Smoothing Estimates for the Wave Equation
and Applications
CHRISTOPHER D. SOGGE*
Mathematics Department
University of California
Los Angeles, CA 90024, USA
The purpose of this paper is to go over some recent results in analysis and partial
differential equations that are related to regularity properties of solutions of the
wave equation
(Du(t,x)=0
\ u(0, x) = f(x), dtu(0, x) = g(x).
Here, D = d2 jdt2 — A, where A is either the Euclidean Laplacian on R n or a
Laplace-Beltrami operator on a compact n-dimensional manifold Mn (without
boundary unless otherwise stated).
Fixed-time Lp -» Lp estimates
In the Euclidean case, the most basic estimate of course is the energy identity:
/ \Vt,Mt,x)fdx=
f (\Vxf(x)\2 + \g(x)\2)dx,
(2)
JR"
jRn
which just follows from integration by parts or a simple application of the Fourier
transform. If one is interested in the L2 norm of the solution, then a related
estimate, which also follows directly from Plancherel's theorem, is
H^oilL^II/ll^ + a + t 2 ) 1 / ^ ^ ^ ,
(3)
if L£ denotes the usual Lp Sobolev space with norm | | / | | L P = \\(I - A ) Q / 2 / | | L P .
Both (2) and (3) easily generalize to the setting of manifolds as well.
The fixed-time Lp, p ^ 2, behavior of the wave equation is much less favorable, and sharp estimates are harder to obtain. For the Euclidean version, Miyachi
[31] and Perai [36] showed that for 1 < p < oo
IN*, • )||Lp < Cp,t ( ||/|| L g p + Wgh*^
), % = (n - 1)1 è - ? I •
(4)
Simple counterexamples show that these estimates are sharp. If one just takes /
to be a nontrivial cutoff times either \x\ap~n/p or (1 — | ^ | ) Q p _ 1 / p , depending on
The author was supported in part by the National Science Foundation.
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
Smoothing Estimates for the Wave Equation and Applications
897
whether p is < 2 or > 2, respectively, then w(±l, •) g Lp(Rn), while / G L£ if
a < Qp. For small times, Beals [1] extended (4) to variable coefficients.
The proofs of these fixed-time estimates relied on the fact that, in the Euclidean setting, or the small-time manifold setting, the kernel of the solution operator has a very simple form; specifically, that it is a conormal distribution. The
techniques of [1], [31], [36] rely on stationary phase and break down when this
is not the case. This sort of situation can of course occur for large times on a
manifold. Using different techniques related to the plane wave decomposition of
the solution of the Euclidean wave equation, Seeger, Stein, and the author [39]
showed that (4) holds for all times on a manifold. In fact, a more general result
concerning Fourier integral operators holds.
To describe this, let us assume that a(x,Ç) G C°° vanishes for x outside of a
compact set and satisfies
\9£SÇa(x,t)\<Ca„{l
+ \Ç\)-™.
Assume also that ip(x,£) is real, homogeneous of degree one in £, smooth away
from £ = 0, and satisfies
det d2tp/dxjd£k ^ 0
(5)
on supp a. Then, if / denotes the Fourier transform of / , and if we let
Faf(x)
= f
e ^ t y x , 0 ( 1 + |Ç|)°/(0 <*£,
(6)
JR™
it was shown in [39] that for 1 < p < oo
Il^a/IUP(R») < CpH/HiPO»»), a = - ( n - 1)| i - Ì |.
(7)
This contains the fixed-time estimates mentioned above because u(t, • ) can always
be decomposed into a finite sum of operators of this type acting on the data.
The examples showing that (4) is sharp can easily be adapted to this context to
show that (7) cannot be improved for conormal operators whose singular support
has codimension one, or, more succinctly, for operators with phases satisfying
rank d2(p/d£jd£k = n — 1 somewhere.
The main step in the proof of (7) is to show that, even though the limiting
L1 estimate with a = —(n — l ) / 2 is false, dyadic versions of this estimate hold.
Specifically, if ß G Cj?( (1/2,2) ), and we set
Txf(x)=
[
c^'O)9(|f|/A)a(x,0(l + | f | ) - ^ / ( 0 d e ,
then one has
||TA/||LI
< C||/||L. ,
A = 2 \ fc = l , 2 , . . . .
(6')
To do this, one breaks up the dyadic operator T\ into "angular pieces," T\ =
Ei=i
ÏA, where
3^/(1)= / e^)X,(0/3(lelM)a(z,0(l + l £ i r V / Ì 0 ^
jRn
898
Christopher D. Sogge
with each \v being homogeneous of degree zero, and satisfying D7xu(0 = 0(A~2~),
|£| = 1. Thus, one should think of { \ v } as a smooth partition of unity of K"\0,
with each term being supported in a cone of aperture 0 ( A - 1 / 2 ) , and the derivatives
of Xv satisfying the natural bounds associated with this decomposition. The reason
for this decomposition is that, in the right scale, £ —> tp(x,£) behaves like a linear
function of £. Based on this, it is not hard to show that one can estimate each
resulting piece in terms of the size of its symbol:
a
L
mf\\v<c\-w \\f\\ i,
which of course yields (6'). This sort of decomposition was first used in the related
context of Riesz means by Fefferman [11], as well as by Christ and the author [7]
and Cordoba [8]. Also, Smith and the author [42] showed that the estimates (4)
also extend to the setting of the wave equation outside of a convex obstacle, and
that estimates related to (7) also hold for Fourier-Airy operators.
Space-time Lp —> Lp estimates
Using (4) and (7), one can apply Minkowski's integral inequality to see that u G
Lpoc(dtdx) if (/,#) G Z/*^ x Lpa _v However, for many applications it is useful to
know whether the regularity assumptions can be weakened, if one considers the
space-time rather than spatial regularity properties of u. If p < 2, one can use the
counterexample for the sharpness of (4) to see that, for this range of exponents,
in general, the local space and space-time regularity properties are the same.
On the other hand, if n > 2 and p > 2 there is an improvement. Specifically,
given p > 2, there is an ep > 0 so that if ST = {(£, x) : 0 < t < T}
ll«(*,*)IU-(Sr)<Ci,.r(||/||LS . +NUs , . )•
(8)
Such a "local smoothing estimate" was first obtained in [45] in the Euclidean case
when n = 2 by the author, using related results from Bourgain [2] and Carbery [5].
Later, a much simpler proof was given by Mockenhaupt, Seeger, and the author
[32], and the extension to variable coefficients and higher dimensions was carried
out in [33]. The first local smoothing estimate for differential equations seems to
go back to Kato [24], who showed that for the K-dV equation there is a local
smoothing of order 1 in L2, due to the dispersive nature of the equation.
As before, (8) generalizes to the setting of Fourier integrals. To be specific,
let us consider operators of the form (6), where now, with an abuse of notation,
x ranges over M 1 + n . Then to improve on trivial consequences of (7), one needs
certain conditions on the phase. First, one needs the "nondegeneracy condition"
that (5) is replaced by the condition that the Hessian appearing there has full rank
n everywhere. This amounts to saying that the projection of the canonical relation
into T*R n \0 is submersive. In addition, one needs a curvature requirement that is
based on the properties of the projection of the canonical relation into the fibers
of the bigger cotangent bundle, T*R 1 + n \0. It turns out that the nondegeneracy
condition implies that the images of these projections must all be smooth conic
hypersurfaces, and they are explicitly given by
r x = { Vx<pOr, 0 : £ G M n \0 } C Tx*M1+n\0 = R 1 + n \ 0 .
Smoothing Estimates for the Wave Equation and Applications
899
Our other requirement, "the cone condition," is that at every rj G Tx there are n—1
nonvanishing principal curvatures. Together the nondegeneracy condition and the
cone condition make up the cinematic curvature condition introduced in [45]. This
condition turns out to be the natural homogeneous version of the Carleson-Sjölin
condition in [6] (see [33], [46]).
If cinematic curvature holds, it was shown in [33] that (7) can be improved
if n > 2 and p > 2, that is, there is an ep > 0 so that
II^a/||LP(Ri+-) < C H / H L ^ ä » ) , a = -(n
- 1)( \ - \ ) -r Ep .
(9)
The proof uses the decomposition employed for the fixed-time estimate, along with
ideas from the related work of Fefferman [11], Carbery [5], and Cordoba [8] to allow
one to exploit the curvature implicit in the cone condition.
The main motivation behind (8) and (9) concerns applications for maximal
theorems. For instance, if in R 2 , Atf(x) denotes the average of / over the circle
of radius t centered at x,
Atf{x)
= f1
Js
f{x-iß)dB,
then Bourgain [2] showed that
|| sup \Atf(x)\
|| LP(R2) < C P | | / | | L P ( R 2 ) , p > 2.
(10)
Earlier, in effect by using an ingenious square function argument based on bounds
that are equivalent to (3), Stein [48] obtained the higher-dimensional version of this
result. Stein's theorem, which inspired much of the work described in this paper,
says that when n > 3 the spherical maximal operator is bounded on Lp(Rn)
if p > n/(n — 1). He also showed that when n > 2 no such result can hold if
p < n/(n — 1). Thus, as 2 is the critical Lebesgue exponent for this problem in two
dimensions, and as At is a Fourier integral operator of order —1/2, one cannot use
fixed-time estimates like (3) or (4) to obtain (10). Hence, if one wishes to use an
argument like Stein's for the two-dimensional setting, harder space-time estimates
like (9) are required.
Using these smoothing estimates one can recover (10) in a straightforward
manner. In fact, if ß G CQ)( (1,2) ), 2 < p < oo, and e > 0, then Sobolev's lemma
yields
|| sup \ß(t)Atf(x)\ || t s < Ce,p\\ (I - d2)e+iß(t)Atf(x)
t
|| L P X .
(11)
'
However, modulo a trivial error, / — > ( / - d2)**+%ß(t)Atf(x)
is the sum of two
operators of the form (6) with ip = x • £ ± t\£\ and a = — \ + -+e. Thus, if e < ep,
(9) implies that the right side of (11) is controlled by the LP norm of / . This
argument thus gives a slightly weaker version of (10), where the supremum in the
left is taken over t E (1,2); however, if one uses Littlewood-Paley theory (see [33],
[46]), a simple variation yields the full circular maximal theorem of Bourgain.
900
Christopher D. Sogge
This argument of course also applies to certain variable coefficient maximal
operators. For instance, if one takes the supremum over small enough radii, one
can obtain a variant of (10) where one averages over geodesic circles on twodimensional Riemannian manifolds (see [45]). Also, it was shown by Iosevich [19]
that there is a natural extension of (10) to averages over finite type curves. Also,
in [47], the local smoothing estimates (9) were used to show that (10) extends to
averages in higher dimensions over hypersurfaces with at least one nonvanishing
principal curvature.
Another application was pointed out to us by Wolff. If in the plane
A6f(t,x) = ô-1 [
f(y)dy,
J\x-y\e(t,t+6)
then the above arguments yield for 0 < 6 < 1
|| sup \A6f(t,x)\
kl<i
|| LP([li2]) < CS''1'*
||/|| L P(R2)
, Ve < ep.
(12)
It turns out that (9) holds for any ep < 1/8 if p = 4. From (12) for p = 4, one can
then deduce, using ideas from Bourgain [3], that a set in the plane containing a
translate of a circle of every radius must have Hausdorff dimension > 3/2. It was
shown by Besicovitch (see [9]) that there are such sets of measure zero; however,
it is felt that their Hausdorff dimension must always be 2. Showing this is related
to a deeper problem concerning the Hausdorff dimension of Besicovitch (3,1) sets:
sets of measure zero in R 3 containing a line in every direction. It is conjectured
that such sets must have full Hausdorff dimension. Bourgain [3] showed that they
must have dimension > 7/3, and this result has recently been improved by Wolff
[54], who showed that, as in the previous case, the Hausdorff codimension must
always be < 1/2, that is, every Besicovitch (3,1) set must have dimension > 5/2.
It was conjectured in [45] that for p > 2n/(n — 1) there should be local
smoothing of order 1/p, that is, that (9) should hold for all e < 1/p for this range of
exponents. This conjecture would imply the Bochner-Riesz conjecture, so in higher
dimensions it seems presently unattainable. However, in view of the Carleson-Sjölin
theorem [6], there might be some hope of verifying the conjecture in (1 + 2)dimensions. If true here, it would imply the conjecture that sets of measure zero
in the plane containing a circle of every radius must have full Hausdorff measure.
In the radially symmetric case the conjecture for solutions of the Euclidean wave
equation was verified in all dimensions by Müller and Seeger [34].
L2 —> Lq estimates
It is much easier to obtain sharp estimates involving L2. For instance, using L2
bounds for Fourier integrals of Hörmander [17] and the Hardy-Littlewood-Sobolev
inequality one obtains the sharp estimate
II ^a/IU^R») < Cq\\f\\L2{Rn) , a = - f + ^ , 2 < g < o o ,
(13)
Smoothing Estimates for the Wave Equation and Applications
901
if T& is as in (7). However, if T& sends functions of n-variables to functions of
(1 + n)-variables there is local smoothing of order 1/q for a range of exponents if
the cinematic curvature condition holds:
l|-F«/IU.(Ri+») < Cq\\f\\L*(*«), oc = - § + a ± l , ygp-
<q<oo.
(14)
This result was obtained by Mockenhaupt, Seeger, and the author [33], and it
generalizes the important special case of Strichartz [52] for the Euclidean wave
equation, which says that if q is as in (14)
\Ht,x)\\Lq(Rl-n)
< Cq ( ||/|| Ä „ + \\g\\Èa-, ) , a = f - 2 ± 1 .
(15)
Here H1 denotes the homogeneous L2 Sobolev space with norm ||(—A) 7 / 2 /||^2.
Earlier partial results go back to Segal [40]. The dual version of (15) is equivalent
to the following restriction theorem for the Fourier transform:
/ \mu)\2^mr2{n+mp-1)/p
< cP\\F\\iHR1+n),
i<p<2J$n.
JR"-
This in turn is related to the earlier I? restriction theorem of the Fourier transform
for spheres of Stein and Tomas [53]. Inequality (14) contains a local extension of
(15) to variable coefficients and the latter was independently obtained by Kapitanski [22]. Strichartz estimates were also obtained by Smith and the author [43]
for the wave equation outside of a convex obstacle.
Mixed-norm estimates and minimal regularity for nonlinear equations
Mixed-norm estimates, where different exponents are used for the time and spatial norms, are often useful for applications in analysis and partial differential
equations. If we write (x, y) G W1 x R m = R n + m , then mixed-norms are given by
\\F(x,y)\\L%L?(Rn+m)
= ( /
JRn
( /
\F(x,y)\"dy)q/pdx) 1/q
JRm
In analysis, typical applications arise from the case where m = 1 and p = 2,
and the estimates involving norms of this types are called square function estimates. We already pointed out that square function estimates for solutions of the
Euclidean wave equation (1) lead to Stein's spherical maximal theorem. Related
harder estimates that are equivalent to L^L 2 (R 2 + 1 ) estimates for (1) were used
by Carbery [5] to obtain estimates for maximal Bochner-Riesz operators in the
critical space L^(R 2 ).
Square function variants of (15) can also be used to obtain sharp estimates
for eigenfunctions. In fact, it was shown in [33] that if one uses L2 instead of L\
one can improve (15) and get for ST = {(x, t) : 0 < t < T}
IMIL*L?(S.) < CT( U/H*. + Hflll*,-! ) , OC = Syl - 2 , «gp.
< q < oo .
(16)
902
Christopher D. Sogge
Thus, there is a gain of 1/2 of a derivative over the fixed-time estimates (13). In
the Euclidean case, the bounds are independent of T, whereas they may not be
on a compact manifold. In [43], Smith and the author showed that these estimates
also hold on compact manifolds with concave (i.e., diffractive) boundary. When
q = co a dyadic version of (16) holds. Using these square function estimates one
can obtain "sharp" bounds for eigenfunctions. In fact, if —Ae\ = A2e^ on either
a smooth compact boundaryless manifold or a (relatively) compact manifold with
concave boundary, one has
||eA||L,(M") < C(l + \)*M ||e A || L 2 ( M n ) ,
(16')
where a(q) = *=± - * if ^
< q < oo, and a(q) = (»-»£-*) if 2 < g <
^ .
The bounds for the first range follow from (16) because if / = e\ and g = 0,
u(t,x) = cost\e\(x),
and hence the left side of (16) is comparable to the left side
of (16'). The bounds for the other range follow via interpolation. Although the
bounds in (16') are not sharp in general, a slightly stronger result holds that is
always sharp. It says that (16') holds if e\ is replaced by a function with spectrum
in [A2, (A + l) 2 ]. In the boundaryless case these estimates were first proved by the
author [44] using a different, but related approach, based on proving estimates for
the standing wave operators A + A2. For the case of manifolds with diffractive
boundaries, the estimates are due to Grieser [14] and Smith and the author [43].
It would be very interesting to know to what extent these bounds carry over
to the setting of general (relatively) compact manifolds with boundary. Results of
Ivrii [20] imply that the L°° bounds always hold. However, Grieser [14] showed
that one cannot have (16') with a(q) = ^^ — ^ for the full range ™-\ — Q — °°
if the second fundamental form of the metric has a negative eigenvalue at some
point of the boundary. One might expect, though, that in this case they might hold
if q > | ^ | , which is the largest range allowable by Grieser's counterexample.
Mixed-norm estimates also have important applications for semilinear wave
equations. This observation first seems to have been made by Pécher [35], where
a variant of the energy estimate (2) was proved, with the right side dominating
mixed-norms that scale like the left side of (2). These estimates are in the spirit
of the Strichartz estimates (15), and they were used by Pécher to prove scattering
theorems for equations like Du = {ul^^ on R 1 + n involving data with small energy.
Subsequently, these type of mixed-norm estimates, along with (15), were used to
prove Grillakis' theorem saying that there are global C°° solutions of the critical
wave equation Du + u5 = 0 in R 1 + 3 with arbitrary C°° Cauchy data (see [15], [41],
as well as [43] for the extension to the obstacle case).
Mixed-norm estimates and smoothing estimates are also useful for proving
existence theorems with minimal regularity for other powers. In [28], Lindblad and
the author showed that if \3w = F in R 1 + 3 and if w has vanishing Cauchy data at
t = 0, then
HI
JU,
L,9"2 L^R 1 * 3 )
<Cq\\F\\
_*_ ^_
,2<q<oc.
(17)
L?~l L2+2 (R!+3)
This inequality is related to earlier Besov space estimates of Ginibre and Velo [12],
[13] and Kapitanski [22]. Using it and estimates for the linear Cauchy problem (1)
Smoothing Estimates for the Wave Equation and Applications
903
that are related to (15), it was shown in [28] that if K > 2 then there are local
(weak) solutions of
Du = \u\K , u(0,x) = f G i P ( R 3 ) , dtu(0,x) =ge
iJ7"1^3),
(18)
provided that 7 = max( § — ^ - , 1 - -^ ) . In the superconformai range, K > 3,
there is also global existence for small data. Scaling arguments show that 7 must
always be > | — ^ j , while a counterexample related to an example giving the
sharpness of (15) shows that 7 must also be > 1 — -^j. The local existence results
were also obtained independently by Kapitanski [23].
If one assumes radial symmetry, it turns out that (15) holds for a larger range
of exponents, namely, q > ^ j . For related reasons, the counterexamples in [28]
no longer apply, and, consequently, one expects better local and global existence
results if / and g in (18) are assumed to be radially symmetric. In fact, recently,
Lindblad and the author [29] have established a stronger theorem under these
assumptions, using this fact about (15) along with a stronger version of (17) that
only holds under the assumptions of radial symmetry, and is proved using ideas
from [26] and [34]. Specifically, if T£ = 00 for K > 1 + \/2, T£ = cs"*-*«-* for
2 < K < 1 + V% and Te = e x p ^ - ^ * - 1 ) ) for K = 1 + \fl, and if c,e > 0 are
sufficiently small there is a weak solution of (18) in [0, T£) x R 3 provided that
the data has H1 x H1~l norm < e with 7 = max( | — ^ y , | — £ ). We already
remarked on the necessity of the first condition. The second is also needed because
if UQ is the solution to the linear wave equation with this data, \UQ\K need not be
a distribution if 7 < \ — \. Notice that, for positive powers,
Also, if « < 1 + y/2, it was shown in [21], [27], and [55] that the lifespan bounds
in the formula for T£ are optimal.
This result is of course related to John's [21] existence theorem, which says
that if compactly supported data (/,<?) G C 3 x C2 are fixed, then Du = \u\K,
(u(0,x),dtu(0,x))
= e(f,g) always has a global C2 solution for small e > 0 if and
only if « > 1 + \/2. The positive part of John's theorem is proved by an ingenious
argument based on iterating in the space with norm
|| (1 + t)(l + 11 - \x\ \)K-2
sup \u(t, \x\u>)\ l u » .
John's argument, following an idea going back to Keller [25], only involves radial
estimates because the positivity of the forward fundamental solution for D in R 1 + 3
allows one to reduce matters to only proving estimates for radially symmetric functions by first taking supremums over the angular variables UJ G S2. If one iterates
instead in L^Lj^.L^, the arguments used to prove the sharp radial existence theorem mentioned above also allow one to recover John's existence theorem, and
the extensions of Lindblad [27] and Zhou [55], which say that there is a solution
u G C 2 ([0, T£) x R 3 ) if 2 < K < 1 + \/2 and K = 1 + y/2, respectively.
904
Christopher D. Sogge
References
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[II]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
M. Beals, Lp boundedness of Fourier integrals, Mem. Amer. Math. Soc. 264 (1982).
J. Bourgain, Averages in the plane over convex curves and maximal operators, J.
Analyse Math. 47 (1986), 69-85.
, Besicovitch type maximal operators and applications to Fourier analysis,
Geom. Functional Anal. 1 (1991), 69-85.
P. Brenner, On Lp — Lp> estimates for the wave equation, Math. Z. 145 (1975),
251-254.
A. Carbery, The boundedness of the maximal Bochner-Riesz opearator on L 4 ( R 2 ) ,
Duke Math. J. (1983), 409-416.
L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disk,
Studia Math. 44 (1972), 287-299.
F. M. Christ and C D . Sogge, The weak type L convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math. 94 (1988), 421-451.
A. Cordoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), 505-511.
K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, Cambridge,
1985.
C. Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971),
330-336.
, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44-52.
J. Ginibre and G. Velo, Conformai invariance and time decay for nonlinear wave
equations, II, Ann. Inst. H. Poincaré Phys. Théor. 47 (1987), 263-276.
, Scattering theory in the energy space for a class of nonlinear wave equations,
Comm. Math. Phys. 123 (1989), 535-573.
D. Grieser, Lp bounds for eigen]"unctions and spectral projections of the Laplacian
near concave boundaries, thesis, UCLA (1992).
M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm.
Pure Appi. Math. 45 (1992), 749-774.
L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968),
193-218.
, Fourier integral operators I, Acta Math. 127 (1971), 79-183.
, Oscillatory integrals and multipliers on FLP, Ark. Mat. 11 (1971), 1-11.
A. Iosevich, Maximal operators associated to the families of flat curves in the plane,
Duke Math. J. 76 (1994), 633-644.
V. Ivrii, Precise spectral asymptotics for elliptic operators, Lecture Notes in Math.
1100, Springer-Verlag, Berlin and New York, 1984.
F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions,
Manuscripta Math. 28 (1979), 235-265.
L. Kapitanski, Some generalizations of the Strichartz-Brenner inequality, Leningrad
Math. J. 1 (1990), 693-726.
, Weak and yet weaker solutions of semilinear wave equations, Brown Univ.
preprint, Providence, RI.
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,
Studies in Applied Math., vol. 8, Academic Press, New York and San Diego, 1983,
pp. 93-128.
J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appi. Math. 10
(1957), 523-530.
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local
existence theorem, Comm. Pure and Appi. Math. 46 (1993), 1221-1268.
Smoothing Estimates for the Wave Equation and Applications
905
[27] H. Lindblad, Blow-up for solutions of Du = \u\p with small initial data, Comm.
Partial Differential Equations 15 (1990), 757-821.
[28] H. Lindblad and C D . Sogge, On existence and scattering with minimal regularity
for semilinear wave equations, J. Funct. Anal., to appear.
[29]
, forthcoming.
[30] W. Littman, The wave operator and IP-norms, J. Math. Mech. 12 (1963), 55-68.
[31] A. Miyachi, On some estimates for the wave operator in Lp and Hp, J. Fac. Sci.
Univ. Tokyo Sect. IA Math. 27 (1980), 331-354.
[32] G. Mockenhaupt. A. Seeger, and C D . Sogge, Wave front sets, local smoothing and
Bourgain's circular maximal theorem, Ann. Math. (2) 136 (1992), 207-218.
[33]
, Local smoothing of Fourier integrals and Carleson-Sjölin estimates, J. Amer.
Math. Soc. 6 (1993), 65-130.
[34] D. Müller and A. Seeger, Inequalities for spherically symmetric solutions of the wave
equation, Math. Z., to appear.
[35] H. Pécher, Nonlinear small data scattering for the wave and Klein-Gordon equations,
Math. Z. 185 (1984), 261-270.
[36] J. Perai, Lp estimates for the wave equation, J. Funct. Anal. 36 (1980), 114-145.
[37] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157.
[38] A. Seeger and C D . Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), 669-682.
[39] A. Seeger, C D . Sogge, and E. M. Stein, Regularity properties of Fourier integral
operators, Ann. Math. (2) 134 (1991), 231-251.
[40] I. E. Segal, Space-time decay for solutions of wave equations, Adv. in Math. 22
(1976), 304-311.
[41] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of
Math. (2) 138 (1993), 503-518.
[42] H. Smith and C D . Sogge, IP regularity for the wave equation with strictly convex
obstacles, Duke Math. J. 73 (1994), 123-134.
[43]
, On the critical semilinear wave equation outside convex obstacles, Amer.
Math. Soc, to appear.
[44] C D . Sogge, Concerning the Lp norm of spectral clusters for second order elliptic
operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-134.
[45]
, Propagation of singularities and maximal functions in the plane, Invent.
Math. 104 (1991), 349-376.
[46]
, Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge,
1993.
[47]
, Averages over hypersurfaces with one non-vanishing principal curvature,
Fourier analysis and partial differential equations (J. Garcia-Cuerva, ed.), CRC
Press, Boca Raton, FL, 1995, pp. 317-323.
[48] E. M. Stein, Maximal functions: Spherical means, Proc. Nat. Acad. Sci. 73 (1976),
2174-2175.
[49]
, Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, NJ, 1988, pp. 307-356.
[50]
, Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993.
[51] R. Strichartz, A priori estimates for the wave equation and some applications, J.
Funct. Anal. 5 (1970), 218-235.
[52]
, Restriction of Fourier transform to quadratic surfaces and decay of solutions
to the wave equation, Duke Math. J. 44 (1977), 705-714.
906
Christopher D. Sogge
[53] P. Tomas, Restriction theorems for the Fourier transform, Proc. Sympos. Pure Math.
35 (1979), 111-114.
[54] T. Wolff, An improved bound for Kakeya type maximal functions, preprint.
[55] Y. Zhou, Blow up of classical solutions to Du = \u\ + a in three space dimensions,
J. Partial Differential Equations Ser. A 5 (1992), 21-32.