Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t_{n}$\end{document} that converge to zero. Our main result is that bounds on the maximal function supn|eitn(−Δ)α/2f|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup_{n} |e^{it_{n}(-\Delta )^{\alpha /2}} f| $\end{document} can be deduced from those on sup0<t≤1|eit(−Δ)α/2f|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup_{0< t\le 1} |e^{it(-\Delta )^{\alpha /2}} f|$\end{document}, when {tn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{t_{n}\}$\end{document} is contained in the Lorentz space ℓr,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell ^{r,\infty}$\end{document}. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou and Seeger, and Li, Wang, and Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.


Introduction
Let α > 0. We consider the fractional Schrödinger operator A classical problem posed by Carleson [5] is to determine the optimal regularity s for which lim t→0 e it(−∆) α/2 f = f a.e.∀f ∈ H s , (1. 2) where H s denotes the inhomogeneous Sobolev spaces of order s with its norm The case α = 2 has been extensively studied until recently.When d = 1, it was shown by the work of Carleson [5] and Kenig-Dahlberg [10] that (1.2) holds true if and only if s ≥ 1/4.In higher dimensions, the problem turned out to be more difficult.Progress was made by contributions of numerous authors.Sjölin [33] and Vega [40] independently obtained (1.2) for s > 1/2.In particular, further improvement on required regularity was made by Moyua-Vargas-Vega [26], Tao-Vargas [39] when d = 2, and convergence for s > (2d − 1)/4d was shown by Lee [19] for d = 2 and Bourgain [3] in higher dimensions.Bourgain [4] showed that (1.2) holds only if s ≥ d/2(d + 1).The lower bound was shown to be sufficient for (1.2) by Du-Guth-Li [12] for d = 2, and Du-Zhang [14] for d ≥ 3 except for the endpoint case s = d/2(d + 1) (also, see [23] for earlier results and references therein).
Maximal estimates.In the study of pointwise convergence the associated maximal functions play important roles.By a standard argument (1.2) follows if we have sup which is, in fact, essentially equivalent to (1.3) by Stein's maximal principle.Our first result shows that the maximal estimate (1.5) can be deduced from (1.4) when Thanks to Theorem 1.1 we can improve the previous results and obtain seemingly optimal results for the convergence of sequential Schrödinger means in higher dimensions.For {t n } ∈ ℓ r,∞ (N) and d = 1, the known estimates (1.4) ( [5,41]) and Theorem 1.1 give (1.5) for s ≥ min(rα/(4r + 2), 1/4) when α > 1, and for s > rα/(4r + 2) when 0 < α < 1.This recovers the result (sufficiency part except the endpoint case when 0 < α < 1) in [11] without the assumption that {t n } decreases.
This improves the previous results in [21,37].We expect that the regularity exponents given in (1.7) is sharp up to the endpoint case.However, we are not able to verify it for the moment.
Remark 1.4.For the wave operator, i.e., α = 1, (1.3) holds true if and only if s ≥ r 2(r+1) for {t n } ∈ ℓ r,∞ (N).When d = 1, this was shown in [11].In higher dimensions, one can show it using Theorem 1.1 (also Corollary 3.2).The sharpness can be obtained by following the argument in [11].We remark that (1.3) is closely related to L p boundedness of the spherical maximal operator given by taking the supremum over more general sets (see [32,1,30]).
Localization argument.The proof of Theorem 1.1 relies on a localization argument.We briefly explain our approach.Via Littlewood-Paley decomposition, the proof of (1.3) can be reduced to showing sup where f is supported in A R := {ξ : R ≤ |ξ| ≤ 2R} (see Section 3).In the previous work [11,35,36,37] the estimate (1.8) was obtained by relying on the kernel estimates.In contrast, we deduce (1.8) directly from (1.4).Clearly, (1.4) Using the estimate and a localization argument, we first obtain from (1.9) a temporally localized maximal estimate sup Moreover, the converse implication from (1.10) to (1.9) is also true as long as R −α < |I| ≤ R 1−α (see Lemma 2.2 for detail).Once we have (1.10), we can obtain (1.5) by following the argument in [11].
If the exponent s in the estimate (1.9) is sharp, then the same is true for the estimate (1.10).For instance, when α = 2, (1.9) holds for s > d/2(d + 1), which is optimal up to the endpoint case, and hence so does (1.10) for the same s.When α > 1 and |I| ≥ R 1−α , one can see the exponent s in (1.10) can not be smaller than that in (1.9) using the localization lemma in [19] (cf.[8,20,29]).
To show the implication from (1.9) to (1.10), we adapt the idea of temporal localization lemma in [19,8].We establish a spatial localization lemma (Lemma 2.4), which plays a crucial role in proving Theorem 1.1.More precisely, we show that the local-in-spatial estimate (1.9) can be extended to the global-in-spatial estimate with the same regularity exponent.After a suitable scaling, we obtain the temporal localized estimate (1.10) from the global-in-spatial estimate.
Extension to fractal measure.Maximal estimates relative to general measures (instead of the Lebesgue measure) have been used to get more precise description on the pointwise behavior of the Schrödinger mean e it(−∆) α/2 f .For a given sequence {t n } converging to zero, we consider where dim H denotes the Hausdorff dimension.One can compare D α,d (s, r) with the dimension of the divergence set The bounds on D α,d (s) can be obtained by the maximal estimate relative to general measures (see, for example, [2,14,17]), to which the fractal Strichartz estimates studied in [6,18,13] are closely related (also see [24,43,15]).The implication in Theorem 1.1 also extends to the maximal estimates relative to general fractal measures, so we can make use of the known estimates for the L 2 -fractal maximal estimates and the fractal Strichartz estimates to obtain upper bounds on D α,d (s, r), 0 < r < ∞.We discuss it in detail in Section 3.2.
Organization of the paper.In Section 2, we deduce from (1.4) temporally localized maximal estimates in Lemma 2.2 (relative to general measure) which are to be used to prove Theorem 1.1.We prove Theorem 1.1 and discuss upper bounds on the dimension of divergence sets in Section 3.
Notations.Throughout this paper, a generic constant C > 0 depends only on dimension d, which may change from line to line.If a constant depends on some other values (e.g.ǫ), we denote it by C ǫ .The notation A B denotes A ≤ CB for a constant C > 0, and we denote by A ∼ B if A B and B A. We often denote L 2 (R d ) by L 2 , and similarly H s (R d ) by H s .

Temporally localized maximal estimates
In this section, we prove that the estimates (1.9) and (1.10) are equivalent.For later use, we consider the equivalence in a more general setting, that is to say, in the form of estimates relative to fractal measures (Lemma 2.2).To do this, we recall the following.Definition 2.1.Let 0 < γ ≤ d and let µ be a nonnegative Borel measure.We say By µ γ we denote the infimum of such a constant C µ .
We first deduce a temporally localized maximal estimate from the estimate sup Remark 2.3.By a simple modification of our argument, Lemma 2.2 can be extended to a class of evolution operators e itP (D) as long as and |∇P (ξ)| |ξ| α−1 hold (see [8]).Hence, an analogue of Theorem 1.1 holds true for e itP (D) .A typical example of such an operator is the non elliptic Schrödinger operator e it(∂ 2 ) .
The rest of the section is devoted to the proof of Lemma 2.2, for which we first consider spatial localization.
2.1.Spatial localization.By adapting the argument in [19,8], we prove a spatial localization lemma exploiting rapid decay of the kernel.
holds for some s ∈ R whenever f is supported in A 1 .Then, there exists a constant holds whenever f is supported in A 1 .
Proof.Let P be a projection operator defined by ) and β = 1 on [1,2].Let {B} be a collection of finitely overlapping balls of radius R which cover Then, we note that F 2 . Since P f = f , by Minkowski's inequality we have where Note that e it(−∆) α/2 P f = K(•, t) * f where K is given by By integration by parts, it is easy to see Therefore, we only need to consider L 1 .
Applying (2.4) on each B, we obtain The last inequality follows since the balls B overlap finitely.This completes the proof.

Proof of Lemma 2.2.
To prove Lemma 2.2, we invoke an elementary lemma.
For a givne γ-dimensional measure µ, we denote by µ R the measure defined by the relation 1 for some C > 0. Changing variables (x, t) → (R −1 x, R −α t) and ξ → Rξ, we see that (2.2) is equivalent to We claim that the estimate (2.7) is equivalent to the seemingly weaker estimate for R ≥ 1 whenever g is supported on A 1 .To show this, we only need to prove that (2.8) implies (2.7) since the converse is trivially true.When α > 1, the implication from (2.8) to (2.7) was shown in [8] (also, see [19,25]) when µ R is the Lebesgue measure and α is an integer.It is easy to see that the argument in [8] works for general γ-dimensional measure µ R .When 0 < α ≤ 1, using Lemma 2.4 with R replaced by R α , we get (2.7) from (2.8).This proves the claim.

Maximal estimate for sequential Schrödinger means
In this section, we prove Theorem 1.1 and obtain results regarding upper bounds on the dimension of the divergence set of e itn(−∆) α/2 f .The results are consequence of extension of the maximal estimates to general measure.See Section 3.2.
3.1.L 2 -maximal estimates.Making use of Lemma 2.2, we deduce the maximal estimates for the sequential Schrödinger mean from the estimate (2.2).
holds for s ≥ min{s * , s * } whenever µ is γ-dimensional and supp f ⊂ A R .
The estimate (3.5) clearly implies (2.2).However, to prove Corollary 3.2, we need to remove the frequency localization in the estimate (3.2) so that the right hand side of (3.2) is replaced by C µ +s .This can be achieved by adapting the argument in [11].For ℓ ≥ 0, we set For each ℓ ≥ 0, we write f = 0≤k<ℓ P k f + k≥0 P ℓ+k f .So, we have .
We recall the maximal estimate (2.2) for the wave operator shown in [2,17].(See also [8,18] for the fractal Strichartz estimates (3.9)).Proposition 3.5.Let I be a subinterval in [0, 1].Then (2.2) holds with α = 1 for s > s 1 (γ, d) where for d = 2; By Proposition 3.1, the estimates in Proposition 3.4 and 3.5 give the corresponding estimates for sup n |e itn(−∆) α/2 f | with {t n } ∈ ℓ r,∞ relative to γ-dimensional measures.Then, by a standard argument (see [2]), one can obtain upper bounds on the Hausdorff dimension of the divergence sets.We summarize the results as follows: a positive Borel measure by the Riesz representation theorem.