates (1.1) and(1.2). One can deduce the estimate (1.1) from (1.4) if it holds with a uniform C forall d-dimensional measures with Cν ≤ 1. This can be shown by the Kolmogorov-Seliverstov-Plessner linearization argument and the Riesz representation theorem.(See, for example, Lemma 3.2.) By the same argument (1.2) follows if (1.4) withLp replaced by Lpǫ holds uniformly for all 1-dimensional measures.

Since the averaging operator A is translation-invariant in x, it is impossible tohave (1.4) unless 1 ≤ p ≤ q ≤ ∞. Also, taking a specific α-dimensional measure, itis easy to see that (1.4) holds only if αp ≥ q. (See (i) in Proposition 4.1.) Thus itis natural for the estimate (1.4) to assume α > 1 and p ≤ q.

When d ≥ 3 the estimate (1.4) is relatively easier to obtain. There are variousestimates which are straightforward consequences of the fractal Strichartz estimatesfor the wave equation ([8, 13]). We discuss the matter in Section 3. However, whend = 2, such estimates are not enough to prove the estimate (1.4). In this paper wefocus on the more interesting case d = 2.

The circular average. For 1 < α ≤ 3, we define P2(α) ⊂ [0, 1]2 as follows. For1 < α ≤ 2,

P2(α) :=(1

p,1

q

)∈ [0, 1]2 :

1

q≤ 1

p<α

q,

3

p< 1 +

α− 1

q

,

and for 2 < α ≤ 3 we define P2(α) to be the set of ( 1p ,1q ) ∈ [0, 1]2 which satisfies

1

q≤ 1

p<α

q,

3

p< 1 +

2α− 3

q,

2

p< 1 +

α− 2

q.

CIRCULAR AVERAGE 3

In Section 4, we show that (1.4) fails if ( p−1, q−1) /∈ P2(α) when d = 2. It seemsto be reasonable to conjecture that the range P2(α) determines, possibly except forthe borderline cases, the optimal range of p, q on which (1.4) holds.1

Conjecture 1.2. Let d = 2 and 1 < α ≤ 3. The estimate (1.4) holds for ν ∈ C3(α)if ( p−1, q−1) ∈ P2(α).

The following is our main result, which verifies the conjecture for α ∈ [3−√3, 3].

Theorem 1.3. Let d = 2, 1 < α ≤ 3, and ν ∈ C3(α). Then, the estimate (1.4)holds if

(1.5) ( p−1, q−1) ∈ P2(α) and p > (6− 2α)/α.

Note that p > 4 − α whenever (1/p, 1/q) ∈ P2(α). Thus, (6 − 2α)/α ≤ 4 − α

if α ≥ 3 −√3. As a corollary we obtain Lp(R2)–Lq(dµ) estimate for the circular

maximal function relative to fractal measures µ ∈ C2(α).

Corollary 1.4. Let 1 < α ≤ 2 and µ ∈ C2(α). For p, q satisfying (1.5), we have

(1.6)∥∥ supt∈I

|Af |∥∥Lq(dµ)

≤ C〈µ〉1qα‖f‖Lp(R2).

A modification of examples which give the necessary conditions of (1.4) (Proposi-

tion 4.1) shows that (1.6) holds only if ( p−1, q−1) ∈ P2(α). Consequently, Corollary

1.4 establishes boundedness on sharp range for α ∈ [3−√3, 2].

Lp(Rd)–Lq(dµ) estimate forMf := supt∈I |Af(·, t)| has been used to study prob-

lems in geometric measure theory. Mitsis [24] proved L2(Rd)–L2(dµ) boundednessfor 1 < α ≤ d, d ≥ 3. D. Oberlin and R. Oberlin [27] obtained L2

(1−α)/2+(Rd)–

Lq(dµ) bound on Mf when 1 < q < 2 and 0 < α < (d − 1)/2. Also, for d ≥ 3,Lp(dµ1)–L

p(dµ2) estimate was discussed in Iosevich et al. [15] with αi-dimensionalmeasure µi, 1 < αi ≤ d, i = 1, 2. Those estimates were used to study the spherepacking problem, Hausdorff dimension of the pinned distance set, or the divergenceset of the solution to the wave equation (see also [12]).

Compared with the previous results [24, 27, 15], it is remarkable that Corollary1.4 gives the sharp Lp improving property for the circular maximal function relativeto some fractal measures. As far as the authors are aware, no such results haveappeared before. In [15] a weaker version of local smoothing estimates were com-bined with a duality argument. However, the estimates in Corollary 1.4 can not beobtained by the approach in [15] even if combined with the sharp local smoothingestimate [11]. Nevertheless, such estimates are easier to prove (see Corollary 3.5)in higher dimensions.

Furthermore, one can use the estimate (1.6) (with p = q) to prove the circlepacking theorem which was shown in [37] (see [24, 30]): Suppose F ⊂ R2 is a Borel

set of Hausdorff dimension exceeding 1 and E is a compact set such that E ⊂ R2

contains circles centered at each point in F . Then E has positive Lebesgue measure.

If α = 2, the condition (1.3) ensures that µ is absolutely continuous with respectto the Lebesgue measure. So, by the Lebesgue-Radon-Nikodym theorem we havedµ(x) = f(x)dx for some f ∈ L1. From (1.3) and the Lebesgue differentiationtheorem we see f is bounded. Thus, Corollary 1.4 with α = 2 recovers the resultson the circular maximal estimates except for the endpoint case ([31, 32, 20]).

1As seen in the above, some of the borderline cases are actually not true.


Local smoothing estimates relative to ν ∈ C3(α). In order to prove Theorem 1.3, weconsider Lpγ–L

q(dν) estimate for the wave operator

eit√−∆f(x) =

∫ei(x·ξ+t|ξ|)f(ξ) dξ.

For 2 < α ≤ 3, we obtain the estimate (1.4) by relying mainly on L2γ–L

q(dν)

estimate for eit√−∆, which is to be discussed in Section 3. However, for 1 < α ≤ 2,

such L2γ–L

q(dν) estimates are not enough for our purpose. Instead, we exploit localsmoothing property of the wave operator relative to fractal measure ν:

(1.7)(∫

B2(0,1)×I

∣∣eit√

−∆f(x)∣∣qdν(x, t)

)1/q

≤ C〈ν〉1qα‖f‖Lp

γ(R2).

In [33], Sogge conjectured that the estimate (1.7) holds with dν(x, t) = dxdt forp = q ≥ 2d/(d − 1) if γ > (d − 1)(12 − 1

p ) − 1p . Since then, the conjecture was

studied by various authors [35, 37, 18, 9, 10, 14, 22, 6, 19]. Recently, Guth, Wang,and Zhang [11] confirmed the conjecture in two dimensions while it remains openin higher dimensions. See also [33, 25, 32] for earlier results.

For 1 < α ≤ 2, Theorem 1.3 is a consequence of the next theorem, which weprove by combining the sharp Lp local smoothing estimate [11] and the trilinearrestriction estimate for the cone (see [3, 7, 22]).

Theorem 1.5. Let 1 < α ≤ 2 and ν ∈ C3(α). If p, q ≥ 1 satisfy (1.5), then the

estimate (1.7) holds for some γ < 1/2.

For the proof of Theorem 1.3 we need only to obtain the estimate (1.7) forγ < 1/2 (see Section 3). However, it is of independent interest to characterize p, qand γ for which (1.7) holds. In this regard, we can obtain the sharp estimate on arestricted range of p, q. To state our result we set

(1.8) κ(α) =

3, 2 < α ≤ 3,

α+ 1, 1 < α ≤ 2,

2α, 0 < α ≤ 1.

Theorem 1.6. Let 0 < α ≤ 3 and ν ∈ C3(α). If 1q ≤ min

(1p ,

13p + 1

6 ,23p′

)and

1p +

κ(α)q ≤ 1, then the estimate (1.7) holds for γ > 1

2 + 1p − α

q .

The regularity assumption γ > 12 +

1p − α

q is sharp in that (1.7) fails in general if

γ < 12 + 1

p − αq (see Section 2). It seems to be natural to conjecture that the same

remains valid as long as 1q ≤ 1

p and 1p + κ(α)

q ≤ 1. This in turn implies Conjecture

1.2. When 2 < α ≤ 3 and 1q ≤ 1

p and 1p + 3

q ≤ 1 the estimate (1.7) is basically a

consequence of the sharp local smoothing estimate [11].2

Estimates for the spherical average. In higher dimensions one can obtain resultsanalogous to Theorem 1.3 for the spherical average in Rd+1, d ≥ 3. (See Theorem3.4.) Those results can be obtained by making use of the Strichartz estimates forthe wave equation relative to fractal measure (see Theorem 3.3). It seems thatthe spherical average estimates in Theorem 3.4 is sharp for d = 3, while we onlymanage to verify the optimality for integer α. See Section 4.

2In higher dimensions (d ≥ 3), the local smoothing estimate in [6] and Lemma 2.7 yield an

analogue of (1.7) for d ≤ α ≤ d+ 1 whenever γ > d−1

2+ 1

p− α

q, 1

p+ d+3

d−1

1

q≤ 1, and p ≤ q.

CIRCULAR AVERAGE 5

Organization of the paper. In Section 2, we obtain the local smoothing estimaterelative to fractal measures (Proposition 2.1 and Proposition 2.12) which we use toprove Theorem 1.5 and Theorem 1.6. In Section 3 we prove Theorem 1.5, Theorem1.6, Theorem 1.3, and Corollary 1.4, and obtain the estimates in higher dimensions.We discuss the sharpness of the estimate (1.4) in Section 4.

2. Weighted local smoothing estimates

In this section, we obtain preparatory results which we use to prove Theorem1.5 and Theorem 1.6. The main object is to prove Proposition 2.1 and Proposition2.12. To this end, we adapt the argument due to Vargas and one of the authors[22] which is based on the trilinear restriction estimate.

We decompose the frequency space of eit√−∆f into two parts; one is a sum of

operators with small frequency supports and the other is a product of operatorswith angularly separated frequency supports. For the latter we apply the multi-linear restriction estimate due to Bennett, Carbery, and Tao [3] since the angularseparation implies transversality on the frequency supports of the trilinear opera-tor. For the first part, we rescale the operator and apply the induction assumption(see (2.5)). However, after rescaling, the support of a rescaled measure is not con-tained in a fixed set so that an induction assumption on a bounded set can not besatisfied. To get around this issue, we consider a class of weight functions whichare not necessarily supported in a compact set.

2.1. Weighted local smoothing estimates in R2+1. For 0 < α ≤ 3, we denoteby Ω(α) the class of nonnegative measurable functions ω : R2×I 7→ R which satisfy

(2.1)

∫

B3(z,r)

ω(y)dy ≤ rα, ∀(z, r) ∈ R3 × R+.

By Littlewood-Paley decomposition, it is enough to consider the estimate

(2.2) ‖eit√−∆Pλf‖Lq(R2×I;ω) ≤ Cλγ‖f‖p.

Here Pλ is the standard Littlewood-Paley projection operator given by Pλf(ξ) =

β(λ−1|ξ|)f(ξ) for β ∈ C∞c ((1/2, 2)) and ‖f‖qLq(B;ω) :=

∫B|f(x)|qω(x)dx.

Instead of proving (2.2) directly, we consider a modified operator to facilitatethe rescaling argument. We consider

ψ0(θ) :=√1 + θ2 − 1,

which is closely related to the phase function of the wave operator. Note that ψ0

can be regarded as a small perturbation of θ2/2.For λ ≥ 1 and J ⊂ [−1, 1], let

Aλ(J) = (η, ρ) : λ/4 ≤ η ≤ 4λ, θ = ρ/η ∈ J.

We define the operator Tλ = T φλ by

(2.3) T φλ g(x, t) =

∫

Aλ([−1,1])

ei(x,t)·(η,ρ,ηφ(ρ/η))β(λ−1η)β0(|ρ|/η)g(η, ρ)dηdρ,

where β0 is a smooth function supported in [0, 2].


Proposition 2.1. Let 1 < α ≤ 2 and ω ∈ Ω(α). If p, q ≥ 1 satisfy (1.5) and

λ≫ 1, then there is a γ < 1/2 such that

(2.4) ‖Tψ0

λ g‖Lq(R2×I;ω) ≤ Cλγ‖g‖p.We deduce the estimate (2.2) from (2.4) via change of variables (see the proof of

Theorem 1.5 in Section 3).

Normalization of phase functions. For our induction argument, we begin with nor-malizing the phase function of Tλg. Let

φ(θ) =1

2θ2.

For a given ǫ > 0, we define a class S(ǫ) of C3 functions by

S(ǫ) = φ ∈ C3([−1, 1]) : φ(0) = φ′(0) = 0, ‖φ− φ‖C3([−1,1]) ≤ ǫ,where ‖φ‖Cn([−1,1]) =

∑j≤n supθ∈[−1,1]

∣∣∂jθφ(θ)∣∣. For θ ∈ (−1, 1), let h be a real

number such that [θ−h, θ+h] ⊂ [−1, 1]. For φ ∈ C3([−1, 1]) satisfying φ′′(θ) 6= 0,we define

φθ,h(θ) = (h2φ′′(θ))−1

(φ(hθ + θ)− φ(θ)− φ′(θ)hθ

).

Then we have the following.

Lemma 2.2. Suppose φ ∈ S(ǫ) for some 0 < ǫ < 1/8. Then, there exists

0 < h = h(ǫ) such that φθ,h ∈ S(ǫ) whenever 0 < h ≤ h and [θ−h, θ+h] ⊂[−1, 1].

Throughout the paper, ǫ denote a positive constant such that 0 < ǫ < 1/8.

Proof. By Taylor expansion, we have

φ(hθ + θ)− φ(θ)− φ′(θ)hθ =φ′′(θ)h2θ2

2+ E(θ, h, θ),

where ‖E(θ, h, ·)‖C3([−1,1]) ≤ Ch3 with a constant C independent of θ ∈ [−1, 1].Since φ ∈ S(ǫ), φ′′(θ) ∈ [3/4, 5/4]. Then we see

‖φθ,h − φ‖C3([−1,1]) ≤ (φ′′(θ)h2)−1‖E(θ, h, ·)‖C3([−1,1]) ≤ Ch.

Thus, there exists h ∈ (0, ǫ] such that φθ,h ∈ S(ǫ) for 0 < h ≤ h.

Induction quantity. As mentioned before, our proof is based on an induction onscale argument. To this end, we need to introduce a quantity which makes itpossible to carry out the induction.

For 1 ≤ p ≤ q ≤ ∞, we define Q(λ) = Q(λ, ǫ, p, q, α) by

(2.5)Q(λ) = sup

‖T φλ g‖Lq(R2×I;ω) : supp g ⊂ Aλ([−1, 1]), ‖g‖p ≤ 1,

φ ∈ S(ǫ), ω ∈ Ω(α).

It is easy to see Q(λ) <∞. Indeed, one can show a rough bound

(2.6) Q(λ) . λ2, λ ≥ 1

for 1 ≤ p ≤ q ≤ ∞. We write Tλg(x, t) = Kλ(·, t) ∗ g, where the kernel is given by

Kλ(x, t) =

∫ei(x,t)·(η,ρ,ηφ(ρ/η))β(λ−1η)β0(|ρ/η|)dηdρ.(2.7)

CIRCULAR AVERAGE 7

In order to prove (2.6), we use an easy estimate:

(2.8) |Kλ(x, t)| . K(x) := λ2(1 + |x|

)−M, t ∈ I

for anyM ≥ 1, which follows by routine integration by parts (for example, see (3.6)below). Using (2.1), it is easy to see

∥∥∥∫

K(x − y)g(y) dy∥∥∥Lq(R2×I,ω)

. λ2‖g‖p(2.9)

for 1 ≤ p ≤ q. Indeed, since∫I

∫K(x− y)ω(x, t) dxdt . λ2 and

∫K(x− y)dy . λ2,

(2.9) holds with p = q by Schur’s test. Also, by Holder’s inequality, we get (2.9)for q = ∞ and 1 ≤ p ≤ ∞. Interpolating these two estimates, we have (2.9) andtherefore (2.6) for 1 ≤ p ≤ q.

In what follows, we show Q(λ) ≤ Cǫλ12−ǫ for some ǫ > 0.

Rescaling. We first observe that the Lp–Lq bound on T φλ improves on a certainrange of p, q if g is supported in a narrow conic neighborhood. More precisely, wehave the following.

Lemma 2.3. For 0 < α ≤ 3, let ω ∈ Ω(α) and φ ∈ S(ǫ). Also, let Jh ⊂ [−1, 1] bea subinterval of length 2h ≥ 2λ−1/2. Suppose that supp g ⊂ Aλ(Jh). Then, there is

an h = h(ǫ) such that

(2.10) ‖T φλ g‖Lq(R2×I;ω) ≤ Ch2α−κ(α)

q− 3

pQ(h2λ)‖g‖pfor h ≤ h.

We recall the definition of κ(α) = min3, α+ 1, 2α (see (1.8)).

Proof. We write Jh = [θ − h, θ + h] ⊂ [−1, 1] for some θ ∈ (−1, 1). Since g(η, ρ)is supported in Aλ(Jh), we have λ/4 ≤ η ≤ 4λ, θ = ρ/η ∈ Jh i.e., |ρ/η − θ| ≤ h.We set

Dθ,h =1

φ′′(θ)

h−2 h−2θ h−2φ(θ)0 h−1 h−1φ′(θ)0 0 φ′′(θ)

,

and

Lθ,h(η, ρ) =1

φ′′(θ)(h−2η, h−1ρ+ h−2θη).

Let z = (x, t). Note that z·(η, ρ, ηφ(ρ/η)) → Dθ,h z·(η, ρ, ηφθ,h(ρ/η)) under thetransformation (η, ρ) → Lθ,h(η, ρ). Thus, changing variables (η, ρ) → Lθ,h(η, ρ)we obtain

T φλ g(z) = Tφθ,h

h2λ gθ,h(Dθ,hz),

where gθ,h is given by

gθ,h(η, ρ) = det(Lθ,h)g(Lθ,h(η, ρ)).

Clearly, gθ,h is supported in Ah2λ([−1, 1]). By Lemma 2.2 there exists h = h(ǫ)such that φθ,h ∈ S(ǫ) for h ≤ h.

Let us define

ωθ,h(z) = (1 + C)−1hκ(α)−2α+3ω(D−1θ,h

z),


where C > 0 is a constant to be chosen later. We claim that ωθ,h ∈ Ω(α) if C > 0is large enough. To show this, we first note

∫

B3(y,r)

ω(D−1θ,h

z)dz = h−3 detDθ,1

∫

D−1θ,h

B3(y,r)

ω(z)dz.

Thus, to prove our claim it is sufficient to show∫

D−1θ,h

B3(y,r)

ω(z)dz ≤ Crαh2α−κ(α).(2.11)

We now observe that D−1θ,h

B3(z, r) is contained in a box of dimensions ch2r×chr×cr for a constant c > 0. We cover D−1

θ,hB3(z, r) with as many as Ch−3, Ch−1, and

C balls of radii h2r, hr, and r, respectively. Thus, using (2.1) we see that the lefthand side of (2.11) is bounded above by Crαh2α−min3,α+1,2α for some C. Thus,(2.11) follows.

By change of variables z 7→ D−1θ,hz, we obtain

‖T φλ g‖qLq(R2×I;ω) ≤ (1 + C)h2α−κ(α)

∫

R2×I

∣∣T φθ,h

h2λ gθ,h(z)∣∣qωθ,h(z)dz.(2.12)

Since φθ,h ∈ S(ǫ), ωθ,h ∈ Ω(α), and gθ,h is supported in Ah2λ([−1, 1]), by thedefinition of Q, we have

∫

R2×I

∣∣T φθ,h

h2λ gθ,h(z)∣∣qωθ,h(z)dz ≤

(Q(h2λ)‖gθ,h‖p

)q.

Therefore, we get (2.10) since ‖gθ,h‖p = h−3p ‖g‖p.

2.2. Trilinear estimates. In this section, we obtain a weighted Lp–Lq estimate

for the trilinear operator∏3i=1 T

φλ gi by interpolating the multilinear restriction

estimate due to Bennett, Carbery and Tao [3], the local smoothing estimate in [11],and an easy L1–L∞ estimate.

We also show that the weighted trilinear estimate holds globally by using decay-

ing property of the kernel Kλ. Throughout this subsection, we write Tλ = T φλ forsimplicity.

For φ ∈ S(ǫ) and an interval J ⊂ [−1, 1], we define a truncated conic surface

Γλ(J) = Γφλ(J) by

Γλ(J) := (η, ρ, τ) ∈ R3 : τ = ηφ(ρ/η), (η, ρ) ∈ Aλ(J).For θ = ρ/η, let N(θ) be the normal vector to Γλ at η(1, θ, φ(θ)). By a computationit is easy to see

(2.13) N(θ) = (φ(θ) − θφ′(θ), φ′(θ),−1).

Let J1, J2, J3 be intervals contained in [−1, 1]. We say Γλ(J1), Γλ(J2), Γλ(J3)are δ-transversal if

(2.14) det(N(θ1),N(θ2),N(θ3)) ≥ δ

holds for some δ > 0 whenever θi ∈ Ji, (1 ≤ i ≤ 3).The following is a consequence of scaling and the multilinear restriction esti-

mate due to Bennett-Carbery-Tao [3]. We denote by dσλ the surface measure onΓλ([−1, 1]).

CIRCULAR AVERAGE 9

Theorem 2.4. Let φ ∈ S(ǫ) and J1, J2, J3 ⊂ [−1, 1] be intervals. Suppose that

fi is supported on Γλ(Ji), i = 1, 2, 3, and suppose (2.14) holds whenever θi ∈ Ji,i = 1, 2, 3. If ǫ is sufficiently small and λ≫ δ−1, then for ǫ > 0

∥∥∥3∏

i=1

fidσλ

∥∥∥L1(B3(0,10))

≤ Cǫλǫ

3∏

i=1

‖fi‖L2(Γλ(Ji)).

By interpolation with the local smoothing estimate in [11] and an easy L1 −L∞

estimate, we get the following.

Proposition 2.5. Let φ ∈ S(ǫ) and J1, J2, J3 ⊂ [−1, 1] be intervals. Suppose that

gi is supported in Aλ(Ji), i = 1, 2, 3, and (2.14) holds whenever θi ∈ Ji, i = 1, 2, 3.If ǫ is sufficiently small and λ≫ δ−1, then we have

(2.15)∥∥∥

3∏

i=1

Tλgi

∥∥∥L

q3 (B2(0,1)×I)

≤ Cλ3γ3∏

i=1

‖gi‖p

provided that 1q ≤ min( 1p ,

13p + 1

6 ,23p′ ), 1 ≤ p ≤ ∞, and γ > 1

2 + 1p − 3

q .

Proof. By interpolation it is enough to show that (2.15) holds for ( 1p ,1q ) = (0, 0),

(14 ,14 ), (12 ,

13 ), and (1, 0). For the first two cases, (2.15) follows from the local

smoothing estimate (1.7) with dν = dxdt which holds for 4 ≤ p = q ≤ ∞ andγ > 1

2 − 2p . By Theorem 2.4 and Plancherel’s theorem, we have (2.15) with ( 1p ,

1q ) =

(12 ,13 ). Also, (2.7) and van der Corput’s lemma yield ‖Kλ‖∞ ≤ λ3/2 (e.g. (3.6)),

which gives ‖Tλgi‖L∞(B2(0,1)×I) . λ32 ‖gi‖1. Thus, we have (2.15) with ( 1p ,

1q ) =

(1, 0).

From (2.8), we see that Kλ(·, t) decays rapidly away from B2(0, 1). Using thisproperty, we can extend the local estimate (2.15) to a global one.

Lemma 2.6. Under the same assumptions as in Proposition 2.5 we have

(2.16)∥∥∥

3∏

i=1

Tλgi

∥∥∥L

q3 (R2×I)

≤ Cλ3γ3∏

i=1

‖gi‖p.

Proof. We follow a standard localization argument (e.g., see [21, Proposition 2.10]).For a constant r satisfying λ−1 ≪ r ≪ δ, let Nrλ(Aλ(Ji)) be a rλ-neighborhood ofAλ(Ji), i = 1, 2, 3. Let ϕ be a smooth function supported in Nrλ(Aλ(Ji)) such that

ϕ = 1 on Aλ(Ji) and |∂mξ ϕ(ξ)| . |ξ|−|m|. Then Tλgi(x, t) = Kλ(·, t) ∗ gi(x), whereFx(Kλ(·, t)) = Fx(Kλ(·, t))ϕ. Here we denote by Fx the Fourier transform in x.

We consider a collection B of finitely overlapping unit balls which cover R2.

Let us fix ε > 0 and let B be a ball of radius λε with the same center as B. For eachi, we decompose gi = gi,B + gi,Bc where gi,B := giχB and gi,Bc = giχBc . Then, weget

∥∥∥3∏

i=1

Tλgi

∥∥∥q3

Lq3 (R2×I)

.∑

B

∑

(m1,m2,m3);mj∈B,Bc

∫

I

∫

B

3∏

i=1

∣∣Kλ(·, t) ∗ (gi,mi)(x)

∣∣ q3 dxdt.

Since the Fourier transform of Kλ(·, t) ∗ (gi,B) is supported in Nrλ(Aλ(Ji)), thetransversality condition (2.14) holds for 2−1δ in place of δ by taking a sufficiently


small r. Thus, we may apply Proposition 2.5 to the case of m1 = m2 = m3 = B toget

∑

B

∫

B×I

3∏

i=1

∣∣Kλ(·, t) ∗ (gi,B)(x)∣∣ q3 dxdt . λqγ

∑

B

3∏

i=1

‖giχB‖q3p . λqγ+cε

3∏

i=1

‖gi‖q3p

for any γ > 12 + 1

p − 3q with some c > 0. The second inequality follows by Holder’s

inequality and the inclusion ℓp ⊂ ℓq for p ≤ q since B overlap at most Cλ2ε.

For the other cases i.e., mi = Bc for some i, we use the decay property of Kλ.

As before, it is easy to show |Kλ(x, t)| . λ2(1+ |x|)−M for any integer M ≥ 1 (c.f.,

(2.8)). Setting K(x) = (1 + |x|)−3, we see |Kλ(·, t) ∗ gi,B(x)| . λ2K ∗ |gi|(x) and∣∣Kλ(·, t) ∗ gi,Bc(x)∣∣ . λ2λε(3−M)

K ∗ |gi|(x) if x ∈ B.Thus, if mi = Bc for some i, we have

3∏

i=1

∣∣Kλ(·, t) ∗ (gi,mi)(x)

∣∣ . λc1−c2εM3∏

i=1

K ∗ |gi|(x), (x, t) ∈ B × I,

for some constants c1, c2 > 0. Thus, by Holder’s inequality and Young’s convolutioninequality, there are positive constants c1, c2 such that

∑

B

∑

(m1,m2,m3) 6=(B,B,B)

∫∫

B×I

3∏

i=1

∣∣Kλ(·, t) ∗ (gi,mi)(x)

∣∣ q3 dxdt . λc1−c2εM

3∏

i=1

‖gi‖q3p

for p ≤ q. Combining the estimates for the two cases, we obtain

∥∥∥3∏

i=1

Tλgi

∥∥∥q3

Lq3 (R2×I)

.(λqγ+cε + λc1−c2εM

) 3∏

i=1

‖gi‖q3p

for γ > 12 + 1

p − 3q . Taking ε > 0 small enough and then a large M , we obtain the

desired estimate (2.16).

We now generalize (2.16) to the estimates relative to weights ω ∈ Ω(α) via thefollowing lemma.

Lemma 2.7. Let ω ∈ Ω(α), 0 < α ≤ 3. Suppose F is supported on B3(0, λ). Then,we have ‖F‖Lq(R3;ω) . λ(3−α)/q‖F‖Lq(R3) for q ≥ 1.

Proof. Let ϕ be a Schwartz function such that ϕ = 1 on B3(0, 1), and let ϕλ =

λ3ϕ(λ·). Since F = F ∗ ϕλ, we have ‖F‖Lq(R3;ω) ≤ ‖F‖Lq(R3)‖|ϕλ| ∗ ω‖1/q∞ . Byrapid decay of ϕ, we see

|ϕλ| ∗ ω(y) ≤ CNλ3∑

ℓ≥0

2−Nℓ∫

B3(y,2ℓλ−1)

ω(z)dz

for any N > 3. Since ω ∈ Ω(α), by (2.1) it follows that ‖|ϕλ| ∗ω‖∞ . λ3−α. Thus,we get the desired bound.

Proposition 2.8. Let φ ∈ S(ǫ) and let ω ∈ Ω(α), 0 < α ≤ 3. Suppose (2.16)holds for some γ. Then, if λ−1 ≪ δ ≪ 1, we have

(2.17)∥∥∥

3∏

i=1

Tλgi

∥∥∥L

q3 (R2×I;ω)

. λ3(γ+3−αq

)3∏

i=1

‖gi‖p.

CIRCULAR AVERAGE 11

Proof. Let χ ∈ C∞0 ((1/2, 4)) such that χ = 1 on I = [1, 2]. We only have to show

(2.17) with Tλ replaced by χ Tλ. Since χ is compactly supported, the support ofspace-time Fourier transform of χ(t)Tλgi(x, t) is unbounded. So, in order to applyLemma 2.7, we decompose the frequency space of χ Tλf into a bounded set and itscomplement. Let us define a frequency localized operator T 1

λ by

Fx,t(T 1λg

)(η, ρ, τ) = Fx,t

(χ Tλg

)(η, ρ, τ)β0((cλ)

−1|τ |)

where c > 0 is a constant to be chosen later. We also set T 2λg = (χ Tλ − T 1

λ)g.Then, we have

∥∥∥3∏

i=1

χ Tλgi

∥∥∥L

q3 (R3;ω)

≤ I + II,

where

I =∥∥∥

3∏

i=1

T 1λgi

∥∥∥L

q3 (R3;ω)

,

II =∑

(m1,m2,m3) 6=(1,1,1);mj∈1,2

∥∥∥3∏

i=1

Tmi

λ gi

∥∥∥L

q3 (R3;ω)

.

First, we show II . λ−M∏3i=1 ‖gi‖p for any M ≫ 1. Indeed, we claim

‖T 2λg‖Lq(R3;ω) . λ−M‖g‖p(2.18)

for anyM > 0 if p ≤ q. Once we have (2.18), it is clear that ‖T 1λg‖Lq(R3;ω) . λ2‖g‖p

since |T 1λg| ≤ |χ Tλg|+ |T 2

λg| and ‖χ Tλg‖Lq(R3;ω) . λ2‖g‖p if p ≤ q (see (2.8) and(2.9)). Thus, the desired bound on II follows by Holder’s inequality.

To prove (2.18), let us set

Φ(η, ρ, τ) =1

2π

∫e−it(τ−ηφ(ρ/η))χ(t)dt β(λ−1η)β0(|ρ/η|).

Recalling the definition of Tλ ((2.3)), we haveFx,t(χ Tλg)(η, ρ, τ) = Φ(η, ρ, τ)g(η, ρ).Since T 2

λ = (χ Tλ − T 1λ), we have T 2

λg(x, t) = K(·, t) ∗ g(x), where

K(x, t) =

∫∫∫ (1− β0((cλ)

−1|τ |))Φ(η, ρ, τ)ei(x·(η,ρ)+tτ)dηdρdτ.

By integration by parts one can easily show

(2.19) |K(x, t)| . λ−M (1 + |x|)−M (1 + |t|)−M

for M ≥ 1. by taking a sufficiently large c. So, by integration by parts we alsosee the partial derivatives of Φ are bounded by CMλ

−M (1 + |τ |)−M for any M ≥ 1since all partial derivatives of ηφ(ρ/η) are bounded by a constant depending only onthe order of derivatives. Thus repeated integration by parts in η, ρ, τ gives (2.19).Using (2.19), we get (2.18) with p = q by Schur’s test. Also, (2.18) holds withq = ∞ by Young’s convolution inequality. By interpolation, we get (2.18) for any1 ≤ p ≤ q ≤ ∞.


We now turn to the term I. Since the Fourier transform of T 1λgi is supported in

a ball of radius 2cλ, we can apply Lemma 2.7 to I. Thus, we get

I . λ3(3−α)

q

∥∥∥3∏

i=1

T 1λgi

∥∥∥L

q3 (R3)

.

Since |T 1λg| ≤ |χTλg| + |T 2

λg|, using ‖χ Tλg‖Lq(R3) . λ2‖g‖p and (2.18), we see

that∥∥∏3

i=1 T1λgi

∥∥q/3

is bounded by ‖∏3i=1 χ Tλgi‖q/3 + Cλ−N

∏3i=1 ‖gi‖p for any

N > 0. Now we apply Lemma 2.6 to ‖∏3i=1 χ Tλgi‖q/3 expanding the interval I

slightly. Thus, choosing N large enough, we obtain

I . λ3(γ+3−αq

)3∏

i=1

‖gi‖p.

Combining all the bounds on I and II, we get the desired estimate (2.17) by takingM,N sufficiently large.

The estimate (2.17) plays a crucial role in proving Theorem 1.6. However, inorder to prove Theorem 1.5 (or Proposition 2.1), it suffices to obtain the followingestimate which is not necessarily sharp.

Corollary 2.9. Let φ ∈ S(ǫ) and ω ∈ Ω(α), 1 < α ≤ 2. If λ−1 ≪ δ ≪ 1 and

(2.20)1

q≤ 1

p<α

qand p >

2α+32α , 3

2 < α ≤ 2,6−2αα , 1 < α ≤ 3

2 ,

then for some γ < 12 we have

(2.21)∥∥∥

3∏

i=1

Tλgi

∥∥∥L

q3 (R2×I;ω)

≤ Cλ3γ3∏

i=1

‖gi‖p.

Proof. By Proposition 2.8, we have

(2.22)∥∥∥

3∏

i=1

Tλgi

∥∥∥L

q3 (B2(0,1)×I;ω)

≤ Cλ3γ′

3∏

i=1

‖gi‖p

for γ′ > 12 + 1

p − αq provided that 1

q ≤ min( 1p ,13p + 1

6 ,23p′ ). Thus, the estimate

(2.22) holds for some γ′ < 12 as long as 1

p <αq . Then, by Holder’s inequality, (2.22)

continues to be true in the range of p, q satisfying (2.20). Repeating the argumentin the proof of Lemma 2.6, we can extend (2.22) to the global estimate (2.21) aslong as q ≥ p. We omit the detail.

2.3. Proof of Proposition 2.1. In this subsection, we prove Proposition 2.1 bycombining (2.21) and (2.10). To this end, we decompose the frequency support of

T φλ g independent of particular choice of φ.

Decomposition. For dyadic numbers Ki, i = 0, 1, 2 which satisfy

λ−1/3 ≪ K2 ≪ K1 ≪ K0 = 1,

let Ji denote a collection of dyadic intervals J i ⊂ [−1, 1] of length 2Ki. Let gJ =g χAλ(J). Then, we have

T φλ g(x, t) =∑

Ji∈Ji

T φλ gJi(x, t)

CIRCULAR AVERAGE 13

for each i = 0, 1, 2. Note that gJi is supported in a rectangle of dimensions cλ×cλKi

for a constant c > 0.Following the argument in [22, Section 3], we have the next lemma.

Lemma 2.10. For each (x, t) ∈ R2× I, there exists a constant C > 0, independentof Ki and (x, t), such that

(2.23)

|T φλ g(x, t)| ≤2∑

i=1

CK−2i−1 max

Ji∈Ji|T φλ gJi(x, t)|

+CK−42 max

(J21 ,J

22 ,J

23 )∈J2(K2)

3∏

i=1

|T φλ gJ2i(x, t)| 13 ,

where J2(K2) := (J21 , J

22 , J

23 ) : J

21 , J

22 , J

23 ∈ J2, mink 6=l dist (J2

k , J2l ) ≥ K2.

If (J21 , J

22 , J

23 ) ∈ J2(K2), then the δ-transversality condition (2.14) for θk ∈ Jk,

k = 1, 2, 3. More precisely, we have the following.

Lemma 2.11. Let J1, J2, J3 be subintervals of [−1, 1] such that mink 6=l dist (J2k , J

2l ) &

δ13 . Suppose that φ ∈ S(ǫ). If ǫ ∈ (0, 1/8), then | det

(N(θ1),N(θ2),N(θ3)

)| & δ

for θi ∈ Ji.

Proof. By a generalization of the mean value theorem (see [28, Part V, Problem95]), there are uj for j = 1, 2, 3 such that min1≤i≤3 θi < uj < max1≤i≤3 θi and

det(N(θ1),N(θ2),N(θ3)) = det(N(u1),N′(u2),N

′′(u3))∏

1≤k<ℓ≤3

|θk − θℓ|.

From (2.13) we note

det(N(u1),N′(u2),N

′′(u3)) = det

(u2φ

′′(u2) φ′′(u3) + u3φ(3)(u3)

φ′′(u2) φ(3)(u3)

).

Since φ ∈ S(ǫ) and ǫ ∈ (0, 1/8), we have 7/8 ≤ |φ′′(θ)| ≤ 9/8 and |φ(3)(θ)| ≤ 1/8for θ ∈ [−1, 1], thus it follows that | det(N(u1),N′(u2),N′′(u3))| ≥ 27/82. So weobtain the desired bound since

∏1≤k<ℓ≤3 |θk − θℓ| & δ.

We are now ready to prove Proposition 2.1.

Proof of Proposition 2.1. We begin by making a primary decomposition by which

we reduce the matter to obtaining estimate for T φλ , φ ∈ S(ǫ) relative to ω ∈ Ω(α).Let ǫ ∈ (0, 1/8). Then, following the argument in the proof of Lemma 2.2, for

each θ ∈ [−1, 1] we see that there is an h = h(ǫ) such that (ψ0)θ,h ∈ S(ǫ). Thus,we can fix a small positive dyadic number h∗ such that (ψ0)jh∗,h∗

∈ S(ǫ) as longas θj := jh∗ ∈ [−1, 1].

We set Jj =[θj , θj+h∗

]and decompose [−1, 1] =

⋃ℓ−1j=−ℓ Jj . Denoting gj(η, ρ) =

g(η, ρ)χJj(ρ/η), we get

‖Tψ0

λ g‖Lq(R2×I;ω) ≤ℓ−1∑

j=−ℓ‖Tψ0

λ gj‖Lq(R2×I;ω).

As in (2.12), change of variables (η, ρ) → Lθj,h∗(η, ρ) gives

‖Tψ0

λ gj‖qLq(R2×I;ω) ≤ Ch2α−κ(α)∗

∥∥T (ψ0)θj,h∗

h2∗λ gθj,h∗

∥∥qLq(R2×I;ωθj,h∗

).(2.24)


Then, we note that ‖gθj,h∗‖p = ‖g‖p and ωθj ,h∗

∈ Ω(α). Since h∗ is a fixed constantdepending only on ǫ, in order to prove Proposition 2.1 it suffices to show

(2.25) ‖T φλ g‖Lq(R2×I;ω) ≤ Cλγ‖g‖pfor some γ < 1/2 if φ ∈ S(ǫ) and ω ∈ Ω(α).

We note that (2.21) holds since (p, q) satisfies (2.20). Thus we have the estimate(2.21) with γ = γ0 < 1/2. Since α−1

q − 3p +1 > 0, we can choose γ0 < γ < 1/2 such

that

(2.26)α− 1

q− 3

p+ 2γ > 0.

For λ′ ≥ 1 we set

Q(λ′) = Q(λ′, p, q, α) := sup1≤λ≤λ′

λ−γQ(λ),

whereQ(λ) is defined by (2.5). Then, the estimate (2.25) follows if we showQ(λ′) ≤C.

Let h be a small positive so that Lemma 2.2 and Lemma 2.3 hold. Let K1, K2

be positive dyadic numbers such that 0 < K2 ≪ K1 ≤ h. We also set K0 = 1 asbefore. Using the decomposition (2.23), we have

(2.27) ‖T φλ g‖Lq(R2×I;ω) ≤ S1 + S2 + S3,

where

Si = CK−2i−1

∥∥ maxJi∈Ji

|T φλ gJi |∥∥Lq(R2×I;ω), i = 1, 2

S3 = CK−42

∥∥∥ max(J2

1 ,J22 ,J

23 )∈J2(K2)

3∏

i=1

|T φλ gJ2i| 13∥∥∥Lq(R2×I;ω)

.

By Lemma 2.3 with κ(α) = α+ 1, for i = 1, 2 we have

Sqi ≤ CqK−2qi−1

∑

Ji∈Ji

‖T φλ gJi‖qLq(R2×I;ω) ≤ C′K−2qi−1

(K

α−1q

− 3p

i Q(K2i λ)

)q‖g‖qp

since∑

Ji∈Ji ‖gJi‖qp ≤ ‖g‖qp by the imbedding ℓp ⊂ ℓq for p ≤ q.

By Lemma 2.11, the transversality condition (2.14) with δ = (K2)3 holds if

(J21 , J

22 , J

23 ) ∈ J2(K2). Thus, by the estimate (2.21) we have S3 ≤ CK−C

2 λγ‖g‖pfor λ≫ K−3

2 . On the other hand, if λ . K−32 , then it is easy to see S3 ≤ CK−C

2 ‖g‖pfor some constant C > 0 using the easy estimate (e.g. (2.9)) and Holder’s inequality.So we have

S3 ≤ CK−C2 λγ‖g‖p.

Combining all the estimates for S1, S2, and S3, which hold uniformly for φ ∈S(ǫ) and ω ∈ Ω(α), we obtain

Q(λ) ≤2∑

i=1

CK−2i−1K

α−1q

− 3p

i Q(K2i λ) + CK−C

2 λγ .(2.28)

Note that (K2i λ)

−γQ(K2i λ) ≤ Q(λ) if K2

i λ ≥ 1. Otherwise, i.e., if 1 ≤ λ ≤ K−2i ,

we have Q(K2i λ) ≤ CK−C

i for some C > 0. Thus, we have (K2i λ)

−γQ(K2i λ) ≤

Q(λ)+CK−Ci ≤ Q(λ′)+CK−C

i if λ ≤ λ′. Here, the constant C may differ at eachoccurrence.

CIRCULAR AVERAGE 15

We now multiply λ−γ to both sides of (2.28). Then, using the above observationwe obtain

λ−γQ(λ) ≤2∑

i=1

CK−2i−1K

α−1q

− 3p+2γ

i Q(λ′) + CK−C2

for λ ≤ λ′. Taking supremum over λ ≤ λ′, we get

Q(λ′) ≤2∑

i=1

CK−2i−1K

α−1q

− 3p+2γ

i Q(λ′) + CK−C2 .

Since (2.26) holds, we can successively choose dyadic numbers K1,K2 such that

CK−2i−1K

α−1q

− 3p+2γ

i ≤ 3−1, i = 1, 2. Therefore, we obtain Q(λ′) ≤ 23Q(λ′)+CK−C

2 ,

which clearly gives Q(λ′) ≤ CK−C2 for λ′ ≥ 1. This completes the proof.

2.4. Weighted local smoothing estimate with sharp regularity. By a slightmodification, one can obtain (2.4) with γ > 1

2 +1p − α

q , which implies Theorem 1.6.

We also show that the regularity exponent is sharp.

Proposition 2.12. Let 0 < α ≤ 3 and ω ∈ Ω(α). If 1q ≤ min

(1p ,

13p +

16 ,

23p′

)and

1p +

κ(α)q ≤ 1, then (2.4) holds for γ > 1

2 + 1p − α

q .

Proof. The proof is essentially identical to that of Proposition 2.1, so we shall bebrief. The main difference is that we use Proposition 2.8 instead of Corollary 2.9.For a fixed γ > 1

2 + 1p − α

q , we need to show Q(λ′) ≤ C.

As in the proof of Proposition 2.1, we use the decomposition (2.23). By Lemma2.3 and the inclusion ℓp ⊂ ℓq, we have

∥∥ maxJi∈Ji

|TλgJi |∥∥Lq(R2×I;ω) ≤ CK

2α−κ(α)q

− 3p

i Q(K2i λ)‖g‖p.

By Proposition 2.8 we also have

∥∥∥ max(J2

1 ,J22 ,J

23 )∈J2(K2)

3∏

i=1

∣∣TλgJ2i

∣∣ 13

∥∥∥Lq(R2×I;ω)

≤ CK−C2 λγ‖g‖p

for γ > 12 + 1

p − αq if 1

q ≤ min( 1p ,13p + 1

6 ,23p′ ) and 1 ≤ p ≤ ∞.

By following the same lines of argument as in the proof of Proposition 2.1 weobtain

Q(λ′) ≤ C

2∑

i=1

K−2i−1K

2α−κ(α)q

− 3p+2γ

i Q(λ′) + CK−C2

for λ′ ≥ 1. We take γ sufficiently close to 12 + 1

p − αq such that 2α−κ(α)

q − 3p + 2γ >

1− 1p −

κ(α)q . If 1− 1

p −κ(α)q ≥ 0, 2α−κ(α)

q − 3p +2γ > 0. Thus, we can choose K1,K2

such that CK−2i−1K

2α−κ(α)q

− 3p+2γ

i ≤ 13 for i = 1, 2. Therefore, we see Q(λ′) ≤ C

holds for p, q such that 1q ≤ min( 1p ,

13p +

16 ,

23p′ ) and 1− 1

p −κ(α)q ≥ 0, as desired.


We discuss the sharpness of the regularity exponent in (1.7). We show (1.7)holds only if

γ ≥ 1

2+

1

p− α

q,(2.29)

γ ≥ 3

2p+κ(α)− 2α

2q.(2.30)

The two lower bounds on γ coincide on the line 1p + κ(α)

q = 1. Thus, we see (1.7)

holds for γ > 12 + 1

p − αq only if 1

p + κ(α)q ≤ 1.

Proof of (2.29). For λ≫ 1, let us take f(x) = ϕ(λ(1− |x|)

)for a smooth function

ϕ such that ϕ ≥ 0, ϕ = 1 on [1, 2], and supp ϕ ⊂ [1/2, 4]. Using polar coordinatesand changing variables r → 1− r, we have

f(ξ) =

∫ ∞

0

∫

S1

e−i(1−r)θ·ξdσ(θ)ϕ(λr)(1 − r)dr.

By asymptotic expansion of the Fourier transform of the spherical measure, itfollows that

f(ξ) =∑

j≥0

∑

±a±j |ξ|−

12−je±i|ξ|

∫e∓ir|ξ|ϕ(λr)|1 − r| 12−j dr

for some constant a±j for a sufficiently large |ξ|. Since ϕ(λr) decreases rapidly for

|r| & λ−1, we may write f =∑

± f±, where

f±(ξ) =N∑

j=0

a±j |ξ|−12−je±i|ξ|λ−1ϕ(λ−1|ξ|) +O(|ξ|−N ).

Since supp f is supported in |ξ| ∼ λ, it is easy to see ‖f‖Lpγ. λγλ−1/p. Note

that |eit√−∆f−(x)| & λ

12 and |eit

√−∆f+(x)| . λ−N if |x|, |t− 1| ≤ λ−1. If we take

dν(z) = χB3(0,3)(z)|z|α−3dz, z = (x, t) ∈ R3,

then it is easy to see 〈ν〉α ≤ C for some C. Thus, (1.7) gives that λ12−α

q . λγλ−1p .

So, taking λ→ ∞, we get (2.29).

Proof of (2.30). Also, we can show (2.30) by a Knapp type example. Let us take

f such that f(ξ) = λ−32 ζ(λ−1ξ1, λ

− 12 ξ2) for some Schwartz function ζ supported

on 1 ≤ |ξ| ≤ 2. Then |eit√−∆f(x)| & 1 for (x, t) ∈ Rλ, where

Rλ := (x, t) : |x1 − t| ≤ Cλ−1, |x2| ≤ Cλ−12 , t ∈ I.

Let us take

dν(x, t) = λ3−2α+κ(α)

2 χRλ(x, t) dxdt.

It is easy to show 〈ν〉α . 1. For 0 < r ≤ λ−1, we have ν(B3(z, r)) . λ3−2α+κ(α)

2 r3 ≤rα since r

32 ≤ r

κ(α)2 . The other cases λ−1 < r ≤ λ−

12 and λ−

12 < r can be handled

similarly. Thus, the inequality (1.7) gives λ−2α+κ(α)

2q . λγλ−32p , which implies

(2.30).

CIRCULAR AVERAGE 17

3. Proofs of Theorem 1.3, Theorem 1.5, and Theorem 1.6

We first prove Theorem 1.5 and Theorem 1.6, which we deduce from Proposition2.1 and Proposition 2.12, respectively. Combining Littlewood-Paley decompositionand Theorem 1.5, we prove Theorem 1.3, and then we obtain Corollary 1.4. Wealso discuss results for the spherical average in higher dimensions.

3.1. Proof of Theorem 1.5 and Theorem 1.6. By Littlewood-Paley decompo-

sition, it suffices to consider the operator eit√−∆Pλf with λ ≥ 1. We first observe

that (2.2) is a simple consequence of the estimate (2.4) via change of variables.Let U(x, t) = (x1 − t, x2, t) where x = (x1, x2). Setting ξ = (η, ρ), we note that

|ξ| − η = η√1 + ρ2/η2 − η = ηψ0(ρ/η). Thus,

U(x, t) · (ξ, |ξ|) = (x, t) · (η, ρ, ηψ0(ρ/η)).

By finite decomposition into finite angular sectors and rotation, we may assume

f is supported in a small conic neighborhood of e1. Thus, by change of variables(x, t) → U(x, t) we see

(3.1) ‖eit√−∆Pλf‖Lq(R2×I;ω) ≤ C‖Tψ0

λ f‖Lq(R2×I;ω′),

where ω′(x, t) := ω(U(x, t)) and f(η, ρ) is supported on 4−1λ ≤ η ≤ 4λ and |ρ/η| ≤2. Since U is an invertible linear map, it is clear that ω′ ∈ Ω(α). Therefore, (2.2)holds for the same γ and p, q as in Proposition 2.1.

The following lemma shows that estimates relative to fractal measures can bededuced from weighted estimates if the Fourier transform is compactly supported.See [23, Section 3] for example.

Lemma 3.1. Let 0 < α ≤ 3 and ν ∈ C3(α). Set ϕλ = λ3ϕ(λ·) for a Schwartz

function ϕ such that ϕ = 1 on B3(0, 4) and ϕ = 0 outside B3(0, 8). Then, there is

a constant C1 = C1(ν) > 0 such that C−11 |ϕλ| ∗ ν ∈ Ω(α).

Proof of Lemma 3.1. Let us define νλ := |ϕλ| ∗ ν. By rapid decay of ϕ, we have

νλ(z) ≤ CNλ3∑

ℓ≥0

2−Nℓν(B3(z, 2ℓλ−1)

)

for any N > 0. Thus, it suffices to show∫B3(z,r)

νλ(y)dy ≤ Crα for a constant C.

To show this, we consider two cases r ≤ λ−1 and r > λ−1, separately. If r ≤ λ−1,note that ‖νλ‖∞ ≤ C〈ν〉αλ3−α. Then, we have

∫B3(z,r)

νλ(z)dz . 〈ν〉αλ3−αr3 ≤C′N 〈ν〉αrα. To handle the case r > λ−1, we note

∫

B3(z,r)

ν(B3(z, 2ℓλ−1)

)dz ≤

∫

B3(z,(2ℓ+1)r)

∫χB3(z,2ℓλ−1)(z

′)dz′dν(z).

If r > λ−1, taking a sufficiently large N we get∫

B3(z,r)

νλ(y)dy ≤ CNλ3∑

ℓ≥0

2−Nℓ(2ℓλ−1)3(2ℓ + 1)αrα ≤ C′′N r

α.

Therefore, taking C1 = maxC′N , C

′′N, we get C−1

1 νλ ∈ Ω(α).

We are ready to prove Theorem 1.5 and Theorem 1.6.


Proof of Theorem 1.5 and Theorem 1.6. By Littlewood-Paley decomposition, it isenough to show

(3.2) ‖eit√−∆Pλf‖Lq(R2×I;ν) ≤ Cλγ‖f‖p, λ ≥ 1.

Let ϕ be the function given in Lemma 3.1. Then, we have Fx,t(eit√−∆Pλf) =

Fx,t(eit√−∆Pλf)ϕλ. So, it follows that |eit

√−∆Pλf |q . |eit

√−∆Pλf |q ∗ |ϕλ| by

Holder’s inequality. Thus, we have

(3.3) ‖eit√−∆Pλf‖Lq(R2×I;ν) ≤ C‖eit

√−∆Pλf‖Lq(R2×I;|ϕλ|∗ν).

By Lemma 3.1, C−1|ϕλ| ∗ ν ∈ Ω(α) for some C > 0. Since we are assuming f issupported in a small conic neighborhood of e1, using (3.3) and (3.1), we see thatthe estimate (2.4) implies (3.2) for the same p, q and γ as in Proposition 2.1 andProposition 2.12. Therefore, applications of Proposition 2.1 and Proposition 2.12respectively prove Theorem 1.5 and Theorem 1.6.

Nextly, we show Theorem 1.3 and Corollary 1.4.

3.2. Estimates for the circular average. For later use, we temporarily considerthe average A in Rd, d ≥ 2. Let us set P0 =

∑j≤0 P2j . Then, the kernel K(·, t)

of f → AP0f(·, t) is rapidly decaying, i.e., |K(·, t)| ≤ C(1 + |x|)−N for any N .Indeed, by Schur’s test, one can easily see ‖A0f‖Lq(R2×I;dν) ≤ C‖f‖p for 1 ≤ p ≤ q.Therefore, it is sufficient to consider the contribution from

∑j≥1 AP2jf .

The Fourier transform of the spherical measure dσt on tSd−1 can be expressed

as follows:

(3.4) dσt(ξ) = eit|ξ|a+(tξ) + e−it|ξ|a−(tξ),

where a± are smooth functions satisfying |∂mξ a±(ξ)| ≤ Cγ(1 + |ξ|)− d−12 −|m| for any

m. It is sufficient to consider eit|ξ|a+(tξ) since the contribution from e−it|ξ|a−(tξ)can be handled similarly by reflection t→ −t. We set

Af(x, t) =

∫ei(x·ξ+t|ξ|)(1 + |ξ|2)− d−1

4 f(ξ)dξ

andAjf = AP2jf.

Since t ∈ I, to prove the estimate (1.4) it suffices to show ‖∑j≥1 Ajf‖Lq(Rd×I;dν) .

〈ν〉1/qα ‖f‖p. Thus, the matter is reduced to showing

(3.5)∥∥Ajf

∥∥Lq(Rd×I;dν) . 〈ν〉

1qα2

−ǫj‖f‖p, j ≥ 1

for some ǫ > 0.Let Kj(·, t) denote the kernel of f → Ajf(·, t), it is easy to see

|Kj(x, t)| . 2j(1 + 2j ||x| − t|)−N(3.6)

for any N ≥ 1. (See, for example, [20].) Using the estimate we obtain

‖Ajf‖L∞(Rd×I;dν) . 2j‖f‖1,(3.7)

‖Ajf‖L1(Rd×I;dν) . 〈ν〉αmin2j, 2(d+1−α)j‖f‖1.(3.8)

In fact, (3.7) is clear from (3.6). To show (3.8), we note that Kj is essentiallysupported in O(2−j) neighborhood of the truncated cone with height ∼ 1. Sincethe neighborhood can be covered by O(2dj) balls of radius 2−j , using (2.1) we see

CIRCULAR AVERAGE 19

‖Kj‖L1(dν) . 〈ν〉α2(d+1−α)j. On the other hand, we have |Kj | . 2j(1 + |x|)−Nsince t ∈ I. Therefore, ‖Kj‖L1(dν) . 〈ν〉αmin2j, 2(d+1−α)j. So, we get (3.8) byFubini’s theorem.

Proof of Theorem 1.3. As discussed in the above, we need only to show (3.5) forsome ǫ > 0 if p, q satisfy (1.5). When 1 < α ≤ 2, the desired estimate follows fromTheorem 1.5 since

(3.9) Ajf = 2−j2 eit

√−∆P2j f

for some f satisfying ‖f‖p . ‖f‖p.To obtain the estimate for 2 < α ≤ 3, we use (3.7), (3.8), and

‖Ajf‖Lr(R2×I;dν) . 〈ν〉1rα2

( 3−αr

− 12+ǫ)j‖f‖r, 2 ≤ r ≤ 4, ǫ > 0.(3.10)

To show (3.10), we use the estimate ‖Ajf‖Lr(R2×I) . 2(−12+ǫ)j‖f‖Lr for ǫ > 0

and 2 ≤ r ≤ 4, which follows from interpolation between the estimates for r = 2and r = 4. Indeed, the first is clear from Plancherel’s theorem and (3.9). Thesecond follows by the sharp local smoothing estimate (1.7) and (3.9) combinedwith the localization argument which gives the global estimate. Then, modifyingthe argument in proof of Proposition 2.8, we get the estimate (3.10) relative to themeasure ν.

For the case of 2 < α ≤ 3, we note that P2(α) is the closed quadrangle withvertices (0, 0),

P1 :=( α

α+ 3,

1

α+ 3

), P2 :=

(α− 1

α,1

α

), P3 :=

( 1

4− α,

1

4− α

).

Interpolating the estimates (3.8) and (3.10) with r = 2, we have

‖Ajf‖Lq(R2×I;dν) . 〈ν〉1qα2

( 4−αq

−1)j‖f‖qfor 1 ≤ q ≤ 2. Since ‖Ajf‖L∞(R2×I;dν) ≤ C‖f‖∞, this gives (3.5) for 4 − α <p = q <∞. Therefore, to complete the proof, by interpolation it is enough to show(3.5) for (1/p, 1/q) arbitrarily close to P1 and P2. Interpolating (3.7) and (3.10)yields

‖Ajf‖Lq(R2×I;dν) . 〈ν〉1qα2

( 32p+

3−2α2q − 1

2+ǫ)j‖f‖pfor ǫ > 0 if 1 − 3

q ≤ 1p ≤ 1 − 1

q and p ≤ q. A simple computation shows that

this gives (3.5) for (1/p, 1/q) contained in the closed quadrangle with vertices(1/4, 1/4),P1,P2, and (1/2, 1/2) excluding the closed line segment [P1,P2].

Corollary 1.4 is a direct consequence of Theorem 1.3 and the following lemma.For 0 < k ≤ d+1, let V be a k-dimensional plane in Rd+1. We write (x, t) = (z1, z2)where z1 ∈ V , z2 ∈ V ⊥.

Lemma 3.2. Let V be a k-dimensional plane in Rd+1 for k = 1, . . . , d + 1. Let

µ be an α-dimensional measure on V for some 0 < α ≤ k. If (1.4) holds with a

uniform constant C independent of ν ∈ Cd+1(α), then

∥∥ supz2∈V ⊥

|χBd(0,1)×IAf(z1, z2)|∥∥Lq(V,dµ)

≤ C〈µ〉1qα‖f‖p.

By Lemma 3.2, Lp–Lq(dµ) estimate for f → supt∈I |Af(·, t)| can be deduced

from (1.4) by taking V = Rd × 0 and ν ∈ Cd+1(d).


Proof. By Kolmogorov-Seliverstov-Plessner linearization it suffices to show

(3.11) ‖χBd(0,1)×IAf(z1, z2(·))‖Lq(dµ) ≤ C〈µ〉1qα‖f‖p

for any measurable function z2 : V ⊥ → V with C independent of z2. For anycompactly supported continuous function F , we define a linear functional ℓ onCc(R

d+1) by ℓ(F ) =∫F (z1, z2(z1))dµ(z1). Then, by the Riesz representation

theorem, there is a unique Radon measure ν on Rd+1 such that

ℓ(F ) =

∫F (x, t)dν(x, t) =

∫F (z1, z2(z1))dµ(z1).

Since ν(Bd+1((y1, y2), r)

)≤

∫χBk(y1,r)(z1) dµ(z1) ≤ 〈µ〉αrα, the measure ν belongs

to Cd+1(α) and 〈ν〉α ≤ 〈µ〉α. Hence ‖χBd(0,1)×IAf(z1, z2(·))‖Lq(dµ) = ‖Af‖Lq(dν),and then (3.11) follows by (1.4).

3.3. Estimate for the spherical average. We conclude this section with dis-cussion on results in higher dimensions. As mentioned in Section 1, we can obtainsome estimates for the spherical average using the fractal Strichartz estimates forthe wave equation in Rd+1.

Theorem 3.3 ([8, Theorem 3.1]). Let d ≥ 3, 2 ≤ q ≤ ∞, and 1 < α ≤ d + 1.Suppose ν ∈ Cd+1(α). Then, we have

(3.12) ‖eit√−∆f‖Lq(Rd×I;dν) . 〈ν〉

1qα‖f‖Hs

holds with s > s(α, q, d), where

s(α, q, d) =

maxd2 − α

q ,d+14 + 1−α

2q ,3d+1−2α

8

, 1 < α ≤ d,

maxd2 − α

q ,d+14 + d+1−2α

2q , d+1−α2

, d < α ≤ d+ 1.

In fact, the estimate in [8, Theorem 3.1] is a local one. However, it can be easilyextended to a global estimate (3.12) by the argument in Lemma 2.6. The regularityexponents in Theorem 3.3 are sharp when d = 3 or α ∈ [1, d−1

2 ]∪ [d, d+1] in higherdimensions.

By using (3.7) and (3.8), we obtain an analogue of Theorem 1.3 for the sphericalaverages. To state the result, let us set

pd(α) =

α+1α , 1 < α ≤ d−1

2 ,

d+3+2αd−1+2α ,

d−12 < α ≤ d.

Theorem 3.4. Let d ≥ 3 and 1 < α ≤ d+ 1. Suppose ν ∈ Cd+1(α).

(i) If 1 < α ≤ d, then (1.4) holds for p, q satisfying p > pd(α),

(3.13)q−1 ≤ p−1 < αq−1,

(d+ 1) p−1 < d− 1 + (α − 1) q−1.

(ii) If d < α ≤ d+ 1, then (1.4) holds for p, q satisfying

q−1 ≤ p−1 < αq−1,

(d+ 1)p−1 < d− 1 + (2α− d− 1)q−1,

dp−1 < d− 1 + (α− d)q−1.

CIRCULAR AVERAGE 21

When d < α ≤ d+1, the results are sharp (see Proposition 4.1). We also expectthe result to be sharp when d = 3, but the necessity of the condition p > p3(α) for1 < α ≤ 3 is verified only for α = 2, 3 (see (iv) and (v) in Proposition 4.1). Therecent result due to Harris [13], which improves the exponent s in (3.12) for q = 2and d+1

2 < α ≤ d, d ≥ 4, makes it possible to obtain a slightly better result.

Proof of Theorem 3.4. As in the proof of Theorem 1.3, it suffices to show (3.5) for

some ǫ > 0. Note that Ajf = 2−d−12 jeit

√−∆P2j f for some f such that ‖f‖p . ‖f‖p.

By (3.12), we have

(3.14) ‖Ajf‖Lq(Rd×I;dν) . 〈ν〉1qα2

(s− d−12 )j‖f‖2

for q ≥ 2 and s > s(α, q, d).We first consider the case 1 < α ≤ d. We interpolate (3.14) and (3.7), and then

(3.14) and the trivial estimate ‖Ajf‖L∞(Rd×I;dν) . ‖f‖∞. Consequently, for p, q

satisfying q−1 ≤ p−1 ≤ 1− q−1 we have the estimate

(3.15) ‖Ajf‖Lq(Rd×I;dν) . 〈ν〉1qα2

sj‖f‖pif

s > max1

p− α

q,d+ 1

2p+

1− α

2q− d− 1

2,d+ 3 + 2α

4p− d− 1 + 2α

4

.

Therefore, (3.5) holds for some ǫ > 0 if s < 0 i.e., p, q satisfy (3.13), p > (d + 3 +2α)/(d− 1 + 2α), and p−1 ≤ 1− q−1.

On the other hand, if p, q satisfy 1− q−1 ≤ p−1 ≤ q−1, then we have (3.15) for

s > maxα+ 1

p− α,

d+ 3 + 2α

4p− d− 1 + 2α

4

,

which is a consequence of interpolation between (3.7), (3.8), and (3.14) with q =2 and s > s(α, 2, d) = max

d−α2 , 3d+1−2α

8

. So, (3.5) holds for p, q satisfying

1 − q−1 ≤ p−1 ≤ q−1 and p > pd(α). Combining the results for the cases q−1 ≤p−1 ≤ 1 − q−1 and 1 − q−1 ≤ p−1 ≤ q−1, we have (3.5) for p, q satisfying (3.13)and p > pd(α). This proves the first part of Theorem 3.4 (when 1 < α ≤ d).

We next consider the case d < α ≤ d + 1. We need to show that (3.5) holdsprovided that (1/p, 1/q) is contained in the union of the open quadrangle withvertices (0, 0),

Q1 :=(

(d−1)α(d−1)α+(d+1) ,

d−1(d−1)α+(d+1)

), Q2 :=

(α−1α , 1

α

), Q3 :=

(d−12d−α ,

d−12d−α

),

and the line segment [(0, 0),Q3). Interpolating (3.14), (3.7), and the trivial L∞–L∞(Rd × I; dν) estimate, we obtain (3.15) for p ≤ q if

s > max1

p− α

q,d+ 1

2p+d+ 1− 2α

2q− d− 1

2,α

p− (α− 1)

.

Thus, (3.5) holds if s < 0 i.e., ( p−1, q−1) is contained in the interior of the quad-rangle with vertices (0, 0), Q1, Q2, and (1/2, 1/2).

By interpolation we need only to obtain (3.5) for (1/p, 1/q) ∈ [(1/2, 1/2),Q3).To see this, we interpolate two estimates (3.8) and (3.14) with q = 2 and s >(d+ 1− α)/2 to obtain (3.15) for 1 ≤ p = q ≤ 2 and s > 2d−α

p − (d− 1). Thus, we

see (3.5) holds on the line segment [(1/2, 1/2),Q3), as desired.


Similarly as in Corollary 1.4, Lp − Lq(dµ) estimate for the spherical maximalfunction holds for the range of p, q as in Theorem 3.4.

Corollary 3.5. Let 1 < α ≤ d and µ ∈ Cd(α). If p > pd(α) and (3.13) holds, thenwe have ∥∥ sup

t∈I|Af |

∥∥Lq(dµ)

≤ C〈µ〉1qα‖f‖Lp(Rd).

4. Necessary conditions on p, q for (1.4)

In this section, we show necessary conditions on p, q, α for (1.4) to hold. In fact,we obtain those conditions for the local estimate

(4.1) ‖Af‖Lq(Bd(0,1)×I;dν) ≤ C〈ν〉1qα‖f‖Lp(Rd),

which is clearly necessary for (1.4). We obtain the conditions by taking specific αdimensional measures and particular functions. A modification of these examplesalso gives necessary conditions for the maximal estimate (1.6) when 1 < α ≤ d.

Proposition 4.1. Let d ≥ 2 and 1 < α ≤ d + 1. Suppose (4.1) holds for ν ∈Cd+1(α). Then, we have the following:

(i) p−1 ≤ α q−1,

(ii) (d+ 1) p−1 ≤d− 1 + (α− 1) q−1, 1 < α ≤ d,

d− 1 + (2α− d− 1) q−1, d < α ≤ d+ 1,

(iii) d p−1 ≤d− 1, 1 < α ≤ d,

d− 1 + (α− d) q−1, d < α ≤ d+ 1.

If d ≥ 3 and k < α ≤ k + 1 for an integer k = [1, d − 1], then the following are

necessary for (4.1) to hold:

(iv)

(d+ k)p−1 ≤ d+ k − 2 + (α− k)q−1,

p−1 ≤ d+ k − 1

d+ k + 1,

(v)

(k + 1)p−1 ≤ k + (α − k)q−1,

p−1 ≤ k + 1

k + 2.

Moreover, when d = 2 and 1 < α ≤ 3, (4.1) holds only if

(vi) p ≥ 4− α.

It seems natural to expect that (4.1) holds for ν ∈ Cd+1(α) only if, instead of(iv) and (v),

p−1 ≤d+α−2d+α , d− 2 < α ≤ d,

αα+1 , 1 < α ≤ d− 2.

(4.2)

When α = k+1 (k = 1, . . . , d− 1), (4.2) matches with the second parts of (iv) and(v). In view of the global estimate (1.4) with d = 2, the condition (vi) is redundantsince (1.4) holds only if p ≤ q. In fact, the other necessary conditions combinedwith p ≤ q show that (1.4) with d = 2 fails unless (vi) holds.

CIRCULAR AVERAGE 23

Proof of Proposition 4.1. Let us assume that (4.1) holds for all ν ∈ Cd+1(α)with 1 < α ≤ d + 1. We show necessity of the conditions (i)–(v) by consideringspecific measures contained in Cd+1(α) and functions which show the failure of (4.1)unless the conditions hold.

In what follows 0 < δ ≪ 1 and c is a small positive constant.

Proof of (i). Let Nδ = x : ||x| − 1| < δ and set f = χNδ. Clearly, Af(x, t) &

1 if (x, t) ∈ Bd(0, cδ) × [1, 1 + cδ]. Let dν(x, t) = ψ(x, t)|x|α−d−1dxdt for ψ ∈C∞

0 (Bd+1(0, 3)). Then, it is easy to see 〈ν〉α . 1. Thus, (4.1) implies δαq . δ

1p .

Then, letting δ → 0, we get 1/p ≤ α/q.

Proof of (ii). Let us set

Vδ = (x1, x′, t) ∈ R× Rd−1 × [1, 2] : |x1 − t| ≤ cδ2, |x′| ≤ cδ.Then, we define a measure ν by

dν(x, t) =

δα−d−2χVδ

dxdt, 1 < α ≤ d,

δ2(α−d−1)χVδdxdt, d < α ≤ d+ 1.

Then, it is easy to see 〈ν〉α . 1. Indeed, when 1 < α ≤ d, ν(Bd+1(z, r)) .δα−d−2rd+1 if 0 < r < δ2, ν(Bd+1(z, r)) . δα−drd if δ2 < r < δ, and ν(Bd+1(z, r)) .δα−1r if δ < r. Thus, we see ν(Bd+1(z, r)) . rα for each case. The case d < α ≤d+ 1 can be handled similarly.

We consider f = χUδwhere Uδ = [0, δ2] × [0, δ]d−1. Then, we have Af(x, t) &

δd−1 if (x, t) ∈ Vδ. Note ν(Vδ) ∼ δα−1 for 1 < α ≤ d, and ν(Vδ) ∼ δ2α−d−1 for

d < α ≤ d + 1. Since ‖f‖p = δd+1p , using (4.1), we get δd−1δ

α−1q . δ

d+1p for

1 < α ≤ d, which gives (ii) for 1 < α ≤ d. When d < α ≤ d+ 1, we get (ii) in thesame manner.

Proof of (iii). Let us take f = χBd(0,δ) so that ‖f‖p . δd/p. Then we have Af &

δd−1χSδwhere Sδ = (x, t) : ||x| − t| ≤ cδ, t ∈ [1, 2] for a small c > 0. Let

dν(x, t) =

δ−1χSδ

dxdt, 1 < α ≤ d,

δα−d−1χSδdxdt, d < α ≤ d+ 1.

Then, it is not difficult to see 〈ν〉α . 1. Indeed, for d < α ≤ d+ 1, ν(Bd+1(z, r)) .δα−d−1rd+1 if 0 < r ≤ δ, and ν(Bd+1(z, r)) . δα−drd if δ < r. The case 1 < α ≤ dcan be treated similarly, so we omit the detail. Since |Sδ| ∼ δ, we have ν(Sδ) ∼ 1for 1 < α ≤ d, and ν(Sδ) ∼ δα−d for d < α ≤ d + 1. So, (4.1) implies δd−1 . δd/p

if 1 < α ≤ d, and δd−1+(α−d)/q . δd/p if d < α ≤ d + 1. Letting δ → 0 yields(iii).

Proof of (iv). We prove (iv) by modifying the example used to show (ii). For agiven α ∈ (1, d], let k ∈ [1, d − 1] be an integer such that k < α ≤ k + 1. DenoteUkδ = [0, δ2]k × [0, δ]d−k. Then, we see

(4.3) AχUkδ& δ2(k−1)+d−kχV k

δ,

where

V kδ :=(x′, x′′, t) ∈ Rk × Rd−k × [1, 2] : ||x′| − t| ≤ cδ2, |x′′| ≤ cδ

.


Let us set dν = δα−d−2χV kδdxdt. Then, in the same manner as before it is easy

to show 〈ν〉α . 1. Since ν(V kδ ) ∼ δα−k and ‖χUkδ‖p . δ(d+k)/p, (4.1) and (4.3) yield

δd+k−2δ(α−k)/q . δ(d+k)/p. This gives the first part of (iv).For the second part of (iv), taking f = χUk+1

δinstead of χUk

δ, we note Af &

δd+k−1χV k+1δ

. We consider dν = δk−d−1χV k+1δ

dxdt. Then, similarly as before, one

can easily see 〈ν〉α . 1 and ν(V k+1δ ) & 1. Since ‖χUk+1

δ‖p . δ(d+k+1)/p, (4.1) gives

δd+k−1 . δ(d+k+1)/p. This gives the second part of (iv).

Proof of (v). We show (v) combining the examples which are considered to show(i) and (iii). Let Uk be the δ-neighborhood of the set (0, z) ∈ Rk×Rd−k : |z| = 1.We set f = χUk

, then ‖f‖p . δk+1p . By parametrizing Sd−1, we have

Af(x, t) ≥∫

|y|≤1/2

χUk

(x− t(y,

√1− |y|2)

)dy.

Denoting x = (x′, x′′, xd) ∈ Rk × Rd−k−1 × R, we set

Vk =(x, t) :

∣∣|x′| −√t2 − 1

∣∣ ≤ cδ, |x′′|, |xd| ≤ cδ, t ∈ [3/2, 2].

We write y = (y′, y′′) ∈ Rk × Rd−k−1. Then, we claim

x− t(y′, y′′,√1− |y|2) ∈ Uk

for t ∈ [3/2, 2] and |y′′| ≤ 1/2 if (x, t) ∈ Vk and |x′ − ty′| ≤ cδ. Indeed, since

|t−1x′ − y′| ≤ cδ and ||x′| −√t2 − 1| ≤ cδ, it is clear that ||y′| − t−1

√t2 − 1| < 3cδ.

This gives t|(y′′,√1− |y|2)| = t

√1− |y′|2 = 1 + O(cδ). Thus, for |x′′|, |xd| ≤ cδ

we have

|(x′′, xd)− t(y′′,√1− |y′|2 − |y′′|2)| ∈ (1− 2−1δ, 1 + 2−1δ)

if c is small enough. This proves our claim. Since y′ = x′/t+ O(cδ) and |y′′| . 1,from the claim we see

Af & δkχVk.

We consider dν = δα−d−1χVkdxdt. Then, we have ν(Vk) & δα−k and 〈ν〉α .

1. For the latter, we note that ν(Bd+1(z, r)) . δα−d−1rd+1 if 0 < r ≤ δ and

ν(Bd+1(z, r)) . δα−krk if δ < r. So (4.1) gives δkδα−k

q . δk+1p . Hence the first part

of (v) follows if we take δ → 0.To prove the second part of (v), by taking f = χUk+1

similarly as in the above,

we have ‖f‖p . δk+2p , and AχUk+1

& δk+1 whenever (x, t) ∈ Vk+1. Taking dν =

δk−dχVk+1dxdt, we have 〈ν〉α . 1 and ν(Vk+1) & 1 since ν(Bd+1(z, r)) . δk−drd+1

for 0 < r ≤ δ, and ν(Bd+1(z, r)) . δrk for δ < r. Therefore, (4.1) implies δk+1 .

δk+2p for 0 < δ ≪ 1. This gives the second part of (v).

Proof of (vi). Let us consider the case 1 < α ≤ 2 first. Let θℓ be a collection ofCδ1−α points on S1 which are separated by δα−1. For each θℓ, we set

Tℓ(δ) =(x, t) ∈ R2 × [1, 2] : ||x| − t| ≤ cδ2,

∣∣x/|x| − θℓ∣∣ ≤ cδ

.

Let Tδ = ∪ℓTℓ(δ). We consider

dν(x, t) = δα−4χTδ(x, t) dxdt,

then 〈ν〉α . 1. In fact, one can easily see ν(B3(z, r)) ≤ Cδα−4r3 if 0 < r ≤ δ2,ν(B3(z, r)) ≤ Cδα−2r2 if δ2 < r ≤ δ, ν(B3(z, r)) ≤ Cδα−1r if δ < r ≤ δ1−α,

CIRCULAR AVERAGE 25

ν(B3(z, r)) ≤ Cδα−1r ×(r/δα−1

)if δα−1 < r < 1, and ν(B3(z, r)) ≤ Cδα−1 ×(

r/δα−1)if 1 < r. Therefore, we have ν(B3(z, r)) . rα. Let R(θ, δ) be a rectangle

of side length δ2 × δ centered at the origin such that the short side is parallel toθ ∈ S1. Setting Rδ := ∪ℓR(θℓ, δ), we have AχRδ

& δχTδ. Since ‖χRδ

‖p . δ(4−α)/p

and ν(Tδ) & 1, we get δ . δ(4−α)/p which implies p ≥ 4− α.Now, we consider the case 2 < α ≤ 3. Let ωj ⊂ R2 be a collection of Cδ2−α

points on the x1-axis which are separated by δα−2. For each ωj , we denote

Cj(δ) = (x, t) ∈ R2 × [1, 2] : ||x− ωj| − t| ≤ cδ, |x2| ≤ 2−3.We set Tδ = ∪jCj(δ) and

dν(x, t) = δα−3χTδ(x, t) dxdt.

Then, it is easy to see that 〈ν〉α . 1 and ν(Tδ) ∼ 1. Indeed, ν(B3(z, r)) .δα−3r3 if 0 < r ≤ δ, ν(B3(z, r)) . δα−2r2 if δ < r < δα−2, ν(B3(z, r)) .

δα−3(r/δα−2)(δr2) = r3 if δα−2 < r < 1, and ν(B3(z, r)) . δα−3(r/δα−2)δ = rif 1 < r. So, it follows that ν(B3(z, r)) . rα. It is easy to see

Aχ∪jB2(ωj ,cδ)(x, t) & δ, (x, t) ∈ Tδ.

Since | ∪j B2(ωj , cδ)| . δ4−α, (4.1) implies δ . δ4−αp . Thus, we see p ≥ 4− α.

Acknowledgement. This work was supported by the NRF (Republic of Ko-rea) grant No. 2017R1C1B2002959 (Ham), 2019R1A6A3A01092525 (Ko), and2021R1A2B5B02001786 (Lee).

References

[1] T. Anderson, K. Hughes, J. Roos and A. Seeger, Lp → Lq bounds for spherical maximal

operators, Math. Z. 297 (2021), 1057–1074.[2] D. Beltran, R. Oberlin, L. Roncal, A. Seeger and B. Stovall, Variation bounds for spherical

averages, arXiv:2009.07366.[3] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures,Acta Math. 196 (2006), 261–302.

[4] A. S. Besicovitch and R. Rado, A plane set of measure zero containing circumferences of every

radius, J. London Math. Soc., 43 (1968), 717–719.[5] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse.Math. 47 (1986), 69–85.

[6] J. Bourgain and C. Demeter, The proof of the l2 decoupling conjecture, Ann. of Math. 182(2015), 351–389.

[7] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear esti-

mates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295.[8] C.-H. Cho, S. Ham, and S. Lee, Fractal Strichartz estimate for the wave equation, NonlinearAnal. 150, (2017), 61–75.

[9] G. Garrigos and A. Seeger, On plate decompositions of cone multipliers, Proc. Edinb. Math.Soc. 52 (2009), 631–651.

[10] G. Garrigos, A. Seeger, and W. Schlag, A mixed norm variant of Wolff’s inequality for

paraboloids, Harmonic analysis and partial differential equations, Contemp. Math., vol. 505,Amer. Math. Soc., Providence, RI, 2010, pp. 179–197.

[11] L. Guth, H. Wang, and R. Zhang, A sharp square function estimate for the cone in R3, Ann.of Math. (2) 192 (2020), no. 2, 551–581.

[12] S. Ham, H. Ko, and S. Lee, Dimension of divergence set of the wave equation,arXiv:2102.12701.

[13] T. L. J. Harris, Improved decay of conical averages of the Fourier transform, Proc. Amer.Math. Soc. 147 (2019), 4781–4796.

[14] Y. Heo, F. Nazarov, and A. Seeger, Radial Fourier multipliers in high dimensions, ActaMath. 206 (2011), 55–92.

http://arxiv.org/abs/2009.07366



[15] A. Iosevich, B. Krause, E. Sawyer, K. Taylor, and I. Uriarte-Tuero, Maximal operators:

scales, curvature and the fractal dimension, Analysis Math., 45 (1) (2019), 63–86.[16] J. R. Kinney, A thin set of circles, Amer. Math. Monthly, 75 (1968), 1077–1081.[17] L. Kolasa and T. Wolff, On some variants of the Kakeya problem, Pacific J. Math. 190

(1999), 111–154.[18] I. Laba and T. Wolff, A local smoothing in higher dimensions, J. Anal. Math. 88 (2002),149–171.

[19] J. Lee, A trilinear approach to square function and local smoothing estimates for the wave

operator, Indiana Univ. Math. J. 69 (2020), no. 6, 2005–2033.[20] S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc. 131(2003), 1433–1442.

[21] , Square function estimates for the Bochner-Riesz means, Anal. PDE, 11 (2018),1535–1586.

[22] S. Lee and A. Vargas, On the cone multiplier in R3, J. Funct. Anal. 263 (2012), 925–940.[23] P. Mattila, Fourier analysis and Hausdorff dimension, Cambridge University Press, Cam-bridge, United Kingdom (2015).

[24] T. Mitsis, On a problem related to sphere and circle packing, J. London Math. Soc. 60 (1999),501–516.

[25] G. Mockenhaupt, A. Seeger, and C. D. Sogge, Wave front sets, local smoothing and Bourgain’s

circular maximal theorem, Ann. of Math. 136 (1992), 207–218.[26] D. Oberlin, Packing spheres and fractal Strichartz estimates in Rd for d ≥ 3, Proc. Amer.Math. Soc., 134 (2006), 3201–3209.

[27] D. Oberlin and R. Oberlin, Spherical means and pinned distance sets, Commun. KoreanMath. Soc. 30 (2015), 23–34.

[28] G. Polya and G. Szego, Problems and theorems in analysis, Die Grundlehren der mathema-tischen Wissenschaften, Band 216, Springer-Verlag, New York-Heidelberg, 1976.

[29] J. Roos and A. Seeger, Spherical maximal functions and fractal dimensions of dilation sets,arXiv:2004.00984.

[30] W. Schlag, Lp → Lq estimates for the circular maximal function, Ph.D. Thesis. CaliforniaInstitute of Technology, 1996.

[31] W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc.10 (1997), 103–122.

[32] W. Schlag and C. D. Sogge, Local smoothing estimates related to the circular maximal theo-

rem, Math. Res. Let. 4 (1997), 1–15.[33] C. D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math.104 (1991), 349–376.

[34] E. M. Stein, Maximal functions: spherical means, Proc. Nat. Acad. Sci. USA 73 (1976),2174–2175.

[35] T. Tao and A. Vargas, A bilinear approach to cone multipliers II: Application, Geom. Funct.Anal. 10 (2000), 216–258.

[36] T. Wolff, A Kakeya type problem for circles, Amer. J. Math. 119 (1997), 985–1026.[37] , Local smoothing estimates on Lp for large p, Geom. Funct. Anal. 10 (2000), 1237–1288.

[38] J. Zahl, L3 estimates for an algebraic variable coefficient Wolff circular maximal function,Rev. Mat. Iberoam. 28 (2012), 1061–1090.

[39] , On the Wolff circular maximal function, Illinois J. Math. 56 (2012), 1281–1295.

Department of Mathematical Sciences and RIM, Seoul National University, Seoul

08826, Republic of Korea

Email address: [email protected]




arXiv:2110.11185v1 [math.CA] 21 Oct 2021

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