Small Time Behavior and Summability for the Schrödinger Equation
Brian Choi
math.AP 40A30, 28A20, 35Q55, 42A24

Towards Model-based Reinforcement Learning for Industry-near Environments
    Per-Arne Andersen0000-0002-7742-4907 
	Morten Goodwin0000-0001-6331-702X
	Ole-Christoffer Granmo0000-0002-7287-030X
    December 30, 2023
=======================================================================================================================


We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schrödinger equation, both linear and nonlinear on ℝ. It is shown, via the smoothing effect of the Schrödinger flow, that the (sharp) result proved by Dahlberg and Kenig for initial data in Sobolev spaces still holds when one considers the full Schrödinger equation with a certain class of potentials. As for s<1/4, the failure of L^p-boundedness of the (localized) maximal operator is investigated.



40A30,28A20,35Q55,42A24


§ INTRODUCTION.


Consider the Cauchy problem

    i∂_t u + Δ u =0,  (x,t) ∈ℝ^n ×ℝ
    
        u(0) = u_0∈ L^2(ℝ^n)

noting that  -Δ is a non-negative operator. A straightforward computation with the Fourier transform yields

    u(x,t)= e^itΔu_0 (x) = ∫û_0(ξ)e^-it|ξ|^2 + iξ· xdξ,
 
where

    f̂(ξ) = ∫ f(x)e^-ix·ξdx, F(x) = (2π)^-n∫ F(ξ)e^ix ·ξdξ.

In this paper we continue to build upon a question initially posed by <cit.>: what is the minimal Sobolev regularity s_∗ for which  e^itΔf  f almost everywhere (a.e.) with respect to the Legesgue measure, for all f ∈ H^s_∗(ℝ)? Carleson originally proved a positive result, that any f ∈ H^s(ℝ) for s ≥1/4 exhibits almost everywhere (a.e.) convergence. Soon <cit.> showed that Carleson's result is sharp. In higher dimensions, this problem is closed except at the endpoint s=n/2(n+1). In n=2, <cit.> showed sufficiency for s>1/3 while <cit.> and <cit.> independently showed sufficiency in n ≥ 3 for s>1/2. <cit.> showed sufficiency for s > 2n-1/4n for n ≥ 2, and though it had long been believed that s>1/4 is the sharp sufficient condition in higher dimensions, <cit.> showed necessity for s ≥n/2(n+1) in n ≥ 2. Recently <cit.> showed sufficiency for s>n+1/2(n+2) for n≥ 3, which was subsequently improved to the sharp condition s>n/2(n+1) by <cit.>. Many of these results generalise nicely to i∂_t u + Φ(D)u=0 where Φ is a Fourier multiplier satisfying |D^γΦ (ξ)| ≲ |ξ|^α-|γ| and |∇Φ (ξ)|≳ |ξ|^α-1 where α≥ 1 and γ is a multi-index, which in particular involves the fractional Schrödinger operator e^-it(-Δ)^α/2; see <cit.> and <cit.>.

Meanwhile further generalizations were established using geometric measure theory. Though Carleson's problem has an affirmative answer for a.e. convergence when s ∈ [1/4,1/2] for n=1, the divergence set (points x∈ℝ where divergence occurs), which is of Lebesgue measure zero for such s, can still be big. <cit.> shows that the divergence set is of Hausdorff dimension at most 1-2s for s ∈ [1/4,1/2]. On the other hand, <cit.> generalizes the necessity result of <cit.> from the Lebesgue measure to the set of α-dimensional non-negative measures μ on ℝ^n for n ≥ 2; here a non-negative Borel measure μ is α-dimensional if c_α(μ) = sup_x∈ℝ^n,r>0μ(B(x,r))/r^α<∞. It is shown that if α∈ [3n+1/4,n] and μ{ x∈ℝ^n: e^itΔu_0 u_0  fails.}=0 for all u_0 ∈ H^s(ℝ), then s ≥(n-1)(n-α)/2(n+1)+n/2(n+1); since the Lebesgue measure on ℝ^n is n-dimensional, the result of <cit.> is recovered by letting α=n. For recent results regarding the size of divergence set in higher dimensions, see <cit.>.

It offers some insight to view this convergence problem in the context of summation methods. These originated in the study of alternative ways of summing Fourier series  such as Abel or Riesz summability.   Summation methods for Fourier series or transforms, in modern terms, involve a family of operators ϕ(-tΔ) (with ϕ a Borel-measurable or continuous function satisfying ϕ(0)=1) forming an approximate identity as t→ 0.
Questions of convergence in this context translate into strong convergence (as t→ 0) of such operator families.  Abel summability corresponds to ϕ(x)=e^-x, while other methods correspond to different choices of ϕ with ϕ(0)=1. Our current (Schrödinger) problem chooses ϕ(x)=e^ix, while the original result of Carleson for a.e. convergence of Fourier series <cit.> made the analogous statement for ϕ(x)=sin(x)/x.

The main purpose of this paper is to answer a variant of Carleson's problem, not for the free Schrödinger equation, but for the Schrödinger equation with a nonzero potential or nonlinearity. One motivation of this note comes from <cit.> that whenever |H|^α u_0 ∈ L^2(X) where X is a measure space and H is some self-adjoint operator on L^2(X) with |H| given by the polar decomposition, we obtain e^-itHu_0u_0 a.e. if α∈ (1/2,∞). Another motivation comes from <cit.>, where given the following Cauchy problem, a.k.a. the quantum harmonic oscillator,

    i∂_t u = -∂_xxu + x^2u, (x,t) ∈ℝ×ℝ
    
        u(0)=u_0 ∈ H^s(ℝ),

pointwise convergence to initial data holds for every s ≥1/4 and fails for s<1/4. Typically, a standard strategy in proving such positive result is to show that the Schrödinger maximal operator satisfies either a strong-type or weak-type estimate, from which pointwise convergence follows by a now-standard approximation argument. For the quantum harmonic oscillator, <cit.> takes advantage of the closed, analytic expression for the fundamental solution associated with the quadratic Schrödinger propagator, also known as the Mehler kernel:

    K_it(x,y) = (2π i sin2t)^-n/2 e^i/2(2t· |y-x/cos2t|^2 - tan2t·|x|^2), ∀ x,y ∈ℝ^n, t ∈ℝ∖π/2ℤ.


For a general potential, we have to work with analytic properties of the unitary group generated by the Hamiltonian -Δ+V; note that the semigroup generated by this operator has been studied extensively, for example, by <cit.>. In fact, an orbit of a square-integrable function generated by e^tΔ, viewed as a spacetime function, solves the heat equation, and by exploiting the exponential decay of the corresponding Green's function, one can easily show pointwise convergence to initial data (the Green's function corresponding to e^itΔ has no such spatial decay). More generally, {e^tΔ}_t>0 defines a holomorphic C_0-semigroup, and the strong convergence e^tΔI is an example of standard Abel summability traditionally studied for Fourier series on an interval.  For complex t→ 0 such convergence occurs in a sector symmetric about the positive t axis. However under the Wick rotation t ↦ it, our sector of convergence is now symmetric about the imaginary t axis, and our case of real t→ 0 constitutes a boundary case of the known region of Abel summability.  Therefore Abel summation is an insufficient tool to answer our problem. To this end, we summarise the main results of this paper:


Suppose s≥1/4 and V∈ L^2(ℝ)∪(W^1,∞(ℝ)∩⋃_ρ∈ [1,∞) L^ρ(ℝ)). Then the solutions to the linear Schrödinger equation converge a.e. to initial data in H^s(ℝ). On the other hand, if s<1/4 and V∈ L^2(ℝ), then there exists a compactly supported initial data f∈ H^s(ℝ) and a measurable set (of positive measure) E_f ⊆ℝ∖ supp(f) such that _t→ 0 |e^it∂_xxf|≥ c>0 on E_f.


It is clear from <ref> that if V ∈ L^2(ℝ), then the pointwise convergence of interest holds for s ≥1/4 sharply. Moreover the main theorem of <cit.> is contained in the previous statement by taking V=0. This class of potentials V contains some well-studied examples in physics such as the finite square well.

The above results related to the linear Schrödinger equation are naturally related to corresponding non-linearizations (<cit.>,<cit.>), for which, perhaps as expected, the corresponding results hold.


The solutions to quadratic nonlinear Schrödinger equation (qNLS) with nonlinearities 


    N_1(u,u)=u^2; N_2(u,u)=uu; N_3(u,u)=u^2  

 
converge a.e. to initial data for s≥1/4, s>1/4 and s≥1/4, respectively. On the other hand, the convergence fails for qNLS with nonlinearities N_1 and N_3 in H^s(ℝ) with s∈ [0,1/4).


We outline the organization of this article. In <ref>, useful notations are introduced. In <ref>, we prove a positive pointwise convergence result for the linear Schrödinger equation with potential using restricted Fourier space methods and Trotter-Kato product formula. In fact the class of potentials investigated does not include the quadratic case V(x)=x^2; our choice of potentials should be thought of as small perturbations to the free case V=0. In <ref>, the quadratic nonlinearities are treated. In <ref>, we prove the negative result that for i∂_t u = -∂_xxu +Vu, with an appropriate potential function, to exhibit pointwise convergence to initial data, it is necessary that u_0 ∈ H^s(ℝ) where s≥1/4.



§ NOTATION AND PRELIMINARIES.

The spaces 𝒮(ℝ^n) and C^∞_c(ℝ^n) denote the Schwartz class of rapidly decaying smooth functions and the set of smooth functions with compact support, respectively. We fix η∈ C^∞_c(ℝ) to be a smooth cutoff function that is identically one on [-1,1] with a compact support in [-2,2]. The inhomogeneous and homogeneous differential operators are:

    ⟨ξ⟩ = (1+|ξ|^2)^1/2; ⟨∇⟩^s f = ℱ^-1(⟨ξ⟩^s f); | ∇ |^s f = ℱ^-1(| ξ |^s f),

and ⟨∂_x⟩, |∂_x|,⟨∂_t⟩ and |∂_t| are defined similarly.

The L^2-based Sobolev space and (dispersive) Sobolev spacess (also known as Fourier restriction space or Bourgain space in the literature) are:

    H^s(ℝ^n)    = ⟨∇⟩^-s L^2(ℝ^n); ‖ f ‖_H^s = ‖⟨∇⟩^s f ‖_L^2
    
        X^s,b   = {u:‖ u ‖_X^s,b<∞}; ‖ u ‖_X^s,b = ‖û(ξ,τ) ⟨ξ⟩^s⟨τ + ξ^2 ⟩^b ‖_L^2_ξ,τ.


To do a local-in-time argument, where t ∈ [-δ,δ] for some δ∈ (0,1], we will need a restricted version of X^s,b as well. We denote such a space by X^s,b_δ and its restricted norm as:

    ‖ u ‖_X^s,b_δ = inf_u=u, t ∈ [-δ,δ]‖u‖_X^s,b.


For k∈ℕ, the inhomogeneous and homogeneous L^∞-based Sobolev spaces are:

    W^k,∞   = {f: ‖ f ‖_W^k,∞<∞}; ‖ f ‖_W^k,∞ = ∑_|α|≤ k‖∂^α f ‖_L^∞
    Ẇ^k,∞   = { f: ‖ f ‖_Ẇ^k,∞<∞}; ‖ f ‖_Ẇ^k,∞ = ∑_|α|= k‖∂^α f ‖_L^∞.

For δ>0, the Banach space of continuous spacetime functions u:[-δ,δ]⊆ℝ→ H^s(ℝ^n) are denoted by C^0_tH^s_x([-δ,δ]×ℝ^n) where ‖ u ‖_C^0_tH^s_x = sup_t∈ [-δ,δ]‖ u(t)‖_H^s(ℝ^n). Let ℋ = p.v.(1/x) be the Hilbert transform on ℝ. By Fourier analysis, ℋ= ℱ^-1(-i sgn(ξ) )ℱ and hence defines a unitary operator on L^2(ℝ), and moreover |∂_x| = ∂_x ℋ, again by considering their Fourier multipliers.

We say A ≲ B if A is bounded above by B multiplied by a universal constant, i.e., if there exists C>0 such that A ≤ CB. Similarly, say A ∼ B if A ≲ B and B ≲ A. For a measurable set E⊆ℝ, we let |E| be the Lebesgue measure of E. We define s+ = s+ϵ for some universal ϵ <<1; s- is defined similarly. We assume s≥ 0 unless stated otherwise.

Lastly, some well-known properties of X^s,b space and basic calculus facts are stated.

 Let δ∈ (0,1]. Then

    
  * <cit.>: For every b > 1/2 and δ>0, the following continuous embedding holds: X^s,b_δ↪ C^0_tH^s_x([-δ,δ]×ℝ).
    
  * <cit.>: Linear estimate: ‖ e^it∂_xxf‖_X^s,b_δ≲_s,b‖ f ‖_H^s whenever the right-hand side is finite.
    
  * <cit.>: Let b ∈ (1/2,1] and s ∈ℝ. Then, ‖η(t) ∫_0^t e^i(t-τ)∂_xx F(τ)dτ‖_X^s,b_δ≲‖ F ‖_X^s,b-1_δ.
    
  * <cit.>: For s ≥ 0 and 1/2=1/p_1+1/q_1=1/p_2+1/q_2 where p_1,q_2 ∈ (2,∞], we have the following Leibniz rule for the L^2-based Sobolev space: ‖ fg ‖_H^s≲‖ f ‖_L^p_1‖⟨∇⟩^s g ‖_L^q_1+‖⟨∇⟩^s f ‖_L^p_2‖ g ‖_L^q_2
    
  * <cit.>: If β≥γ≥ 0 and β + γ>1, then
    
    ∫dx⟨ x-a_1 ⟩^β⟨ x-a_2 ⟩^γ≲⟨ a_1-a_2 ⟩^-γϕ_β (a_1-a_2),

    where

    

        ϕ_β (a) ∼
    1,   β > 1
log (1+⟨ a ⟩),                 β=1
⟨ a ⟩^1-β,    β<1
    

    
  * <cit.>: Let -1/2<b^'≤ b<1/2, s∈ℝ and δ∈ (0,1]. Then, ‖ u‖_X^s,b^'_δ≲_s,b^',bδ^b-b^'‖ u ‖_X^s,b_δ.
    
  * For all a,b∈ℝ, we have ⟨τ-a ⟩⟨τ - b ⟩≳⟨ a-b ⟩.





§ LINEAR OPERATOR ESTIMATES: POSITIVE RESULTS.

Let H = -∂_xx+V denote the Hamiltonian operator on  ℝ, where V=V(x) is a real-valued time-independent multiplication operator. Note that H is self-adjoint on D(H)=D(-∂_xx)∩ D(V), if V ∈ L^2(ℝ)∪  L^∞(ℝ), where D(-∂_xx)= H^2(ℝ) and D(V)={f∈ L^2(ℝ): Vf∈ L^2(ℝ)}; see <cit.>. Therefore, e^-itH gives a family of unitary actions on L^2(ℝ). It is of interest to ask whether known positive results for pointwise convergence of the free Schrödinger equation as t→ 0 can be recovered with an addition of a potential.


Let s≥1/4 and 1≤ρ <∞, and suppose a time-independent potential V satisfies the following hypothesis:

    V ∈ L^2(ℝ)∪(W^1,∞(ℝ)∩ L^ρ(ℝ)).

Then for all u_0 ∈ H^s(ℝ), e^-itHu_0 → u_0 as t→ 0 almost everywhere with respect to Lebesgue measure. More precisely,

    |{x∈ℝ:_t→ 0 |e^-itHu_0-u_0|>0}|=0.



By virtue of V being time-independent, the conclusion holds in the limit when t→ t_0 for any t_0 ∈ I⊆ℝ, i.e., e^-itHu_0e^-it_0 Hu_0 a.e.
 
By Stone's theorem on a Hilbert space, a time-evolution operator for non-relativistic quantum mechanics is in one-to-one correspondence with a self-adjoint operator. However, the self-adjointness of H generally fails on H^s(ℝ) for s>0, and therefore, e^-itH defines a family of unitary operators on H^s(ℝ) only if s=0. In fact, it is not clear whether we have persistence of regularity for e^-itH on H^s(ℝ) for s>0, and so this shall be proved. Some of these results are likely to be known; however the lemmas below contain some estimates that will be of use later. We begin with definitions (see <cit.>).


For δ>0, u∈ C^0_tH^s_x([-δ,δ],ℝ) is a strong solution of

    i∂_t u    = -∂_xxu + Vu, (x,t) ∈ℝ× [-δ,δ]
    
        u(0)   =u_0 ∈ H^s(ℝ),

if u satisfies the following Duhamel integral formula for all t∈ [-δ,δ]:

    u(t)=e^it∂_xxu_0-i∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'.

 


The Cauchy problem <ref> is well-posed in H^s(ℝ) if for every g∈ H^s(ℝ), there exists δ>0, an open ball B⊆ H^s(ℝ) containing g, and a subset X⊆ C^0_tH^s_x([-δ,δ]×ℝ) such that for every u_0∈ B, there exists a unique strong solution u∈ X whose map u_0↦ u is continuous. If δ>0 can be arbitrarily large, then we say the well-posedness is global.
 


For u_0∈ L^2(ℝ), we claim that the notion of strong solution as in above, where we treat the potential term as a perturbation, coincides with that of an orbit generated by the unitary group. Though this seems intuitive, some care is needed if V is not sufficiently regular. At least when u_0∈ D(H), u(t)=e^-itHu_0 satisfies the Duhamel integral formula for each t, which is an immediate consequence of the following product rule:

    ∂_t (e^-it∂_xxe^-itHu_0 ) = -ie^-it∂_xx(Ve^-itHu_0).


For u_0 ∈ L^2(ℝ)∖ D(H), let u_0^(n)→ u_0 as n→∞ where u_0^(n)∈ D(H). Then we have,

    e^-itHu_0^(n)=e^it∂_xxu_0^(n)-i∫_0^t e^i(t-t^')∂_xx(Ve^-it^' Hu_0^(n))dt^'.


As n→∞, we have e^-itHu_0^(n)→ e^-itHu_0 and e^it∂_xxu_0^(n)→ e^it∂_xxu_0 by unitarity. We claim

    ∫_0^t e^i(t-t^')∂_xx(Ve^-it^' Hu_0^(n))dt^'→∫_0^t e^i(t-t^')∂_xx(Ve^-it^' Hu_0)dt^',

in L^2(ℝ) as n→∞. Firstly for V∈ L^∞(ℝ), we have

    ||∫_0^t e^i(t-t^')∂_xx(V(e^-it^' Hu_0^(n)-e^-it^' Hu_0))dt^'||_L^2   ≤∫_0^t ‖(V(e^-it^' Hu_0^(n)-e^-it^' Hu_0))‖_L^2
       ≤∫_0^t ‖ V ‖_L^∞‖ u_0^(n)-u_0‖_L^2dt^'→ 0,

where the last inequality is by the Hölder's inequality.

Secondly for V∈ L^2(ℝ), we apply the following form of inhomogeneous Strichartz estimate (see <cit.>): 

    || ∫_0^t e^i(t-t^')∂_xxF(t^')dt^'||_L^∞_t L^2_x≲‖ F ‖_L^4/3_t L^1_x.


Let χ be a characteristic function on t^'∈ [0,T] where 0<t<T. Then we have,

    || ∫_0^t e^i(t-t^')(V e^-it^' H(u_0^(n)-u_0))dt^'||_L^2_x   =|| ∫_0^t e^i(t-t^')(Vχ(t^') e^-it^' H(u_0^(n)-u_0))dt^'||_L^2_x
       ≤|| ∫_0^t e^i(t-t^')(Vχ(t^') e^-it^' H(u_0^(n)-u_0))dt^'||_L^∞_t L^2_x
       ≲‖ Vχ e^-itH(u_0^(n)-u_0) ‖_L_t^4/3L^1_x≲‖ V ‖_L^2‖ u_0^(n)-u_0‖_L^2→ 0.

 


The key idea of our proof is the Bourgain space estimate of the potential term.


Let s∈ [0,3/4). Then, there exists b ∈ (1/2,1], γ∈ [0,1/2) and a∈ [0,1/2) that satisfy

    s+a/2≤γ <min(s/2+1/4,1/2); max(s/2+1/4,1/2)< b <1-γ.

Furthermore for every such (s,b,γ,a), we have ‖ Vu ‖_X^s+a,-γ_δ≲_s,b,γ,a‖ V ‖_L^2‖ u ‖_X^s,b_δ.


The first statement is a  straightforward algebra exercise. As for the second, it suffices to prove the statement neglecting the δ-dependence, for if u = u on t∈[-δ,δ], we have

    ‖ Vu ‖_X^s,-γ_δ≤‖η(t/δ)Vu‖_X^s,-γ≲_η‖ Vu‖_X^s,-γ≲‖ V ‖_L^2‖u‖_X^s,b.

Taking infimum over u, we derive the desired result. We argue as in the proof of <cit.>.
Define

    F(ξ) = |V̂(ξ)|;  G(ξ,τ) = ⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^b |û(ξ,τ)|,

and

    W(ξ, τ, ξ_1) = ⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γ⟨ξ_1 ⟩^s⟨τ + ξ_1^2 ⟩^b.

Noting that ℱ[Vu](ξ,τ) = ∫V̂(ξ-ξ_1)û(ξ_1,τ)dξ_1, we have ‖ Vu ‖_X^s,-γ^2

    = || ∫⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γV̂(ξ-ξ_1)û(ξ_1,τ)dξ_1||_L^2_ξ,τ^2
        ≤|| ∫⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γ⟨ξ_1 ⟩^s⟨τ + ξ_1^2 ⟩^b F(ξ-ξ_1)G(ξ_1,τ)dξ_1||_L^2_ξ,τ^2
       ≤|| (∫ W^2 d ξ_1 )^1/2(∫ F(ξ-ξ_1)^2G(ξ_1,τ)^2dξ_1 )^1/2||_L^2_ξ,τ^2
        = || ∫ W^2 dξ_1 ·∫ F(ξ-ξ_1)^2 G(ξ_1,τ)^2dξ_1||_L^1_ξ,τ
       ≤|| ∫ W^2 dξ_1||_L^∞_ξ,τ·‖ F^2 ∗_ξ_1 G^2‖_L^1_ξ,τ≤|| ∫ W^2 dξ_1||_L^∞_ξ,τ·‖ V ‖_L^2^2 ‖ u‖_X^s,b^2,

where the second inequality is due to the Cauchy-Schwarz inequality, the third by the Hölder's inequality and the fourth by the Young's inequality. It remains to prove that ‖∫ W^2 dξ_1‖_L^∞_ξ,τ is finite. Changing variable z=ξ_1^2,

    ∫ W^2 dξ_1 ≃⟨ξ⟩^2s+2a⟨τ + ξ^2 ⟩^-2γ∫dξ_1/⟨ξ_1^2 ⟩^s⟨τ+ ξ_1^2 ⟩^2b≃⟨ξ⟩^2s+2a⟨τ + ξ^2 ⟩^-2γ∫_0^∞dz/⟨ z ⟩^s⟨ z+τ⟩^2bz^1/2.

Note that sup_τ∈ℝ can be replaced by sup_|τ|>1 without loss of generality, for if |τ|≤ 1, then ⟨τ+ξ^2⟩^-2γ≤⟨ξ^2-1⟩^-2γ for |ξ|≥ 1, and 

    sup_|τ|≤ 1∫_0^∞dz/⟨ z ⟩^s⟨ z+τ⟩^2bz^1/2<∞.


Hence sup_|ξ|≥ 1,|τ|≤ 1∫ W^2 dξ_1≲sup_|ξ|≥ 1⟨ξ⟩^2s+2a-4γ<∞, whereas sup_|ξ|≤ 1,|τ|≤ 1∫ W^2dξ_1<∞ follows from extreme value theorem. Now suppose |τ|>1. Then,

    ∫_0^∞dz/⟨ z ⟩^s⟨ z+τ⟩^2bz^1/2   =∫_0^1/2dz/⟨ z ⟩^s⟨ z+τ⟩^2bz^1/2+∫_1/2^∞dz/⟨ z ⟩^s⟨ z+τ⟩^2bz^1/2
       ≲⟨τ⟩^-2b + ⟨τ⟩^-(s+1/2)≲⟨τ⟩^-(s+1/2),

since s+1/2<2b. Moreover since 2γ < s+1/2,

    sup_ξ∈ℝ,|τ|>1⟨ξ⟩^2s+2a⟨τ + ξ^2 ⟩^-2γ⟨τ⟩^-(s+1/2)≲sup_ξ∈ℝ⟨ξ⟩^2s+2a-4γ<∞.



When V ∈ L^2(ℝ), the following Bourgain space estimate is obtained with ease via Fourier analysis.


Suppose V ∈ L^2(ℝ). The Cauchy problem <ref> is globally well-posed in H^s(ℝ) for s ∈ [0,3/4). In particular if u is the strong solution with the initial data u_0 ∈ H^s(ℝ), then there exists δ = δ(‖ V ‖_L^2)>0 such that ‖ u ‖_X^s,b_δ≲_s,b,‖ V ‖_L^2‖ u_0‖_H^s for some b ∈ (1/2,1].
 


Assuming that <ref> holds, let s,b,γ be as in <ref>, δ∈ (0,1] and fix C>0 that satisfies ‖ e^it∂_xxf‖_X^s,b_δ≤ C‖ f ‖_H^s for all f∈ H^s(ℝ) by <ref>. Let 

    X = {u∈ X^s,b_δ:‖ u ‖_X^s,b_δ≤ 2C‖ u_0‖_H^s}.


Define Γ u = e^it∂_xxu_0 - i ∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'. Then by <ref> we have,

    ‖Γ u ‖_X^s,b_δ   ≲‖ u_0 ‖_H^s + ‖ Vu ‖_X^s,b-1_δ≲‖ u_0 ‖_H^s+δ^1-(b+γ)‖ Vu ‖_X^s,-γ_δ
       ≲‖ u_0 ‖_H^s+δ^1-(b+γ)‖ V‖_L^2‖ u ‖_X^s,b_δ.
    ⇒‖Γ u ‖_X^s,b_δ   ≤ C ‖ u_0‖_H^s+CCδ^1-(b+γ)‖ V ‖_L^2‖ u_0‖_H^s.


By choosing δ≤ (C‖ V ‖_L^2)^-1/1-(b+γ), it is shown that Γ:X→ X. Similarly, we obtain

    ‖Γ u-Γ v ‖_X^s,b_δ≤ C_0 δ^1-(b+γ)‖ V ‖_L^2‖ u-v ‖_X^s,b_δ,

from which it is shown that Γ is a contraction map by shrinking δ>0 if necessary, and the resulting unique fixed point is the desired strong solution. Since the time step only depends on the norm of V, this local result can be iterated infinitely many times, and hence our solution is global in time.

Continuous dependence on initial data follows similarly, for if T>0, u_0^(n)→ u_0 in H^s(ℝ) and u^(n),u denote the strong solution corresponding to u_0^(n), u_0, respectively, then for t≤ T,

    ‖ u^(n)(t)-u(t) ‖_H^s(ℝ)   ≤‖ u_0^(n)-u_0‖_H^s+|| ∫_0^t e^i(t-t^')∂_xx(V(u^(n)-u))(t^')dt^'||_H^s
       ≲‖ u_0^(n)-u_0‖_H^s+T^1-(b+γ)‖ V ‖_L^2‖ u^(n)-u‖_X^s,b_T
       ≲‖ u_0^(n)-u_0‖_H^s+T^1-(b+γ)‖ V ‖_L^2‖ u^(n)_0-u_0‖_H^s,

where the implicit constant may depend on T. Taking sup_t∈ [0,T] both sides and taking n→∞, we obtain the desired result.


So far, the dispersive estimate of e^it∂_xx was used to control the Duhamel contribution of V. Now we directly study the dispersive estimate of e^-itH. If V and ∂_xx commute, then

    e^-itH=e^it∂_xxe^-itV,

and therefore, the operator e^-itH would obey the same maximal operator estimate of e^it∂_xx as in <cit.>, and our problem would be trivial. Generally the exponential map does not take addition into multiplication. If t is small, however, it is feasible to believe that <ref> holds approximately, and the following lemma quantifies this intuition: 

<cit.>
Let A and B be self-adjoint operators on a Hilbert space ℋ. If A+B is self-adjoint on D(A) ∩ D(B), then

    lim_n→∞ (e^it/nAe^it/nB)^nϕ = e^it(A+B)ϕ, ∀ϕ∈ℋ.



We apply this Trotter-Kato product formula to obtain persistence of regularity when the derivative of V is bounded.


Suppose t∈ℝ and s∈[0,1]. If ‖ V ‖_Ẇ^1,∞(ℝ^n)<∞, then we obtain

    ‖ e^-itHf‖_H^s≤ e^st√(n)‖ V ‖_Ẇ^1,∞‖ f ‖_H^s, ∀ f∈ C^∞_c(ℝ^n).




We first show ‖ e^-itV‖_H^1 → H^1≤ 1+ t√(n)‖ V ‖_Ẇ^1,∞, ∀ t ∈ℝ. Let f ∈ C_c^∞(ℝ^n). Then we have


    ‖ e^-itV f ‖_H^1^2    = ‖ e^-itV f ‖_L^2^2 + ∑_j=1^n ‖∂_j (e^-itVf)‖_L^2^2
       ≤‖ f ‖_L^2^2 + ∑_j=1^n (t‖∂_j V · f‖_L^2+‖∂_j f ‖_L^2)^2≤ (1+t√(n)‖ V ‖_Ẇ^1,∞)^2‖ f ‖_H^1^2.

Hence, the best constant C(t) ≤ 1+t√(n)‖ V ‖_Ẇ^1,∞,∀ t ∈ℝ.


Let ϕ = e^-itHu_0 for a fixed u_0 ∈ H^1(ℝ^n) and ϕ_m = (e^-it/mV e^i t/m∂_xx)^m u_0. Then, ϕ_m →ϕ in L^2(ℝ^n) by <ref>. By the estimate on ‖ e^-itV‖_H^1 → H^1, we obtain ‖ e^-itV e^it∂_xx f‖_H^1≤ (1+t√(n)‖ V ‖_Ẇ^1,∞)‖ f ‖_H^1. Then we have

    ‖ϕ_m ‖_H^1≤ (1+√(n)‖ V ‖_Ẇ^1,∞t/m)^m‖ u_0 ‖_H^1.

Hence for t ∈ [0,T] for T>0, we have a bounded sequence {ϕ_m }_m ⊂ H^1(ℝ^n), a reflexive Banach space. Then, there exists a weakly convergent subsequence {ϕ_m_k}_k where ϕ_m_k⇀ϕ∈ H^1(ℝ^n). Since H^1(ℝ^n)↪ L^2(ℝ^n), ϕ_m_k⇀ϕ in L^2(ℝ^n) and since ϕ_m →ϕ in L^2(ℝ^n), the convergence holds in the weak topology, and by the uniqueness of weak-limit in Banach space, ϕ = ϕ; in particular, e^-itHu_0 ∈ H^1(ℝ^n). Since norm is lower semicontinuous with respect to weak topology,

    ‖ e^-itHu_0 ‖_H^1≤_k →∞‖ϕ_m_k‖_H^1≤ e^t√(n)‖ V ‖_Ẇ^1,∞‖ u_0‖_H^1.

Since the bound above holds for all t ∈ [0,T] uniformly in T, it holds for all t ∈ℝ. Then by complex interpolation, it follows that for s ∈ [0,1],

    ‖ e^-itHf ‖_H^s≤ e^st√(n)‖ V ‖_Ẇ^1,∞‖ f‖_H^s, ∀ f ∈ H^s(ℝ^n).



Recall the estimate ‖ e^-itHu_0 ‖_X^s,b_δ≲‖ u_0‖_H^s for s∈ [1/4,1/2] and b=1/2+ was obtained in <ref>. A similar estimate via a different approach - the Trotter-Kato product formula and the fractional Gagliardo-Nirenberg interpolation - is obtained in <ref>. Let s∈ (0,1) and  max(1-s/s,2)<ρ<∞. Then for every f∈ C^∞_c(ℝ)[For a more general statement, see <cit.>.],

    ‖ f ‖_W^s,ρ/1-s≲‖ f ‖_L^ρ^1-s‖ f ‖_W^1,∞^s.


Let s∈ (0,1),b ∈ (1/2,1] and δ∈ (0,1]. Then, ‖ e^-itH f ‖_X^s,b_δ≲_s,b,η‖ f ‖_H^s, ∀ f ∈ H^s(ℝ).



Let F(ξ,τ) = ℱ[η(·)e^-i· Hf](ξ,τ) and F(ξ,t) = ℱ^-1_τF where ℱ^-1_τ is the inverse Fourier transform in τ variable, and let ℱ_x be defined similarly. Then we obtain

    ‖ e^-itH f ‖_X^s,b_δ   ≤‖η(t) e^-itHf ‖_X^s,b= ‖⟨ξ⟩^s ⟨τ + ξ^2 ⟩^b F(ξ,τ) ‖_L^2_ξ,τ
       =‖⟨ξ⟩^s ‖ e^itξ^2F(ξ,t)‖_H^b_t‖_L^2_ξ≤‖⟨ξ⟩^s ‖ e^itξ^2F(ξ,t)‖_H^1_t‖_L^2_ξ
       ≲_s‖⟨ξ⟩^s F(ξ,t) ‖_L^2_ξ,t + ‖⟨ξ⟩^s(|∂_t| (e^itξ^2F(ξ,t)) ) ‖_L^2_ξ,t.

For the first term, integrate in ξ variable first using Plancherel's theorem, followed by the estimate for the operator norm ‖ e^-itH‖_H^s → H^s and followed by the t-integral as follows:

    ‖⟨ξ⟩^s F(ξ,t) ‖_L^2_ξ,t =‖η(t)·‖ e^-itH f  ‖_H^s_x‖_L^2_t≤‖η(t) e^s‖ V^'‖_L^∞t‖_L^2_t·‖ f ‖_H^s≲_η‖ f ‖_H^s.

As for the second term,

    ‖⟨ξ⟩^s(|∂_t| (e^itξ^2F(ξ,t)) ) ‖_L^2_ξ,t   = ‖⟨ξ⟩^s ·‖ |∂_t| (e^itξ^2F(ξ,t)) ‖_L^2_t‖_L^2_ξ=‖⟨ξ⟩^s ·‖ℋ∂_t e^itξ^2F(ξ,t) ‖_L^2_t‖_L^2_ξ
       =‖⟨ξ⟩^s ·‖∂_t (η(t)e^itξ^2ℱ_x[e^-itHf](ξ,t) )‖_L^2_t‖_L^2_ξ
       ≤‖⟨ξ⟩^s ·‖∂_t η· e^itξ^2ℱ_x[e^-itHf] ‖_L^2_t‖_L^2_ξ + ‖⟨ξ⟩^s ·‖η(t) ∂_t (e^itξ^2ℱ_x[e^-itHf] ) ‖_L^2_t‖_L^2_ξ.

For the first term, switching the order of integration and recalling that the family e^it∂_xx is unitary on H^s(ℝ),

    ‖⟨ξ⟩^s ·‖∂_t η· e^itξ^2ℱ_x[e^-itHf] ‖_L^2_t‖_L^2_ξ   = ‖∂_t η·‖ e^-it∂_xxe^-itHf ‖_H^s_x‖_L^2_t
       = ‖∂_t η·‖ e^-itHf ‖_H^s_x‖_L^2_t≤‖∂_t η· e^s‖ V^'‖_L^∞t‖_L^2_t·‖ f ‖_H^s≲_η‖ f ‖_H^s.

For the second term, use product rule in t to obtain

    ‖⟨ξ⟩^s ·‖η(t) ∂_t (e^itξ^2ℱ_x[e^-itHf] ) ‖_L^2_t‖_L^2_ξ = ‖η(t)·‖∂_t (e^-it∂_xxe^-itHf ) ‖_H^s_x‖_L^2_t= ‖η(t) ·‖ V e^-itHf ‖_H^s_x‖_L^2_t.

where the second equality follows from ∂_t (e^-it∂_xxe^-itHf ) = -ie^-it∂_xx(Ve^-itHf). Then with q ∈ (2,∞) defined as follows,

    1/q+1-s/ρ=1/2,

apply the following particular form of Leibniz rule for Sobolev space to obtain[Unfortunately, the Leibniz rule generally fails when the L^∞ norm is applied to the Bessel potential term. Had this been true, the decay condition on V could have been removed.]

    ‖ V e^-itHf ‖_H^s   ≲_s ‖ V ‖_L^∞‖ e^-itHf ‖_H^s +  ‖ V ‖_W^s,ρ/1-s‖ e^-itHf ‖_L^q
       ≲(‖ V ‖_L^∞+ ‖ V ‖_W^s,ρ/1-s)‖ e^-itHf ‖_H^s≤ (‖ V ‖_L^∞+‖ V ‖_W^s,ρ/1-s)e^s ‖ V^'‖_L^∞ t‖ f ‖_H^s.


Since the first factor of the RHS is finite by <ref>, the proof is complete by integrating the upper bound in t against the smooth bump η.



In the case of V=0, <ref> reduces to <ref> where the proof heavily depends on the fact that the time-evolution operator defines a Fourier multiplier. However, if V is not identically zero, then the linear action by e^-itH defines a Fourier integral operator. The linear estimate as above, therefore, is not entirely obvious for e^-itH.




§ PROOF OF <REF>.



For initial data in H^s(ℝ) for s>1/2, the solution for each t∈ℝ can be identified with a continuous function by Sobolev embedding, and therefore, the conclusion follows immediately. Let V ∈ L^2(ℝ). Suppose u_0∈ H^s(ℝ) for s∈ [1/4,1/2]. We have

    |{x:_t→ 0|e^-itHu_0 - u_0|>0}|    ≤|{x:_t→ 0|e^it∂_xxu_0 - u_0|>0}|+|{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'|>0}|
       =|{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'|>0}|,

where the equality holds due to <cit.>. Then there exists a>0,b>1/2 and γ<1/2 such that

    s+a>1/2;  s+a≤ 2γ;  b-1<-γ,

and for δ∈ (0,1]:

    ‖ Vu‖_X^s+a,-γ_δ≲‖ V‖_L^2‖ u ‖_X^s,b_δ.


By <ref>,

    || ∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'||_C^0_tH^s+a_x([-δ,δ]×ℝ)   ≲|| ∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'||_X^s+a,b_δ≲‖ Vu‖_X^s+a,b-1_δ
       ≲‖ V‖_L^2‖ u ‖_X^s,b_δ≲‖ V ‖_L^2‖ u_0‖_H^s<∞.


Hence another application of Sobolev embedding implies

    |{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(Vu)(t^')dt^'|>0}|=0.


Now let V∈ W^1,∞∩ L^ρ. Fix an open cover {(k/2,k/2+1)}_k∈ℤ of ℝ and let {ψ_k}_k be a smooth partition of unity subordinate to the open cover. Then, V = ∑_k∈ℤ V_k where V_k = V ψ_k∈ L^2(ℝ). For u_0∈ H^s(ℝ) where s∈ [1/4,1/2], we have

    |{x:_t→ 0|e^-itHu_0 - u_0|>0}|
       ≤|{x:_t→ 0|e^it∂_xxu_0 - u_0|>0}|
        +∑_k∈ℤ|{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(V_ke^-it^' Hu_0)dt^'|>0}|
       =∑_k∈ℤ|{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(V_ke^-it^' Hu_0)dt^'|>0}|.

As before for δ∈ (0,1] and b=1/2+, we obtain

    || ∫_0^t e^i(t-t^')∂_xx(V_ke^-it^' Hu_0)(t^')dt^'||_C^0_tH^1/2+_x([-δ,δ]×ℝ)≲‖ V_k‖_L^2‖ e^-it Hu_0 ‖_X^1/4,b_δ≲‖ V_k ‖_L^2‖ u_0‖_H^1/4<∞,

where the second inequality follows from <ref>. By Sobolev embedding,

    |{x:_t→ 0|∫_0^t e^i(t-t^')∂_xx(V_ke^-it^' Hu_0)dt^'|>0}|=0,

for all k∈ℤ and this completes the proof.
 



§ QUADRATIC NONLINEARITIES.

Consider the following qNLS Cauchy problem:


    i∂_t u + ∂_xxu = N_i(u,u), u(0) = u_0 ∈ H^s(ℝ),

    N_1(u,u) = u^2; N_2(u,u) = uu; N_3(u,u) = u^2.



The well-posedness of qNLS above is studied in <cit.>. By X^s,b method, the qNLS for N_1 and N_3 are well-posed in H^s(ℝ) for s>-3/4 whereas that for N_2 is well-posedness for s>-1/4; the well-posedness associated to N_1 was improved to H^-1(ℝ) and was shown to be sharp in <cit.>. In the integral form, the solution satisfies

    u(t) = e^it∂_xxu_0 -i ∫_0^t e^i(t-τ)∂_xxN_i(u)(τ)dτ, t ∈ [-δ,δ],

and the goal is to prove smoothing estimates for N_i, i=1,2,3 as in <ref> from which convergence to initial data follows by the Sobolev embedding.


Let s ≥ 0, a ∈ [0,1/2) and δ∈ (0,1]. Then there exists b=1/2+ and γ=1/2- such that b<1-γ and the following estimates hold for i=1,3:

    ‖ N_i(u,u) ‖_X^s+a,-γ_δ   ≲_s,a,b,γ‖ u ‖_X^s,b_δ^2.




Let s >1/4, a ∈ [0,1/2] and δ∈ (0,1]. Then there exists b=1/2+, γ=1/2- such that b<1-γ and the following estimate holds:

    ‖ N_2(u,u) ‖_X^s+a,-γ_δ   ≲_s,a,b,γ‖ u ‖_X^s,b_δ^2.




The N_3-estimate will be shown to be an easy consequence of the N_1-estimate, and therefore we focus on the former. Denote

    F(ξ,τ) = |û(ξ,τ)|⟨ξ⟩^s ⟨τ + ξ^2 ⟩^b;  W(ξ,τ,ξ_1,τ_1) = ⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γ/⟨ξ-ξ_1⟩^s ⟨τ - τ_1 + (ξ - ξ_1)^2⟩^b⟨ξ_1 ⟩^s ⟨τ_1 + ξ_1^2 ⟩^b.

Neglecting δ-dependence as before, we have

    ‖ u^2 ‖_X^s+a,-γ^2    = || ∫⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γû(ξ-ξ_1,τ-τ_1)û(ξ_1,τ_1)dξ_1 dτ_1||_L^2_ξ,τ^2
       ≤|| ∫⟨ξ⟩^s+a⟨τ + ξ^2 ⟩^-γ/⟨ξ-ξ_1⟩^s ⟨τ - τ_1 + (ξ - ξ_1)^2⟩^b⟨ξ_1 ⟩^s ⟨τ_1 + ξ_1^2 ⟩^b F(ξ-ξ_1,τ-τ_1)F(ξ_1,τ_1)dξ_1 dτ_1||_L^2_ξ,τ^2
       = || ∫ W^2 dξ_1dτ_1 ·∫ F(ξ-ξ_1,τ-τ_1)^2F(ξ_1,τ_1)^2dξ_1 dτ_1||_L^1_ξ,τ
       ≤|| ∫ W^2 dξ_1 dτ_1||_L^∞_ξ,τ·‖ F^2 ∗ F^2‖_L^1_ξ,τ=|| ∫ W^2 dξ_1 dτ_1||_L^∞_ξ,τ·‖ u ‖_X^s,b^4.


Hence, it suffices to prove that ‖∫ W^2 dξ_1 dτ_1‖_L^∞_ξ,τ<∞.

By <ref>, we have

    ∫dτ_1/⟨τ - τ_1 + (ξ - ξ_1)^2⟩^2b⟨τ_1 + ξ_1^2 ⟩^2b≲⟨τ + (ξ_1-ξ)^2+ξ_1^2 ⟩^-2b.


Similarly,

    ⟨τ + (ξ_1-ξ)^2+ξ_1^2 ⟩^-2b⟨τ + ξ^2 ⟩^-2γ≲⟨ξ_1(ξ_1-ξ)⟩^-2γ,

and

    ⟨ξ⟩^2s+2a/⟨ξ_1-ξ⟩^2s⟨ξ_1⟩^2s≲⟨ξ⟩^2a.


Altogether we have

    sup_ξ,τ∫ W^2 dξ_1 dτ_1≲sup_ξ(⟨ξ⟩^2a∫dξ_1/⟨ξ_1(ξ_1-ξ)⟩^2γ).


Note that the integral is symmetric with respect to ξ_1 = ξ/2, and therefore ∫dξ_1/⟨ξ_1 (ξ_1-ξ)⟩^2γ = 2∫_ξ/2^∞dξ_1/⟨ξ_1 (ξ_1-ξ)⟩^2γ. Henceforth, assume ξ≥ 0 without loss of generality. On the region of integration, change variable η = ξ_1 (ξ_1-ξ)= ξ_1^2 - ξξ_1 to obtain:

    ξ_1 = ξ + √(ξ^2 + 4 η)/2;  dξ_1 = dη/√(ξ^2 + 4 η).

and so the integral becomes

    2∫_ξ/2^∞dξ_1/⟨ξ_1 (ξ_1-ξ)⟩^2γ   = 2 ∫_-ξ^2/4^∞dη/√(ξ^2 + 4 η)⟨η⟩^2γ= ∫_0^∞dη/√(η)⟨η - ξ^2/4⟩^2γ.


Since this integral is bounded for all ξ∈ [0,1), it suffices to assume ξ≥ 1 and show ∫_0^∞dη/√(η)⟨η - ξ^2/4⟩^2γ≲1/⟨ξ⟩^4γ-1. Then with a <1/2, it follows immediately that ‖∫ W^2 dξ_1 dτ_1‖_L^∞_ξ,τ<∞, provided b>1/2 is chosen sufficiently small.


Let c = ξ^2/4 and estimate the integral in three different regions: i) η∈ [2c,∞); ii) η∈ [c/2,2c); iii) η∈ (0,c/2).


    i)    : ∫_η≥ 2c≲∫_2c^∞dη/√(η)η^2γ≃1/ξ^4γ-1.
    
        ii)    : ∫_c/2^2c≲∫_c/2^2cdη/√(η) |η-c|^2γ≲ c^-1/2∫_c/2^2cdη/|η-c|^2γ≲ c^-1/2· c^1-2γ≃1/ξ^4γ-1.
    
        iii)    : ∫_0^c/2dη/√(η)⟨η -c ⟩^2γ≲1/⟨ c ⟩^2γ∫_0^c/2dη/√(η)≲c^1/2/⟨ c ⟩^2γ≃1/ξ^4γ-1.


Bringing all three cases together, we obtain the desired estimate, and this proves the first smoothing estimate.

As for the second estimate, for a general spacetime function u,

    ‖u‖_X^s.b = ‖û(ξ,τ)⟨ξ⟩^s ⟨τ - ξ^2 ⟩^b ‖_L^2_ξ,τ.


Arguing as before, one obtains

    ‖u^2 ‖_X^s+a,-γ≤|| ∫Ω^2 dξ_1 dτ_1 ||_L^∞_ξ,τ‖ u ‖_X^s,b^4.

where

    Ω(ξ,τ,ξ_1,τ_1) = ⟨ξ⟩^s+a⟨τ - ξ^2 ⟩^-γ/⟨ξ-ξ_1⟩^s ⟨τ - τ_1 + (ξ - ξ_1)^2⟩^b⟨ξ_1 ⟩^s ⟨τ_1 + ξ_1^2 ⟩^b,

and therefore it suffices to show ‖∫Ω^2 dξ_1 dτ_1 ‖_L^∞_ξ,τ<∞. As before,


    || ∫Ω^2 dξ_1 dτ_1 ||_L^∞_ξ,τ   ≲|| ∫⟨ξ⟩^2s+2a⟨τ - ξ^2 ⟩^-2γ/⟨ξ_1 - ξ⟩^2s⟨ξ_1 ⟩^2s⟨τ + (ξ_1-ξ)^2 + ξ_1^2 ⟩^2b dξ_1 ||_L^∞_ξ
       ≤|| ∫⟨ξ⟩^2s+2a/⟨ξ_1-ξ⟩^2s⟨ξ_1 ⟩^2s⟨ξ_1^2 -ξ_1ξ +ξ^2 ⟩^2γdξ_1||_L^∞_ξ≲|| ⟨ξ⟩^2a∫dξ_1/⟨ξ_1^2-ξ_1ξ +ξ^2⟩^2γ||_L^∞_ξ,

where these inequalities are direct applications of <ref>. Then by a direct computation,

    |ξ_1^2-ξ_1ξ +ξ^2|   ≥ |ξ_1 (ξ_1-ξ)|.
    ⇒⟨ξ_1^2-ξ_1ξ +ξ^2⟩   ≥⟨ξ_1 (ξ_1-ξ)⟩, ∀ξ,ξ_1 ∈ℝ.

Then ‖∫Ω^2 dξ_1 dτ_1 ‖_L^∞_ξ,τ<∞ follows from our previous result:
 
    || ⟨ξ⟩^2a∫dξ_1/⟨ξ_1 (ξ_1-ξ)⟩^2γ||_L^∞_ξ<∞.




Arguing as before, it suffices to prove

    sup_ξ,τ(⟨ξ⟩^2s+2a⟨τ+ξ^2⟩^-2γ∫dξ_1dτ_1/⟨ξ-ξ_1⟩^2s⟨ξ_1⟩^2s⟨τ_1-(τ+(ξ-ξ_1)^2)⟩^2b⟨τ_1-ξ_1^2⟩^2b)<∞.

For |ξ|<1,

    ⟨ξ⟩^2s+2a⟨τ+ξ^2⟩^-2γ∫dξ_1dτ_1/⟨ξ-ξ_1⟩^2s⟨ξ_1⟩^2s⟨τ_1-(τ+(ξ-ξ_1)^2)⟩^2b⟨τ_1-ξ_1^2⟩^2b   ≲∫dξ_1/⟨ξ_1 ⟩^4s⟨τ + ξ^2 - 2 ξξ_1⟩^2b
       ≲_s,a∫dξ_1/⟨ξ_1 ⟩^4s≤ C<∞,

where the upper bound C is independent of τ. For |ξ|≥ 1, changing variable z=2ξξ_1-(τ+ξ^2),

    ⟨ξ⟩^2s+2a⟨τ+ξ^2⟩^-2γ∫dξ_1dτ_1/⟨ξ-ξ_1⟩^2s⟨ξ_1⟩^2s⟨τ_1-(τ+(ξ-ξ_1)^2)⟩^2b⟨τ_1-ξ_1^2⟩^2b   ≲⟨ξ⟩^2a/⟨τ+ξ^2⟩^2γ∫dξ_1/⟨ 2ξξ_1-(τ+ξ^2)⟩^2b
       ≃⟨ξ⟩^2a-1/⟨τ+ξ^2⟩^2γ∫dz/⟨ z ⟩^2b≲⟨ξ⟩^2a-1.



As for the smoothing estimate for N_2, the condition s>1/4 is necessary to make certain integrals converge; in fact if ξ=τ=0, then the expression inside the sup_ξ,τ (see the proof for <ref>) is

    ∫dξ_1 dτ_1/⟨ξ_1 ⟩^4(1/4)⟨τ_1-ξ_1^2⟩^4b=∞.




The positive statements are consequences of <cit.> and Duhamel nonlinear terms being continuous in space and time via the smoothing estimates followed by the Sobolev embedding. We focus on the negative part.

For s∈ (0,1/4) we know from <cit.> that there exists u_0∈ H^s(ℝ) such that convergence to initial data fails on some set E of positive measure. By Lemma 3.1, we choose a=1/2- to obtain

    DN(x,t)∫_0^t e^i(t-t^')∂_xxN_i(u,u)(t^')dt^'∈ C^0_t H^1/2+_x ([0,δ]×ℝ).

By the triangle inequality,

    |u(t)-u_0|≥ |e^it∂_xxu_0-u_0|-|DN(x,t)|.


By continuity, lim_t→ 0|DN|=0 a.e., and therefore

    |{x∈ E: _t→ 0|u(t)-u_0|>0}|≥|{x∈ E: _t→ 0|e^it∂_xxu_0-u_0|>0}|>0.


Since H^s(ℝ)↪ L^2(ℝ) for s ∈ (0,1/4), a.e. pointwise convergence cannot hold for initial data in L^2(ℝ), and this finishes the proof.




§ NEGATIVE RESULT: BAIRE CATEGORY APPROACH.

Note that if a.e. pointwise convergence does not hold for s<1/4, then it also fails for 0≤ s^'≤ s. Define D(s) to be the collection of f∈ H^s(ℝ) with a compact support such that _t→ 0|e^-itHf|≳ 1 uniformly on some measurable set (of positive measure) E_f⊆ℝ∖ supp(f). Define D_0(s) similarly via e^it∂_xx. One motivation for considering functions in D(s) comes from Sjölin's work on localisation of Schrödinger means.  
<cit.> Let s<1/4. There exists f = ∑_n=1^∞ f_n ∈ H^s(ℝ) supported in (-δ/4,δ/4) for some δ>0 where f_n's are smooth and _t→ 0 |e^it∂_xxf(x)|≥ c>0 uniformly on a measurable set E ⊆ (δ/2,δ) of positive measure.


Given K⊆ℝ, a compact subset, one can modify the arguments of <cit.> to explicitly construct f ∈ H^s(ℝ) with its support in K such that e^it∂_xxf → f as t→ 0 fails in a.e. sense on K^c.
 
We show that D_0(s) = D(s) for s<1/4, or i.e., that in the short-time limit, the potentials play no role in the convergence of solutions.

Let V∈ L^2(ℝ) and 0<s<1/4. Then, D_0(s) = D(s). 


Writing u(t)=e^-itHf, the Duhamel formula yields

    u(t)-f = e^it∂_xxf-f -i∫_0^t e^i(t-τ)∂_xx(Vu)(τ)dτ.

As in the proof of Theorem 2.1, we apply the smoothing estimate (<ref>) on Vu by choosing a=1/2- and the well-posedness result (<ref>) to obtain that the Duhamel integral term is continuous in time and H^1/2+ in space, from which D_0(s)=D(s) follows immediately.
 

In the proof, note that our smoothing estimate is insufficient to conclude D_0(0)=D(0).

Motivated by <ref>, we restrict the collection of counterexamples to f∈ D(s)=D_0(s), assuming V ∈ L^2, and therefore e^-itH can be replaced by e^it∂_xx. Fix J=(-1,1), ϕ∈ C^∞_c(K) where K⊆ℝ∖ J is compact. It turns out that it is not an easy task to explicitly find such examples. Another more commonly-used approach is via the Stein-Nikisin maximal principle (<cit.>), which states:

e^itΔff as t→ 0 for all f∈ H^s(ℝ^n) if and only if

    || sup_0<t<1 |e^itΔf|||_L^2(B(0,1))≲_s,n‖ f ‖_H^s(ℝ^n), ∀ f ∈ C^∞_c(ℝ^n).


Note that the L^2(B(0,1)) cannot be strengthened to L^p(B(0,1)) for p∈ (2,∞] when s<1/4, n=1 due to the Hölder's inequality. For such p, we ask whether the f on the LHS of <ref> can be replaced by fϕ, i.e., whether the H^s norm controls the ϕ-localised maximal operator. It turns out that this fails for a big class of functions.

For s<1/4 and p∈ (2,∞), the following strong-type estimate fails:

    || sup_0<t<1|e^it∂_xx(fϕ)|||_L^p(J)≲‖ f ‖_H^s(ℝ).

 
Note that if <ref> fails for p, then it fails for p≥ p. On the other hand, Sjölin in <cit.> showed that for every f∈ H^s(ℝ^n) with a compact support, e^itΔf 0 for all x∈ℝ^n∖ supp(f) if and only if s≥n/2. Since the free Schrödinger operator is given by the convolution e^itΔf = K_t ∗ f where K_t(x) = (4π it)^-n/2e^i|x|^2/4t, it is evident that e^itΔf∈ C^∞_x(ℝ^n) for each t∈ℝ∖{0} since K_t ∈ C^∞_x(ℝ) and f has a compact support, and hence it makes sense to evaluate e^itΔf pointwise. Sjölin showed, via Baire category approach, that for s<n/2 there exists f∈ H^s(ℝ^n) with a compact support in S={|x|∈ (1,2)} such that e^itΔf(0)→∞ as t→ 0. Hence ‖ e^itΔf‖_L^∞ (B(0,1))→∞ as t→ 0 since e^itΔf is smooth. Here we are interested in the L^p-behavior of solutions in the short-time limit. For p∈ [1,2], ‖ e^itΔf‖_L^p(B(0,1)) stays bounded due to the L^2-conservation of solutions. For p∈ (2,∞), it is unclear whether the solution blows up or stays bounded. We show a weaker result that the L^p-norm of solutions diverges in some time-averaged sense:

Let {t_k}_k=1^∞ be a real-sequence contained in (0,1] that tends to zero as k→∞ and p∈ (2,∞). Then there exists a dense, G_δ residual set 𝒞⊆ H^s(ℝ) such that for every f∈𝒞, {e^it_k∂_xx(fϕ)}∉ l^q L^p(ℕ× J) for all q∈ [1,p].[Recall that a measurable set is G_δ if it can be realised as a countable intersection of open sets. A set is meager if it can be realised as a countable union of nowhere dense sets, and its complement is called a residual.]

Our proof is a simple application of the Banach-Steinhaus theorem. Given a sequence t_k0, define 

    S_n f = sup_k≤ n|e^it_k∂_xx(fϕ)|,  S f =sup_k|e^it_k∂_xx(fϕ)|,

for f∈𝒮(ℝ). It is straightforward to verify
 
    S_n(f+g)≤ S_nf+S_n g,  S_n(λ f) = |λ| S_n f,∀λ∈ℂ,

and in particular, S_n is not linear. For this reason, the following extension of the Banach-Steinhaus Theorem, traditionally studied in the context of linear operators, is applied where its proof could be done as <cit.>.

Let {T_α}_α∈ A be a family of continuous operators on X into Z=L^p(Y,ν) for p∈ [1,∞] where X is a Banach space, (Y,ν) is a σ-finite measure space and A is some directed set.

    ‖ T_α (x+y)‖≤‖ T_α x ‖ + ‖ T_α y ‖, ‖ T_α (λ x)‖ = λ‖ T_α x ‖,∀λ≥ 0.

Then either lim_x → 0‖ T_α x ‖ = 0 uniformly in α, i.e., {T_α} is equicontinuous at the origin, or
 { x ∈ X: T_α x is unbounded in Z} forms a residual set that is dense G_δ in X.
 


We first claim that {S_n} defines a family of continuous sublinear operators on H^s(ℝ) into L^p(J) that satisfies the hypotheses of <ref>. By the triangle inequality,

    |S_n f- S_ng|≤ S_n(f-g).

Hence it suffices to show that S_n is a bounded map to show continuity. Since l^p(ℕ)↪ l^∞(ℕ), we obtain

    ‖ S_n f‖_L^p(J)   = ‖ e^it_k∂_xx(fϕ)‖_L^p(J) l^∞_k≤ n≤‖ e^it_k∂_xx(fϕ)‖_ l^p_k≤ nL^p(J)
       ≲|| |t_k|^-(1/2-1/p)‖ f ϕ‖_L^p^'(J)||_l^p_k≤ n≲_p,ϕ,n‖ f ‖_H^s(ℝ),

and hence the continuity. From <ref>, <ref> could be verified. It is shown, by contradiction, that the {S_n} cannot be equicontinuous at the origin. Assume it is. Then S is continuous in measure at the origin. Suppose f_j→ 0 in H^s(ℝ) as j→∞ and let λ>0. Let ϵ>0 for which there exists δ>0 such that ‖ S_n f‖_L^p(J)<ϵ^1/pλ for all but finitely many n∈ℕ and all f such that ‖ f ‖_H^s<δ. Then let j ≥ N, some N∈ℕ sufficiently big, such that ‖ f_j‖_H^s<δ for all j≥ N, and let n be sufficiently large such that |{|Sf_j -S_nf_j|≥λ/2}|≲ϵ; recall that S_n f_j Sf_j implies S_nf_j → Sf_j in measure on a finite measure space as n→∞. Then we obtain

    |{|S f_j|≥λ}|≤|{|Sf_j -S_n f_j|≥λ/2}|+|{|S_n f_j|≥λ/2}|≲ϵ,

where the second term is bounded above by ϵ up to a constant by the Chebyshev's inequality.

Now it is shown that convergence a.e. to initial data holds for all f∈ H^s(ℝ) with a compact support, which is a contradiction since s<1/4 and due to the explicit construction of an initial datum with a compact support in <cit.>. Pick f_n→ f∈ H^s(ℝ) where f_n ∈𝒮(ℝ). Then,

    |{x∈ J:_k→∞|e^it_k∂_xx(fϕ)|>λ}|   =|{x∈ J:_k→∞|e^it_k∂_xx((f-f_n)ϕ)|>λ}|
       ≤| {x∈ J:S (f-f_n)>λ}| 0,

for all λ>0 where the last limit follows from the continuity in measure of S. Hence the supposed equicontinuity fails and there exists a dense G_δ set 𝒞⊆ H^s(ℝ) such that if f ∈𝒞, then {S_n f } is unbounded in L^p(J). By monotonicity, ‖ S_n f‖_L^p(J)≤‖ Sf‖_L^p(J)≤|| sup_0<t<1|e^it∂_xx(fϕ)|||_L^p(J), and therefore, <ref> cannot hold for every f∈𝒞. By the right-most estimate in <ref>, we obtain that {e^it_k∂_xx(fϕ)}∉ l^p L^p(ℕ× J) for all f∈𝒞.




§ ACKNOWLEDGEMENTS.


The author would like to appreciate his doctoral advisor Mark Kon for insightful comments.



    



    




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