We prove some weighted reﬁnements of the classical Strichartz inequalities for initial data in the Sobolev spaces $H^s (R^n)$. We control the weighted $L^2$ –norm of the solution of the free Schrödinger equation when the weight is in a Morrey–Campanato type space adapted to that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular solutions of the free Schrödinger equation

Alberto Ruiz, Maricruz

Barceló, Juan Antonio

Bennett, Jonathan M.

Carbery, Anthony

Collectanea mathematica

Revistes Catalanes amb Accés Obert

Collect. Math. 61, 1 (2010), 49–56
c© 2010 Universitat de Barcelona
Strichartz inequalities with weights in Morrey–Campanato classes
J.A. Barcelo´
ETSI de Caminos, Universidad Polite´cnica de Madrid, 28040, Madrid, Spain
E-mail: juanantonio.barcelo@upm.es
J.M. Bennett
School of Mathematics, The Watson Building, University of Birmingham
Edgbaston, Birmingham, B15 2TT, England
E-mail: J.Bennett@bham.ac.uk
A. Carbery
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh
JCMB, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland
E-mail: A.Carbery@ed.ac.uk
A. Ruiz
Departamento de Matema´ticas, Universidad Auto´noma de Madrid, 28049, Madrid, Spain
E-mail: alberto.ruiz@uam.es
M.C. Vilela
Departamento de Matema´tica Aplicada, Universidad de Valladolid
Plaza Santa Eulalia 9 y 11, 40005 Segovia, Spain
E-mail: maricruz@dali.eis.uva.es
Received November 5, 2008. Revised April 24, 2009
Abstract
We prove some weighted refinements of the classical Strichartz inequalities for
initial data in the Sobolev spaces H˙s(Rn). We control the weighted L2-norm
of the solution of the free Schro¨dinger equation whenever the weight is in a
Morrey–Campanato type space adapted to that equation. Our partial positive
results are complemented by some necessary conditions based on estimates for
certain particular solutions of the free Schro¨dinger equation.
The first and fourth authors were supported by Spanish Grant MTM2008-02568, the fifth by
Spanish Grant MTM2007-62186, the second by EPSRC Grant EP/E022340/1 and the third by EC
project “Pythagoras II” and a Leverhulme Study Abroad Fellowship.
Keywords: Schrodinger equations, Morrey–Campanato Spaces.
MSC2000: 34A12, 42B35.
49
50 Barcelo´, Bennett, Carbery, Ruiz, and Vilela
1. Introduction
Consider the initial value problem associated to the free Schro¨dinger equation:{
i∂tu+ ∆xu = 0 (x, t) ∈ Rn × R
u(x, 0) = f(x)
(1)
where f ∈ H˙s(Rn), the usual homogeneous L2-Sobolev space.
As usual, we denote the solution u of (1) by eit∆f(x). The purpose of this note
is to establish some weighted refinements of the classical Strichartz inequalities∗
‖eit∆f‖Lrx,t(Rn×R) . ‖f‖H˙s(Rn), (2)
where 0 ≤ s < n2 and r = 2(n+2)n−2s . This inequality follows in a straightforward manner
from the Hardy–Littlewood–Sobolev inequality along with well-known mixed-norm
Strichartz estimates (see [16, 8, 11] and [13]). The range 0 ≤ s < n2 is the best-possible
as one can see by using classical counterexamples.
In order to describe the refinements that we have in mind, we must first recast
the inequalities (2) in the L2-weighted form
‖eit∆f‖2L2x,t(V ) . ‖V ‖Lpx,t(Rn×R)‖f‖
2
H˙s(Rn), (3)
where V is an arbitrary element of Lpx,t(Rn × R) with p = n+22s+2 and 0 ≤ s < n2 .
This weighted formulation, which is more adapted to the study of perturbations of the
equation given in (1) by time-dependent potentials, naturally leads one to consider the
possibility of the existence of more subtle positive functionals V 7→ Cs(V ), for which
one might have control of eit∆f in the form
‖eit∆f‖2L2x,t(V ) . Cs(V )‖f‖
2
H˙s(Rn), (4)
for certain 0 ≤ s < n2 . Questions of this nature have been posed many times before in
related settings; see for example [15, 3] and [4].
The functionals that we shall introduce here are variants of the so-called Morrey–
Campanato norms, and are motivated by the closely related work of Ruiz and Vega
in [15] in the context of the stationary Schro¨dinger operator, and also by the works
in [9, 10] and [17]. For α > 0 and 1 ≤ p ≤ n+2α let
Lα,ppar =
{
F ∈ Lploc(Rn × R) : ‖F‖Lα,ppar <∞
}
,
where
‖F‖Lα,ppar = sup
(x,t)∈Rn×R,r>0
rα
(
1
rn+2
∫
C(x,t,r)
|F (y, s)|pdy ds
)1/p
,
∗ For non-negative quantities X and Y we use X . Y (X & Y ) to denote the existence of a
positive constant C, depending on at most n, p and s, such that X ≤ CY (X ≥ CY ). We write
X ∼ Y if both X . Y and X & Y .
Strichartz inequalities with weights in Morrey–Campanato classes 51
and C(x, t, r) denotes the “parabolic box” B(x, r) × (t − r2, t + r2). We observe that
these norms have the parabolic homogeneity
‖F (λ ·, λ2 ·)‖Lα,ppar = λ−α‖F‖Lα,ppar .
We notice also that Lα,ppar = Lp(Rn × R) when p = n+2α , and for p < n+2α the Lorentz
space Lp,∞(Rn×R) ⊂ Lα,ppar. Finally, for q < p we have the strict inclusion Lα,ppar ⊂ Lα,qpar.
These observations and scaling considerations raise the possibility that the inequalities
‖eit∆f‖2L2x,t(V ) . ‖V ‖L2s+2,ppar (Rn×R)‖f‖
2
H˙s(Rn), (5)
might hold for some p < n+22s+2 and 0 ≤ s < n2 , thus yielding improvements on (3).
Our main theorem states that these inequalities (5) do indeed hold, at least when
the number of derivatives s is sufficiently large.
Theorem 1.1
The estimate (5) holds if n4 ≤ s < n2 and 1 < p ≤ n+22s+2 .
Remark. Theorem 1.1 allows potentials strongly depending on time of the form
V (x, t) = |x|−a|t|−b with ap < n, bp < 1, a + 2b = 2s + 2 + n+2p , and p and s un-
der the conditions of the theorem. Notice that these potentials are never in Lebesgue
spaces.
In Section 2 we give a proof of Theorem 1.1 using a bilinear interpolation technique
due Keel and Tao [13] in the context of Strichartz mixed-norm inequalities.
In Section 3 we use the special solutions of the Schro¨dinger equation from [1] to
give the following necessary conditions on the exponents p and s for (5) to hold.
Proposition 1.2
The estimate (5) is false if 0 ≤ s < n4 and p < n+44(s+1) .
Figure 1 summarises all the results that we present here. The non-convexity
of the regions (s, 1/p) may be possibly due to the bad interpolation properties of
Morrey–Campanato classes, see [5].
In particular, we leave unanswered the perhaps more difficult question of
whether (5) might hold for functions f ∈ Hs for 0 ≤ s < n4 and some n+44s+4 ≤ p < n+22s+2 .
Remark. As we have already mentioned, the inequalities (5) were inspired by similar in-
equalities that are known to hold for the stationary Schro¨dinger operator, with weights
belonging to the classical Morrey–Campanato classes Lα,p (see [15]). As is well-known
(again see for example [15]), norm estimates for the stationary Schro¨dinger operator
imply related norm estimates for the time-dependent operator, which in this setting
leads to the validity of
‖eit∆f‖2L2x,t(V ) . ‖V ‖L2s+2,px (L∞t )‖f‖
2
H˙s(Rn), (6)
within a certain range of the exponents p and s (see [15] for the specific exponents
corresponding to s = 0, and the methods that extend in a straightforward manner
to s 6= 0). It should be pointed out that although the parabolic norm ‖ · ‖Lα,ppar is in
52 Barcelo´, Bennett, Carbery, Ruiz, and Vilela
n/ 2n/ 4
/p1
4/(   + 2)n
4/(   + 4)n
s
1
Figure 1: The region with the lightest shade of grey corresponds to where
the estimates (5) are known to be true, and the remaining darker shade
corresponds to where it is known to be false.
general much smaller than the mixed norm ‖ · ‖Lα,px (L∞t ), the interesting feature of the
latter is that it continues to be relevant for certain s < 0, yielding smoothing estimates
for eit∆f for f ∈ L2(Rn).
We remark that the examples we give in Section 3, yielding necessary conditions
for (5) to hold, are also effective in producing necessary conditions for (6) to hold
(see [2]).
2. The proof of Theorem 1.1
We begin by observing that (5) is equivalent to the inequality
‖Iseit∆f‖2L2x,t(V ) . ‖V ‖L2s+2,ppar (Rn×R)‖f‖
2
L2(Rn), (7)
where Is denotes the fractional integral operator of order s. By duality, this is in turn
equivalent to the bilinear inequality∣∣∣∣〈I2s ∫ +∞−∞ ei(t−τ)∆F (·, τ) dτ,G
〉∣∣∣∣ . ‖V ‖L2s+2,ppar ‖F‖L2x,t(V −1)‖G‖L2x,t(V −1).
Using the Fourier transform we can express the linear operator in the previous inequa-
lity as the convolution on Rn+1 of F with the kernel K, where
K(x, t) =
∫
Rn
e−it|ξ|
2+ix·ξ dξ
|ξ|2s . (8)
If we take a partition of unity by smooth functions {ϕj}j∈Z defined on Rn+1 such that
for each j
supp (ϕj) ⊆ B(0, 2j)× (−22j, 22j) \ B(0, 2j−2)× (−22(j−2), 22(j−2)),
Strichartz inequalities with weights in Morrey–Campanato classes 53
it is enough to prove that∑
j∈Z
|〈(ϕjK) ∗ F,G〉| . ‖V ‖L2s+2,ppar ‖F‖L2x,t(V −1)‖G‖L2x,t(V −1).
If we write {〈(ϕjK) ∗ F,G〉}j∈Z = T (F,G), then the previous inequality is equivalent
to a bound for
T : L2x,t(V −1)× L2x,t(V −1)→ `01(C). (9)
Here, in general, for a ∈ R and 1 ≤ p ≤ ∞, `ap(C) denotes the Banach space of
sequences x¯ = {xk}k∈Z such that
‖x¯‖`ap =

(∑
k∈Z
(
2ka|xk|
)p)1/p
<∞, if p 6=∞
sup
k∈Z
(
2ka|xk|
)
<∞, if p =∞.
Following the ideas of Keel and Tao [13], we now appeal to a bilinear interpolation
result (see [6, Section 3.13, Exercise 5(b)]).
Lemma 2.1
If A0, A1, B0, B1, C0 and C1 are Banach spaces, and T is a bilinear operator such
that
T : A0 ×B0 −→ C0,
T : A0 ×B1 −→ C1,
T : A1 ×B0 −→ C1,
then, whenever 0 < θ0, θ1 < θ = θ0 +θ1 < 1, 1 ≤ p, q ≤ ∞ and 1 ≤ 1p + 1q , we have that
T : (A0, A1)θ0,p × (B0, B1)θ1,q −→ (C0, C1)θ,1.
Here (C0, C1)θ,1 denotes the Banach space obtained from C0 and C1 by the real inter-
polation method. Implicit in this is that the bounds interpolate nicely.
We will prove that whenever n4 ≤ s < n2 and 1 < p ≤ n+22s+2 , the bilinear vector-
valued operator T satisfies
‖T (F,G)‖
`
β0∞ (C)
. ‖V ‖pL2s+2,ppar (Rn×R)‖F‖L2x,t(V −p)‖G‖L2x,t(V −p), (10)
‖T (F,G)‖
`
β1∞ (C)
. ‖V ‖p/2L2s+2,ppar (Rn×R)‖F‖L2x,t(V −p)‖G‖L2x,t , (11)
‖T (F,G)‖
`
β1∞ (C)
. ‖V ‖p/2L2s+2,ppar (Rn×R)‖F‖L2x,t‖G‖L2x,t(V −p), (12)
with β0 = (2s+ 2)(p− 1) and β1 = (2s+ 2)
(p
2 − 1
)
.
Using these estimates (10), (11) and (12), we may apply Lemma 2.1 to obtain an
appropriate interpolated bound for
T : (L2x,t(V −p), L2x,t)θ0,q0 × (L2x,t(V −p), L2x,t)θ1,q1 → (`β0∞(C), `β1∞(C))θ,1,
whenever 0 < θ0, θ1 < θ = θ0+θ1 < 1 and 1 ≤ 1q0 + 1q1 . The proof will then be completed
on taking θ0 = θ1 = 1− 1p , q0 = q1 = 2, and using the following real interpolation space
identities (see [6, Theorems 5.6.1 and 5.4.1]),
54 Barcelo´, Bennett, Carbery, Ruiz, and Vilela
(i) (`β0∞(C), `β1∞(C))θ,1 = `01(C) if β0 6= β1 and (1− θ)β0 + θβ1 = 0 and
(ii) (L2x,t(V
−p), L2x,t)θ0,q0 = L2x,t(V −1) and (L2x,t(V −p), L2x,t)θ1,q1 = L2x,t(V −1).
We now turn to the proofs of (10), (11) and (12). In order to obtain (10) we have to
prove that for each j ∈ Z,∣∣〈(ϕjK) ∗ F,G〉∣∣ . 2−j(2s+2)(p−1)‖V ‖pL2s+2,ppar (Rn×R)‖F‖L2x,t(V −p)‖G‖L2x,t(V −p).
For j fixed we decompose Rn+1 into a grid of rectangles {Qν}ν∈Zn+1 with disjoint
interiors and dimensions 2j × · · · × 2j × 22j , so that∣∣〈(ϕjK) ∗ F,G〉∣∣ ≤ ∑
ν∈Zn+1
∑
µ∈Zn+1
∣∣〈(ϕjK) ∗ (FχQν ), GχQµ〉∣∣.
Due to the disjointness of the supports of (ϕjK) ∗ FχQν and GχQµ , we have that∣∣〈(ϕjK) ∗ F,G〉∣∣ ≤ ∑
ν∈Zn+1
∣∣〈(ϕjK) ∗ (FχQν ), GχQ˜ν 〉∣∣,
where Q˜ν denotes the rectangle with the same centre as Qν but dilated three times
about its centre. From here, using Young’s inequality, the definition of K given in (8),
the case γ ≥ n/2 of Lemma 2.2 (see below) and the Cauchy–Schwarz inequality, we
obtain that∣∣〈(ϕjK) ∗ F,G〉∣∣ ≤ ∑
ν∈Zn+1
‖ϕjK‖L∞x,t‖FχQν‖L1x,t‖GχQ˜ν‖L1x,t
. 2−j(n−2s)
∑
ν∈Zn+1
‖FχQν‖L1x,t‖GχQ˜ν‖L1x,t (13)
. 2−j(2s+2)(p−1)‖V ‖pL2s+2,ppar ‖F‖L2x,t(V −p)‖G‖L2x,t(V −p).
This completes the proof of (10). Finally, inequalities (11) and (12) (which are
equivalent by symmetry) follow by a similarly straightforward application of the
Cauchy–Schwarz inequality in (13). 
Lemma 2.2
Let 0 < γ < n. Then if t 6= 0
∣∣∣∣∫
Rn
e−piit|ξ|
2+2piix·ξ dξ
|ξ|γ
∣∣∣∣ .

|t|−(n/2−γ)
(|x|2 + |t|)γ/2 if 0 ≤ γ ≤
n
2
,
1
(|x|2 + |t|)(n−γ)/2 if
n
2
≤ γ < n.
This estimate can be reduced to the case t = 1 by a change of variables, and is then
the content of [7, Lemma A.1].
Remarks
Strichartz inequalities with weights in Morrey–Campanato classes 55
(1) Following the same argument used to prove (10) and (11), but using Young’s
inequality in (13) with more general exponents, we obtain similar such estimates
when s is smaller than n4 , with the space L
2
x,t(V
−p) replaced by L2x,t(V −q) for
0 ≤ q < p, β0 = (2s+ 2)(q − 1) + n
(
1− q
p
)
and
β1 = (2s+ 2)
(q
2
− 1
)
+ n
(
1− q
p
)
.
However, the desired estimate (5) for s < n4 fails to follow from this since the
interpolation method that we use requires that (1−θ)β0 +θβ1 = 0 with θ = 2− 2q ,
and thus q = p. We include the relevant oscillatory integral estimate (for γ < n2 )
in the statement of Lemma 2.2 merely for the sake of completeness.
(2) We note that for γ = n2 in Lemma 2.2 we have an exact formula,∫
Rn
e−piit|ξ|
2+2piix·ξ dξ
|ξ|n/2 =
Cn
|t|1/2|x|(n−2)/2 e
i|x|2/|t|J(n−2)/4
( |x|2
8|t|
)
.
This follows using polar coordinates and the identities (6.686-1-2) given on [12,
page 759]. Lemma 2.2, in this case, is easily seen to be sharp by appealing to
standard Bessel function asymptotics.
3. The proof of Proposition 1.2
Following [1], let 0 < δ << 1 and 0 < σ < 12 . We consider the function of one variable
g =
∑
`∈N, 1≤`≤δ−σ
χ
(`δσ−δ,`δσ+δ) . (14)
Note that g is simply the characteristic function of a union of disjoint, equally spaced
subintervals of [0, 1] of equal size. (If preferred, these characteristic functions may be
replaced by smooth compactly supported bump functions.) We now define f by
f̂(ξ) =
n∏
j=1
g(ξj), (15)
where ξ = (ξ1, . . . , ξn).
We now set X = {pδ−σ : p ∈ N with p . δσ−1}, Ω to be an O(1)-neighbourhood
of
Λ =
{
(x, t) ∈ Rn × R : x ∈ Xn and t = 2qδ−2σ where q ∈ N and q . δ2σ−1},
and V the characteristic function of the set Ω.
Arguing as in [1], for (x, t) ∈ Ω we have the uniform bound∣∣eit∆f(x)∣∣ = ∣∣∣∣∫
Rn
e−it|ξ|
2+ix·ξ f̂(ξ) dξ
∣∣∣∣ (16)
∼
∫
Rn
f̂(ξ) dξ ∼ δn(1−σ),
56 Barcelo´, Bennett, Carbery, Ruiz, and Vilela
and hence,
‖eit∆f‖L2x,t(V ) ∼ δ
n(1−σ)|Ω|1/2 = δ(1−σ)n/2+σ−1/2. (17)
On the other hand,
‖f‖H˙s(Rn) = ‖| · |sf̂‖L2(Rn) . δ(1−σ)n/2, (18)
and for 1 ≤ p ≤ n+22s+2 ,
‖V ‖L2s+2,ppar ∼ max
{
1, δ(σ(n+2))/p−s−1, δ(σ(n+2)+1)/p−2s−2
}
.
Letting δ → 0 now leads the required necessary condition for (7) (and hence (5)). 
References
1. J.A. Barcelo´, J.M. Bennett, A. Carbery, A. Ruiz, and M.C. Vilela, Some special solutions of the
Schro¨dinger equation, Indiana Univ. Math. J. 56 (2007), 1581–1593.
2. J.A. Barcelo´, J.M. Bennett, A. Carbery, A. Ruiz, and M.C. Vilela, A note on weighted estimates
for the Schro¨dinger operator, Rev. Mat. Complut. 21 (2008), 481–488.
3. J.A. Barcelo´, A. Ruiz, and L. Vega, Weighted estimates for the Helmholtz equation and some
applications, J. Funct. Anal. 150 (1997), 356–382.
4. J.M. Bennett, A. Carbery, F. Soria, and A. Vargas, A Stein conjecture for the circle, Math. Ann.
336 (2006), 671–695.
5. O. Blasco, A. Ruiz, and L. Vega, Non-interpolation in Morrey–Campanato and block spaces, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 31–40.
6. J. Bergh and J. Lo¨fstro¨m, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New
York, 1976.
7. T. Cazenave, L. Vega, and M.C. Vilela, A note on the nonlinear Schro¨dinger equation in weak Lp
spaces, Commun. Contemp. Math. 3 (2001), 153–162.
8. T. Cazenave and F.B. Weissler, The Cauchy problem for the nonlinear Schro¨dinger equation in
H1, Manuscripta Math. 61 (1988), 477–494.
9. S. Chanillo and E. Sawyer, Unique continuation for ∆ + V and the C. Fefferman–Phong class,
Trans. Amer. Math. Soc. 318 (1990), 275–300.
10. C. Fefferman and D.H. Phong, Lower bounds for Schro¨dinger equations, Conference on Partial
Differential Equations (Saint Jean de Monts), Conf. No. 7, Soc. Math. France, Paris, 1982.
11. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schro¨dinger equation
revisited, Ann. Inst. H. Poincare´ Anal. Non Line´are 2 (1985), 309–327.
12. A. Jeffrey (Ed.) Table of Integrals, Series and Products, Academic Press, New York-London,
1965.
13. M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.
14. A. Ruiz and L. Vega, Unique continuation for Schro¨dinger operators with potential in Mor-
rey spaces, Conference on Mathematical Analysis (El Escorial, 1989), Publ. Mat. 35 (1991),
291–298.
15. A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent poten-
tials, Duke Math. J. 76 (1994), 913–940.
16. R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of
wave equations, Duke Math. J. 44 (1977), 705–714.
17. T.H. Wolff, Unique continuation for |∆u| ≤ V |∇u| and related problems, Rev. Mat. Iberoame-
ricana 6 (1990), 155–200.


Strichatz inequalities with weights in Morrey-Campanato classes

Abstract

Similar works

Full text

Available Versions

Revistes Catalanes amb Accés Obert