Weighted gradient inequalities and unique continuation problems
aa r X i v : . [ m a t h . A P ] A p r Weighted gradient inequalities and uniquecontinuation problems
Laura De Carli, Dmitry Gorbachev, and Sergey Tikhonov
Abstract.
We use Pitt inequalities for the Fourier transform to provethe following weighted gradient inequality k e − τℓ ( · ) u q f k q ≤ c τ k e − τℓ ( · ) v p ∇ f k p , f ∈ C ∞ ( R n ) . This inequality is a Carleman-type estimate that yields unique continua-tion results for solutions of first order differential equations and systems.
1. Introduction
The main purpose of this paper is to prove that the following weightedSobolev gradient inequality holds for every linear function ℓ : R n → R , every f ∈ C ∞ ( R n ) and every τ ≥
0, with suitable weights u , v and exponents1 < p, q < ∞ .(1.1) k e − τℓ ( · ) u q f k q ≤ c τ k e − τℓ ( · ) v p ∇ f k p Here, c τ is a finite constant that may depend on τ but does not depend on ℓ and f . We have denoted with k f k r = (cid:0)R R n | f ( x ) | r dx (cid:1) r the norm in L r ( R n )and with h x, y i = x y + · · · + x n y n and | x | = h x, x i the standard innerproduct and norm in R n .When τ >
0, we prove in Theorem 1.1 that c τ = max ( τ − , C ; hereand throughout the paper, C denotes a generic constant that depends onlyon non-essential parameters, i.e. C = C u,v,p,q,n . In particular, c τ = C when τ ≥
1. Inequalities like (1.1) are often called
Carleman inequalities in literature. In Sections 3 and 4 we will discuss Carleman inequalities andtheir connection with unique continuation problems and we will prove new
Mathematics Subject Classification.
Primary: 42B10; Secondary: 35B60.
Key words and phrases.
Pitt inequalities, weighted Carleman estimates, first ordersystems of PDE.D. G. was supported by the Russian Science Foundation under grant 18-11-00199.S. T. was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCAProgramme of the Generalitat de Catalunya. unique continuation results for systems of partial differential equations andinequalities.When τ = 0 in (1.1), we obtain a standard weighted Sobolev gradientinequality (also called weighted Poincar´e-Sobolev inequality )(1.2) k u q f k q ≤ c k v p ∇ f k p , f ∈ C ∞ ( R n ) . These inequalities have deep applications in partial differential equations.For example, the case p = 2 < q of (1.2) arises in Harnack’s inequalityand regularity estimates for degenerate second order differential operatorsin divergence form. They also have applications in the study of the stablesolutions of the Laplace and the p -Laplace operators in the Euclidean space,the Laplace-Kohn operator in the Heisenberg group, the sub-Laplace oper-ator in the Engel group, etc.; see e.g. [
49, 22, 57 ] and the references citedin these papers; see also [ ].Conditions on the weights u and v and the exponents p , q for which(1.2) holds have been investigated by several authors. The most naturalapproach to study (1.2) is based on the following pointwise inequality (seee.g. [
19, 46 ]) | f ( x ) | ≤ CI ( |∇ f | )( x ) , x ∈ R n , where I α φ ( x ) = R R n φ ( y ) | x − y | n − α dy , α < n , is the Riesz potential . This inequal-ity follows from the classical Sobolev integral representation and is provede.g. in [ ].If the weighted inequality(1.3) k u q I f k q ≤ C k v p f k p holds for the weights u and v , we also have k u q f k q ≤ C k u q I ( |∇ f | ) k q ≤ C k v p |∇ f |k p . E. Sawyer obtained in [ ] a complete characterization of the weights u and v for which the gradient inequality (1.3) holds with p ≤ q . However, in somecases, the conditions in [ ] are difficult to verify. When p = q = 2, a fullcharacterization of the weights for which (1.2) holds is also in [ ], but alsothe conditions in this paper are difficult to verify.H. P. Heinig showed in [ ] that weighted norm inequalities for theFourier transform (or: Pitt-type inequalities ) in the form of(1.4) k b f u q k q ≤ C k f w p k p , f ∈ C ∞ ( R n ) , can be used to prove weighted gradient inequalities. The Fourier transformis defined as b f ( y ) = R R n f ( x ) e − i h x, y i dx .To prove (1.2) from (1.4), we observe that d I α f ( y ) = c α | y | − α b f ( y ) , where c α is an explicit constant; we can see at once that (1.3) is equivalent to k u q ( | y | − b f ) ˇ k q ≤ C k v p f k p EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 3 where ˇ denotes the inverse Fourier transform. We can apply Pitt’s in-equality twice (with a suitable weight w and an exponent γ ∈ (1 , ∞ )) toobtain k u q ( | y | − b f ) ˇ k q ≤ C k w γ | y | − b f k γ ≤ C k v p f k p . Taking w = | y | γ and γ = q and assuming conditions on the weights thatensure that both Pitt’s inequalities hold we obtain the main theorem in [ ],which was proved differently; see Theorem 2.1 in Section 2. Throughout this paper, we will often write A . B when A ≤ CB with a constant C >
0. We will also write A ≍ B whenthere exists a constant C >
0, called the constant of equivalence , such that C − A ≤ B ≤ CA . As usual, we let g ∗ be the non-increasing rearrangementof g . We let p ′ = pp − be the dual exponent of p ∈ (1 , ∞ ).Our main result can be stated as follows. Theorem . Let u and v + ∞ be weights on R n , n ≥ . (a) Let < p ≤ q < ∞ . If there exists γ > that satisfies max ( p, p ′ ) ≤ γ ≤ q , for which (1.5) A qu (0) := sup s> s − q ( γ ′ − n ) u ∗ ( s ) < ∞ , n < γ ′ ≤ n + q ,A qu ( τ ) := sup s> Z s u ∗ ( t ) dt (cid:16)Z s ( t + τ n ) − γ ′ n dt (cid:17) qγ ′ < ∞ , τ > , and (1.6) A pv := sup s> s pγ ′ − (1 /v ) ∗ ( s ) < ∞ , the inequality (1.7) k e − τℓ ( · ) u q f k q ≤ c τ k e − τℓ ( · ) v p ∇ f k p , f ∈ C ∞ ( R n ) , holds for every τ ≥ and every linear function ℓ ( x ) = h a , x i + b , a ∈ R n , | a | = 1 , b ∈ R , with the constant (1.8) c τ = CA u ( τ ) A v , where C = C p,q,γ,n is some positive constant. Moreover, (1.9) A u ( τ ) ≤ max ( τ − , A u (1) , τ > . (b) Let < q < p < ∞ . If there exists γ > that satisfies ( nn − < γ ≤ q, τ = 0 , < γ ≤ q, τ > , for which (1.5) holds and ˜ A rv := Z ∞ s − rγ − (cid:16)Z s (1 /v ) ∗ ( t ) p − dt (cid:17) rp ′ ds < ∞ , LAURA DE CARLI, DMITRY GORBACHEV, AND SERGEY TIKHONOV with r = γ − p , the inequality (1.7) holds with the constant (1.10) c τ = CA u ( τ ) ˜ A v . Remark . When τ = 0 and γ = q we obtain Theorem 2.4 in [ ]with simplified conditions on u and v . The proof of Theorem 1.1 shows thatthe assumptions γ ′ ≤ n + q and p ′ ≤ γ are to rule out the trivial weights u ≡ v ≡ + ∞ . Remark . For the applications of Theorem 1.1 it is important to havethe uniform boundedness of c τ as τ → ∞ . From (1.8), (1.10) and (1.9), wehave c τ ≤ c ≍ A u (1) whenever τ ≥
1; thus, to prove the boundedness of c τ , it is sufficient to verify that A u (1) < ∞ . Remark . It is interesting to compare our weighted gradient inequal-ities with those proved by G. Sinnamon in [ ]. In that paper, a weightednorm inequality in the form of(1.11) k f u q k q ≤ C kh∇ f, x i w p k p , f ∈ C ∞ ( R n )is considered. If we denote with ∂ r f = h x | x | , ∇ f i the radial derivative of f ,the inequality (1.11) is equivalent to k f u q k q ≤ C k | x | w p ∂ r f k p , f ∈ C ∞ ( R n ) , and implies (1.2) with v = | x | p w .In [ , Theorem 4.1], (1.11) is only proved for p = q and q < p undersome conditions on u , w ; moreover, in [ , Theorem 3.4] it is proved thatwhen 1 ≤ p < q < ∞ and the weight w is locally integrable on R n , theinequality (1.11) holds for every f ∈ C ∞ ( R n ) if and only if u ≡ f is radial, ∇ f ( x ) = x | x | ∂ r f ( x ), and so |∇ f ( x ) | = | ∂ r f ( x ) | . Thus,our Theorem 1.1 yields (1.11) for radial functions with a nontrivial weight u and with w = | x | − p e − pτℓ ( x ) v . We proved in Corollary 1.2 below that we canconsider piecewise power weights v = | x | ( β , β ) , with 0 ≤ β ≤ n (cid:0) pγ ′ − (cid:1) (seedefinition (1.13)). For example, if β = n (cid:0) pγ ′ − (cid:1) , then w is locally integrablefor n < γ ′ because − p + β > − n . We remark that the counterexample in[ , Theorem 3.4] is not radial. Remark . The inequality (1.7) is equivalent to(1.12) k u q f k q ≤ c τ k v p ( τ a f + ∇ f ) k p . To see this, it is enough to use the substitution f = e − τℓ ( · ) f and ∇ ( e τℓ ( · ) f ) = e τℓ ( · ) ( τ a f + ∇ f ).Let β , β ∈ R ; we define the piecewise power function t t ( β ,β ) asfollows:(1.13) t ( β ,β ) := ( t β , < t ≤ ,t β , t ≥ . EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 5
In the following corollary of Theorem 1.1 we consider the important case ofpiecewise power weights.
Corollary . Let < p ≤ q < ∞ ; let γ > that satisfies max ( p, p ′ ) ≤ γ ≤ q and n < γ ′ ≤ n + q .With the notation and the assumptions of Theorem , the inequality (1.7) holds with u ( x ) = | x | ( − α , − α ) , v ( x ) = | x | ( β ,β ) , with α j , β j ≥ ,provided that (1.14) α ≤ n (cid:16) − qγ ′ + qn (cid:17) , ( α ≥ n (cid:0) − qγ ′ + qn (cid:1) when τ = 0 ,α ≥ when τ > , (1.15) β ≤ n (cid:16) pγ ′ − (cid:17) , β ≥ n (cid:16) pγ ′ − (cid:17) . In particular, for power weights u ( x ) = | x | − α , v ( x ) = | x | β the inequality (1.7) holds if α = n (cid:0) − qγ ′ + qn (cid:1) ≥ when τ = 0 , ≤ α ≤ n (cid:0) − qγ ′ + qn (cid:1) when τ > , β = n (cid:16) pγ ′ − (cid:17) ≥ . Moreover, the conditions (1.16) αq + βp = n (cid:0) q − p (cid:1) + 1 when τ = 0 , αq + βp ≤ n (cid:0) q − p (cid:1) + 1 when τ > , are necessary for the validity of (1.7) . Letting τ = α = β = 0, 1 < p < n , and γ = q in (1.16), we obtain q = npn − p and Corollary 1.2 yields the classical Sobolev inequality k f k q ≤ C k∇ f k p ; see also [ , Corollary 2.5].When τ = 0, we obtain the inequality (cid:16)Z R n | f | q | x | ( − α , − α ) dx (cid:17) q ≤ C (cid:16)Z R n |∇ f | p | x | ( β ,β ) dx (cid:17) p , which was proved by Maz’ya [ ] and Caffarelli, Kohn, and Nirenberg [ ]for power weights. In [ , Sect. 2.1.6] it was proved that if 1 < p < n , p ≤ q ≤ pnn − p , and − αq = βp − n (cid:16) p − q (cid:17) > − nq , then(1.17) (cid:16)Z R n | f | q | x | − α dx (cid:17) q ≤ C (cid:16)Z R n |∇ f | p | x | β dx (cid:17) p . In [ , Lemma 2.1], this inequality was proved for n ≥
2, 1 < p < + ∞ ,0 ≤ p − q = n (cid:16) − βp − αq (cid:17) and − nq < − αq ≤ βp . Note that the conditionsin [ ] and [ ] are the same, except for the extra condition p < n in [ ].From Corollary 1.2 with τ = 0 we have that α = n (cid:0) − qγ ′ + qn (cid:1) ≥ β = n (cid:0) pγ ′ − (cid:1) ≥
0, where max ( p, p ′ ) ≤ γ ≤ q and n < γ ′ ≤ n + q . These LAURA DE CARLI, DMITRY GORBACHEV, AND SERGEY TIKHONOV inequalities imply p − q = n (cid:0) − βp − αq (cid:1) , − nq < − αq ≤ βp , but we also haveto assume α ≥ β ≥ ] and also in [ ] for special values of α and β . Our Theorem 1.1 can be used to proveunique continuation results for weak solutions (also called solutions in dis-tribution sense ) of systems of differential equations and inequalities; seeSection 3 for definitions and preliminary results.We consider solutions in weighted Sobolev spaces of distributions: givena domain D ⊂ R n , we let W m,p,v ( D ) be the closure of C ∞ ( D ) with respectto the norm k f k W m,p,v ( D ) = m X | α | =0 k v p ∂ αx f k L p ( D ) where α = ( α , . . . , α n ) ∈ N n and the ∂ αx f = ∂ α x · · · ∂ α n x n f are the partialderivatives of f . In Section 3 we prove the following Theorem . Let p , q , γ , u and v be as in Theorem . Let r = p − q . Let f ∈ W ,p,v ( R n ) be a solution of the differential inequality (1.18) |∇ f | ≤ V | f | with V ∈ L r (supp f, v rp u − rq dx ) . If, for some linear function ℓ : R n → R , wehave that supp f ⊂ { x : ℓ ( x ) ≥ } , necessarily f ≡ . Note that the condition V ∈ L r (supp f, v rp u − rq dx ) follows from ei-ther V ∈ L r ( R n , v rp u − rq dx ) if supp f is unbounded, or from V ∈ L r loc ( R n , v rp u − rq dx ) if f has compact support. In particular, for powerweights u , v as in Corollary 1.2, the differential inequality (1.18) does nothave solutions with compact support if V ≍ | x | − ǫ for some ǫ >
0; seeRemark 3.1.To prove Theorem 1.3 we use a method developed by T. Carleman in [ ].A brief discussion on unique continuation problems and Carleman’s methodis in Sections 3 and 4.When D is measurable and v is a suitable weight we consider the Dirich-let problem(1.19) ( − div ( v ∇ f |∇ f | p − ) = v V f | f | p − ,f ∈ W ,p,v ( D ) , where div (( g , . . . , g n )) = ∂ x g + · · · + ∂ x n g n and the potential V is ina suitable L r space. The operator div ( v ∇ f |∇ f | p − ) is known as weighted p -Laplacian in the literature (see e.g. [
23, 31 ]) and is denoted by ∆ p when v ≡
1. The weighted p -Laplacian is nonlinear when p = 2 and is linear when p = 2. EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 7
When v ≡
1, (1.19) can be compared to the Sturm-Liouville problemin the form of − ∆ p f = ( λm − V ) f | f | p − (see e.g. [ ]). When n = 1 and p = 2 we have − ( vf ′ ) ′ = vV f . This problem is related to the classicalSturm-Liouville problem − ( vf ′ ) ′ = ( λw − q ) f . See [ ].We prove the following Theorem . Let f ∈ W ,p,v ( D ) be a solution of the Dirichlet problem (1.19) . Let V + = max { V, } . Assume that | V | p ∈ L r ( D, v rp u − rq dx ) , where u , v are as in Theorem and r = p − q . Then, either c k u − q v p V p + k L r ( D ) ≥ , where c is as in (1.2) , or f ≡ in D . Thus, the Dirichlet problem (1.19) has the unique solution f ≡ L r norm of V p + on D is small enough.To the best of our knowledge, the method of proof of Theorem 1.4 hasbeen used for the first time in [ ]; it is extensively used in [ ] and [ ].
2. Proof of Theorem 1.1
In this section we prove our main theorem and a few corollaries.
We will use the following theorem due toHeinig [ ], Jurkat-Sampson [ ], and Muckenhoupt [ ]. Theorem . Let n ≥ . If < p ≤ q < ∞ and the weights u and w satisfy sup s> (cid:16)Z s u ∗ ( t ) dt (cid:17) q (cid:16)Z s ((1 /w ) ∗ ( t )) p − dt (cid:17) p ′ =: A < ∞ , or if < q < p < ∞ , and sup s> (cid:18)Z ∞ (cid:16)Z s u ∗ ( t ) dt (cid:17) rq (cid:16)Z s ((1 /w ) ∗ ( t )) p − dt (cid:17) rq ′ ((1 /w ) ∗ ( s )) p − ds (cid:19) r (2.1) =: A < ∞ where r = qpq − p , then Pitt’s inequality k b f u q k q ≤ C j k f w p k p , f ∈ C ∞ ( R n ) , j = 1 , , holds with C j ≤ C p,q,j A j . Recall that the non-increasing rearrangement of a measurable radiallydecreasing function f ( x ) = f ( | x | ) is defined as follows: let for λ > µ f ( λ ) = µ { x : | f ( x ) | > λ } = µ { x : | x | < f − ( λ ) } = ( f − ( λ )) n V n , LAURA DE CARLI, DMITRY GORBACHEV, AND SERGEY TIKHONOV where V n is the volume of the unit ball B n = { x ∈ R n : | x | ≤ } . Then for t > f ∗ ( t ) = inf { λ > µ f ( λ ) < t } = f (( t/V n ) n ) . Note that the conditions on u and w are also necessary when u and w are radial, i.e., u = u ( | x | ) and w ( x ) = w ( | x | ), with u ( r ) non-increasingand w ( r ) non-decreasing. See [ ] and also [ , Theorem 1.2 ] for simplerand more general necessary conditions on the weight u and w . We shouldalso mention [ , Theorem 2.1] where a necessary condition similar to thatin [ ], with u replaced by a measure dµ , was proved.We also need the following Lemma . Let ψ be a non-increasing non-negative function; let β , β > and let β ′ = min ( β , . If either A = sup s> s ( − β , − β ) Z s ψ ( t ) dt < ∞ , or B = sup s> s (1 − β , − β ′ ) ψ ( s ) < ∞ , then β ≤ and A ≍ B . Proof.
Assume
A < ∞ ; then, for every s >
0, we have that R s ψ ( t ) dt ≤ As ( β ,β ) . Since ψ is non-increasing, sψ ( s ) ≤ R s ψ ( t ) dt , so ψ ( s ) ≤ As ( β − ,β − . If β >
1, then lim s → + ψ ( s ) = 0 and consequently ψ ≡
0; since we assumed ψ
0, necessarily β ≤ ψ ( s ) ≤ ψ (1) for s ≥ ψ ( s ) . As β ′ − and so B . A .If we assume B < ∞ , for every s > ψ ( s ) ≤ Bs ( β − ,β ′ − As above we conclude that β ≤
1. For 0 < s ≤ R s ψ ( t ) dt . Bs β .If s ≥
1, then Z s ψ ( t ) dt = Z ψ ( t ) dt + Z s ψ ( t ) dt . B + B Z s t β ′ − dt . Bs β ′ ≤ Bs β . Thus, sup s ≥ s − β R s ψ ( t ) dt . B and A . B . (cid:3) We can assume ℓ ( x ) = h a , x i , | a | = 1,without loss of generality.(a) Let p ≤ γ ≤ q .Step 1. For τ ≥ ξ ∈ R n , define w τ ( ξ ) = | ξ − iτ a | γ = ( | ξ | + τ ) γ . By Theorem 2.1 (a), the inequality(2.2) (cid:16)Z R n | b g ( x ) | q u ( x ) dx (cid:17) q . A u,w τ (cid:16)Z R n w τ ( ξ ) | g ( ξ ) | γ dξ (cid:17) γ EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 9 holds with A u,w τ = sup s> (cid:16)Z s u ∗ ( t ) dt (cid:17) q (cid:16)Z s ((1 /w τ ) ∗ ( t )) γ − dt (cid:17) γ ′ < ∞ . The weight w τ is radially increasing, so(1 /w τ ) ∗ ( t ) = (( t/V n ) n + τ ) − γ ≍ ( t + τ n ) − γn with the constant of equivalence independent of τ . This implies Z s ((1 /w τ ) ∗ ( t )) γ − dt ≍ Z s ( t + τ n ) − γ ′ n dt, s > . Therefore, for τ ≥ A qu,w τ ≍ sup s> Z s u ∗ ( t ) dt (cid:16)Z s ( t + τ n ) − γ ′ n dt (cid:17) qγ ′ = A qu ( τ ) . Since ( t + τ n ) − ≤ max ( τ − n , t + 1) − for t, τ >
0, from (2.3) weconclude that A u ( τ ) ≤ max ( τ − , A u (1) , τ > . We can give a simple expression for A qu (0). Observing that I := R /s t − γ ′ n dt is finite when − γ ′ n > − nn − < γ , we havethat I ≍ s γ ′ n − . Therefore, (2.3) can be rewritten as(2.4) A qu (0) ≍ sup s> s − q ( γ ′ − n ) Z s u ∗ ( t ) dt. By (2.4) and Lemma 2.2 with β = β = q ( γ ′ − n ), there holds that q ( γ ′ − n ) ≤ γ ′ ≤ n + q and we can redefine A qu (0) as follows. A qu (0) = sup s> s − q ( γ ′ − n ) Z s u ∗ ( t ) dt ≍ sup s> s − q ( γ ′ − n ) u ∗ ( s ) . Step 2. Let g ( x ) = e −h τ a ,x i f ( x ). Then g ∈ C ∞ ( R n ) and(2.5) b g ( ξ ) = Z R n g ( x ) e − i h ξ, x i dx = Z R n f ( x ) e − i h ξ, x i−h τ a , x i dx = b f ( ξ − iτ a ) . Since for g ∈ C ∞ ( R n ) the Fourier inversion formula holds, (2.2) and(2.3) imply (cid:16)Z R n | g ( x ) | q u ( x ) dx (cid:17) q . A u ( τ ) (cid:16)Z R n | ξ − iτ a | γ | b g ( ξ ) | γ dξ (cid:17) γ = A u ( τ ) (cid:16)Z R n (cid:12)(cid:12) ( ξ − iτ a ) b f ( ξ − iτ a ) (cid:12)(cid:12) γ dξ (cid:17) γ . (2.6)Note that b f is entire analytic (and so it is defined at ξ − iτ a ) because f has compact support. Since c ∇ f ( ξ ) = iξ b f ( ξ ) , from (2.5) with h ( x ) =( h ( x ) , . . . , h n ( x )) = e −h τ a ,x i ∇ f ( x ) we get b h ( ξ ) = c ∇ f ( ξ − iτ a ) = i ( ξ − iτ a ) b f ( ξ − iτ a ) . Hence (cid:16)Z R n (cid:12)(cid:12) ( ξ − iτ a ) b f ( ξ − iτ a ) (cid:12)(cid:12) γ dξ (cid:17) γ = (cid:16)Z R n | b h ( ξ ) | γ dξ (cid:17) γ = (cid:16)Z R n (cid:16) n X j =1 | b h j ( ξ ) | (cid:17) γ dξ (cid:17) γ ≤ (cid:16)Z R n (cid:16) n X j =1 | b h j ( ξ ) | (cid:17) γ dξ (cid:17) γ ≤ n X j =1 (cid:16)Z R n | b h j ( ξ ) | γ dξ (cid:17) γ , where the first inequality holds trivially and the second is Minkowski’s in-equality.Let us use Pitt’s inequality with p ≤ γ :(2.7) k b f k γ ≤ C p,γ A ,v k f v p k p , where(2.8) A ,v := sup s> (cid:16)Z s dt (cid:17) γ (cid:16)Z s (1 /v ) ∗ ( t ) p − dt (cid:17) p ′ < ∞ . As in Step 1, we apply Lemma 2.2 with β = β = p ′ γ . We obtain p ′ γ ≤ A p ′ ,v ≍ sup s> s − p ′ γ Z s (1 /v ) ∗ ( t ) p − dt ≍ sup s> s − p ′ γ (1 /v ) ∗ ( s ) p − . It follows that A p ,v = A p ′ ( p − ,v ≍ sup s> s pγ ′ − (1 /v ) ∗ ( s ) = A pv < ∞ . Applying (2.7) with f replaced by h j , j = 1 , . . . , n , we gather (cid:16)Z R n | b h j ( ξ ) | γ dξ (cid:17) γ . A v (cid:16)Z R n | h j ( x ) | p v ( x ) dx (cid:17) p . A v (cid:16)Z R n (cid:16) n X k =1 | h k ( x ) | (cid:17) p v ( x ) dx (cid:17) p = A v (cid:16)Z R n | e −h τ a ,x i ∇ f ( x ) | p v ( x ) dx (cid:17) p . (2.9)This, together with (2.6) proves part (a) of the theorem.(b) Let 1 < γ ≤ q < p . We proceed as in the proof of part (a) to obtain(2.2), provided that (2.3) holds. We note that we assume nn − < γ when τ = 0. EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 11
Analogously, we get (2.9), but instead of (2.8) we use (2.1) with u = 1, w = v and γ < p . Then we have A r ,v = Z ∞ s − rγ (cid:16)Z s (1 /v ) ∗ ( t ) p − dt (cid:17) rγ ′ (1 /v ) ∗ ( s ) p − ds ≍ Z ∞ s − rγ dds (cid:16)Z s (1 /v ) ∗ ( t ) p − dt (cid:17) rγ ′ +1 ds, where r = γ − p . After integrating by parts, we get A r ,v ≍ Z ∞ s − rγ − (cid:16)Z s (1 /v ) ∗ ( t ) p − dt (cid:17) rp ′ ds = ˜ A rv < ∞ . This proves part (b) of the theorem.
Let us first discuss the conditions on γ in Theorem 1.1. We recall that in part (a) of Theorem 1.1 we assume1 < p ≤ q < ∞ and max ( p, p ′ ) ≤ γ ≤ q ; when τ = 0 we assume also n < γ ′ .Note that this extra assumption on γ is not necessary when n ≥ p, p ′ ) ≤ γ ≤ q follows that 2 ≤ γ ≤ q and q ′ ≤ γ ′ ≤ n < γ ′ whenever n ≥ n = 1, the inequality n < γ ′ (or: γ > nn − ) can never be satisfiedby γ ′ and only the case n ≥ γ ′ is possible. In fact, the condition max ( p, p ′ ) ≤ γ ≤ q always implies ≤ γ ′ < n = 2 we can either have n < γ ′ or n ≥ γ ′ . Note that ≥ γ ′ impliesthat p = γ = 2.For applications, it is important to simplify the expression for A qu (1) in(1.5). Recall that, when τ > A u ( τ ) ≤ max ( τ − , A u (1) (see Remark1.2). We prove the following Corollary . Let < p ≤ q < ∞ and let max ( p, p ′ ) ≤ γ ≤ q . (i) If n ≥ and n < γ ′ , then γ ′ ≤ n + q and A qu (1) ≍ sup s> s (1 − q ( γ ′ − n ) , u ∗ ( s ) . (ii) If n = 2 and p = γ = 2 , then A qu (1) = sup s> (cid:0) ln ( s − + 1) (cid:1) q/ Z s u ∗ ( t ) dt. (iii) If n = 1 , then A qu (1) ≍ sup s> s (0 , − qγ ′ ) Z s u ∗ ( t ) dt. Proof. (i) Recall that A qu ( τ ) = sup s> Z s u ∗ ( t ) dt (cid:16)Z s ( t + τ n ) − γ ′ n dt (cid:17) qγ ′ , τ > . For τ = 1 and n < γ ′ , we have (cid:16)Z s ( t + 1) − γ ′ n dt (cid:17) qγ ′ ≍ (cid:0) ( s − + 1) − γ ′ n − (cid:1) qγ ′ ≍ s ( − q ( γ ′ − n ) , − qγ ′ ) , s > . Hence A qu (1) ≍ sup s> s ( − q ( γ ′ − n ) , − qγ ′ ) Z s u ∗ ( t ) dt, where q ( γ ′ − n ) > qγ ′ ≥
1, since γ ′ ≤ ≤ q .Now we can apply Lemma 2.2 with β = q ( γ ′ − n ), β = qγ ′ ≥
1. Weobtain q ( γ ′ − n ) ≥ γ ′ ≤ n + q and A qu (1) ≍ sup s> s (1 − q ( γ ′ − n ) , u ∗ ( s ) . Part (ii) is obvious. To prove part (iii), we note that γ ′ >
1, which gives R s ( t + 1) − γ ′ dt ≍ s (0 , − . (cid:3) Proof of Corollary 1.2.
Recall that in this corollary u ( x ) = | x | ( − α , − α ) , v ( x ) = | x | ( β ,β ) with α j , β j ≥
0. We consider the case when1 < p ≤ q < ∞ and γ ∈ [max ( p, p ′ ) , q ], with n < γ ′ ≤ n + q , and we let τ = 0 or τ = 1.Since w ∗ ( s ) ≍ w ( s n ), s >
0, for any non-increasing radial weight func-tion w ( x ) = w ( | x | ) we have u ∗ ( s ) ≍ s ( − α n , − α n ) , (1 /v ) ∗ ( s ) ≍ s ( − β n , − β n ) . whenever α j , β j ≥ s > u ∗ ( s ) . ( s q ( γ ′ − n ) − , τ = 0 ,s ( q ( γ ′ − n ) − , , τ = 1 , (1 /v ) ∗ ( s ) . s − pγ ′ . It is easy to see that when a j , b j ≥
0, the inequality s ( − a , − a ) . s ( − b , − b ) , holds if and only if a ≤ b , a ≥ b . It follows that α ≤ n (cid:16) − qγ ′ + qn (cid:17) , ( α ≥ n (cid:0) − qγ ′ + qn (cid:1) , τ = 0 ,α ≥ , τ = 1 , and 0 ≤ β ≤ n (cid:16) pγ ′ − (cid:17) , β ≥ n (cid:16) pγ ′ − (cid:17) which proves (1.14) and (1.15).To prove (1.16) we use a standard homogeneity argument. Let us con-sider (1.12) (which by Remark 1.4 is equivalent to (1.7)) with f = f λ ( x ) = EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 13 f ( λx ) for some f ∈ C ∞ ( R n ) and λ >
0. We obtain k| x | − αq f λ k q ≤ c τ k| x | βp ( τ a f λ + λ ( ∇ f ) λ k p . After the change of variables x λ − x , we get(2.10) λ αq − nq + βp + np − k| x | − αq f k q ≤ c τ k| x | βp ( λ − τ a f + ∇ f ) k p . The limits of the two sides of the inequality (2.10), as λ → λ → ∞ ,must be the same. If τ = 0 the right-hand side of (2.10) does not dependon λ , so we must have αq − nq + βp + np − τ >
0, we must have λ αq − nq + βp + np − . ( λ − , λ → , , λ → ∞ . so necessarily αq − nq + βp + np − ≤ (cid:3)
3. Uniqueness problems
In this section and in Section 4 we use the inequality (1.1) to proveuniqueness questions for solutions of partial differential equations and sys-tems. First, we state some definitions and preliminary results.Let α = ( α , . . . , α n ) be a vector with non-negative integer components;we use the notation | α | = α + · · · + α n and ∂ αx f = ∂ α ∂ α x · · · ∂ αn ∂ αnxn f .Let D ⊂ R n open and connected and let 1 ≤ p < ∞ . Recallthat W m,p ( D ) is the closure of C ∞ ( D ) with respect to the Sobolev norm k f k W m,p ( D ) = P m | α | =0 k ∂ αx f k p , When m = 1, and D is bounded in atleast one direction, the classical Poincare’ inequality states that k f k L p ( D ) ≤ C k∇ f k L p ( D ) (see e.g. [ ]); thus, the Sobolev norm in W ,p ( D ) is equivalentto k∇ f k L p ( D ) .Given the weight v : D → [0 , ∞ ] and 1 ≤ p < ∞ , we let W m,p,v ( D )be the closure of C ∞ ( D ) with respect to the norm k f k W ,p,v ( D ) = P m | α | =0 k v p ∂ αx f k p . We use the standard notation L p,v ( D ) or L p ( D, v dx )for the closure of C ∞ ( D ) with respect to the norm k v p f k p .Let P ( ∂ ) = P m | α | =0 a α ∂ αx be a linear partial differential operator oforder m > P ( − ∂ ) u = P m | α | =0 a α ( − | α | ∂ αx u .A weak solution (or: a solution in distribution sense ) of the equation P ( ∂ ) f = 0 on a domain D ⊂ R n is a distribution f ∈ W m,p ( D ) that satisfies R D f ( x ) P ( − ∂ ) φ ( x ) dx = 0 for every φ ∈ C ∞ ( D ). Weak solutions for nonlinear partial differential operators can be defined on a case-by-case basis.See e.g. [ ] or other standard textbooks on partial differential equationsfor details. We will often consider differential inequalities in the form of | P ( ∂ ) f | ≤ | V f | on a given domain D ; by that we mean that the inequality | P ( ∂ ) f ( x ) | ≤ | V f ( x ) | is satisfied a.e. in D , i.e., it is satisfied pointwise withthe possible exception of a set of measure zero. Let P ( ∂ ) be ahomogeneous partial differential operator of order m ≥
1. Clearly, f ≡ P ( ∂ ) f = 0 on any domain D ⊂ R n . It is natural toask whether this equation has also nontrivial solutions, i.e., distributions insome suitable Sobolev space that satisfy the equation in distribution senseand are not identically = 0. In particular it is natural to ask whether (1),(2) or (3) below are satisfied or not on a given domain D .(1) Uniqueness for the Dirichlet problem.
The only solution of theDirichlet problem ( P ( ∂ ) f = 0 ,f ∈ W m,p ( D ) is f ≡ Weak unique continuation property (or: unique continuation froman open set). Every solution of the equation P ( ∂ ) f = 0 which is ≡ D is ≡ Strong continuation property (or: unique continuation from apoint). Let x ∈ D . Every solution of the equation P ( ∂ ) f = 0that satisfies lim r → r − N Z | x − x |
32, 33, 35, 52 ] and thesurvey papers [
58, 37, 55 ].The inequality (1.7) in Theorem 1.1 can be viewed as a weightedCarleman-type inequality for the operator P ( ∂ ) f = |∇ f | . To the bestof our knowledge, the inequality (1.7) is new in the literature, even when u ( x ) ≍ v ( x ) ≍ In this section we prove Theorem 1.3and some corollary.
Proof of Theorem 1.3.
Assume for simplicity that f ≡ x n < f ≡ S ǫ = { x : 0 < x n < ǫ } , where ǫ > a = (0 , . . . , , τ ≥ c τ ≤ c (see Remark 1.2), the differential inequality (1.18) and H¨older’sinequality with p = q + r , we can write the following chain of inequalities: k e − τx n f u q k L q ( S ǫ ) ≤ c k e − τx n ∇ f v p k L p ( R n ) ≤ c k e − τx n ∇ f v p k L p ( S ǫ ) + c k e − τx n ∇ f v p k L p ( { x n >ǫ } ) ≤ c k e − τx n f V v p k L p ( S ǫ ) + c e − τǫ k∇ f v p k L p ( { x n >ǫ } ) ≤ c k V v p u − q k L r ( S ǫ ∩ supp f ) k e − τx n f u q k L q ( S ǫ ) + C ′ e − τǫ . Here, r = p − q and we have let C ′ = c k∇ f v p k L p ( { x n >ǫ } ) . Note that C ′ does not depend on τ .Since V ∈ L r (supp f, v rp u − rq dx ) we can chose ǫ > c k V v p u − q k L r ( S ǫ ∩ supp f ) < . From the chain of inequalities above, fol-lows that k e − τx n f u q k L q ( S ǫ ) ≤ k e − τx n f u q k L q ( S ǫ ) + C ′ e − τǫ . We gather 12 k e τ ( ǫ − x n ) f u q k L q ( S ǫ ) ≤ C ′ . Since ǫ − x n > S ǫ , if f τ goes to infinity; this is a contradiction because C ′ does notdepend on τ and so necessarily f ≡ S ǫ . (cid:3) Corollary . Let p , q and γ be as in Theorem . Let u = | x | ( − α , − α ) and v = | x | ( β , β ) , with ≤ α ≤ n (cid:0) − qγ ′ + qn (cid:1) , α ≥ and ≤ β ≤ n (cid:0) pγ ′ − (cid:1) , β ≥ n (cid:0) pγ ′ − (cid:1) . Let V = | x | ( s ,s ) , with (3.2) s > − nr − α q − β p , and, if supp f is unbounded, (3.3) s < − α q − β p − nr . Then, every solution of the differential inequality |∇ f | ≤ V | f | is ≡ . Proof.
The weights u and v are as in Corollary 1.2, so the inequality(1.7) holds with τ >
0. By Theorem 1.3, every solution of the differentialinequality |∇ f | ≤ V | f | is ≡ V v p u − q ∈ L r (supp f ). We can seeat once that V v p u − q = | x | ( t ,t ) ∈ L r (supp f ) if and only if t = s + α q + β p > − nr and, if supp f is unbounded, t = s + α q + β p < − nr , which isequivalent to (3.2) and (3.3). This concludes the proof. (cid:3) Remark . From the inequalities above and the assumptions on α j , β j , and γ ′ (see Corollary 1.2) follows that t ≤ s + nq (cid:16) − qγ ′ + qn (cid:17) + np (cid:16) pγ ′ − (cid:17) = s − nr + 1 .t ≥ s + np (cid:16) pγ ′ − (cid:17) = s + nγ ′ − np > s − np + 1 . The condition t > − nr yields s > −
1. We can see at once that t < − nr yields s < nq −
1. In particular, V = | x | − ǫ with 0 < ǫ < nq , satisfies theassumptions of Corollary 3.1. If f has compact support, then we can omitthe condition on t and assume only ǫ > V ( x ) = C | x | − s , with s, C > Hardy potentials in the literature. They appear in the relativistic Schr¨odinger equations andin problem of stability of relativistic matter in magnetic fields. See e.g. [ ]and the introduction to [ ] and [ ], just to cite a few.It is proved in [ ] that when L is the Dirac operator in dimension n ≥ |L f | ≤ C | x | − | f | has the strongunique continuation property from the point x = 0 whenever C ≤
1. Weconjecture that also the differential inequalities |∇ f | ≤ C | x | − | f | has thestrong unique continuation property from the origin when C is sufficientlysmall. Recall that the solution f of the Dirichletproblem (1.19) is intended in distribution sense, i.e., f satisfies(3.4) Z D h∇ ψ, ∇ f i|∇ f | p − v dx = Z D ψ V f | f | p − v dx for every ψ ∈ C ∞ ( D ). To prove Theorem 1.4 we need two important lem-mas: EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 17
Lemma . Suppose that the weighted gradient inequality (3.5) k u q f k q ≤ c k v p ∇ f k p , f ∈ C ∞ ( D ) holds with exponents ≤ p, q < ∞ . Then the space W ,p,v ( D ) embeds into L q ( D, u dx ) and k f k L q ( D, u dx ) ≤ c k∇ f k L p ( D, v dx ) . Proof.
Fix f ∈ W ,p,v ( D ); let { f n } n ∈ N ⊂ C ∞ ( D ) be a sequence thatconverges to f in the Sobolev norm k · k W ,p,v ( D ) . Thus, { f n } is a Cauchysequence in W ,p,v ( D ); for every ǫ > N > k f n − f m k W ,p,v ( D ) = k v p ( f n − f m ) k L p ( D ) + k v p ∇ ( f n − f m ) k L p ( D ) < ǫ whenever n, m > N ; thus, k v p ∇ ( f n − f m ) k L p ( D ) < ǫ . By (3.5), k u q ( f n − f m ) k L q ( D ) ≤ c k v p ∇ ( f n − f m ) k L p ( D ) < c ǫ. We have proved that { f n } is a Cauchy sequence in L q ( D, u dx ) (which iscomplete) and so it converges to f also in L q ( D, u dx ). We gather k f k L q ( D, u dx ) = lim n →∞ k f n k L q ( D, u dx ) ≤ c lim n →∞ k∇ f n k L p ( D, v dx ) = c k∇ f k L p ( D, v dx ) as required. (cid:3) Lemma . Suppose that the weighted gradient inequality (3.5) holdswith < p < q . Let f be a solution to the Dirichlet problem (1.19) , with | V | p ∈ L r ( D, v rp u − rq dx ) . We have Z D |∇ f | p v dx = Z D V | f | p v dx. Proof.
Let { ψ n } be a sequence of functions in C ∞ ( D ) that convergesto f , the complex conjugate of f , in W ,p,v ( D ). We show first thatlim n →∞ R D h∇ ψ n , ∇ f i|∇ f | p − v dx = R D |∇ f | p v dx . Indeed, Z D (cid:0) h∇ ψ n , ∇ f i|∇ f | p − − |∇ f | p (cid:1) v dx = Z D (cid:0) h∇ ψ n , ∇ f i|∇ f | p − − h∇ f , ∇ f i|∇ f | p − (cid:1) v dx = Z D h∇ ψ n − ∇ f , ∇ f |∇ f | p − i v dx ≤k ( ∇ ψ n − ∇ f ) v p k p k|∇ f | p − v p ′ k p ′ = k∇ ( ψ n − f ) v p k p k |∇ f | v p k pp ′ p and lim n →∞ k∇ ( ψ n − f ) v p k p = 0, as required. In view of (3.4), we have that Z D h∇ ψ n , ∇ f i|∇ f | p − v dx = Z D ψ n V f | f | p − v dx ;to complete the proof it suffices to show thatlim n →∞ Z D ψ n V f | f | p − v dx = Z D V | f | p v dx when | V | p ∈ L r ( D, v rp u − rq dx ). By Lemma 3.2, ψ n converges to ¯ f in L q ( D, u dx ). Using Holder’s inequality with pr + pq = 1, we gather Z D (cid:0) ψ n V f | f | p − − V | f | p (cid:1) v dx ≤ Z D | V vu − pq | | f | p − | ψ n − ¯ f | u pq dx ≤ (cid:16)Z D | vV u − pq | rp dx (cid:17) pr (cid:16)Z D | f | ( p − qp | ψ n − ¯ f | qp u dx (cid:17) pq = (cid:16)Z D ( | V | p v p u − q ) r dx (cid:17) pr (cid:16)Z D | f | qp ′ | ψ n − ¯ f | qp u dx (cid:17) pq . We let C = (cid:0)R D ( | V | p v p u − q ) r dx (cid:1) pr and we apply H¨older’s inequality (with p + p ′ = 1) to the remaining integral. We obtain Z D (cid:0) V f | f | p − ψ n − V | f | p (cid:1) v dx ≤ C (cid:16)Z D | f | q u dx (cid:17) pqp ′ (cid:16)Z D | ψ n − ¯ f | q u dx (cid:17) q = C k f u q k p − q k ( ψ n − ¯ f ) u q k q . (3.6)By assumption, lim n →∞ k ( ψ n − ¯ f ) u q k q = 0; by Lemma 3.2, k f u q k q < ∞ ,and so the right-hand side of (3.6) goes to zero when n → ∞ as required. (cid:3) Proof of Theorem 1.4.
Since the weights u and v are as in Theo-rem 1.1, the weighted gradient inequality (3.5) holds. By Lemma 3.3 andH¨older’s inequality (with pq + pr = 1) we have the following chain of inequal-ities k f u q k pL q ( D ) ≤ c p k∇ f v p k pL p ( D ) = c p Z D v |∇ f | p dx = c p Z D V v | f | p dx ≤ c p Z D V + vu − pq | f | p u pq dx ≤ c p (cid:16)Z D V rp + v rp u − rq dx (cid:17) pr (cid:16)Z D | f | q u dx (cid:17) pq ≤ c p k V p + k pL r ( D, v rp u − rq dx ) k f u q k pL q ( D ) . EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 19
We obtain k f u q k L q ( D ) (cid:0) − c p k V p + k L r ( D, v rp u − rq dx ) (cid:1) ≤
0; this inequality ispossible only if either c p k V p + k L r ( D, v rp u − rq dx ) ≥ f ≡ D . (cid:3)
4. Linear systems of PDE and the Dirac operator
We use the following notation: If ~p = ( p , . . . , p m ) ∈ R m , we let | ~p | =( p + . . . + p m ) . If A is a matrix with rows A , , . . . , A N , we will let | A | =( | A | + . . . + | A N | ) . Note that, by Cauchy Schwartz inequality, | A ~p | = ( h A , ~p i + . . . + h A N , ~p i ) ≤ ( | A | + . . . + | A N | ) | ~p | = | A | | ~p | . Let ~F = ( f , . . . , f N ) ∈ C ∞ ( R n , R N ). We denote with ∇ ~F the N × n matrix whose rows are ∇ f , . . . , ∇ f N .Unless otherwise specified, we assume that p , q , u and v are as in The-orem 1.1 (a) and that r = p − q .In this section we use the Carleman inequality (1.1) to prove uniquecontinuation properties of systems of linear partial differential equations ofthe first order. Most of the first order systems consid-ered in the literature are in the form of(4.1) n X j =1 L j ( x ) ∂ x j ~F = V ( x ) ~F , where ~F = ( f , . . . , f N ) and the L j ( x ) and V are M × N matrices definedin a domain D ⊂ R n . We let L ( x )( ~F ) = P nj =1 L j ( x ) ∂ x j ~F . Differentialinequalities in the form of(4.2) | L ( x ) ~F | ≤ | V ( x ) ~F | are also considered. In some of early papers on the subject, it is proved thatsolutions of elliptic systems in the form of (4.1) that vanish of sufficientlyhigh order at the origin are ≡
0; see [
7, 15, 47 ] and the references cited inthese papers for definitions of elliptic systems. A classical method of proofis to reduce the systems to (quasi-) diagonal form; this approach requiresconditions on the regularity and the multiplicity of the eigenvalues of thesystem that are often difficult to check; see [
9, 24, 29, 56 ]. The strongcontinuation properties of systems of complex analytic vector fields in theform of ~Lu = 0 defined on a real-analytic manifold is proved in [ ].We have found only a few papers in the literature where the Carle-man method is used to prove unique continuation properties of first-ordersystems. The Carleman method often allows to prove unique continuationresults for the differential inequality (4.2), often with a singular potential V .In [ , Theorem 4.1] Carleman estimates are used to prove that (4.2) has the weak unique continuation property when ~L is a system of vector fields ona pseudoconcave Cauchy-Riemann (CR) with some specified conditions and V is bounded. In [ ] and [ ] T. Okaji considers systems in two indepen-dent variables, Maxwell’s equations, and the Dirac operator; he proved thatthe differential inequalities (4.2) with | V ( x ) | ≍ | x | − has the strong uniquecontinuation property using sophisticated L → L Carleman estimates. Seealso [ ], which improves results in [ ].We prove the following Theorem . Let ~F ∈ W ,p,v ( R n , R N ) be a solution of the differentialinequality (4.2) . Assume that ~F satisfies also (4.3) |∇ ~F | . | L ( x ) ~F | . If | V | ∈ L r (supp ~F , u − rq v rp dx ) , with p = r + q and ~F vanishes on one sideof a hyperplane, then ~F ≡ . In particular, for power weights u , v as in Remark 3.1, the differentialinequality (4.2) does not have solutions with compact support support thatsatisfy also (4.3) if V ≍ | x | − ǫ for some ǫ > L r spacesthat, to the best of our knowledge, have not been considered in other pa-pers.Before proving Theorem 4.1 we prove the following Lemma, which is aneasy consequence of Theorem 1.1. Lemma . Let A be a N × N invertible matrix. Under the assumptionsof Theorem , the following inequality holds for all ~F ∈ C ∞ ( R n , R N ) and τ ≥ k e − τℓ ( x ) u q ~F k q ≤ c τ,N, A k e − τℓ ( x ) v p A ∇ ~F k p , where c τ,N, A = N C A c τ and c τ is the constant in (1.7) . Proof.
Using Theorem 1.1 (a), the elementary inequalities | ~F | = ( f + . . . + f N ) ≤ | f | + . . . + | f N | , | f j | ≤ | ~F | , and Minkowsky’s inequality, we obtain k e − τℓ ( x ) u q ~F k q ≤ N X j =1 k e − τℓ ( x ) u q f j k q ≤ c τ N X j =1 k e − τℓ ( x ) v p ∇ f j k p ≤ c τ N k e − τℓ ( x ) v p ∇ ~F k p . EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 21 If A is invertible, then, for every ξ ∈ R n , we have that | A ~ξ | ≥ C − A | ξ | for some C A >
0; thus, k e − τℓ ( x ) u q ~F k q ≤ c τ N C A k e − τℓ ( x ) v p A ∇ ~F k p as required. (cid:3) Proof of Theorem 4.1.
We argue as in the proof of Theorem 1.3.Without loss of generality, we can assume that ~F ≡ x n < A = I , where I is the N × N identity matrix. For simplicity of notation, wedenote with c the constant c ,N, I in Lemma 4.2. We show that ~F ≡ S ǫ = { x : 0 < x n < ǫ } , for some ǫ > ℓ ( x ) = x n and τ ≥
1, the differential inequality (4.3),H¨older’s inequality and Remark 1.2, we obtain k e − τx n ~F u q k L q ( S ǫ ) ≤ c k e − τx n ∇ ~F v p k L p ( R n ) ≤ c k e − τx n ∇ ~F v p k L p ( S ǫ ) + c k e − τx n ∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C k e − τx n L ( x )( ∇ ~F ) v p k L p ( S ǫ ) + c e − τǫ k∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C k e − τx n V ~F v p k L p ( S ǫ ) + c e − τǫ k∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C k| V | v p u − q k L r ( S ǫ ∩ supp ~F ) k e − τx n ~F u q k L q ( S ǫ ) + C ′ e − τǫ , where we have let C ′ = c k∇ ~F v p k L p ( { x n >ǫ } ) .Since | V | ∈ L r (supp ~F , u − rq v rp dx ) we can chose ǫ > c C k| V | v p u − q k L r ( S ǫ ∩ supp ~F ) < . We have obtained k e − τx n ~F u q k L q ( S ǫ ) ≤ k e − τx n ~F u q k L q ( S ǫ ) + C ′ e − τǫ . In view of ǫ − x n > S ǫ , the left-hand side of this inequality goes toinfinity with τ unless ~F ≡ S ǫ ; this is a contradiction because C ′ doesnot depend on τ , and so ~F ≡ S ǫ . (cid:3) Let G ( x ) , . . . , G n ( x ) be N × n matrices defined on a domain D ⊂ R n .We consider the operator G ( ~F ) = G ( f , . . . , f N ) = N X j =1 G j ( x ) f j with f j ∈ C ∞ ( D ).In [ ], systems in the form of ∇ F = G ~F are considered. These systemscan be used to model linear elasticity (in curvilinear coordinates) of linearlyelastic shells. See [ ] and the references cited there. We prove the following Theorem . Let ~F ∈ W ,p,v ( D, R N ) be a solution of the differentialinequality (4.5) |∇ ~F | . | G ~F | . If | G | ∈ L r (supp ~F , u − rq v rp dx ) , with p = r + q , and ~F vanishes on one sideof a hyperplane, then ~F ≡ . Proof.
Assume for simplicity that ~F ≡ x n < ~F ≡ < x n < ǫ , for some ǫ > A = I , ℓ ( x ) = x n and τ ≥ j = 1 , . . . , N , we use the differential inequality (4.5) and H¨older’sinequality in the following chain of inequalities k e − τx n ~F u q k L q ( S ǫ ) ≤ c k e − τx n ∇ ~F v p k L p ( R n ) ≤ c k e − τx n ∇ ~F ~v p k L p ( S ǫ ) + c k e − τx n ∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C k e − τx n G ~F v p k L p ( S ǫ ) + c e − τǫ k∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C N X j =1 k e − τx n | G j | f j v p k L p ( S ǫ ) + c e − τǫ k∇ ~F v p k L p ( { x n >ǫ } ) ≤ c C N X j =1 k| G j | v p u − q k L r ( S ǫ ∩ supp ~F ) k e − τx n f j u q k L q ( S ǫ ) + C ′ e − τǫ ≤ c CN k| G | v p u − q k L r ( S ǫ ∩ supp ~F ) k e − τx n | ~F | u q k L q ( S ǫ ) + C ′ e − τǫ , where we have let C ′ = c k∇ ~F v p k L p ( { x n >ǫ } ) .We chose ǫ > c CN k| G | v p u − q k L r ( S ǫ ∩ supp ~F ) < . We gather k e − τx n ~F u q k L q ( S ǫ ) ≤ k e − τx n ~F u q k L q ( S ǫ ) + C ′ e − τǫ which gives 12 k e τ ( ǫ − x n ) ~F u q k L q ( S ǫ ) ≤ C ′ , and we can conclude the proof as in Theorem 4.1. (cid:3) Remark . It is shown in [ ] that the W , ( D, R n ) solutions of thesystem ∇ ~F = G ~F , with G ∈ L ( D, R ( n × n ) × n ), cannot vanish on an openset. The proof in [ ] does not use Carleman inequalities. EIGHTED GRADIENT INEQUALITIES AND UNIQUE CONTINUATION 23
Let α j , j = 0 , . . . , n , be N × N matriceswhich satisfy the following relations.(4.6) α ∗ j = α j , α j = I, α j α k + α k α j = 0 , j = k (we also say that the α j form a basis of a Clifford algebra). It is known thatfor (4.6) to hold, N must be in the form 2 [ n +12 ] m , with m > n -dimensional) Dirac operator associated to the matrices α j is amatrix value operator, initially defined on C ∞ ( R n , R N × N ) as follows. L U = − i n X j =1 α j ∂ x j U. Here, ∂ x i U is a matrix whose entries are the partial derivative of the en-tries of U . We can use (4.6) to show that L ◦ L U = − ∆ U I , where I isthe identity matrix. When U = f I , where f ∈ C ∞ ( R n ), we can see atonce that ( L ( f I )) = − I |∇ f | , Thus, a Dirac operators can be viewed as ageneralization of the gradient operator and a square root of the Laplacian.There is a lot of literature on the Dirac operator and its role in severaldomains of mathematics and physics See e.g. [ ]. For example, the Diracequation which describes free relativistic electrons is represented by i ~ ∂ t ψ ( t, x ) = H ψ ( t, x ) , where H is given explicitly by the 4 × H = − i ~ c X j =1 α j ∂ x j + α mc . Here, c is the speed of light, m is a mass of a particle and ~ is the Planck’sconstant.In [ ] is proved that the the differential inequality(4.7) |L U | ≤ | V U | where V ( x ) is a N × N matrix, has the strong unique continuation propertyfrom the origin whenever V ( x ) ≤ C | x | − , with 0 ≤ C ≤
1. It is also provedin [ ] that the condition C ≤ ] and thereferences cited there. We prove the following Theorem . Let f ∈ W ,p,v ( D ) be a solution of the differential in-equality (4.7) . If | V | ∈ L r (supp f, u − rq v rp dx ) with p = r + q and f vanisheson one side of a hyperplane, then f ≡ . Proof.
Since L ( f I ) · L ( f I ) = − I |∇ f | , we can see at once that |∇ f | = |L ( f I ) · L ( f I ) | ≤ |L ( f I ) | With this observation, the proof of Theorem 4.4 is almost a line-by-linerepetition of the proof of Theorem 4.1. We leave the details to the reader. (cid:3)
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L. De Carli, Florida International University, Department of Mathemat-ics, Miami, FL 33199, USA
E-mail address : [email protected] D. Gorbachev, Tula State University, Department of Applied Mathemat-ics and Computer Science, 300012 Tula, Russia
E-mail address : [email protected] S. Tikhonov, ICREA, Centre de Recerca Matem`atica, and UAB, Campusde Bellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain.
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