### Abstract

The unique continuation theorems for elliptic differential-operator equations with
variable coefficients in vector-valued *L _{p}*-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*L*

_{p}-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:

where

here *a _{ij }*are real numbers,

*A*=

*A*(

*x*),

*A*=

_{k }*A*(

_{k }*x*) and

*V*(

*x*) are the possible linear operators in a Banach space

*E*.

Jerison and Kenig started the theory of *L _{p }*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*∈

*L*

_{n/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 A comprehensive introductions and historical references to Carleman estimates and unique continuation properties may be found, e.g., in [5].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 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 *L _{p }*spaces are studied. We will prove that if

*V*∈

*L*(

_{μ }*R*;

^{n}*L*(

*E*)),

*p*,

*μ*∈ (1, ∞) and

*u*is identically zero if it vanishes in a nonempty open subset, where

*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

In the Hilbert space *L*_{2 }(*R ^{n}*;

*H*), we derive the following Carleman estimate

Any of these inequalities would follow from showing that the adjoint operator *L _{t }*(

*x*;

*D*) =

*e*(

^{tw}L*x*;

*D*)

*e*satisfies the following relevant local Sobolev inequalities

^{-tw }

uniformly to *t*, where *L*_{0t } = *e ^{tw}L*

_{0}

*e*. In application, putting concrete Banach spaces instead of

^{-tw}*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

Let *E *and *E*_{1 }be two Banach spaces, and *L *(*E*, *E*_{1}) denotes the spaces of all bounded linear operators from *E *to *E*_{1}. For *E*_{1 }= *E *we denote *L *(*E*, *E*_{1}) by *L *(*E*). A linear operator *A *is said to be a *φ*-positive in a Banach space *E *with bound *M > *0 if *D *(*A*) is dense on *E *and

with *λ *∈ *S _{φ}*,

*φ*∈ (0,

*π*],

*I*is identity operator in

*E*. We will sometimes use

*A*+

*ξ*or

*A*instead of

_{ξ }*A*+

*ξI*for a scalar

*ξ*and (

*A*+ ξI)

^{-1 }denotes the inverse of the operator

*A*+

*ξI*or the resolvent of operator

*A*. It is known [19, §1.15.1] that there exist fractional powers

*A*of a positive operator

^{θ }*A*and

We denote by *L _{p }*(Ω;

*E*) the space of all strongly measurable

*E*-valued functions on Ω with the norm

By *L*_{p,q }(Ω) and
*p*, *q*)-integrable function space and Sobolev space with mixed norms, where 1 *≤ p*, *q < ∞*, see [20].

Let *E*_{0 }and *E *be two Banach spaces and *E*_{0 }is continuously and densely embedded *E*.

Let *l *be a positive integer.

We introduce an *E*-valued function space
*u *∈ *L _{p }*(Ω;

*E*

_{0}) such that the generalized derivatives

The Banach space *E *is called an *UMD*-space if the Hilbert operator
*dy *is bounded in *L _{p }*(

*R*,

*E*),

*p*∈ (1,

*∞*) (see e.g., [21,22]).

*UMD*spaces include, e.g.,

*L*,

_{p}*l*spaces and Lorentz spaces

_{p }*L*,

_{pq}*p*,

*q*∈ (1,

*∞*).

Let *E*_{1 }and *E*_{2 }be two Banach spaces. Let *S *(*R ^{n}*;

*E*) denotes a Schwartz class, i.e., the space of all

*E*-valued rapidly decreasing smooth functions on

*R*. Let

^{n}*F*and

*F*

^{-1}denote Fourier and inverse Fourier transformations, respectively. A function Ψ ∈

*C*(

^{m }*R*;

^{n}*L*(

*E*

_{1},

*E*

_{2})) is called a multiplier from

*L*(

_{p }*R*;

^{n}*E*

_{1}) to

*L*(

_{q }*R*;

^{n}*E*

_{2}) for

*p*,

*q*∈ (1,

*∞*) if the map

*u → Ku*=

*F*

^{-1 }Ψ (

*ξ*)

*Fu*,

*u*∈

*S*(

*R*;

^{n}*E*

_{1}) is well defined and extends to a bounded linear operator

We denote the set of all multipliers from *L _{p }*(

*R*;

^{n}*E*

_{1}) to

*L*(

_{q }*R*;

^{n}*E*

_{2}) by

*E*

_{1 }=

*E*

_{2 }=

*E*and

*q*=

*p*we denote

*M*(

_{p }*E*). The

*L*-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].

_{p}A set *K *⊂ *L *(*E*_{1}, *E*_{2}) is called *R*-bounded [22,23] if there is a constant *C *such that for all *T*_{1}, *T*_{2}, . . . , *T _{m }*∈

*K*and

*u*

_{1,}

*u*

_{2}, . . . ,

*u*∈

_{m }*E*

_{1},

*m*∈

*N*

where {*r _{j}*} 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*).

Let

For any *r *= (*r*_{1}, *r*_{2}, . . . , *r _{n}*),

*r*∈ [0,

_{i }*∞*) the function (

*iξ*)

*,*

^{r}*ξ*∈

*R*will be defined such that

^{n }

where

**Definition 2.1. **The Banach space *E *is said to be a space satisfying a multiplier condition with respect to *p*, *q *∈ (1, *∞*) (with respect to *p *if *q *= *p*) when for Ψ ∈ *C*^{(n) }(*R ^{n}*;

*L*(

*E*

_{1},

*E*

_{2})) if the set

is *R*-bounded, then

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

is *R*-bounded.

**Remark 2.1**. By virtue of [29] or [30] UMD spaces satisfy the multiplier condition with respect to *p *∈ (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 *L _{q}*, 1

*≤ q ≤ ∞*[31],

*A*has the bounded imaginary powers with

*A*is a generator of a semigroup with Gaussian bound in

*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 *∈ (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
*H _{k }*is a uniform collection of multipliers if there exists a constant

*M >*0, independent on

*h*∈

*K*, such that

for all *h *∈ *K *and *u *∈ *S *(*R ^{n}*;

*E*

_{1}).

We set

In view of [17, Theorem A_{0}], we have

**Theorem 2.0. **Let *E*_{1 }and *E*_{2 }be two UMD spaces and let

If

uniformly with respect to *h *∈ *K *then Ψ* _{h }*(

*ξ*) is a uniformly collection of multipliers from

*L*(

_{p }*R*;

^{n}*E*

_{1}) to

*L*(

_{q }*R*;

^{n}*E*

_{2}).

Let

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 *∈ (1, *∞*) and *A *is a *R*-positive operator on *E*;

(2) *l *is a positive and *α _{k }*are nonnegative integer numbers such that 0

*≤ μ ≤*1

*- ϰ*,

*t*and

*h*are positive parameters.

Then the embedding

is continuous and there exists a positive constant *C _{µ }*such that for

the uniform estimate holds

Moreover, for

### 3 Carleman estimates for DOE

Consider at first the equation with constant coefficients

where
*A *is the possible unbounded operator in a Banach space *E*.

Let
*t *is a positive parameter.

**Remark 3.1**. It is clear to see that

where
*L*_{0t }(*x*, *ξ*) is the principal operator symbol of *L*_{0t }(*x*, *D*) on the domain *B*_{0}, i.e.,

where

Our main aim is to show the following result:

**Remark 3.2**. Since *Q*(*ξ*) ∈ *S *(*φ*) for all *φ *∈ [0, *π*) due to positivity of *A*, the operator function *A *+ *|ξ*^{|}*|*^{2}, *ξ *∈ *R ^{n }*is uniformly positive in

*E*. So there are fractional powers of

*A*+

*|ξ*

^{|}

*|*

^{2 }and the operator function

*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)

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

for

To prove the Theorem 3.1, we shall show that *L*_{0t }(*x*, *D*) has a right parametrix *T*, with the following properties.

**Lemma 3.1**. For *t > *0 there are functions *K *= *K _{t }*and

*R*=

*R*so that

_{t }

where *δ *denotes the Dirac distribution. Moreover, if we let *T *= *T _{t }*be the operator with kernel

*K*, i.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

**Proof**. By Remark 3.2 the operator function
*E *for all *ξ *∈ *R ^{n}*. Since

*tw*

_{1 }+

*iξ*

_{1 }∈

*S*(

*φ*), due to positivity of

*A*, for

*ξ*∈

*R*,

^{n}*t*> 0 and

Therefore, we call *G _{t }*(

*x, ξ*) the regular factor. Consider now the second factor

By virtue of operator calculus and fractional powers of positive operators (see e.g.,
[19, §1.15.1] or [35]) we get that *- *[*tw*_{1 }+ *iξ*_{1}] ∉ *S *(*φ*) for *ξ*_{1 }= 0 and *tw*_{1 }= *|ξ ^{|}|*, i.e., the operator

*B*(

_{t }*x*,

*ξ*) does not has an inverse, in the following set

So we will called *B _{t }*the singular factor and the set Δ

*call singular set for the operator function*

_{t }*B*. The operator

_{t}*. Nevertheless, the operator*

_{t}*x*,

*ξ*) is sufficiently far from Δ

*. For instance, if we define*

_{t}

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

where the constant *C *is independent of *x*, *ξ*, *t *and * ^{c}*Γ

*denotes the complement of Γ*

_{t }*.*

_{t}Let
*β*(*ξ*) = 0 if
*β *(*ξ*) = 0 near the origin. We then define

and notice that *β*_{0 }(*ξ*) = 0 on Γ* _{t}*. Hence, if we define

and recall (11), then by [31] it follows from standard microlocal arguments that

where *R*_{0t }belongs to a bounded subset of *S*^{-1 }which is independent of *t*. Since operator

By reasoning as in [31] we get that *tR*_{0t }belongs to a bounded subset of *S*^{0}. So, we have the following estimate

Moreover, the Remark 3.2, positivity properties of *A *and, (11) and (12) imply that, the operator functions
*T*_{0 }be the operator with kernel *K*_{0 }(*x*, *y*), by using the Minkowski integral inequality and Plancherel's theorem we obtain

For inverting *L*_{0t }(*x*, *D*) on the set Γ* _{t }*we will require the use of Fourier integrals with complex phase. Let

*β*

_{1 }(

*ξ*) = 1

*- β*

_{0 }(

*ξ*). We will construct a Fourier integral operator

*T*

_{1 }with kernel

so that the analogs of (16) and the estimates (7)-(9) are satisfied. Since
* _{t}*, we should expect to construct the phase function Φ in (13) using the factor

*B*(

_{t }*x*,

*ξ*). Specifically, we would like Φ to satisfy the following equation

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

Since *w*_{1 }(*x*) = 1 + *x*_{1}, *w*_{1 }(*y*) = 1 + *y*_{1}, we have

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

Next, if we set

then it follows from *L*_{0t }(*x*, *ξ*) = *G _{t }*(

*x*,

*ξ*)

*B*(

_{t }*x*,

*ξ*) and (14) that

Consequently, (16)-(18) imply that

By reasoning as in [3] we obtain that the first and second summands in (19) belong to a bounded subset of
*S*^{0}. So, we see that the equality (5) must hold. Now we let *K *(*x*, *y*) = *K*_{0 }(*x*, *y*) + *K*_{1 }(*x*, *y*) and *R *(*x*, *y*) = *R*_{0 }(*x*, *y*) + *R*_{1 }(*x*, *y*), where

Due to regularity of kernels, by using of Minkowski and Hölder inequalities we get
the analog estimate as (7) and (9) for the operators *T*_{0 }and *R*_{10}. Thus, in order to finish the proof, it suffices to show that for *f *∈*L*_{2 }(*B*_{0}; *E*) one has

However, since *R*_{1,1 }≈ *tT*_{1}, we need only to show the following

By using the Minkowski inequalities we get

where

where

Consequently, it follows from Plancherel's theorem that

Note that for every *N *we have

Since *A *is a positive operator in *E*, we have

when

i.e.,

Moreover, it is clear that

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

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

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 *L*_{p}-Carleman estimates and unique continuation for equation with variable coefficients

Consider the following DOE

where
*A *is the possible unbounded operator in a Banach space *E *and *a _{ij }*are

real-valued smooth functions in *B _{ε }*= {

*x*∈

*R*,

^{n}*|x| < ε*}.

**Condition 4.1**. There is a positive constant *γ *such that
*ξ *∈ *R ^{n}*,

*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 *R*-positive operator in *E*. Suppose the Condition 4.1 holds, *n *≥ 3, *p *=
*p' *is the conjugate of *p*, *w * =
*a _{ij }*∈

*C*

^{∞ }(

*B*)

_{ε}_{. }Then for

*B*;

_{ε}*E*(

*A*)) and

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

where,

Consequently, since *w*_{1 }≃ 1 on *B _{ε}*, it follows that, if we let

*Q*(

_{t }*εx*,

*D*) be the differential operator whose adjoint equals

then it suffices to prove the following

The desired estimates will follow if we could constrict a right operator-valued parametrix
*T*, for *Q _{t}* *(

*εx*,

*D*) satisfying

*L*estimates. these are contained in the following lemma.

_{p }**Lemma 4.1**. For *t > *0 there are functions *K *= *K _{t }*and

*R*=

*R*, so that

_{t}

where *δ *denotes the Dirac distribution. Moreover, if we let *T *= *T _{t }*be the operator with kernel

*K*(

*x*,

*y*) and

*R*be the operator with kernel

*R*(

*x*,

*y*), then if

*ε*and

**Proof**. The key step in the proof is to find a factorization of the operator-valued symbol
*L *(*x*, *D*) is of the form

therefore, we can expressed

where

where

The ellipticity of *Q*(*x*, *D*) and the positivity of the operator *A*, implies that the factor *G _{t }*(

*x*,

*ξ*) never vanishes and as in the proof of Theorem 3.1 we get that

i.e., the operator function *G _{t }*(

*εx*,

*ξ*) has uniformly bounded inverse for (

*x*,

*ξ*) ∈

*B*. One can only investigate the factor

_{ε }×R^{n}*B*(

_{t }*εx*,

*ξ*). In fact, if we let

then the operator function *B _{t }*(

*x*,

*ξ*) is not invertible for (

*x*,

*ξ*) ∈ Δ

*. Nonetheless,*

_{t}*B*(

_{t }*εx*,

*ξ*) and

*x*,

*ξ*) is sufficiently far away. For instance, if we define

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

As in § 3, we can use (38) to microlocallity invert
* _{t }*
. To do this, we first fix

It is clear that *β*_{0 }(*ξ*) = 0 on Γ* _{t}*. Consequently, if we define

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

where *R*_{0 }belongs to a bounded subset of *S*^{-1 }that independent of *t*. Since the adjoint operator

Moreover, the positivity properties of *A *and the estimate (38) imply that the operator functions
*T*_{0 }be the operator with kernel *K*_{0}. Then in a similar way as in [31] we obtain that

which also the first estimate is stronger than the corresponding inequality in Lemma
4.1. Finally, since *T*_{0 }∈ *S*^{-2 }and

Thus, we have shown that the microlocal inverse corresponding to ^{c}Γ* _{t}*, satisfies the desired estimates.

Let *β*_{1 }(ξ) = 1*-β*_{0 }(ξ). To invert
*x*, ξ) ∈ Γ* _{t}*, we have to construct a Fourier integral operator

*T*

_{1}, with kernel

such that the analogs of (39) and (32)-(35) are satisfied. For this step the factorization
(36) of the symbol
*G _{t }*(

*εx*, ξ) has a bounded inverse for (

*x*, ξ) ∈ Γ

*, the previous discussions show that we should try to construct the phase function in (46) using the factor*

_{t}*B*(

_{t }*εx*, ξ). We would like Φ (

*x*,

*y*, ξ) to solve the complex eikonal equation

Since *B _{t }*(

*εx*, Φ

*)*

_{x}*- B*(

_{t }*εy*, ξ) is a scalar function (it does not depend of operator

*A*), by reasoning as in [3, Lemma 3.4] we get that

where *ϕ *is real and defined as

while

and

Then we obtain from the above that

Next, if we set

then it follows from (36) and (48) that

for every *N *when *β*_{1 }(*ξ*) ≠ 0. Consequently, (49), (50) imply that

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

where

while *R*_{10 }belongs to a bounded subset of *S*^{-1 }and *tR*_{10 }belongs to a bounded subset of *S*^{0}. In view of this formula, we see that if we let *K *(*x*, *y*) = *K*_{0 }(*x*, *y*) + *K*_{1 }(*x*, *y*) and *R *(*x*, *y*) = *R*_{0 }(*x*, *y*)+*R*_{1 }(*x*, *y*), where *R*_{1 }= *R*_{10 }+*R*_{11}, then we obtain (31). Moreover, since *R*_{10 }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

where

To prove the above estimates we need the following prepositions for oscillatory integral
in *E*-valued *L _{p }*spaces which generalize the Carleson and Sjolin result [36].

**Preposition 4.1**. Let *E *be Banach spaces and
*C ^{∞ }*satisfies

*|*∇Φ

*| ≥*γ

*>*0 on supp

*A*. Then for all

*λ >*1 the following holds

where *C _{N}*-depends only on

*γ*if Φ and

*A*(

*x*) belong to a bounded subset of

*C*and

^{∞ }*C*(

^{∞ }*R*,

^{n}*L*(

*E*)) and

*A*is supported in a fixed compact set.

**Proof**. Given *x*_{0 }∈ supp *A*. There is a direction *ν *∈ *S*^{n-1 }such that *|*(*ν*, ∇Φ)*| ≥ *
*x*_{0}. Thus, by compactness, we can choose a partition of unity
*ν _{j }*such that

*A*and

*φ*. For

_{j}*A*=

_{j }*φ*it suffices to prove that for each

_{j}A*j*

After possible changing coordinates we may assume that *ν _{j }*= (1, 0, . . . , 0) which means that

*φ*. If let

_{j}

Since our assumption imply that (*L**)* ^{N }A_{j }*(

*x*) =

*O*(

*λ*), the result follows.

^{-N}**Preposition 4.2**. Suppose Φ ∈ *C ^{∞ }*is a phase function satisfying the non-degeneracy condition det

Then for

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

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

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