Dimension of divergence set of the wave equation
aa r X i v : . [ m a t h . A P ] F e b DIMENSION OF DIVERGENCE SET OFTHE WAVE EQUATION
SEHEON HAM, HYERIM KO, AND SANGHYUK LEE
Abstract.
We consider the Hausdorff dimension of the divergence set onwhich the pointwise convergence lim t → e it √ − ∆ f ( x ) = f ( x ) fails when f ∈ H s ( R d ). We especially prove the conjecture raised by Barcel´o, Bennett, Car-bery and Rogers [1] for d = 3, and improve the previous results in higher di-mensions d ≥
4. We also show that a Strichartz type estimate for f → e it √ − ∆ f with the measure dt dµ ( x ) is essentially equivalent to the estimate for the spher-ical average of b µ which has been extensively studied for the Falconer distanceset problem. The equivalence provides shortcuts to the recent results due toLiu [11] and Rogers [18]. introduction Let d ≥
1. We consider the wave operator e it √ − ∆ f ( x ) = 1(2 π ) d Z R d e i ( x · ξ + t | ξ | ) b f ( ξ ) dξ. In this note we are mainly concerned with the pointwise behavior of e it √ − ∆ f as t →
0, from which we can deduce the pointwise convergence to the initial data ofthe solution u to the Cauchy problem: ∂ t u − ∆ u = 0 , u ( · ,
0) = u , ∂ t u ( · ,
0) = u , where ( u , u ) ∈ H s ( R d ) × H s − ( R d ). Here, H s ( R d ) denotes the inhomogeneousSobolev space of order s which is equipped with the norm k f k H s = k (1+ |·| ) s b f k L .It is well-known that e it √ − ∆ f converges to f almost everywhere as t → f ∈ H s ( R d ) if s > / s ≤ . Theconvergence follows from the maximal estimate(1.1) k sup
2, which is an easy consequence of the Sobolev imbedding and Plancherel’stheorem. (See for exmaple [3].) The estimate fails for s ≤ / s ≤ /
2. This can be shown using Stein’smaximal theorem [19], by which almost everywhere convergence for all f ∈ H s implies H s – L , ∞ bound on the maximal operator sup Key words and phrases. divergence set, fractal Strichartz estimate, spherical average. hand, it is not difficult to see the H / – L , ∞ estimate fails by a construction (seeLemma A.2 in Appendix).We study a more refined pointwise behavior of e it √ − ∆ f by considering the Hausdorffdimension of the divergence set D d ( f ) := { x : e it √ − ∆ f ( x ) f ( x ) as t → } . Dimension of D d ( f ) naturally relates to the regularity exponent s for which f ∈ H s .Almost everywhere convergence only tells the divergence set is of Lebesgue measurezero. However, if the initial datum f possesses an additional regularity, one mayexpect that the set D d has smaller dimension. A natural question is how big the(Hausdorff) dimension of D d can be depending the regularity of f .To study the question, let us set D d ( s ) := sup f ∈ H s ( R d ) dim H D d ( f ) . The problem of determining D d ( s ) was first considered by Barcel´o, Bennett, Car-bery and Rogers [1] for the more general dispersive equations and the result waslater extended by Luc`a and Rogers [13]. Since | e it √ − ∆ f | . k f k H s for s > d/ 2, itfollows D d ( s ) = 0 for s > d/ 2. On the other hand, D d ( s ) = d if s ≤ / H / – L , ∞ bound (1.1) generally fails. Thus it is sufficient to consider1 / < s ≤ d/ . So, there is nothing interesting if d = 1. For d ≥ 2, the problem is no longer trivial.When d = 2 Barcel´o et. al. [1] obtained the optimal result: D ( s ) = 2 − s for3 / < s ≤ D ( s ) = 4 − s for 1 / < s ≤ / 4. In higher dimensions thefollowing was conjectured to be true ([1, Theorem 5.1]). Conjecture 1.1. For d ≥ , D d ( s ) = ( d − s, ≤ s ≤ d ,d + 2 − s, < s ≤ . (1.2)The lower bound was verified by considering specific functions and measures whichshow the sharpness of maximal estimate against the measures. As is typical withthe problems of similar nature, proving the upper bound is more difficult.Let B d ( x, ρ ) ⊂ R d denote the closed ball centered at x of radius ρ . For α ∈ (0 , d ],we denote by M d ( α ) the collection of non-negative Borel measures µ supported in B d (0 , 1) such that µ ( B d ( x, ρ )) ≤ C µ ρ α for all x ∈ R d and ρ > C µ . If µ ∈ M d ( α ), we define h µ i α = sup x ∈ R d , ρ> ρ − α µ ( B d ( x, ρ )) . An approach to obtain the upper bound on D d ( s ) is to exploit the decay order β = β ( α ) of the estimate for L average of b µ over the sphere :(1.3) Z S d − | b µ ( λθ ) | dθ . λ − β k µ kh µ i α , µ ∈ M d ( α ) , where k µ k is the total variation of µ . IVERGENCE SET OF THE WAVE EQUATION 3 The estimate (1.3) was previously studied by various authors ([21, 7, 8]) being re-lated to the Falconer distance set problem, and further improvements were recentlyobtained ([13, 6]). These improvements rely on sophisticated argument which hasbeen developed in the study of Fourier restriction problem.An upper bound on D d ( s ) can be obtained using (1.3). In fact, it was shown in[1] that D d ( s ) ≤ α if (1.3) holds with β > d − s . Thus, making use of the bestestimate so far (see [14, 6]), we have, for d ≥ D d ( s ) ≤ ( d − s, d +14 < s ≤ d , d − dsd − , < s ≤ d +14 . (1.4)The bound in the case of d +14 < s ≤ d coincides with the sharp upper bound of β = α for 0 < α ≤ d − (see [14]). However, when < s ≤ d +14 and d ≥ 4, thecurrent approach based on (1.3) does not seem to be efficient enough to prove theoptimal upper bound for D d ( s ). In fact, considering the known upper bound on β (for example see [4]) for which (1.3) holds, the conjectured optimal upper bound D d ( s ) = d + 2 − s which corresponds to the estimate (1.3) with β = d + α − cannot be achieved by the aforementioned implication from (1.3).In this paper we take an alternative approach which relies on the fractal Strichartzestimate with respect to a measure (see (1.7) below), which was previously studiedby some authors (see [22, 7, 2, 10, 18]). Via the approach we prove the conjecture(1.2) when d = 3 and improve the previously known results (see (1.4)) for higherdimensions d ≥ < s ≤ d +14 . Theorem 1.2. Let d ≥ . Then D d ( s ) ≤ d − s, d +14 ≤ s ≤ d , d +12 − s, d ≤ s ≤ d +14 , d − d − d − − d − sd − , < s ≤ d . In Figure 1 we compare the result in Theorem 1.2, the conjectured optimal bound(1.2) and the previously known result (1.4).1.1. Maximal estimates with respect to general measures. To examine thesize of the divergence set, we consider the maximal estimate(1.5) k sup 2] let us define α ∗ ( s ) to be the infimum of α such that the estimate(1.5) holds. Then, by a standard argument (see, for example, [1, Appendix B] or[15, Section 17]), it follows D d ( s ) ≤ α ∗ ( s ) . Thus Theorem 1.2 is a consequence ofthe following. SEHEON HAM, HYERIM KO, AND SANGHYUK LEE D d ( s ) s d d +14 d d − d d +12 d − d Previous results (1.4)Theorem 1.2Conjecture 1.1 Figure 1. Dimension of the divergence set, d ≥ Theorem 1.3. Let d ≥ and µ ∈ M d ( α ) . Then we have (1.5) if s > s ( α, d ) := d − α , < α ≤ d − , d +18 − α , d − < α ≤ d +12 , d − α + α − d − , d +12 < α ≤ d. (1.6)Especially s ( α, 3) = − α for α ∈ (1 , α -dimensional measure in R d can be deduced from afamily of estimates with respect to α -dimensional measures in R d +1 . This can beshown by the Kolmogorov-Seliverstov-Plessner linearization argument. Thus theSobolev exponent s in (1.5) is closely related to the exponent γ = γ ( α ) for whichthe fractal Strichartz estimate (1.7) k e it √− ∆ f k L ( dν ) . h ν i α k f k H γ ( R d ) holds for any ν ∈ M d +1 ( α ). For d ≥ 2, this type of estimates was studied inconnection with geometric measure theory (e.g., [22, 7, 16, 17, 2, 10, 18]). Inparticular, the sharp exponent γ was established by Wolff [22], Erdogan [7] when d = 2 and by Cho and two of the authors [2] when d = 3. We discuss the estimatefurther in Section 2 before the proof of Theorem 1.2. IVERGENCE SET OF THE WAVE EQUATION 5 L estimate for e it √− ∆ with a product measure. The average decayestimate (1.3) has been of interest in relation to not only the Falconer distance setproblem but also its refined variant: the pinned distance set problem. We refer thereader to [6, 11, 9, 12, 5] for the most recent progresses related to the distance setproblems.By the argument due to Mattila [14], it can be shown that if E is a Borel set, then(1.8) dim E > α ⇒ |{| x − y | : x, y ∈ E }| > α + β > d .On the other hand, Rogers [18] proposed an approach to Falconer’s distance setproblem which is based on, instead of (1.3), a Strichartz type estimate for e it √ − ∆ f with respect to fractal measures. Using the Riesz representation theorem, he showedthat (1.8) holds for d − < α < d + 1 if the estimate(1.9) k e it √ − ∆ P λ f k L ( R d × I ; dµdt ) . λ d − − γ k f k L ( R d ) , µ ∈ M d ( α )holds true for γ > d − α . Here, I = (1 , 2) and P λ is the standard Littlewood-Paley projection operator which is given by d P λ f ( ξ ) = ψ ( | ξ | /λ ) b f ( ξ ) where ψ ∈ C ∞ c ((2 − , Theorem 1.4. Let λ ≫ and µ ∈ M d ( α ) for < α ≤ d . Then (1.9) holds for γ < β if the estimate (1.3) holds. Conversely, we have the estimate (1.3) with β = γ if (1.9) holds. Recently, Liu [11] deduced the similar implication as (1.8) for the pinned distanceset: there exists x ∈ E such that D x ( E ) = {| x − y | : y ∈ E } has positive Lebesguemeasure if the estimate (1.3) holds and α + β > d . The estimate(1.10) k f ∗ σ t k L ( B d (0 , × I ; dµdt ) . k f k H − β ǫ played a crucial role in Liu’s argument where σ t is the normalized surface measureon the sphere { x : | x | = t } and I = (1 , L identity, he showed that(1.3) implies (1.10) for any ǫ > 0. The estimate (1.10) is an easy consequence of(1.9) since c dσ t = c + e it | ξ | a + ( tξ ) + c − e − it | ξ | a − ( tξ ) with a ± ( tξ ) = O ( | tξ | − d − ) when | ξ | ≥ 1. Using Theorem 1.4 and the argument due to Mattila [14], one can deducethe results in Liu [11] and Rogers [18].2. Proof of Theorem 1.3 and Theorem 1.4 To prove Theorem 1.3, we use the fractal Strichartz estimate (1.7). Concerningthe Lebesgue measure the estimate (1.7) is well understood but for the generalmeasure ν the sharp regularity is not known when d ≥ 4. When d = 2, Wolff [22]obtained the estimate (1.7) with α < α ≥ d ≥ d = 3, the sharp estimatewas proved by using the sharp bilinear restriction estimate for the cone. Recently, SEHEON HAM, HYERIM KO, AND SANGHYUK LEE Harris [10] improved the bound when d ≥ d +12 < α ≤ d . These results canbe summarized as follows. Theorem 2.1 ([2, 10]) . Let d ≥ . For < α < d + 1 , let ν ∈ M d +1 ( α ) . Then (1.7) holds with γ > γ ( α, d ) where γ ( α, d ) = d − α , < α ≤ d − , d +18 − α , d − < α ≤ d +12 , d − α + α − d − , d +12 < α ≤ d, d +1 − α , d < α ≤ d + 1 . (2.1)Now we prove Theorem 1.3 making use of Theorem 2.1. Proof of Theorem 1.3. In order to prove Theorem 1.3, by Kolmogorov-Seliverstov-Plessner linearization argument, it suffices to show that(2.2) k e i t ( · ) √− ∆ f k L ( dµ ) ≤ C h µ i α k f k H s ( R d ) for any measurable function t : B d (0 , → (0 , 1) with C independent of t .Let us define a linear functional ℓ by ℓ ( F ) = Z F ( x, t ( x )) dµ ( x )for any continuous function F in C ( R d +1 ). By the Riesz representation theoremit follows that there is a unique Radon measure ν on R d +1 such that ℓ ( F ) = Z F ( y, s ) dν ( y, s ) = Z F ( x, t ( x )) dµ ( x ) . Clearly ν belongs to M d +1 ( α ) and h ν i α ≤ h µ i α . Since k e i t ( · ) √− ∆ f k L ( dµ ) = k e it √ − ∆ f k L ( dν ) , applying Theorem 2.1, we see that (1.5) holds if (1.6) is satis-fied. (cid:3) We now proceed to prove Theorem 1.4. The relation of (1.3) and (1.9) is basicallydue to the following lemma. Lemma 2.2. Let q ≥ and λ ≫ . For µ ∈ M d ( α ) , suppose that (2.3) (cid:16) Z | b F | q dµ (cid:17) q . λ s k F k whenever supp F ⊂ λ S d − + O (1) . Then we have (2.4) (cid:16) Z Z | b G | q dµdt (cid:17) q . λ s + − q k G k whenever supp G ⊂ λ Γ d + O (1) . Here Γ d := { ( ξ, | ξ | ) ∈ R d +1 : 2 − ≤ | ξ | ≤ } . Inparticular, when q = 2 , the estimates (2.3) and (2.4) are equivalent. It is clear that | ( x − y, t ( x ) − s ) | ≤ r implies | x − y | ≤ r and ν (cid:0) B d +1 (( y, s ) , r ) (cid:1) = Z χ B d +1 (( y,s ) ,r ) (cid:0) x, t ( x ) (cid:1) dµ ( x ) ≤ Z χ B d ( y,r ) ( x ) dµ ( x ) ≤ h µ i α r α . IVERGENCE SET OF THE WAVE EQUATION 7 Proof. Assume that (2.3) holds. Let ψ be a smooth function such that b ψ & , 2] and ψ is supported in [ − , Z Z (cid:12)(cid:12) b G ( x, t ) (cid:12)(cid:12) q dµ ( x ) dt . Z Z (cid:12)(cid:12) b G ( x, t ) b ψ ( t ) (cid:12)(cid:12) q dtdµ ( x ) . We consider the inverse Fourier transform in t and observe that (cid:0) b G ( x, · ) b ψ ( · ) (cid:1) ∨ ( τ ) = F ξ G ( x, · ) ∗ ψ ( τ ). Here F ξ denotes the Fourier transform in ξ , so that b G ( x, t ) = R F ξ G ( x, τ ) e itτ dτ . For each x , it is easy to see that F ξ G ( x, · ) ∗ ψ is supported in[ λ/c, cλ ] for some c > 1. Thus, by Bernstein’s inequality, we obtain(2.6) Z Z (cid:12)(cid:12) b G ( x, t ) b ψ ( t ) (cid:12)(cid:12) q dtdµ ( x ) . λ ( − q ) q Z (cid:16) Z (cid:12)(cid:12) b G ( x, t ) b ψ ( t ) (cid:12)(cid:12) dt (cid:17) q dµ ( x ) . By Plancherel’s theorem in t and Minkowski’s inequality, we have(2.7) Z (cid:16)Z (cid:12)(cid:12) b G ( x, t ) b ψ ( t ) (cid:12)(cid:12) dt (cid:17) q dµ ( x ) = Z (cid:16)Z |F ξ G ( x, · ) ∗ ψ ( τ ) | dτ (cid:17) q dµ ( x ) . (cid:16)Z (cid:16)Z (cid:12)(cid:12) F ξ G ( x, · ) ∗ ψ ( τ ) (cid:12)(cid:12) q dµ ( x ) (cid:17) q dτ (cid:17) q . Note that F ξ G ( x, · ) ∗ ψ ( τ ) = F ξ (cid:16) Z G ( ξ, ρ ) ψ ( τ − ρ ) dρ (cid:17) . Since ψ is supported in [ − , 1] and supp G ⊂ λ Γ d + O (1), we see that for each fixed τ , R G ( ξ, ρ ) ψ ( τ − ρ ) dρ is supported in the set || ξ | − τ | = O (1) and τ ∼ λ . Thus, by(2.3) it follows that, for each τ , Z (cid:12)(cid:12) F ξ G ( x, · ) ∗ ψ ( τ ) (cid:12)(cid:12) q dµ . λ sq (cid:13)(cid:13)(cid:13) Z G ( ξ, ρ ) ψ ( τ − ρ ) dρ (cid:13)(cid:13)(cid:13) qL ξ . Combining this with (2.5), (2.6) and (2.7), by Minkowski’s and Young’s convolutioninequality we have (cid:16) Z Z (cid:12)(cid:12) b G (cid:12)(cid:12) q dµdt (cid:17) q . λ s + − q (cid:16) Z (cid:13)(cid:13)(cid:13) Z G ( ξ, ρ ) ψ ( τ − ρ ) dρ (cid:13)(cid:13)(cid:13) L ξ dτ (cid:17) . λ s + − q k G k . Therefore we get (2.4).Now let q = 2. Let F be a function with supp F ⊂ λ S d − + O (1) and consider G ( ξ, τ ) = F ( ξ ) χ [ λ,λ +1] ( τ ) . Then, since R | \ χ [ λ,λ +1] ( t ) | dt & λ , by Fubini’s theorem it is clear Z (cid:12)(cid:12) b F (cid:12)(cid:12) dµ . Z Z (cid:12)(cid:12) b G (cid:12)(cid:12) dµdt. By (2.4) with q = 2 and Plancherel’s theorem, it follows that Z (cid:12)(cid:12) b F (cid:12)(cid:12) dµ . λ s k G k = λ s k F k . Hence (2.3) and (2.4) are equivalent when q = 2. (cid:3) Using this lemma, we prove Theorem 1.4. SEHEON HAM, HYERIM KO, AND SANGHYUK LEE Proof of Theorem 1.4. Since µ has compact support, by duality and the uncertaintyprinciple, one can easily see that (1.3) is equivalent to(2.8) (cid:0) Z | b F | dµ (cid:1) . λ d − − β k µ kh µ i α k F k , when supp F ⊂ λ S d − + O (1) and µ ∈ M d ( α ). By the argument in [1] the estimatecan be strengthened to(2.9) k b F k L , ∞ ( dν ) . λ ( d − − β ) / h ν i / α k F k for ν ∈ M d ( α ). To see this, we consider a measure dν E = ν ( E ) − χ E dν for ν ∈ M d ( α ) and a Borel set E . Then ν E ∈ M d ( α ), k ν E k ≤ 1, and h ν E i α ≤ ν ( E ) − h ν i α .Applying (2.8) with ν E , we see that ν ( E ) − / R E | b F | dν = ν ( E ) / R E | b F | dν E . λ d − − β h ν i α k F k . So, we get (2.9) by taking E = { x : | b F | > ω } .Interpolating (2.9) with the trivial estimate k b F k L ∞ ( dν ) . λ d − k F k we obtain L → L q ( dν ) estimate. Since supp ν ⊂ B d (0 , q arbitrarily close to 2, weget Z (cid:12)(cid:12) b F (cid:12)(cid:12) dν . λ d − − γ h ν i α k F k (2.10)for γ < β . By Lemma 2.2, we can see that (2.10) is equivalent to Z Z (cid:12)(cid:12) b G (cid:12)(cid:12) dνdt . λ d − − γ h ν i α k G k , (2.11)where supp G ⊂ λ Γ d + O (1). Let ϕ ∈ S ( R d ) such that ϕ ≥ B d (0 , 1) andsupp b ϕ ⊂ B d (0 , ϕe it √ − ∆ P λ f is sup-ported in λ Γ d + O (1). By (2.11) and Plancherel’s theorem we get Z Z (cid:12)(cid:12)(cid:12) ϕ ( x ) e it √ − ∆ P λ f ( x ) (cid:12)(cid:12)(cid:12) dνdt . λ d − − γ h ν i α k f k , (2.12)which gives (1.9) with γ < β as desired.To show the converse implication we note that (1.9) implies (2.11). It can be seeneasily by Plancherel’s theorem and the standard slicing argument decomposing theconic neighborhood into a family of cones. Thus by Lemma 2.2 we have (2.10)which clearly implies (cid:12)(cid:12) R b F dµ (cid:12)(cid:12) . λ d − − γ k µ kh µ i α k F k . Therefore by duality andthe uncertainty principle we get (1.3) with β = γ . (cid:3) Appendix A.In this section, we discuss necessary conditions for the estimate(A.1) k sup Let d ≥ . The estimate (A.1) holds for all µ ∈ M d ( α ) only if (A.2) s ≥ ( max( d − αq , d +14 ) , < α ≤ , max( d − αq , d +14 − α − q , d +2 − α , d − α ) , < α ≤ d. IVERGENCE SET OF THE WAVE EQUATION 9 Proof of Lemma A.1. We first show s ≥ d − αq . Let b f ( ξ ) = χ B d (0 ,λ ) ( ξ ). If wetake t = 1 /λ , then we have | e it √− ∆ f ( x ) | & λ d for | x | ≤ λ − , which impliessup 1. A simple computation gives λ d λ − αq . λ s + d . Thisshows s ≥ d − αq .Next, we show that(A.3) s ≥ ( d +14 , < α ≤ , d +14 − α − q , < α ≤ d. Let P = { ξ = ( ξ , ξ ′ ) ∈ R × R d − : λ ≤ ξ ≤ λ, | ξ ′ | ≤ λ / } . Then we consider f which is given by b f = χ P . Then the set { ( ξ, | ξ | ) : ξ ∈ P } is contained in a( d + 1)-dimensional rectangle Q of dimensions Cλ × Cλ / × · · · × Cλ / | {z } d − times × C fora constant C > 0. We see that | e it √ − ∆ f | & | P | on the set Q ∗ = { ( x , x ′ , t ) : | x − t | ≤ cλ − , | x ′ | ≤ cλ − / , t ≤ c } for a constant c ≤ 1. It follows that k sup 1) separated by ∼ λ − α − d − . Then wehave | e it √− ∆ f ( x ) | & N − | P | χ S ( x, t ) for S = ∪ Nk =1 ( Q ∗ + (0 , v k , dµ = λ d − α χ P roj x ( S ) dx , then h µ i α . µ ( P roj x ( S )) ∼ 1. Also, k f k . P k k b f k k . | P | ∼ λ d +12 . Therefore, we obtain N − λ d +12 . k sup 1) separated by λ − αd . If we choose t = 1 /λ ,then sup 1. We can conclude that M − λ d . k sup Let q < ∞ and let µ ∈ M d ( α ) . Suppose k µ k 6 = 0 , then the estimate k sup This work was supported by the National Research Founda-tion of Korea (NRF) grant number NRF-2017R1C1B2002959 (Seheon Ham), NRF-2019R1A6A3A01092525 (Hyerim Ko), and NRF-2018R1A2B2006298 (SanghyukLee). IVERGENCE SET OF THE WAVE EQUATION 11 References [1] J. A. Barcel´o, J. Bennett, A. Carbery and K. M. Rogers, On the dimension of divergence setsof dispersive equations , Math. Ann. (2011), no. 3, 599–622.[2] C.-H. Cho, S. Ham and S. Lee, Fractal Strichartz estimate for the wave equation , NonlinearAnal. (2017), 61–75.[3] M. Cowling, Pointwise behaviour of solutions to Schr¨odinger equations. Harmonic Analysis(Cortona, 1982). Lecture Notes in Math., vol. 992, pp. 83–90. Springer, Berlin (1983)[4] X. Du, Upper bounds for Fourier decay rates of fractal measures , J. London Math. Soc. (2020), 1318–1336.[5] X. Du, A. Iosevich, Y. Ou, H. Wang, and R. 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Anal., Birkh¨auser Boston,Boston, MA, 1999.[21] T. Wolff, Decay of circular means of fourier transforms of measures , Int. Math. Res. Not.(1999), 547–567.[22] , Local smoothing estimates on L p for large p , Geom. Funct. Anal. (2000), 1237–1288. Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826,Republic of Korea Email address : [email protected] Email address : [email protected] Email address ::