O Boundary Value Problems

a SpringerOpen Journal

RESEARCH Open Access

Carleman estimates and unique continuation property for abstract elliptic equations

Veli B Shakhmurov

Correspondence: veli. sahmurov@okan.edu.tr Department of Electronics Engineering and Communication, Okan University, Akfirat Beldesi, Tuzla, 34959, Istanbul, Turkey

Abstract

The unique continuation theorems for elliptic differential-operator equations with variable coefficients in vector-valued Lp-space are investigated. The operator-valued multiplier theorems, maximal regularity properties and the Carleman estimates for the equations are employed to obtain these results. In applications the unique continuation theorems for quasielliptic partial differential equations and finite or infinite systems of elliptic equations are studied. AMS: 34G10; 35B45; 35B60.

Keywords: Carleman estimates, unique continuation, embedding theorems, Banach-valued function spaces, differential operator equations, maximal Lp-regularity, operator-valued Fourier multipliers, interpolation of Banach spaces

1 Introduction

The aim of this article, is to present a unique continuation result for solutions of a differential inequalities of the form:

\\P(x, D) u (x) \\e < || V(x) u (x) IE, (1)

n d 2 u n d u

i,j=1 ' k=1

here fljj are real numbers, A = A (x), Ak = Ak (x) and V (x) are the possible linear operators in a Banach space E.

Jerison and Kenig started the theory of Lp Carleman estimates for Laplace operator with potential and proved unique continuation results for elliptic constant coefficient operators in [1]. This result shows that the condition V e Ln/2 loc is in the best possible nature. The uniform Sobolev inequalities and unique continuation results for second-order elliptic equations with constant coefficients studied in [2]. This was latter generalized to elliptic variable coefficient operators by Sogge in [3]. There were further improvement by Wolff [4] for elliptic operators with less regular coefficients and by Koch and Tataru [5] who considered the problem with gradients terms. A comprehensive introductions and historical references to Carleman estimates and unique continuation properties may be found, e.g., in [5]. Moreover, boundary value problems for

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differential-operator equations (DOEs) have been studied extensively by many researchers (see [6-18] and the references therein).

In this article, the unique continuation theorems for elliptic equations with variable operator coefficients in E-valued Lp spaces are studied. We will prove that if lp + ¥ = ^ = lp-7>p + ¥ = 1>Ve LK (R"-> L№))> p> ^ 0-, and u e Wp(Rn;E(A), E) satisfies (1), then u is identically zero if it vanishes in a nonempty open subset, where W^(Rn; E(A), E) is an E-valued Sobolev-Lions type space. We prove the Carleman estimates to obtain unique continuation. Specifically, we shall see that it suffices to show that if w (x) = xi + % then

Lp(Rn; E) < C

etwL(sx, D)u

1 + 77 — M) II Jw r^a,, || , \\Jwa,, || ,

r n \\e D U ||Lp(Rn;E)+ IIe Au \\Lp(Rn;E) <

i a | <1

C \\ etwL(ex, D) u ¡Lp(Rn;E). In the Hilbert space L2 (Rn; H), we derive the following Carleman estimate

I a I <2

t.2 |etwDaU |li(R'1;H)+ IetWAu ||L2(fl";H) < c \\etWL0U ||L2(fl";H)-

Any of these inequalities would follow from showing that the adjoint operator Lt (x; D) = ewL (x; D) etw satisfies the following relevant local Sobolev inequalities

Lp| (Rn;E) < C \\LtU

f'1+n 1 • IIlkf№";-E)+ WAu ||Lp(R";£) < C IILtU ||L,,(fi";£),

uniformly to t, where L0t = etwL0e-tw. In application, putting concrete Banach spaces instead of E and concrete operators instead of A, we obtain different results concerning to Carleman estimates and unique continuation.

2 Notations, definitions, and background

Let R and C denote the sets of real and complex numbers, respectively. Let S^ = (f e C, | argf | < {0}, <p e [0, n).

Let E and E1 be two Banach spaces, and L (E, E1) denotes the spaces of all bounded linear operators from E to E1. For E1 = E we denote L (E, E1) by L (E). A linear operator A is said to be a ^-positive in a Banach space E with bound M >0 if D (A) is dense on E and

(A + fI)'1 l(E) < M(1 + If I)-1

with l e S^, ^ e (0, n], I is identity operator in E. We will sometimes use A + f or Af instead of A + fl for a scalar f and (A + ^I)-1 denotes the inverse of the operator A + fl or the resolvent of operator A. It is known [19, §1.15.1] that there exist fractional powers AA of a positive operator A and

E(Ae) = {u e D{Ae), IImMbc^^ ) = A u\\e + mull < <d <

We denote by Lp (O; E) the space of all strongly measurable E-valued functions on O with the norm

llullip = Wu\\Lp{Q;E) = ( Jq |u(x)|Ed^ , 1 < p

< 00 -

By Lp,q (O) and Wlp,q(Q,) let us denoted, respectively, the (p, q)-integrable function space and Sobolev space with mixed norms, where 1 < p, q < see [20].

Let E0 and E be two Banach spaces and E0 is continuously and densely embedded E.

Let l be a positive integer.

We introduce an E-valued function space Wlp(Q; E0, E) (sometimes we called it Sobo-lev-Lions type space) that consist of all functions u e Lp (O; E0) such that the generalized derivatives D^u = jj e Lp(Q;E) are endowed with the

WWp (Q,

:;Eo,E) = WuWLp(Q;E0) + J2 |Dku \\Lp(Q;E) < 1 < p <

The Banach space E is called an UMD-space if the Hilbert operator

(H/)(.r) = lim / is bounded in L„ (R, E), p e (1, <x>) (see e.g., [21,22]). UMD

|x-y\>e

spaces include, e.g., Lp, lp spaces and Lorentz spaces Lpq, p, q e (1,

Let E1 and E2 be two Banach spaces. Let S (Rn; E) denotes a Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on Rn. Let F and F-1denote Fourier and inverse Fourier transformations, respectively. A function Y e Cm (Rn; L (E1, E2)) is called a multiplier from Lp (Rn; E1) to Lq (Rn; E2) for p, q e (1, if the map u ® Ku = F1 Y (f) Fu, u e S (Rn; E1) is well defined and extends to a bounded linear operator

K : Lp(Rn;Ei) ^ Lp(Rn;E2).

We denote the set of all multipliers from Lp (Rn; E1) to Lq (Rn; E2) by Mqp(Ei, E2). For E1 = E2 = E and q = we denote Mpp(E1, E2) by Mp (E). The Lp-multipliers of the Fourier transformation, and some related references, can be found in [19, § 2.2.1-§ 2.2.4]. On the other hand, Fourier multipliers in vector-valued function spaces, have been studied, e.g., in [23-28].

A set K c L (Ei, E2) is called R-bounded [22,23] if there is a constant C such that for all Tx, T2, . . . , Tm e K and ui,u2, . . . , um e Ex, m e N

1 m 1 m

/ Erj(y)Tjuj y ia Erj(y)uj dy,

0 j=i E2 0 j=1 E1

where {rj} is a sequence of independent symmetric {-1, 1}-valued random variables on [0,1]. The smallest C for which the above estimate holds is called a R-bound of the collection K and denoted by R (K).

Un = [P = (Pi, Pi, Pn), Pi e {0,1}, i =1,2, ..., n},

ÇP = ÇPi ÇPi ...n, | ÇP | = | Çi | p1 | Çi |Pi In |Pn.

For any r = (ri, r2, . . . , rn), ri e [0, the function (if)r, f e Rn will be defined such that

(iÇ)r =\(iÇi)ri ■■■mrn, Çi,Çi.....Çn =0,

(iÇ) \ 0, Çi, Çi.....Çn = 0,

ysign t.), te (-00, oo), v e [0, oo).

Definition 2.1. The Banach space E is said to be a space satisfying a multiplier condition with respect to p, q e (1, (with respect to p if q = p) when for Y s L (E1, E2)) if the set

Jd^(I) eRn\0, p e L7„J

is R-bounded, then ^ e Mp(E1, E^.

Definition 2.2. The ^-positive operator A is said to be a R-positive in a Banach space E if there exists ^ e [0, n) such that the set

La = (f (A + f I)-1: f e S9} is R-bounded.

Remark 2.1. By virtue of [29] or [30] UMD spaces satisfy the multiplier condition with respect to p e (1,

Note that, in Hilbert spaces every norm bounded set is R-bounded. Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on Lq, 1 < q < ^ [31], A has the bounded imaginary powers with v < j, v < j or if A is a generator of a semigroup with Gaussian bound in E e UMD then those operators are R-positive (see e.g., [24]).

It is well known (see e.g., [32]) that any Hilbert space satisfies the multiplier condition with respect to p e (1, By virtue of [33] Mikhlin conditions are not sufficient for operator-valued multiplier theorem. There are however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition (see Remark 2.1).

Let Hk = {^ e Mp,(E1, E2), h = (h1, h2, ■ ■■, K) e K} be a collection of multipliers in Mp,(E1, E2). We say that Hk is a uniform collection of multipliers if there exists a constant M >0, independent on h e K, such that

lF'1^hFulLp(Rn;E2) < M\\U\\Lp(Rn;E1) for all h e K and u e S (Rn; E1).

We set

Cb(Q;E) = \ u e C(Q;E), lim u(x) exists?.

In view of [17, Theorem A0], we have

Theorem 2.0. Let E1 and E2 be two UMD spaces and let V e C(n)(Rn\0;L(EU E2)) for p,p e (1,

kJ^'+HD^^) : £ eB"\0, J0 6 UnJ <00

uniformly with respect to h e K then Yh (f) is a uniformly collection of multipliers from Lp (Rn; E1) to Lq (Rn; E2).

X = -j-/ a = («1/ «2/ • • •, »„).

Embedding theorems in Sobolev-Lions type spaces were studied in [13-18,32,34]. In a similar way as [17, Theorem 3] we have Theorem 2.1. Suppose the following conditions hold:

(1) E is a Banach space satisfying the multiplier condition with respect to p, q e (1, ro) and A is a R-positive operator on E;

(2) l is a positive and ak are nonnegative integer numbers such that 0 < ^ < 1 - x, t and h are positive parameters.

Then the embedding

DaWlp(Rn; E(A), E) c Lp(Rn; E(A1-X)) is continuous and there exists a positive constant C^ such that for

u e Wlp (Rn; E(A), E) the uniform estimate holds

W^ ulLp (Rn;E(A1-x-*)) < CV [^^WuWWp (Rn;E(A),E) + h (1 ^ Wu||Lp(R";E)] .

Moreover, for u e Wlp(Rn; E(A), E) the following uniform estimate holds

u\\Lp(Rn;E) < CV [^Mw'(Rn;E(A),E) + h-^Wuhp^E)] .

3 Carleman estimates for DOE

Consider at first the equation with constant coefficients

L0u = ^2 D2u + Au = f (x), (2)

where D;, = and A is the possible unbounded operator in a Banach space E.

Let w(x) = X\ + y and t is a positive parameter. Remark 3.1. It is clear to see that

etwLo[e-twu] = Lot(x, D)u = et^ ^D2(e-twu) + e-twAu

V^1 / (3)

E2 d u 2 2

Dku + Au + 2tW\-+ [—t w1 + t\u,

, , d X1

where W\ = f^. Let Lot (x, i) is the principal operator symbol of Lot (x, D) on the domain B0, i.e.,

Lot(x, f) = f2 - 2if1w1t + A + If ||2 - t2w\ = Gt(x, f )Bt(x, f), where

Gt(x, f ) = f1 - i

Bt (x, f ) = f1+ i

(A+|fl|2)2 +twl

(A+ If112)2 - twi

, I ff = £ fk2.

Our main aim is to show the following result:

Remark 3.2. Since Q(f) e S for all ^ e [0, n) due to positivity of A, the operator

function A + |f'|2, f e Rn is uniformly positive in E. So there are fractional powers of

i •> I

and the operator function ||-l|2)2 is positive in E (see e.g., [19, §1. 15.1]).

First, we will prove the following result.

Theorem 3.1. Suppose A is a positive operator in a Hilbert space H. Then the following uniform Sobolev type estimate holds for the solution of Equation (3)

£ t-2 IetwDau |li(R'i;H)+ IetwAu ||t,(K«^) < C \\etwL0u ft^H). (4)

By virtue of Remark 3.1 it suffices to prove the following uniform coercive estimate

II II II II

f2 IIDaU ||l2(]?";H)+ IIAu || L2(Rn;H) < C \\L0tu ¡L2(Rn;H) (5)

for u e W22(Rn; H(A), E).

To prove the Theorem 3.1, we shall show that L0t (x, D) has a right parametrix T, with the following properties. Lemma 3.1. For t >0 there are functions K = Kt and R = Rt so that

L0t(x, D) K(x, y) = S(x - y) + R(x, y), x, y e B0, (6)

where 8 denotes the Dirac distribution. Moreover, if we let T = Tt be the operator with kernel K, i.e.,

Tf (x) = J K(x, y)f (y)dy, f e Co00(B0; E),

and R is the operator with kernel R (x, y), then for large t >0, the adjoint of these operators satisfy the following estimates

t2 w T*f ¡liiboh) < C \\f \\l2(Bo;H), \\AVf \\l2(Bo;H) < C \\f \\l2(Bo;H),

12 11^*/ whibo.h) < c ||/ \\l2{B0;H),

H \\DvR*f ||L2(Bo;H) < c ¡°af lk,(Bo;H), 1 < |v| <2.

(8) (9)

Proof. By Remark 3.2 the operator function ^ + ||-l |2) 2 is positive in E for all f e R". Since tWi + zfi e S(<j>), due to positivity of A, for <p e [f/^r) the factor

has a bounded inverse G° 1(x, % )for all f e Rn, t

Gt(x, %) = -i > 0 and

\\G—1(x, %) |b(H) < C(1 + \twi + i%i\)-1. (10)

Therefore, we call Gt (x, f) the regular factor. Consider now the second factor

Bt (x, %) = i

(A + 112)2 +

By virtue of operator calculus and fractional powers of positive operators (see e.g., [19, §1.15.1] or [35]) we get that - [tw1 + if1] t S ($ for f1 = 0 and tw1 = if1!, i.e., the operator Bt (x, f) does not has an inverse, in the following set

At = {(x, %) e Bo x Rn : %1 = 0, \%\\ = tw1}.

So we will called Bt the singular factor and the set At call singular set for the operator function Bt. The operator B—1 cannot be bounded in the set At. Nevertheless, the operator B—1, and hence L-11, can be bounded when (x, f) is sufficiently far from At. For instance, if we define

%) e Bo x Rn : \%l\ e

-, 4t 4

, \%1 \ <

by properties of positive operators we will get the same estimate of type (10) for the singular factor Bt. Hence, using this fact and the resolvent properties of positive operators we obtain the following estimate

¡Lo^x, %) ||b(b) < C(1 + \%\2 + fy'when (x, %) e cFt, (11)

where the constant C is independent of x, f, t and crt denotes the complement of rt.

Let P e C£'{R) such that, j3(i) = 0 if e [±,4] and ¡3 (Z) = 0 near the origin. We then define

Po(%) = Pot(%)Po(%) = 1 - P(\%l \/t)P(1 - %1/t)

and notice that b0 (f) = 0 on rt. Hence, if we define

K0(x, y) = (2n)-n / №)ei((x-y),f)L^1(y, f )df (12)

and recall (11), then by [31] it follows from standard microlocal arguments that L0t(x, D)K0(x, y) = (2n)-n / 00(f )ei((x-y),f)df + R^(x, y),

where R0t belongs to a bounded subset of S-1 which is independent of t. Since operator R0t also has the same property, it follows that for all f e C0 (B0; H)

DR0f ¡L2(B0;H) < C E \Daf ¡L2(B0;H), 1 <|v |< 2.

|a|<|v|-1

By reasoning as in [31] we get that tR0t belongs to a bounded subset of S0. So, we have the following estimate

t \\DVR0tf \l2(B0;H) < C \\f \l2(B0;H). Moreover, the Remark 3.2, positivity properties of A and, (11) and (12) imply that,

the operator functions £ &0(f )2 |a| f Lot(x f) and &0(f )AL-)t1(x, f) are uniformly

| a|< 2 0t

bounded. Then, if we let T0 be the operator with kernel K0 (x, y), by using the Min-kowski integral inequality and Plancherel's theorem we obtain

£ t2-|a| \DaT0f \L2(B0;H) < C \\f ¡L2(Bo;H), \\AT0f \L2(B0;H) < C \\f ^^H).

For inverting L0t (x, D) on the set rt we will require the use of Fourier integrals with complex phase. Let b1 (f) = 1 - b0 (f). We will construct a Fourier integral operator T1 with kernel

K1(x, y) = (2n)-nj &1(f y*(x,y,f)L-t1 (y, f) df (13)

so that the analogs of (16) and the estimates (7)-(9) are satisfied. Since G-1(x, f) is uniformly bounded on rt, we should expect to construct the phase function F in (13) using the factor Bt (x, f). Specifically, we would like F to satisfy the following equation

Bt(x, ®x) = Bt(y, f), y e B0, (x, f e Ft). (14)

The Equation (14) leads to complex eikonal equation (i.e., a non-linear partial differential equation with complex coefficients).

(A+ |$<:|(x, y, f)|2)2 - [WiWt+iQ^X, y, £)] =

1 (15)

(A + If112)2 - (u>i(y)t+ i£i).

Since w1 (x) = 1 + x1, w1 (y) = 1 + y1, we have

<P = (x — y, §) + .-— +

2(1+y1)

2(1+y1)

is a solution of (15). To use this we get

L0t(x, D)c: - > = e'®L0t(x, <t>r) +e№ -T.

Next, if we set

r(x, y, %) - Gt(y, %) - Gt(x, %) - -i [w1(y) - W1(x)]t (17)

then it follows from L0t (x, f) = Gt (x, f)Bt (x, f) and (14) that

Lot(x, $x) = Lot(y, %) + Bt(y, %)r(x, y, %). (18)

Consequently, (16)-(18) imply that

(2n )nLot (x, D)K1(x, y)-j p1(% )eiOd% + J p1(% )r(x, y, % )G-1(y, % )eiOd%

R R 2 (19)

/f d 2 O

Pi^AL^iy, + j p^—L^iy,

Rn Rn 1

By reasoning as in [3] we obtain that the first and second summands in (19) belong to a bounded subset of S0. So, we see that the equality (5) must hold. Now we let K (x, y) = K0 (x, y) + Ki (x, y) and R (x, y) = R0 (x, y) + Ri (x, y), where

R1(x, y)- R1o(x, y) + Rn(x, y), Rw(x, y) - j P1(%)r(x, y, %)G-1(y, %)eiOd%,

f d2O , Rn{x, y) = J (y< WdH, Tof(x) =

j Ko(x, y)f (y)dy, T1f (x) - j K1(x, y)f (y)dy.

Due to regularity of kernels, by using of Minkowski and Holder inequalities we get the analog estimate as (7) and (9) for the operators T0 and R10. Thus, in order to finish the proof, it suffices to show that for f e L2 (B0; E) one has

t2~M \\DaTlf ¡L2(BO;H)+ ¡ATJ ¡L2(BO;H) < C \\f ¡L2(BO;H), (20)

& \\Rhf hiBo.H) < C 1/ \\Ll(B0,H), (21)

ri IIDvR*nf ||t2(Bo;H) < c ID"f Ikowf), 1 < |V| < 2. (22)

\a\<M-1

However, since R1>1 » tT1, we need only to show the following

t3/2 \\Tlf ¡L2(BO;H) < C \\f \\L2(BO;H). (23)

By using the Minkowski inequalities we get

||Tif ¡L2(Rn-1;E) < j J K(x, y)f (y)dy

where Kj (x, y) = K1(y, x). The estimates (13) and (16) imply that Kj(x,y) = (2n)-n / ei(x-y)m(x1, yu f|)df|,

m(x1, y1, f |) = 0 &1(f )A(x1-y1)f1+(x1-y1)2(i|f|-f1)/2(1+x1)]l-1 (y, f )df1.

Consequently, it follows from Plancherel's theorem that

f Kj(x, y)f(y)dy| < sup | m(x1, yu f|)| J f(y)| 2dy^ . (24)

K1 (x, y)f(y)dy|

Note that for every N we have

e^nf^mu^] < cn [1 + t(x1 - y1)2]-N on supp fo.

Since A is a positive operator in E, we have

HLo^x, f) \b(e) < 1 + | - 2if1w1t + | f | 2 - t2wj|-1

when -2if1 w1t + A + | f | 2 - t2w\ e S(^). Then by using the above estimate it not easy to check that

J ß)e'^i(xi-yi)-(xi-yi)1/2(1+xi)]L-1 (y, |)%!= O(t-1),

Mxi, yi, Ç|)|<Ct-i[i + t(xi - yi)2]-i. Moreover, it is clear that

j (1 + txi^dxi = O ■

Thus from (24) by using the above relations and Young's inequality we obtain the desired estimate

Tf \\l2Îb0h - Ct-1J f [l + t(x1 - y1)2] Iffa ■)lL2dy1

<ct-3/2"-"

WL2(Rn;H)-

Moreover, by using the estimate (10) and the resolvent properties of the positive operator A we have

L2(B0,H) - C\\f IL2(B0;H)-

The last two estimates then, imply the estimates (20)-(22).

Proof of Theorem 3.1: The estimates (7)-(9) imply the estimate (5), i.e., we obtain the assertion of the Theorem 3.1.

4 Lp-Carleman estimates and unique continuation for equation with variable coefficients

Consider the following DOE

L(x, D)u = aij(x)Dju + Au = f (x), x e Rn, (25)

where D;, = and A is the possible unbounded operator in a Banach space E and a¡7-are

real-valued smooth functions in Be = {x e Rn, |x| < e}.

Condition 4.1. There is a positive constant g such that 12 aij(x)?i?j — Y ? I for all f

e Rn, x e B0 = {xe Rn, |.x| < ±}. The main result of the section is the following

Theorem 4.1. Let E be a Banach space satisfies the multiplier condition and A be a

-/^-positive operator in E. Suppose the Condition 4.1 holds, n > 3, p = ^and p' is the conjugate of p, w = xi + y and a^ e C°° (BE). Then for u e Cq '( Be; E(A)) and e> 0, y < j the following estimates are satisfied:

KALpimE) ± C\\etwL(ex, D)u\\LpiRn;Ey i + i = 1, (26)

Et(1+ïïHal)+ htuM\Lp{R,,E) * (27)

C\\etwL(sx, D)u\\

Lp(Rn;E)'

Proof. As in the proof of Theorem 3.1, it is sufficient to prove the following estimates

\\v\\L {Rn.E) < C\\Lt (ex, D) v ||L - + -r = 1, (28)

J2 t(1+" |a|)|Da!;||if(j;„;£) + \\Av\\Lp{R,,E) < C||Lt(ex, DJvf^^ (29)

where,

U (ex,D) = etu'L (ex, D)e~tu' = L(ex, D) + 2- (toi)2 - t2, u>! = —.

dx1 d X1

Consequently, since w1 - 1 on Be, it follows that, if we let Qt (ex, D) be the differential operator whose adjoint equals

Q*(ex, D) = w72L(ex, D) + UwT1--t2,

then it suffices to prove the following

IMlLfl(£<■;£) < C|Qt(ex, D)v\\hmEy - + J\=1'

+ |\Av\|Lp(fi";£) < C\\Qt{ex,D)v\\Lp(Rn-,E), (30)

v e CO(Be; E(A)).

The desired estimates will follow if we could constrict a right operator-valued para-metrix T, for Qt* (ex, D) satisfying Lp estimates. these are contained in the following lemma.

Lemma 4.1. For t >0 there are functions K = Kt and R = Rt, so that

Qj (ex, D) K (x, y) = S (x - y) + R (x, y), x, y e Bs, (31)

where 8 denotes the Dirac distribution. Moreover, if we let T = Tt be the operator with kernel K (x, y) and R be the operator with kernel R (x, y), then if e and y are sufficiently small, the adjoint of these operators satisfy the following uniform estimates

||r/||v(j;„;£)<C||/||if(j;„;£ri + i = l, (32)

£t(1+»-W)||Dar/||if(j;„;£) < C||/||if(j;„;£r (33)

\\AT*f \\lp(R",E) < C\\f hp (R";E)'

* ll**/!^, < C\\f\\L^E)/1 = p,pK (34)

II'Wfkw.m ± Clfl^E, f e • (35)

Proof. The key step in the proof is to find a factorization of the operator-valued symbol Qjj (sx, %) that will allow to microlocally invert Qjj (sx, D) near the set where Qj (sx, %) vanishes. Note that, after making a suitable choice of coordinates, it is enough to show that if L (x, D) is of the form

t 1 d L(x,D) =D21 + y (ljjDjDj, Dj = ——

l d Xj

l,j=2 ' therefore, we can expressed Qj (sx, as

Qj (sx, = Bt (sx, Gt (sx, , (36)

Bt (x, = W-1?1 + i[(A + W- la {ex, ?- t] , Gt (x, ?) = w-1?1 - ^(A + w-1a (ex, ? + t],

a (x, ? ^ = J2 aij (x)?i?j. i, j=2

The ellipticity of Q(x, D) and the positivity of the operator A, implies that the factor Gt (x, f) never vanishes and as in the proof of Theorem 3.1 we get that

G"1 (ex, Ç) ||B(H) < C 1 + |w^1 a (ex, f1) |2 + \t + , (37)

x e Be, ? e Rn,

i.e., the operator function Gt (ex, f) has uniformly bounded inverse for (x, f) e Be xRn. One can only investigate the factor Bt (ex, f). In fact, if we let

At = {(x,?) e Be x Rn : = 0, || = twi},

then the operator function Bt (x, f) is not invertible for (x, f) e At. Nonetheless, Bt (ex, f) and Qj (ex, ?) can be have a bounded inverse when (x, f) is sufficiently far away. For instance, if we define

{(*,£) eBe xRn : If1! e , |fi| <

T = |(x,?) e Be x Rn : ||e

by properties of positive operators we will get the same estimate of type (37) for the singular factor Bt. Hence, we using this fact and the resolvent properties of positive operators we obtain the following estimate

(Qj) 1 (ex, Ç) < C(1 + |Ç 'j + |t + w-1^ |) 1when (x, Ç) e cTt. (38)

As in § 3, we can use (38) to microlocallity invert Qj (ex, D) away from rt. To do this, we first fix P e Cg (R) as in § 3. We then define

Po= Pot =1 - P (\Ç\/t) P (1 - &/t).

It is clear that bo (£) = 0 on rt. Consequently, if we define

Ko (x, y) = (2n)-n J Po (Ç) ei((x-y)'ç) (Qj)-1 (ey, Ç) dÇ (39)

and recall (37), then we can conclude that standard microlocal arguments give that

Qj (ex, D) Ko (x, y) = (2n)-n J Po (Ç) ei((x-y),ç)dÇ + Ro (x, y),

where R0 belongs to a bounded subset of S-1 that independent of t. Since the adjoint operator R0 also is abstract pseudodifferential operator with this property, by reasoning

as in [31, Theorem 6] it follows that

IIVR0f \\lp(r«;e) < c\\f Il^ey f e C0° (bs;E), (41)

tlROf \\L (RnE) < C\\f L(Rn;Ey f e CO0 (Bs;E), (42)

Lq(Rn;E)

4 = + = L

Moreover, the positivity properties of A and the estimate (38) imply that the operator functions E (f)t2-|a| ? (Qj)-1 (sx, f) and Po (f) A (Qj)-1 (sx, f) are uniformly

|a| < 2

bounded. Next, let T0 be the operator with kernel K0. Then in a similar way as in [31] we obtain that

E t(2 M)lDa T0f ¡Lp(Rn;E) < C\\f ¡Lp(Rn;E), (43)

lAT0f \\Lp(R";E) < C\\f ¡Lp(R";E)

which also the first estimate is stronger than the corresponding inequality in Lemma 4.1. Finally, since T0 e S'2 and I - 77 = - it follows from imbedding theorem in abstract Sobolev spaces [17] that

IT f ILpl(Rn;E) < C \\f \ILp(R«.;E), f e C0 (Bs; E). (44)

Thus, we have shown that the microlocal inverse corresponding to crt, satisfies the desired estimates.

Let b © = 1-b0 (Q). To invert Qj (ex, D) for (x, Q) e rt, we have to construct a Fourier integral operator T1, with kernel

K1 (x, y) = (2n)-n J P1 (f) e^(x,y,f)QO-1 (sy, f) df, (45)

such that the analogs of (39) and (32)-(35) are satisfied. For this step the factorization (36) of the symbol Qj (sy, f) will be used. Since the factor Gt (ex, Q) has a bounded inverse for (x, Q) e rt, the previous discussions show that we should try to construct the phase function in (46) using the factor Bt (ex, Q). We would like F (x, y, Q) to solve the complex eikonal equation

Bt (sx, ) = Bt (sy, f), x, y e Bs, f e supp P1, (46)

Since Bt (ex, Fx) - Bt (ey, Q) is a scalar function (it does not depend of operator A ), by reasoning as in [3, Lemma 3.4] we get that

Q(x, y, f) = $(x, y, f') + f (x, y, f),

where j is real and defined as

& (x', y, f 0 = [x1 - yO f1 + o(\x y'\2\f ,

f (x, y, ?) = (xi - yi) ?- + o(\xi - yi\2 \?

Im f (x, y, ?) — cfa - y-i)2 \? '\, c > 0. (47)

Then we obtain from the above that

Qj (ex, D) eiQ(xy?) = eiQQJ (ex, Qx) + eiQw-2L (ex, D) Q. Next, if we set

r (x, y, ?) = Gt (ey, ?) - Gt (ex, ?) = w-1 (y) ? - ia (ey, ?')]

-1 r ( rn (48)

- w- (x) ? - ia [ex, ? jj then it follows from (36) and (48) that

eiQ Qj (ex, Qx) = eiQQj (ey, ?) + eiQBt (ey, ?) r (x, y, ?) + O (t-N) (49)

for every N when b (f) * 0. Consequently, (49), (50) imply that

(2n)nQj (ex, D) Ki (x, y) = j fii (?) eiQd? + j fii (?) r (x, y, ?) G-1 (ey, ?)eiQd?

w-2 f fii(?)Qt\ey, ?) (L(ex, D)Q)eiQd? + O(t-N). (50)

By reasoning as in Theorem 3.1 we obtain from (51) that

Qj(ex,D)Ki(x,y) = (2n)-n J fi-(?)ei(x-y?)d? + Rw{x,y) + Rn(x,y), where

Rii(x, y) = (2n )-nw-2 J fii(? )Qt\ey, ? )(L(ex, D)Q)eiQ d? (51)

while R10 belongs to a bounded subset of S-1 and tR10 belongs to a bounded subset of S0. In view of this formula, we see that if we let K (x, y) = K0 (x, y) + K1 (x, y) and R (x, y) = R0 (x, y)+R1 (x, y), where R1 = R10 +R11, then we obtain (31). Moreover, since R10 satisfies the desired estimates, we see from Minkowski inequality that, in order to finish the proof of Lemma 4.1, it suffices to show that for f e C^0 (Be; E)

\\Tlf ||lp| (Rn;E) < C \\f IILp(Rn;E), (52)

£ t(1+»Ha|) |DaT{f |Lf(J?„;£) < C 1/ ||Lf(J№), (53)

i" \\Kf lk(fi-;i) < C 1/ IL^.E), <1 = p,p], (54)

\\VKf < C 1/ |Lf(R«;E), (55)

where j + j = 1.

To prove the above estimates we need the following prepositions for oscillatory integral in E-valued Lp spaces which generalize the Carleson and Sjolin result [36].

Preposition 4.1. Let E be Banach spaces and A e C^(Rn, L(E)). Moreover, suppose F e C°° satisfies | VF| > g >0 on supp A. Then for all l >1 the following holds

J eiX$(x)A(x)dx

< CNk-N, N =1,2,.

where CN-depends only on g if F and A (x) belong to a bounded subset of C°° and C°° (Rn, L (E)) and A is supported in a fixed compact set.

Proof. Given x0 e supp A. There is a direction v e Sn-1 such that |(v, VF)| > T7 on some ball centered at x0. Thus, by compactness, we can choose a partition of unity Vj e C^0 consisting of a finite number of terms and corresponding unit vectors v, such

that E^jW = 1 on supp A and |(vj, V$)| > \ on supp 4>j. For A;- = 4>jA it suffices to j=i

prove that for each j

eix®(x)Aj(x)dx

< Cnk-N, N = 1,2,..

After possible changing coordinates we may assume that v,- = (1, 0, . . . , 0) which means that |ff|>f on supp If let L(x;D) = (f*)"1^ then

L(.r; D)ea<I>M = ea<t>W. Consequently, if L* = ^ ( f|) j is a adjoint, then

j eiX$(x)A(x)dx = j eix$x)(L*)NAj(x)dx.

Since our assumption imply that (L*)N A- (x) = O (1-N), the result follows. Preposition 4.2. Suppose F e C°° is a phase function satisfying the non-degeneracy

r d2 $ 1

condition det - ^ Oon the support of

_ dxidxj J

A(x, y) e CJ£(Rn x Rn, L(E)). Then for T>f = I eik$ix,y)Aix, y)f (y)dx, 0 the following estimates hold

IM|if(J?,,;£) < '' 1<P<2,

-— 1 1

IM|if(J?,,;£) < CA. ?\\f\\LpiR,,Ey - + -7 = 1.

Proof. In view of [3, Remark 2.1] we have

\Vx[$(x,y) - $(x,z)]\~\y - z| (56)

where |y - z| is small. By using a smooth partition of unity we can decompose A (x, y) into a finite number of pieces each of which has the property that (57) holds on its

support. So, by (57) we can assume

mHx, y) - H(x,z)]| > C|y - z| (57)

on supp A for same C >0. To use this we notice that

\\Tif \\2 = j f Kx(y,z)f(y)f(z)dydz,

Kx(y,z)=f eik[H(x,y)-Hx,z)]A(x,y)A(x,z)dx. (58)

Hence, by virtue of Preposition 4.1 and by (58) we obtain that

\\Kx(y, z)\\m) < CN (1 + |À| |y - z|-N ), for all N.

Consequently, by Young's inequality, the operator with kernel K acts Lp(Rn; E)to Lp(Rn; E).

By (59) we get that

\\Txf \\l2(R";E) < C^ llf llL2(R";E). Moreover, it is clear to see that

\\Txf IIL „ (R";E) < CX n \\f ¡Li(R";E).

Therefore, by applying Riesz interpolation theorem for vector-valued spaces (see e. g., [19, § 1.18]) we get the assertion. In a similar way as in [3, Preposition 3.6] we have. Preposition 4.3. The kernel K1 (x, y) can be written as

tn-2eitjx,y)

where, for every fixed N, the operator functions A;- satisfy

\\DaAj(x,y) I < Ca(1 + t(xi - yi)2)-Nx - УГ|a|,

and moreover, the phase functions <pj are real and the property that when e is small enough, 0 < 8 < e and y1 e [-e, e] is fixed, the dilated functions

(x, y') ^ (-1)j8-1<pj(Sx, y1, Sy')

in the some fixed neighborhood of the function y0(x',y') = \x - y'| in the C°° topology. Then, the following estimates holds

K1(x,y) < Ctn-2(1 + tx - yi\)-1. (59)

Proof. By representation of K1 (x, y) and F (x, y, f) we have

K1(x, y) - tn-2 f fi1(t?)eitQ(x,y'?)Q0- 1(ey, ?) d?.

Then, by using (36) in view of positivity of operator A, by reasoning as in [3, Preposition 3.6] we obtain the assertion. Let us now show the end of proof of Lemma 4.1. Let n e C^(R) be supported in

[±,4] such that £ 1]{2l's) = l,s > 0 and set /;0(s) = 1 - £ ??(2v's). Then

define kernels /<i,v, v = 0, 1, 2, . . . , as follows

K f n(t2-v\x'-y'\)Ki(x,y), v > 0 1,V I no(t\x - y'\)Ki(x,y), v = 0.

Let T1,v denotes the operators associated to these kernels. Then, by positivity properties of the operator A and by Prepositions 4.2, 4.3 we obtain for f e C^(Be; E) the following estimates

|IV(^i) ^ C2~2v/" I\f\\h№E). \ + =

Lp(Rn;E) < C(t2-V) Wt (1+") I/ \\LpiR,,E). (61)

By summing a geometric series one sees that these estimates imply (52) and (53) for case of a =0.

Let us first to show (60). One can check that the estimate (59) implies that the Lr norm of K*10 is O (i"-2i -n/r). But, if we let r = я/я - 2, it is follows from Young inequality and the fact that j — = f that

\\T1,0f Lp (Rn;E) < Ctn-2rn/t \\f || Lp(Rn;E) = C \\f \\lp(R";E)

as desired. To prove the result for v >0, set B'e = {X e Rn-1, \X\ < ej and let Kjv be the kernel of the operator T^ Then, if we fix Xi and yi, it follows that the Lp(B'e; E) ^ Lp(B'e; E) norm of the operator

Tug[x') = J Kl [x, yM-ZW

equal ^»(-i)!»-!)',1 p + p') times the norm of the dilated operator

T1,v g(x,) = J K1v sx, y^ S}/)g{}/W,

where S = 2v t-1. By Preposition 4.3, the kernel in last integral equals the complex conjugate of

ei{tS)S-1Vj{Sy',x1,Sx')

tn 2i](t2 l'\x' - V|) V A,(yi, 8V,xi, 8x')—-, , ,--,

J\t{x'-y')\("-2y2\t{x1,8x',y1,8y')\'

and, consequently by using the Proposition 4.2, for 0 < ô < e and for supp g C B's we obtain that

V = —CO

V = —CO

Tl,vg(x') Lpl(Rn;E) <

C(tS)-(n-2)/p'tn-2(tS)-(n-2)/2t-1[(Xi - n)2 + &2\-1'2 ¡g ¡Lp^E).

This estimate implies

Kiv (x y) g (y')

< Ct~n[(xi — yi) + (2l'/f) ]

v It)2]-1/2

¡g ¡Lp(B's;E).

Lp(B's ;E)

For r = we set

f [(xi - yi)2 + (2V/t)2] r/2dxi\ = C(t/2V )2/n.

Then, the desired estimate (60) follows from the above estimate and Young's inequality. The other inequality (61), follows from a similar argument. Preposition 4.4. The estimates (32)-(34) imply (30). Proof. Indeed, (31) implies that

v(x) = T*(Qt(ex, D)v) - R*v(x), and so Minkowski's inequality, (32) and (34) give that

IIv y,E < ¡T*(Qt(ex,D)v) y,E+ ¡R+v ¡p,e <

||Qt(ex, D)v

+ Ct " llf

which implies that the first inequality in (30) for sufficiently large t. Moreover, in a similar way, using (32) and (33) we get (30) for a = 0. To prove (30) for |a| =1, we use (33), (34) and obtain

IIvv ¡p,E < ¡Vr-(Qt(ex,D)v) ¡p,E+ ¡VR*v ¡p,E <

Ct. " | Qt{ex,D)v

rA-1 ii II p,E + Ct n ||v ¡p,E.

Hence, the result follows.

Now we can show the end of the proof of Theorem 4.1. Really, we obtain the estimate (30), which implies the estimates (26) and (27). That is the assertion of Theorem 4.1 is hold.

Theorem 4.2. Assume all conditions of Theorem 4.1 are satisfied, then for

" e WP,ioc(Bo;-E(A),E) if \\L{x,D)u ||£ < ||Vu ||E and v e Lpoc{B0;E) then u is identically 0 if it vanishes in a nonempty open subset. Proof. Suppose

¡L(x,D) u ||e < IIVu ||e + ¡V'.Vu ||e (62)

in a connected open set 6, where ^e ^!i loc(G',E), V e L,XJ/i0c(G;ii) anj u e W2,1oc(G; E(A), E). Then, after the possibly change of variables, one sees that Theorem 4.2 would follow if we could show that if

supp u n {x e Be, x1 > 0} c {0} (63)

then 0 isupp u. Moreover, by making a proper choice of geodesic coordinate system, we may assume L (x, D) as

L (x,/)) = I)2 + CijDiDj, Dj = -

i,j=2 ' ~ 1

Then argue as in [29], first set ue (x) = u (ex) where e is chosen small enough so that

(26) and (27) hold for Be. Let e C^(Be) be equal to one when |x| <§ and set Ue =

hue. Then if Ve (x) = V (ex) and

L(sx,D)Ue = s2i](Lu)(sx) + V ^-Dai](L{a\ex,D))i

' J DJ 1

which implies that

¡L(ex, D)Ue ¡E < Co(1+ II Ve 11E ) IIU ||e + Co |V Ue 11E , x e Be/2. (64)

S, = {x e Be : -S < x1 < 0, S > 0}. If the condition (63) holds, then we can always choose 8 to be small enough that Ss n supp u c Be/2,

and so that if C is as in (26), (27) and C0 is as in (64) then

CCol j {1 + \\Ve \\E)n/2dx) <i. S 0

Next, (26), (27) imply

Lp(Ss;E) + tVn \\etWVUe |lp(Ss;E)

\\etwUe

< ¡etwL(ex,D)Ue ¡Lp^E) < C ¡etwL(ex,D)Ue ¡Lp(Ss;E) + C ¡etwL(ex,D)Ue ¡l^s,;E).

If we recall that j — y = §, then we see that (64) and Holder's inequality imply

C ¡etWL(ex, D)Ue ¡lp(s, ;E) < CC0 ¡0(1 + ||Ve ||e )etWUe ¡lp(s,;e) + CC0 ¡etwv Ue ¡lp(ss;e)

je ||lp(sj ;E) < n * e UE ^e ||lp(ss;e) f v u e ||lp(ss;e

< i|K"U |L(Sj;£)+CCo ||e""VL/s ||Lp(Sj;£).

Thus, by (63) for sufficiently large t >0 and BS = {x e Be : X\ < —5} we can conclude that

\\etwUe \\lp,(SS,E)+ \\etwvUe \\lp(SS;E) < 2C \\etwL(ex,D)Ue |lp(s5;E).

finally, since w' (x) = 1 + xi>0 on Be , this forces UE (x) = 0 for x e Ss and so 0 € supp u which completes the proof.

Consider the differential operator

P(x, D)u = dijDiDjU + Au + AkDku, i,j=l k=l

where «j are real-valued functions numbers, A = A (x), Ak = Ak (x), V (x) are the possible linear operators in a Banach space £.

By using Theorem 4.2 and perturbation theory of linear operators we obtain the following result

Theorem 4.3. Assume:

(1) all conditions of Theorem 4.1 are satisfied;

(2) AfeA^-"1) e L(X,(B0;L(£)) forO < Mk < j-

Then, for Dau e Lpioc (B0; E) if 11P (x, D) u\\E < \\Vu\\E and v e Lf,ioc(B°;Ê), then u is identically 0 if it vanishes in a nonempty open subset.

Proof. By condition (2) and by Theorem 2.1, for all e >0 there is a C (e) such that

< e llu iiw2(bo;e(a),e) + c(e) wu ||Lp(Bo;E). Lp(Bo;E) 11

Then, by using (29) and the above estimate we obtain the assertion.

5 Carleman estimates and unique continuation property for quasielliptic PDE

Let O c R1 be an open connected set with compact C2m-boundary dO. Let us consider the BVP for the following elliptic equation

Lu = ^2 aij(x)DiDju + ^ dk(x, y)Dku

i,j=l k=l

J2 aa (yDu = f (x, y), x e Rn, y e to c Rl,

\a\<2m

Bju = Y^ bjfi (y)Dp u(x, y) = 0, x e Rn, y e dto, j =1,2,..., m,

\P\<m,

where u = (x, y), Dj = -i—, T = (7i, • • •, Tn+i). Let ù = Rn x ii-

Let Q denotes the operator generated by the problem (64), (65). Theorem 5.1. Let the following conditions be satisfied;

(1) aa e ) for each |a| = 2m and aa e [Lœ + Lrkfor each |a| = k <2m with

rk > q and 2m - k > h

„ -.»„y'

dG, where s = (o1, s2, . . . , ffB) e Rm is a normal to dG ;

\a\=2m

(2) bjP e C2m-mj (dO) for each j, b and mf <2m, £ j(y\)°j = 0, for |b| = mj, y'e

(3) for ye Ù,Ç e R',X e S{<p),<p e (0, f ), + |À| f 0 let À + ^ 7^0;

(4) for each y0 e dQ local BVP in local coordinates corresponding to y0 A + J2 aa(y0)Da&(y) = 0,

\a\=2m

Bj0& =22 1 (y0)D u(y) = hj, j =1,2,..., m

has a unique solution 9 e C0 (R+) for all h = (hi, h2, . . . , hm) e Rm, and for f1 e R1-1 with

H\| + =0;

(5) Condition 4.1 holds, a^ e C°° (BE), n > 3, p = ^and p' is the conjugate of p and

w = x\ + y;

(6) dk e L^ (Rn x Q).

(a) for sufficiently large b >0, t > t0 and for n Q - jr^J < 2,p e (l,oo) the Carle-man type estimate

¡e-twu ¡Lm< C ¡e-tw(Q + b)u ^^

holds for u e W^O, .

(b) for V e LM(&) and y = j — j? the differential inequality ¡(Q + b)u(x .) ¡Lq (O) < ^^Mx .) ¡L,(O)

has a unique continuation property.

Proof. Let E = Lq (Q). Consider the following operator A which is defined by D(A) = W;im(O; Bju = 0), Au =22 aa(y)Dau(y).

\a\<2m

For x e Rn also consider operators

Ak(x)u = dk(x, y)u(y), k =1,2,..., n.

The problem (5.1), (5.2) can be rewritten in the form (4.1), where u (x) = u (x, .), f (x) = f (x, .) are functions with values in E = Lq (Q). Then by virtue of [24, Theorems 3.6 and 8.2] the (1) condition of Theorem 4.1 is satisfied. Moreover, by using the embedding W2m(O) c Lq(O) and interpolation properties of Sobolev spaces (see e.g., [19, §4]) we get that there is e >0 and a continuous function C (e) such that

< e ||u ¡W2m + C(e) ||u ¡Lq.

Due to positive of the operator A, then we obtain that < e ||Au \\lq + C(e) ||u \\lq.

Then it is easy to get from the above estimate that (2) condition of the Theorem 4.3 is satisfied. By virtue of (5) condition, (2) condition of the Theorem 4.1 is fulfilled too. Hence, by virtue of Theorems 4.1 and 4.3 we obtain the assertions.

6 Carleman estimates and unique continuation property for infinite systems of elliptic equations

Consider the following infinity systems of PDE

Yak{x)Dkumix) + (dm (x) + X)um{x)

k=\ » (67)

+ Y £ dkjm(x)DkUj(x) = fm(x), x e Rn, m =1,2,....

k=l j=1

D(x) = {dm(x)}, dm > 0, u = {Um}, Du = |dmumj, m =1,2,...,

lq(D) =

U : U e lq, IlU \\lq(D) = l|Du \\lq = ^ |dmUm|q I <

x e Rn, 1 < q < œ.

Let O denotes the operator generated by the problem (66). Theorem 6.1. Let the following conditions are satisfied:

(1) ak e Cb (Rn), ak (x) * 0, x e Rn, k = 1, 2, . . . , n and the Condition 4.1 holds;

(2) there are 0 < v <j such that

sup£M*Kjl2 <M>

a.e. for x e Rn. Then:

(a) for sufficiently large b >0, t > t0 and for n(- — i) < 2, 1 < p < pl < oo the Carleman type estimate

\\e-tWu Lmiq) < C \\e-tw(O + b)u \\)

holds for U e W2(Rn; lq(D), lq).

(b) for V e Lij L(£)^and y = j; — ^r the differential inequality ¡(O + b)u(x) ¡q < ¡V(x)u(x) ¡k has a unique continuation property.

Proof. Let E = and A, Ak (x) be infinite matrices, such that A = [dm (x)Sjm], Ak(x) = [dkjm (x)], m, j =1,2,..., to.

It is clear to see that this operator A is R-positive in and all other conditions of Theorems 4.1 and 4.3 are hold. Therefore, by virtue of Theorems 4.1 and 4.3 we obtain the assertions.

Competing interests

The author declares that they have no competing interests.

Reeeived: 26 January 2012 Aeeepted: 23 April 2012 Published: 23 April 2012

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doi:10.1186/1687-2770-2012-46

Cite this article as: Shakhmurov: Carleman estimates and unique continuation property for abstract elliptic equations. Boundary Value Problems 2012 2012:46.

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