Strichartz estimates for Schr\"odinger equations in weighted $L^2$ spaces and their applications

We obtain weighted $L^2$ Strichartz estimates for Schr\"odinger equations $i\partial_tu+(-\Delta)^{a/2}u=F(x,t)$, $u(x,0)=f(x)$, of general orders $a>1$ with radial data $f,F$ with respect to the spatial variable $x$, whenever the weight is in a Morrey-Campanato type class. This is done by making use of a useful property of maximal functions of the weights together with frequency-localized estimates which follow from using bilinear interpolation and some estimates of Bessel functions. As consequences, we give an affirmative answer to a question posed in \cite{BBCRV} concerning weighted homogeneous Strichartz estimates, and improve previously known Morawetz estimates. We also apply the weighted $L^2$ estimates to the well-posedness theory for the Schr\"odinger equations with time-dependent potentials in the class.


Introduction
In this paper we consider the following Cauchy problem for Schrödinger equations: i∂ t u + (−∆) a/2 u = F (x, t), u(x, 0) = f (x), (1.1) where (x, t) ∈ R n+1 , n ≥ 2, and (−∆) a/2 is given for a > 1 by means of the Fourier transform F f (= f ) as follows: These equations arise in mathematical physics.Particular interest is granted to the fractional-order cases where 1 < a < 2. This is because fractional quantum mechanics has been recently introduced by Laskin [29] where it is conjectured that physical realizations may be limited to the fractional cases.Of course, the classical case a = 2 has attracted interest from the ordinary quantum mechanics.The higher-order counterpart (a > 2) of it has been also attracted for decades from mathematical physics.Especially when a = 4, (1.1) can be found in the formation and propagation of intense laser beams in a bulk medium ( [19,20]).By Duhamel's principle, we have the solution of (1.1) which can be given by u(x, t) = e it(−∆) a/2 f (x) − i t 0 e i(t−s)(−∆) a/2 F (•, s)ds, where the evolution operator e it(−∆) a/2 is defined by R n e ix•ξ e it|ξ| a f (ξ)dξ.
There has been a lot of work on a priori estimates for the solution which control spacetime integrability of (1.2) in view of those of the Cauchy data f and F .This is because they play central roles in the study of (nonlinear) dispersive equations (cf.[4,43]).In the classical case a = 2, such an estimate was first obtained by Strichartz [41] in L q t,x (R n+1 ) norms.Since then, Strichartz's estimate has been studied by many authors [15,24,5,21,14,46,25,30] naturally in more general mixed norms L q t (R; L r x (R n )).(See also [12,36,31] and references therein for different related norms.)Similar estimates are also well known for the higher-order cases and can be found in [6].In recent years, much attention has been devoted to the fractional-order cases under the radial assumptions that the Cauchy data f, F are radial with respect to the spatial variable x (see [39,23,18,9,8] and references therein).
In this paper we address the problem of obtaining the Strichartz estimates for the solution (1.2) on weighted L 2 spaces of the form L 2 (w(x, t)dxdt), with the radial assumptions on the Cauchy data f, F .More precisely, we want to find conditions on the weight w(x, t) ≥ 0 for which e it(−∆) a/2 f L 2 (w(x,t)) ≤ C w f L 2  (1.3) and t 0 e i(t−s)(−∆) a/2 F (•, s)ds L 2 (w(x,t)) ≤ C w F L 2 (w(x,t) −1 ) (1.4) hold under the radial assumptions on f, F .(For simplicity, we are using the notation L 2 (w(x, t)) instead of L 2 (w(x, t)dxdt).)When a = 2, Strichartz estimates were studied in a weighted L 2 setting as above using weighted L 2 resolvent estimates for the Laplacian [35,2,38].Our method here for general orders a > 1 is entirely different from them and is based on a combination of bilinear interpolation and localization argument which makes use of a property of Hardy-Littlewood maximal functions.Before stating our main results, we need to introduce a function class L α,p a−par of functions V on R n+1 which is defined by the norm r α 1 r n+a Q(x,r)×I(t,r a ) |V (y, s)| p dyds 1/p < ∞ for α > 0, a ≥ 1 and 1 ≤ p ≤ (n + a)/α.Here, Q(x, r) denotes a cube in R n centered at x with side length r, and I(t, l) denotes an interval in R centered at t with length l.
In fact, L α,p a−par with a = 1 is just the same as the so-called Morrey-Campanato class.In this regard, we shall call L α,p a−par a-parabolic Morrey-Campanato class.Also, when a = 2, this class was already appeared in [1] concerning the homogeneous estimate (1.3) (see Remark 1.3 below) and was independently considered by the second author in the study of unique continuation.Following [1], we observe the following properties: The region of (s, 1/p) for (1.7) particularly when a = 2.
Our result on the estimates (1.3) and (1.4) deals with weights in a-parabolic Morrey-Campanato classes, and is stated as follows: Theorem 1.1.Let n ≥ 2 and w ∈ L a,p a−par .Assume that f and F are radial functions with respect to the spatial variable x.Then we have Here we are assuming α = a but this is just needed for the scaling invariance of the estimates (1.5) and (1.6) under the scaling (x, t) → (λx, λ a t), λ > 0.
Remark 1.3.For (1.5), we will prove more generally if a > 1, s ≥ 0 and max{ a a−1+2s , 1} < p ≤ n+a a+2s .When a = 2, it was shown in [1] that (1.7) holds for general functions f ∈ Ḣs if (s, 1/p) lies in the triangle with vertices B, C, D and fails if (s, 1/p) lies in the triangle with vertices A, B, F .See Figure 1.So it was naturally asked in [1] whether (1.7) with a = 2 might hold for the quadrangle with vertices B, D, E, F .With the radial assumption on f , we give an affirmative answer to this question that it can hold on the quadrangle and even on a region off the line BF .
The estimate (1.7) was shown for the wave equation (a = 1) ( [26]) and can be compared with the following smoothing estimates (known for Morawetz estimates) which have been studied by many authors for the wave equation (a = 1) [32] and for the Schrödinger equation (a = 2) [22,42,45].Indeed, since In [27], the estimates (1.5) and (1.6) were obtained particularly for higher-order cases where a > (n + 2)/2, without the radial assumption on f but with a more restrictive class L α,β,p of weights w satisfying ).) From the definition, it is easy to check that when l = r a , L α,β,p becomes equivalent to the a-parabolic Morrey-Campanato class L α+aβ,p a−par .Hence, L α,β,p ⊂ L a,p a−par under the condition a = α + aβ.Now we present a few applications of our estimates to the well-posedness theory for the following Cauchy problem in the radial case: where we assume that u, u 0 , V and F are radial functions with respect to the spatial variable x.The well-posedness for Schrödinger equations has been studied by many authors (see [34,35,13,33,2,38,27,8]). Making use of Theorem 1.1, we obtain here that (1.9) is globally well-posed in the space L 2 (|V |dxdt) with potentials V ∈ L a,p a−par .More precisely, we have the following result.
, there exists a unique solution of the problem (1.9) in the space L 2 (|V |).Furthermore, the solution u belongs to C t L 2 x and satisfies the following inequalities: (1.10) The rest of the paper is organized as follows.In Section 2, we give some preliminary lemmas which will be used in later sections for the proof of Theorem 1.1.In Section 3, we deduce Theorem 1.1 from Proposition 3.1 which is a frequency-localized version of Theorem 1.1.In this step we use Lemma 2.1 which is a useful property of ndimensional maximal functions of weights in a-parabolic Morrey-Campanato classes.Section 4 is devoted to proving Proposition 3.1 whose proof is based on a combination of bilinear interpolation, localization argument and some estimates for Bessel functions.In the final section, Section 5, we make use of the weighted L 2 Strichartz estimates in Theorem 1.1 to obtain the global well-posedness result in Theorem 1.4.
Throughout this paper, we will use the letter C to denote positive constants which may be different at each occurrence.We also denote A B and A ∼ B to mean A ≤ CB and CB ≤ A ≤ CB, respectively, with unspecified constants C > 0.

Preliminary lemmas
In this section we present preliminary lemmas which will be used in later sections for the proof of Theorem 1.1.
Let us first recall that a weight 1 w : (See, for example, [17].)In the following lemma, we give a useful property of weights in a-parabolic Morrey-Campanato classes regarding the Hardy-Littlewood maximal function.Similar properties for Morrey-Campanato type classes can be also found in [7,26,27].Such property has been studied earlier in [7,37,38] concerning unique continuation for Schrödinger equations.
Lemma 2.1.Let w ∈ L α,p a−par be a weight on R n+1 , and let w * (x, t) be the ndimensional maximal function defined by where Q ′ denotes a cube in R n with center x.Then, if α > a/p and p > ρ, we have w * L α,p a−par ≤ C w L α,p a−par , and  w * (x, t) p dxdt Since (x, t) ∈ Q(z, r) × I(τ, r a ), it is obvious that φ * (x, t) = 0. Hence we may consider only the first part in the right-hand side of (2.1).
For the term where k = 0, we use the following well-known maximal theorem where M (f ) is the Hardy-Littlewood maximal function defined by Here, the sup is taken over all cubes Q in R n with center x.Indeed, by applying (2.2) with q = p/ρ in x-variable, we see that if p > ρ r α 1 r n+a Q(z,r)×I(τ,r a ) w (0) * (x, t) p dxdt w(y, t) p dydt 1 p ≤C w L α,p a−par . (2.4) Now we only need to consider the terms where , where, for the last inequality, we used Hölder's inequality since p ≥ ρ.Hence, By combining this and (2.4), we get where M (w) is the Hardy-Littlewood maximal function of w (see (2.3)).Then, (2.5) (See [17] for details.) Also, the following fact can be found in Chapter 5 of [40] (see also Proposition 2 in [11]) with C A1 independent of w.Now we are ready to show that w Let {A 0 , A 1 } be an interpolation couple.Namely, A 0 and A 1 are two complex Banach spaces, both linearly and continuously embedded in a linear complex Hausdorff space.For 0 < t < ∞ and a ∈ A 0 + A 1 , let us set For 0 < θ < 1 and 1 ≤ q ≤ ∞, we denote by (A 0 , A 1 ) θ,q the real interpolation spaces equipped with the norms a See [3,44] for details.
We recall here two existing results concerning the real interpolation spaces.The first one is the following bilinear interpolation lemma (see [3], Section 3.13, Exercise 5(a)).
Lemma 2.2.For i = 0, 1, let A i , B i , C i be Banach spaces and let T be a bilinear operator such that

Then one has
T : For s ∈ R and 1 ≤ q ≤ ∞, let ℓ s q denote the weighted sequence space with the norm Then the second one concerns some useful identities of real interpolation spaces of weighted spaces (see Theorems 5.4.1 and 5.6.1 in [3]): 1 , and for 1 ≤ q 0 , q 1 , q ≤ ∞ and s 0 = s 1 ,

Proof of Theorem 1.1
This section is devoted to proving Theorem 1.1.Let us first consider the multiplier operators P k f for k ∈ Z which are defined by Then we will obtain the following frequency localized estimates in the next section which imply Theorem 1.1 using Lemma 2.1 and the Littlewood-Paley theorem on weighted L 2 spaces.Proposition 3.1.Let n ≥ 2. Assume that f and F are radial functions with respect to the spatial variable x.Then we have To deduce Theorem 1.1 from this proposition, we first observe that we may assume w(•, t) ∈ A 2 (R n ) uniformly in almost every t ∈ R. Indeed, since w ≤ w * and w * L α,p a−par ≤ C w L α,p a−par for α > a/p and p > ρ > 1 (see Lemma 2.1), if we show the homogeneous estimate (1.7) replacing w with w * , we get By this A 2 condition we can use the Littlewood-Paley theorem on weighted L 2 spaces (see Theorem 1 in [28]) to get On the other hand, since if α > 1 + a/p, a > 1 and 1 < p ≤ (n + a)/α.Consequently, by taking α = a + 2s, we get s ≥ 0 and max{ a a−1+2s , 1} < p ≤ n+a a+2s , as desired.The inhomogeneous estimate (1.6) follows also from the same argument.Indeed, by the Littlewood-Paley theorem as before, one can see that .
By using (3.2), the right-hand side in the above is bounded by  4.1.Proof of (3.1).From the scaling (x, t) → (λx, λ a t), it is enough to show the following case where k = 0: where α > 1 + a/p, a > 1 and 1 < p ≤ (n + a)/α.In fact, once we show this estimate, we get as desired.Now, by duality, (4.1) is equivalent to and so it is enough to show the following bilinear form estimate for α > 1+a/p, a > 1 and 1 < p ≤ (n+a)/α.Of course, F and G are assumed here to be radial with respect to the space variable x.For this estimate, we first decompose the involved functions into spacial-localized pieces as follows: and For simplicity, we set and Then, by using this decomposition we are reduced to showing that for α > 1 + a/p, a > 1 and 1 < p ≤ (n + a)/α.To show (4.4), we assume for the moment the following three estimates for a > 1 which will be shown later: where M (j, k) := max(j, k) and m(j, k) := min(j, k).
Lemma 4.1.Let n ≥ 2. For integers j, k ≥ 0, let F j and G k be given as in (4.2) and (4.3), respectively, which are radial functions on R n+1 with respect to the spatial variable x.If a > 1, we then have the following three estimates: Indeed, the estimate (4.5) is just the same as (4.13).From now on, we deduce (4.6) and (4.7) from (4.14) and (4.15), respectively.For fixed j, k ≥ 0, we denote R = max(2 j , 2 k ), and we set and for l ≥ 1 Then we may write To show (4.6) and (4.7), we assume for the moment that for a sufficiently large number M > 0, where ψ l ν stands for ψ l ν+ or ψ l ν− .This will be shown in the end of this subsection.Then by (4.17), we only need to bound where ψ ν (t) := (ψ To show the bound (4.6) for (4.18), from (4.14) we first see that and note that and Since we are assuming |j − k| ≤ 1, R = max(2 j , 2 k ) = C2 j .Hence we see that from the definition of the a-parabolic Morrey-Campanato class.Similarly, Consequently, we get the desired bound using the Cauchy-Schwarz inequality in ν with the trivial estimates and By (4.17) and (4.24), we obtain (4.6).Now we have to show the bound (4.7) for (4.18).For simplicity, we will consider the case j ≥ k only, because the other case j ≤ k can be shown clearly in the same way.From (4.15), we first see that Then by (4.19) and (4.20), it follows that

.27)
Since R = max(2 j , 2 k ) = 2 j , we see that as in (4.22).Now we claim that Indeed, when j − ak ≥ 0, where [2 j−ak − 1] denotes the least integer greater than or equal to 2 j−ak − 1.On the other hand, when j − ak ≤ 0, Using the Cauchy-Schwarz inequality in ν with (4.25) and (4.26), we get as desired.
Using this and the Cauchy-Schwarz inequality as before, we finally get Here, to apply the Cauchy-Schwarz inequality, we have used the following trivial estimates: Since N is large and R = max(2 j , 2 k ) ≥ 2 (j+k)/2 , (4.34) implies directly the estimate (4.17).Then by using the fact (see [40], p. 347) that where J m denotes the Bessel function with order m, it is easy to see that Note here that F j (λy ′ , s) is independent of y ′ ∈ S n−1 since F j is a radial function in the x variable.Hence, by setting F j (λ, s) := F j (λy ′ , s), we may write with K jk (r, λ, t), j, k ≥ 0, which is given as where I 0 = (0, 1), and for j, k ≥ 1, x,t , we are now reduced to showing that First we show (4.36) for the case |j − k| ≤ 1.For n ≥ 2, we see that using the following known estimates for Bessel functions J ν (r) (see [16], pp.429-431): Hence it follows from (4.37) that From the supports of χ I k and χ Ij , this implies now that as desired.Now we turn to (4.36) for the case |j − k| > 1.We divide cases into the case j, k ≥ 1 and the case where j = 0 or k = 0.
The case j, k ≥ 1.In this case, we will decompose K jk into four parts based on the following estimates for Bessel functions (see Lemma 3.5 in [8]): where and Indeed, using this lemma, we first see that and where the letter c n stands for constants different at each occurrence and depending only on n.Then we may write where From this, K jk is now decomposed as K jk = 4 l=1 K jk,l , with e itρ a J l (r, λ, ρ)ρφ(ρ) 2 dρ.
This implies directly that It remains to bound K jk,2 and K jk,3 .We shall consider only for K jk,2 because the same argument used for K jk,2 works clearly for K jk,3 .Since the factor (λρ) − 3 2 in J 2 would give a better boundedness than (λρ) − 1 2 , we only need to show the desired bound for Applying Lemma 4.3 with R = |λ| ∼ 2 j and Φ(ρ) = E n−2
The case where j = 0 or k = 0.In this case, we will use the following known fact (see [16], p. 426): For 0 ≤ r < 1 and Indeed, using this, we easily see that as desired.Now we only need to consider the case where k = 0 and j ≥ 1 since the other case where j = 0 and k ≥ 1 follows clearly from the same argument.Now we have to show that for j ≥ 1 where Recall from (4.40) that χ I0 (r) Now we may consider only the part of K j0 coming from (λρ) − 1 2 , because the factor (λρ) − 3 2 in (4.44) would give a better boundedness than (λρ) − 1 2 .Namely, we have to show the bound (4.43) for From (4.42), it is also easy to see that Hence it follows that (4.13).By Hölder's inequality, we first see that and so we only need to show that for j, k ≥ 0 For this, we consider the operators T k , k ≥ 0, defined by Then the adjoint operator T * k of T k is given by (λρ)H(λ, s)dλds, and so Then, by regarding ρφ 2 (ρ) and F j (λy ′ , s) as ϕ 2 (ρ) and H(λ, s), respectively, in (4.35), we are reduced to showing that for j, k ≥ 0 r , where L 2 r = L 2 (r n−1 dr).To show this, by changing variables ρ = ρ a , we first see that and so we get using Plancherel's theorem in t.On the other hand, by (4.37) we see that for k ≥ 0. Hence it follows that for k ≥ 0 and by the usual T T * argument, we now get for j, k ≥ 0, as desired.which implies (3.2).Indeed, to deduce (3.2) from this, first decompose the L 2 t norm in the left-hand side of (3.2) into two parts, t ≥ 0 and t < 0. Then the latter can be reduced to the former by a change of variables t → −t, and so we only need to consider the first part t ≥ 0. But, since [0, t) = (−∞, t) ∩ [0, ∞), by applying (4.46) with F replaced by χ [0,∞) (s)F , the first part follows directly, as desired.
To show (4.46), by duality we may show the following bilinear form estimate as before: But, once we have Lemma 4.1 replacing R with t −∞ , this estimate follows clearly by repeating the previous argument used for the homogeneous part (3.1).Since (4.36) is obviously valid for this replacement, it does not affect the last two estimates in the lemma.We only need to modify the first estimate (4.13) as uniformly in 0 < ε < 1.This is because the T T * argument used for (4.45) is no longer available in the case of t −∞ .Since ε is arbitrary and may be sufficiently small, it is not difficult to see that this modification is harmless in repeating the previous argument.
which follows from real interpolation between the estimates in Lemma 4.1.Indeed, we first note that from (4.14) and (4.15), R e i(t−s)(−∆) a/2 P 2 0 F j (•, s)ds, G k (x, t) ).Now, by real interpolation between (4.13) and (4.50), with ψ ν F j and ψ 0 ν G k instead of F j and G k , respectively, we get (4.49) after some easy computations.

Proof of Theorem 1.4
In this final section, we deduce the well-posedness (Theorem 1.4) for the Cauchy problem (1.9) from the weighted L 2 Strichartz estimates in Theorem 1.1.
Here we observe that (I − Φ)(u) = e it(−∆) a/2 u 0 (x) − i (5.2) Here, for the last inequality, we have used the smallness assumption on the norm V L α,p a−par .On the other hand, from (5.1), (5.2) and Theorem 1.1, we easily see that which is just the dual estimate of (1.5).First, from (5.1), (5.4) with w = |V |, and the simple fact that e it(−∆) a/2 is an isometry in L 2 , it follows that |) and V L α,p a−par is small enough, from this and (5.3), we now get as desired.This completes the proof.

andQ
(z,r)×I(τ,r a ) a−par f Ḣs as desired.Similarly for the inhomogeneous estimate (1.6).So we may show the estimates (1.7) and (1.6) by replacing w with w * .By this replacement and the property w

4 .
Proof of Proposition 3.1 In this section we prove Proposition 3.1.We first show (3.1) assuming Lemma 4.1 which is proved in Subsection 4.2, and then (3.2) follows from a similar argument in Subsection 4.3.
* (x, t) ≤ k≥0 w (k) * (x, t) + φ * (x, t)1 It is a locally integrable function which is allowed to be zero or infinite only on a set of Lebesgue measure zero.
by applying the Littlewood-Paley theorem again and taking α = a, this is now bounded by C w 2 2 (w(x,t) −1 ) if a > 1 and a/(a − 1) < p ≤ (n + a)/a.Consequently, we get (1.6).Theorem 1.1 is now proved.