Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Carleman Estimates, Uniform Sobolev Inequalities
for Second-Order Differential Operators,
and Unique Continuation Theorems
CARLOS E. KENIG
1. Introduction, background, and history. It is well known that if
P(x,D) is an elliptic differential operator, with real analytic coefficients, and
P(x,D)u = 0 in an open, connected set fi C R n , then u is real analytic in fi.
Hence, if there exists XQ G fi such that u vanishes of co order at xo, u must be
identically 0. If a differential operator P(x, D) has the above property, we say
that P(x,D) has the strong unique continuation property (s.u.c.p.). If, on the
other hand, P(x,D)u = 0 in fi, and u = 0 in fi', an open subset of fi, implies
that u = 0 in fi, we say that P(x,D) has the unique continuation property
(u.c.p.). Finally, if P(x, D)u = 0 in fi, and supp w c i T c Ü implies that u = 0
in fi, we say that P(x,D) has the weak unique property (w.u.c.p.).
Through the work of Hadamard [19] on the uniqueness for the Cauchy problem, and Holmgren's uniqueness theorem (which are strongly related to the
unique continuation property), it became clear that it would be desirable to
establish the unique continuation property for operators whose coefficients are
not necessarily real analytic, or even of class C°°. The first results in this direction are to be found in the pioneering work of T. Carleman [12] in 1939. He
was able to show that P(x,D) = A + V(x) in R 2 has the s.u.c.p. whenever
the function V(x) is in L^C(K2). In order to prove this result he introduced a
method, the so-called Carleman estimates, which has permeated almost all the
subsequent work in the subject. In this context, a Carleman estimate is, roughly
speaking, an inequality of the form
\\e»f\\mo)<C\\e»*f\\mtr),
for all / G CQ°(U), U an open subset of R 2 , and a suitable function 0, where the
constant C is independent of A, for a sequence of real values of À tending to oo.
As we shall see later on, such estimates readily give the s.u.c.p. for A -I- V(x),
Supported by the National Science Foundation.
© 1987 International Congress of Mathematicians 1986
948
CARLEMAN ESTIMATES
949
V G Zq£ c (R 2 ). By now, there is a large literature of applications of Carlemantype estimates to uniqueness questions. (See Hörmander [25].)
Carleman's result was extended to R n by C. Müller [34] in 1954. Subsequently, there was a great flurry of activity in the subject, in the late fifties
and early sixties. Most notable among these works are the contributions of H.
0 . Cordes [15], N. Aronszajn [5], L. Nirenberg [36], the fundamental work of
A. P. Calderón [11] on the uniqueness for the Cauchy problem, the work of L.
Hörmander [20, 21], of N. Aronszajn, A. Krzywicki, and J. Szarski [6], and of S.
Agmon [1]. In the context of the strong unique continuation property, the best
result was the one of Aronszajn, Krzywicki, and Szarski [6], who showed that if
{a>jk(%)} is a real, symmetric, positive definite, n X n matrix that is Lipschitz
continuous for x G R n , and if u verifies the differential inequality
d2
oxjOXk
QOt
H
^
-—u(x)
dx<* K J
(*)
in a connected neighborhood of 0 in R n , and u vanishes of infinite order at 0,
then u must be identically 0 if Ca(x) G Lj£ c (R n ). They established their result
by means of an appropriate Carleman estimate. Moreover, an example of Plié
shows that the regularity assumed on the coefficients {a,jk(x)} is optimal. In fact
A. Plia [37] showed the existence of a nonzero solution u, vanishing in an open
set, whenever {a>jk(%)} are Holder continuous of any order < 1. At this point
we would like to mention the recent work of N. Garofalo and F. H. Lin [17],
who established the result of Aronszajn, Krzywicki, and Szarski without using
Carleman estimates, but using instead real variable methods and the theory of
AQQ weights developed by B. Muckenhoupt a.nd others [33, 14].
Recently, there has been growing interest in establishing unique continuation
results for solutions of differential inequalities such as (*), with IP conditions,
p < oo, on the lower-order coefficients (see Simon [44] and Kenig [29] for surveys of these problems). The reason for interest in these questions comes from
mathematical physics. Suppose, for example, that we consider the Schrödinger
operator H = - A + V(x) as a selfadjoint operator on the Hilbert space L 2 ( R n ) .
Here HQ = —A is the kinetic energy, and V(x) the potential energy. V(x) is
not supposed to be smooth, continuous, or even locally bounded. In fact, the
Coulomb potentials (V(x) = l/\x\) are unbounded, and appear in models for
hydrogen atoms.
A useful division of the spectrum of H (see, for example, Reed and Simon
[38] for the precise definitions) distinguishes among av, the point spectrum,
consisting of eigenvalues, isolated or not, and acont, the continuous spectrum,
which is associated to the restriction of H to the part of L 2 ( R n ) orthogonal to
eigenvectors. The definitions allow for the possibility that 0-cont a n d 0P intersect.
In one-body physics, the potential V tends to 0 at oo, and typically
0"cont = [0, OO),
<7p H O-cont = { 0 } Or 0 .
950
G. E. KENIG
This is expected on physical grounds. The relationship of this decomposition
to quantum physics is that orp comprises the energy of bound states, and the
spectral subspace associated with aCont consists of dynamical states that may
participate in scattering. The reason why one expects on physical grounds that
ffp D <7cont = {0} or 0 is that if the potential V tends to 0 at oo and the energy
of a particle is positive, one expects that quantum fluctuations would eventually
propel the particle to a place where its motion would not be confined, and this
would of course make the bound state large, and hence, not in L 2 ( R n ) . In 1929,
von Neumann and Wigner [54] constructed an example of a one-dimensional
potential V(x) that goes to 0 at oo, but with positive eigenvalues. (See also Reed
and Simon [38] for a study of this example, as well as Reed-Simon [38, Vol. 4,
Ex. 2, XIII] for a different kind of 3-dimensional example.) Thus we see that the
problem of determining for which potentials we have no positive eigenvalues is a
difficult one. The most successful philosophy for eliminating positive eigenvalues
in dimension greater than one has been developed by T. Kato [28], S. Agmon [2,
3], B. Simon [43], and others. To illustrate this philosophy, let us assume that
V has compact support, say supp V C {\x\ < R}. Suppose that u G L 2 ( R n ) ,
E > 0, and [-A + V(z)]w = Eu in R n . By our support assumption on V, we
have
-Au-Eu
= 0 in |z| > R,E>0,ueL2(\x\
> R).
A classical theorem of Rellich [39] now shows that u = 0 for \x\ > R. We
then have {-A + [V(x) - E]}u = 0 in |z| < 2R, supp u C {\x\ < R}. If we
knew that {-A + [V(x) — E]} has the w.u.c.p., we would then conclude that
u = 0, and we would be done. As was mentioned before, V need not be locally
bounded, and this leads us to study the unique continuation property when the
lower-order coefficients are in Lp classes. Similar considerations, also connected
with mathematical physics, lead us to also study operators where we replace the
Laplacian A by the Dirac operator D.
The first results on the unique continuation property for operators of the form
A + V(x), V G £f o c (R n ),p < oo, seem to be due to A. M. Bérthier [8, 9] in
1979 and to V. Georgescu [18] also in 1979. Bérthier proved the w.u.c.p. if p >
max(n — 2, n/2), while Georgescu proved the u.c.p. if p > max(2, (2n —1)/3). M.
Schechter and B. Simon [42] proved the u.c.p. if p > 1 for n = 1,2, p > (2n —1)/3
for n = 3,4,5, and p > n - 2 if n > 6, while Saut and Scheurer [40] proved the
u.c.p. for p > 2n/3. Also, Amrein, Bérthier, and Georgescu [4] proved the
=
xuTc;prförTr^^21fw==^^
Hörmander [24] showed the u.c.p. for solutions of (*), with (a3k(x)) Lipschitz
continuous, Ga[x) G Lf 0 ( ^ a|) (R n ), where p(0) = 2, when n < 4; p(0) > 2 when
n = 4, and p(0) = (An — 2)/7 when n > 4; and p(l) > 2 when n > 2 and
p(l) > (3n - 2)/2 when n > 2.
(See also the work of E. Sawyer [41] in R 3 , under a different kind of assumption on V.)
CARLEMAN ESTIMATES
951
To clarify this myriad of results, let us restrict our attention to A + V(x),
V G Lf 0C (R n ). For n = 2,3,4, the best results are those of Amrein, Bérthier,
and Georgescu [4], who proved the s.u.c.p. for p > n/2. For n large, the best
results are those of Hörmander [24], who proved the u.c.p. when p > (4n — 2)/7.
EXAMPLE [27], Let u(x) = exp(-log l/|a:|) 1 + e , e > 0. Then u vanishes at
0 of oo order, while V(x) = -Au(x)/u(x)
~ (log l/|a:|) 2e • l / | z | 2 G £f o c (R n ) for
all p < n/2. Hence the s.u.c.p. cannot hold for p < n/2.
In 1984, D. Jerison and C. Kenig [27] were able to show that V G
L^(Rn)
implies that A + V(x) has the s.u.c.p. This shows that for the s.u.c.p., L*JC ( R n )
is the sharp class in the scale of Lp spaces.
In 1985, D. Jerison [26] gave an alternative proof of the result of Jerison and
Kenig [27], and was also able to show that the operator D + V(x), where D is
the Dirac operator, and V(x) G L^0C(Kn), 7 = (3n — 2)/2, has the u.c.p. In the
case, n = 3, 7 = 7/2. This improved previous results of Bérthier and Georgescu
(1980) [10], who had obtained 7 = 5. The natural conjecture for the u.c.p. for
D+V(x) is that V G L^oc(Kn). Jerison's result falls short of this conjecture, but
he proved that his result is the best one can obtain by the method of Carleman
estimates.
Finally, in 1986, C. Kenig, A. Ruiz, and C. Sogge [30] observed that to prove
the w.u.c.p. for A + V(x), V(x) G L n / 2 ( R n ) , it sufficed to prove the Carleman
estimate
II* A * B «IILP'(R») < C||e A ^A W || L p ( R n),
- - - = -
(**)
n
P P
whose proof is simpler than the Carleman estimate proved by Jerison and Kenig
in [27], to obtain the s.u.c.p. (See §2 for this result.)
If one makes the substitution eXXnu = v, one sees that (**) is equivalent to
the estimate
1
1
2
2 A - * - + A2
(**')
IMILP'(R») ^ C
oxn
f
LP(Rn)
p
p
n
Thus one is led to the idea of proving Sobolev-type estimates for second-order
constant coefficient operators, which are uniform in the lower-order terms. This
was accomplished for operators P(D) with constant complex coefficients, and
principal part Q(D), where Q(£) is a nonsingular real quadratic form on R n ,
i.e., Q(C) = - f i
tf
+ f-+1 + • • • + Ç*. In this setting, Kenig, Ruiz, and
Sogge [31] proved the uniform Sobolev inequality
IMLP'(R»)
< c||p(i?H L ,( R »),
\ - ^ = £,
p
p
(s)
n
where G depends only on n.
(S) in turn implies Carleman estimates and unique continuation theorems for
operators whose top-order terms are not necessarily elliptic, and include, for
example, the wave operator
• - — - — d2
2
9a;
dx\
dx\ '
952
C. E. KENIG
(S) yields global and local unique continuation theorems. For the global ones,
let p be as in (S), and suppose that (dau/dxot) G L p (R n ), \a\ = 2. Assume
also that u vanishes on one side of a hyperplane, that P(D) is as in (S), and
that \P(D)u\ < \Vu\, where V G L n / 2 ( R n ) . Then u = 0. If the hyperplane is
characteristic for P(D) (and hence we are in the nonelliptic case), there are well
known examples (see, e.g., [25, vol. I, pp. 310-311]) of C°° functions u, with
P(D)u = 0, and which vanish on one side of the hyperplane. These examples
however do not have the growth property (dau/dxa),
\a\ = 2 G L p ( R n ) . As
far as local theorems, one can use a reflection across convex "spheres" as in
Nirenberg [36], when the principal part of P(D) is A or D. In the case when
it is A, this shows that the u.c.p. for A + V(x), V G L^c (R n ), follows from
(**). When it is D, if T is the light cone T = {x:x\ > \x'\,x' — (x<i,... ,xn)},
and (dau/dxa),
\a\ = 2, belong to L*oc(T) (p as in (S)), \P(D)u\ < \Vu\ in T,
with V G Lfoc (T), then, if u vanishes for large x\, it must vanish identically.
Similar results, under the stronger assumption V G L™C(T), were obtained by
Hörmander [23].
Let us now illustrate the mechanism for passing from Carleman estimates to
unique continuation theorems. For example, let us show that if Au G L p ( R n ) ,
1/p - 1/p' = 2/n, V G L n / 2 ( R n ) , \Au\ < \Vu\, and supp u C {xn > 0}, then
u = 0. We will use (**) for A < 0. It is enough to show that there exists p > 0
so that u = 0 in Sp, where Sp = {x G R n : 0 < xn < p}. Choose p so small that
if C is as in (**), G\\V||L«/2(s„) < V 2 - T h e n
< C\\eXx"Vu\\LP{sp)
+C||e A ^A W || L P ( R n X S p )
< h\\eXXnu\\Lp'(sp)
+C\\eXx»Au\\LP{B.n\sp)
The third inequality follows by Holder's inequality, our choice of p, and 1/p —
1/p' = 2/n. Hence, \\ex{Xn-p)u\\LP.{Sp)
< 2C||Au|| Z/P(R n ) , uniformly for all
A < 0, which shows that u = 0 in Sp.
In the rest of the paper, we will attempt to describe some of the main points
in the proofs of the Carleman estimates in the works of Jerison and Kenig [27],
Jerison [26], and Kenig, Ruiz, and Sogge [31].
The main underlying theme is the application of the ideas and methods of
classical Fourier analysis, such as oscillatory integrals [13, 22, 49], restriction
theorems for the Fourier transform [52, 51, 49], complex interpolation [46], and
the uncertainty principle [16] T to problems arising in mathematical physics and
partial differential equations.
The connection between restriction lemmas for the Fourier transform and
Carleman estimates seems to have been first observed by Hörmander [24]. He
used the L2-restriction theorem for R n . His Carleman estimates, unlike (**),
the estimate in [27], and (S), involve L2 norms in the right-hand side (and
the "convex" weights ex^Xn~^x^), and for this reason his unique continuation
theorems involve potentials in "worse" Lp spaces. On the other hand, the use of
CARLEMAN ESTIMATES
953
the L2 norm expédiâtes the passage to variable coefficients. This is accomplished
exploiting the "convexity" of the weights, via the so-called "Treves identity" [53].
This is an example of the uncertainty principle, as was pointed out by Jerison
[26].
The Carleman estimate in [27] was proved by complex interpolation in a
manner resembling the proof of the L2 restriction theorem. Later, D. Jerison
[26] used the discrete restriction lemma of [45] for S""""1, to give a new proof of
[27]. Jerison also combined this with ideas related to the uncertainty principle
and "convex" weights to prove Carleman estimates for the Dirac operator.
Finally, C. Kenig, A. Ruiz, and C. Sogge [30] used the mapping properties of
the Stein-Tomas operator (which is the main tool in proving the L2 restriction
theorem) in R n (as opposed to R n _ 1 ) to prove (**). Similar ideas, involving
Strichartz's [51] generalization of the Stein-Tomas operator for quadratic forms
of arbitrary signature, led to the proof of (S) in [31].
2. Schrödinger operators of the form A + V(x), V G L^(Rn).
The
main point in the work of Jerison and Kenig [27] was the following Carleman
estimate:
THEOREM 2 . 1 . Let n > 3, 1/p - 1/p' = 2/n. Suppose that X > 0 is not
an integer, and let 6 be the distance from A to the nearest integer. There is a
constant C, depending only on 6 and n} such that for every u G Co°(R n \0) ;
\\\X\~Xu\\LP,(Knidx/\X\n)
<
C\\\x\-X+2Au\\Lp(Kntdx/\x\n).
This inequality was proved by complex interpolation. Fix A, and consider the
analytic family of operators Tz, depending on A, given by
T,g =
\x\-xÒT'l\\x\x-g),
modified by a Taylor series of order the integer part of A. Theorem 2.1 follows
from estimates for Tz. By E. Stein's interpolation theorem [46], we need an
L2 —* L2 estimate when Re z = 0. Because of rotation and dilation invariance,
using polar coordinates (r,oS), we study Tz(ririPk(w)), where P& is a spherical
harmonic of degree fc. We have:
2~* r(|(fc - A - iti)) • r(|(n +fc+ A - z + n/))'ir
r >Pk(u).
Lr(af*)r(è(*-A + «-»i?))-r(ì(» +fc+ A + ^)).
When Re z = 0, Stirling's formula shows that the expression in brackets is
bounded independently of fc, A, and r], with bound depending only on 6. This
gives the desired L2 —• L2 estimate.
At the other end point of the interpolation, Re z — n, and Tz is essentially
a logarithmic potential. It was proved in [27] that in this case Tz'.L1 —» X,
where X is an enlargement of L°°, which has the same complex interpolation
properties as L°°, but allows logarithmic singularities. This estimate follows
by a uniform asymptotic estimate for the hypergeometric function. (See [27].)
954
C. E. KENIG
E. M. Stein [48] observed that one could bypass the space X and the hypergeometric function, by noting that when n — 1 < Re z < n, the kernel of Tz is
pointwise dominated by the one for fractional integration of order Re z. This
also gives Lorentz space estimates.
3. The Schrödinger operator revisited, and the Dirac operator. We
will now discuss Jerison's work [26]. Introduce polar coordinates in R n , x = eyuj,
u G S 71 " 1 , y G R. In those coordinates,
A = e" 2 «
dy„2 2 + ( " - 2 ) ^ +
A
s
where As denotes the spherical Laplacian. Let us start out by outlining Jerison's
proof of Theorem 2.1. Because of the above formula, if Pk is a spherical harmonic
of degree fc, then
|z|- A + 2 A(|a;| V ^ P * ( o ; ) ) = -(fc - (A + ir,)) • (fc + n - 2 + A +
i^e^P^uj).
(Notice that this is the same formula as the case Re z = 2 of Tz in §2.)
Let o\(r], fc) = — l/(fc — A + irj)(h + n — 2 + A + ir)), and let &. denote the
projection operator from L 2 ( S n _ 1 ) to the space of spherical harmonics of degree
fc. Also, let f(r]i^) = J_TO eiriyf(y,üj)dy denote the partial Fourier transform
in y. For / G Cg°(R x S71'1), let
0 0
-t
/«OO
*=s27r J-°°
It is then easy to see that Theorem 2.1 is equivalent to
II^A/IILP^RXS«"1) ^ C | | / | | L P ( R X S » - I ) '
for all / G C^ÇRxS71'1).
Now let {9ß}%=Q be a partition of unity of the positive
real axis with supp 0O C {r:r < 1}, supp 0N C {r:r > A/400}, supp Oß C
{r:2ß~2 < r < 2ß), 2N < A/10 < 2 JV + 1 . Consider the operator Rßx, analogous
to R\, but with symbol aß(rj, fc) = 0ß(\k — A + ir]\)ax(r], fc). Note that aß(rj, fc)
for ß < N — 1 is supported where |fc — A + ir\\ < 2ß, and hence there are at most
2 ^ + 1 nonzero terms in the sum over fc which defines Rß, and the value of fc is in
each case comparable to A. We can now apply the following "discrete" version
of the restriction theorem, due to C. Sogge [45].
LEMMA 3 . 1 . There is a constant C such that
ll&ffllLP'^-i^^-^llffllLPCS»-!).
where p is as in Theorem 2.1.
In fact, using the formula
+ß
/
-OO
Kßx(y-y')f(y',-)(u)(dy%
CARLEMAN ESTIMATES
955
where
oo
-,
/»-foe
K
x(s) = E è
^(r),k)e^dr,^
A;=oZ7r J-°°
we see that for ß < N—1, the integration in r\ is over an interval of length < 2 ^ + 1 .
Hence, by Lemma 3.1 and integration by parts in r\, it follows that K\(s) is
a bounded operator from Lp(Sn~1) to Lp (Sn~1) whose norm is bounded by
C2^A~ 2 / n (l + |2^s|)- 1 0 . If we now let 1/r + 1/p = 1/p' + 1, i.e., 1/r = 1 - 2/n,
Minkowski's integral inequality and Young's inequality show that
ll^f/llLP'(RXfl-i) ^ O A - a / w 2 a ^ | | / | | L P ( R x 5 - i ) .
But X^^To1 22/3/iV ^ A 2 / n > w h i l e Ä ^ / c a n b e controlled by ordinary fractional
integration, and hence Theorem 2.1 follows.
The Dirac operator is D = X^?=i otjd/dxj, where ay are skew hermitian
matrices with a* = —OùJ, aja^otk^j
= —l&jk, °tj £ GL(ra, C), where m = 2 n / 2
n+1 2
if n is even, and m = 2( )/ if n is odd. It is easy to see that £>* = D, and
D2 = —A. Jerison observed that the analogue of Theorem 2.1 fails for D. In
fact if
|||^r A w|| L 9 ( t / j c^) <
C\\\x\-x+1Du\\LP(u,cm)
for all u G Cfi°(U,Cm), U = {x G Rn:a < \x\ < b}, 0 < a < b, uniformly for a
sequence of A —• oo, then q < p. Serious difficulties remain even if one replaces
|a;|~A by ex^x\
where 0 is any smooth real-valued function, not identically
0. The corresponding inequality can then only hold if 1/p — 1/q < I / 7 , 7 =
(Sn — 2)/2. This is unfortunate, because the conjectured gap was I / 7 = 1/n.
On the positive side, Jerison proved
THEOREM 3.2. Let 0 < a < b < 1, n > 3, U = {x G Rn:a < \x\ < b}. Let
<j>(x) = (log|a:|) 2 /2, q= ( 6 n - 4 ) / ( 3 n - 6 ) , i.e., 1/2-1/q = I / 7 , 7 = ( 3 n - 2 ) / 2 .
Then there exist G = C(n, a, b) such that, for all A G R,
||eA^||L9(t/jC-) <
C\\ex*Du\\L2{UiCm),
forallueC§°(U,Cm).
The above theorem easily implies the u.c.p. for solutions of Du = Vu, V G
L^ oc (R n , GL(m,C)), 7 = (3n - 2)/2, with Du G L2(Q, C m ) .
In order to prove Theorem 3.2, Jerison considered eXy eyDe~Xy . This equals
aA\, where à is unitary, A\ = d/dy — (Xy + L), and L is a first-order operator
on the cj variables. A\ is now variable coefficient, and this gives the improved estimate A\ > CA 1 / 2 . (This is an instance of the uncertainty principle.) However,
to find a left inverse for A\, one then has to use "pseudodifferential" operators. Jerison found an exact left inverse, using the formula for the left inverse of
(d/dy — y) on R, given in Nagel and Stein [35]. He then bound the left inverse
using Lemma 3.1 and a device of P. Tomas [52] to obtain Lq —• L2 estimates
from Lp —• Lp estimates. See [26] for the details.
956
C. E. KENIG
4. Uniform Sobolev inequalities for second order constant coefficients operators. In this section we will outline some of the ideas in the work
of C. Kenig, A. Ruiz, and C. Sogge [31]. Let Q(fl = - £ 2
# + 3 + i + - • -+C
be a nonsingular real quadratic form on R n . Let P(D) be a constant coefficient
operator, with complex coefficient lower-order terms, and whose principal part
is Q(D).
THEOREM 4 . 1 . Letn>3,
and let 1/p - 1/p' = 2/n. Then there exists a
constant G depending only on n such that, for all u G Co°(R n ), we have
||«||LP(R«)<C||P(I>) U || L P - ( R n ) .
As was remarked in the introduction, Theorem 4.1 yields unique continuation
theorems for operators whose principal part is not necessarily elliptic.
Let H% and H™ be the open subsets of R n on which Q is strictly positive
and negative respectively. Also, let S + _ 1 and S™ -1 be the level sets S ± _ 1 =
{£'-Q(€) — u } - There are canonical measures du± on 5 ± - 1 so that on H±,
d£ = p n _ 1 dpduj±. The key ingredient for Theorem 4.1 is
as above. Then, for f G Cg°(Rn)
LEMMA 4 . 2 . Letn>3,Q,p
(a)
}
Jsi'1
du)±
< C1|/||LP(R»)î
LP'(R")
(b) there exists an absolute constant C, such that, for all z EC,
II«IIL>'(R«) < C\\[Q(D)+z}u\\LP(Rn),
u e C 0 °°(R").
In the Euclidean case (when Q is elliptic), (a) is due to Stein-Tomas [52],
while the other cases are due to Strichartz [51]. (b) does not seem to be in the
literature; however, its proof involves only simple modifications of the proof of
(a).
The difficulty in establishing Theorem 4.1 comes from the fact that the symbol
of P(D) may vanish away from the origin. However, if this is the case, the zero
set of P(£) always lies on a "sphere," which explains the relevance of Lemma
4.2.
It is not hard to see that it is enough to prove Theorem 4.1 in the case when
P(D) = Q(D) +a + e{d/dxj + iß}, where a = ±1, e, ß G R\{0}, j = 1 or n.
We will deal with the case j = n, o = 1, the other ones being similar. We are
_thus jceduced _to_prov ing_the_multiplier__theorem-
/(*)
Q ( 0 + 1 + **(&+/?)
Let m ( 0 = ( Q ( 0 + 1 + ie(Çn + ß))'1, and let x{t) = 1, for \t\ G [1,2], and
0 otherwise. Set Xk(£n) = x(2fc(£n + /?)), and define mk(Ç) = Xk(£n)m(Ç).
Because of Littlewood-Paley theory (see [47]), the fact that p < 2 < p', and
CARLEMAN ESTIMATES
957
Minkowski's integral inequality, it suffices to show that there is a constant G,
independent of fc, e, and ß, for which
\\{mk{i)f{OV\\v<C\\f\\P.
To prove this last estimate, we use (b) in Lemma 4.2, with z = 1 + ie2~k.
We are then reduced to showing that
(Q(fl + 1 + ie(tn + ß))(Q(0
+ 1 + te2-*)
<C\\f\\PP'
Let Tfc be the above multiplier, and use polar coordinates £ = pu associated to
Q. It is easy to see that Minkowski's integral inequality and Lemma 4.2(a) give
l|T*/||p'
<Ef
/
J
Par1
±°
ef(f^)(Cn+ß-2-k)xkUn)ei'""x
duj±
(±p + 1 + ie(Çn + ß))(±p2 + 1 + ie2~k)
2
ef(QXk(tn)(tn+ß-2-k)
(±p + 1 + ie(£n + ß))(±P2 + 1 + ie2~k)
/»OO
2
+ Jo
rJl-l
dp
dp.
Since 1 = n — 1 — 2n/p', the definition of Xfc shows that this last term is bounded
by
Cll/ll, f
Jo
e2-k,
2
(p - I)2 + (e2~k)2
dp,
which gives the desired inequality.
In the case when Q(D) = A, one can prove that
\\u\\Ls(Kn)
<C\\P(D)u\\Lr{nn),
for the optimal range of s and r (see [31]). Also, iîP(D) = Q(D)+Y^=1
ajd/dxj
+6, a / s real, |Re b\ > 1, then ||w|| L r' (R „) < C||P(2?)u|| L r( R n) for 2/(n + 1) <
1/r - 1/r' < 2/n (see [31]). This generalizes some results in [50] and [32].
5. Some open problems. To conclude, we would like to point out a few
open problems.
(a) Does the unique continuation property hold for
D + V,
V G
L?oc(n,GL(m,C)),
where D is the Dirac operator of §3? As was pointed out in §3, the corresponding
Carleman estimates are false.
(b) Does the unique continuation property hold for A + Yi,Vi(x)d/dxi, where
Vi G LJJ)c(Rn)? This question is closely tied to the previous one. It is known
that the u.c.p. holds if the Vi G L £ c ( R n ) , 7 = ( 3 n - 2 ) / 2 (see [24, 7]). However,
it is also known, just as for the Dirac operator, that this is the best exponent
that Carleman estimates can yield.
(c) Variable coefficient problems: For example, does the s.u.c.p. or even the
w.u.c.p. hold for operators of the form P(x,D) = J2ajk(%)(d2/dxj dxk) + V(x),
958
C E . KENIG
where a,jk(x) is an elliptic, Lipschitz continuous real symmetric matrix, and
V(x) G L^c (R n )? We could also ask whether the uniform Sobolev inequalities
of §4 hold for operators with Ylajk(%)(d2/dxjdxk)
as principal part. Part of
the difficulty comes from the need of a very precise knowledge of the left inverses
of such operators.
(d) The last question bears on the distinction between the s.u.c.p., w.u.c.p.,
and u.c.p. As was pointed out in the introduction, there are examples of potentials V G Lf o c (R n ), p < n/2, for which the s.u.c.p. for A + V does not hold. As
far as we know, there are no examples known of potentials V G L 1 1 oc (R n ) such
that the u.c.p. for A + V does not hold. Indeed, it is possible that the u.c.p.
holds for A + V, whenever V G L 1 1 oc (R n ). This would be of interest for the
application to the absence of positive eigenvalues.
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