Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations
aa r X i v : . [ m a t h . A P ] M a y Semismall perturbations,semi-intrinsic ultracontractivity,and integral representations of nonnegativesolutions for parabolic equations ∗ Pedro J. Mendez-HernandezEscuela de Matem´atica, Universidad de Costa RicaSan Jos´e, Costa Ricae-mail: [email protected] MurataDepartment of Mathematics, Tokyo Institute of Technology,Oh-okayama, Meguro-ku, Tokyo, 152-8551 Japane-mail: [email protected]
Abstract
We consider nonnegative solutions of a parabolic equation in acylinder D × I , where D is a noncompact domain of a Riemannianmanifold and I = (0 , T ) with 0 < T ≤ ∞ or I = ( −∞ , D ), we establishan integral representation theorem of nonnegative solutions: In thecase I = (0 , T ), any nonnegative solution is represented uniquely byan integral on ( D × { } ) ∪ ( ∂ M D × [0 , T )), where ∂ M D is the Martinboundary of D for the elliptic operator; and in the case I = ( −∞ , ∗ ny nonnegative solution is represented uniquely by the sum of anintegral on ∂ M D × ( −∞ ,
0) and a constant multiple of a particularsolution. We also show that [SSP] implies the condition [SIU] (i.e.,the associated heat kernel is semi-intrinsically ultracontractive).
This paper is a continuation of [34]. It is concerned with integral repre-sentations of nonnegative solutions to parabolic equations and perturbationtheory for elliptic operators.We consider nonnegative solutions of a parabolic equation( ∂ t + L ) u = 0 in D × I, (1.1)where ∂ t = ∂/∂t , L is a second order elliptic operator on a noncompactdomain D of a Riemannian manifold M , and I is a time interval: I = (0 , T )with 0 < T ≤ ∞ or I = ( −∞ , p ( x, y, t ) for (1.1). Furthermore, he has shown that [IU] implies [SP](i.e., the constant function 1 is a small perturbation of L on D ). It is known( [30]) that [SP] implies [SSP] (i.e., 1 is a semismall perturbation of L on D ).In this paper, we show that [SSP] implies [SIU] (i.e., semi-intrinsic ul-tracontractivity) and give integral representation theorems of nonnegativesolutions to (1.1) under the condition [SSP]. We consider that [SSP] is one ofthe weakest possible condition for getting ”explicit” integral representationtheorems.Now, in order to state our main results, we fix notations and recall severalnotions and facts. Let M be a connected separable n -dimensional smoothmanifold with Riemannian metric of class C . Denote by ν the Riemannianmeasure on M . T x M and T M denote the tangent space to M at x ∈ M andthe tangent bundle, respectively. We denote by End( T x M ) and End( T M )the set of endmorphisms in T x M and the corresponding bundle, respectively.2he inner product on T M is denoted by h X, Y i , where X, Y ∈ T M ; and | X | = h X, X i / . The divergence and gradient with respect to the metric on M are denoted by div and ∇ , respectively. Let D be a noncompact domainof M . Let L be an elliptic differential operator on D of the form Lu = − m − div( mA ∇ u ) + V u, (1.2)where m is a positive measurable function on D such that m and m − arebounded on any compact subset of D , A is a symmetric measurable sectionon D of End( T M ), and V is a real-valued measurable function on D suchthat V ∈ L p loc ( D, mdν ) for some p > max( n , . Here L p loc ( D, mdν ) is the set of real-valued functions on D locally p -th inte-grable with respect to mdν . We assume that L is locally uniformly ellipticon D , i.e., for any compact set K in D there exists a positive constant λ suchthat λ | ξ | ≤ h A x ξ, ξ i ≤ λ − | ξ | , x ∈ K, ( x, ξ ) ∈ T M.
We assume that the quadratic form Q on C ∞ ( D ) defined by Q [ u ] = Z D ( h A ∇ u, ∇ u i + V u ) mdν is bounded from below, and put λ = inf (cid:26) Q [ u ]; u ∈ C ∞ ( D ) , Z D u mdν = 1 (cid:27) . Then, for any a < λ , ( L − a, D ) is subcritical, i.e., there exists the (minimalpositive) Green function of L − a on D . We denote by L D the selfadjointoperator in L ( D ; mdν ) associated with the closure of Q . The minimal fun-damental solution for (1.1) is denoted by p ( x, y, t ), which is equal to theintegral kernel of the semigroup e − tL D on L ( D, mdν ).Let us recall several notions related to [SSP]. [IU] λ is an eigenvalue of L D ; and there exists, for any t >
0, a constant C t > p ( x, y, t ) ≤ C t φ ( x ) φ ( y ) , x, y ∈ D, where φ is the normalized positive eigenfunction for λ .3his notion was introduced by Davies-Simon [13], and investigated exten-sively because of its important consequences (see [7], [8], [9], [10], [12], [23],[24], [31], [34], [42], and references therein). It looks, on the surface, notrelated to perturbation theory. But it has turned out ( [34]) that [IU] impliesthe following condition [SP] for any a < λ . [SP] The constant function 1 is a small perturbation of L − a on D , i.e., forany ε > K of D such that Z D \ K G ( x, z ) G ( z, y ) m ( z ) dν ( z ) ≤ εG ( x, y ) , x, y ∈ D \ K, where G is the Green function of L − a on D .This condition is a special case of the notion introduced by Pinchover [37].Recall that [SP] implies the following condition [SSP] (see [30]). [SSP] The constant function 1 is a semismall perturbation of L − a on D ,i.e., for any ε > K of D such that Z D \ K G ( x , z ) G ( z, y ) m ( z ) dν ( z ) ≤ εG ( x , y ) , y ∈ D \ K, where x is a fixed reference point in D .This condition [SSP] implies that L D admits a complete orthonormal baseof eigenfunctions { φ j } ∞ j =0 with eigenvalues λ < λ ≤ λ ≤ · · · repeatedaccording to multiplicity; furthermore, for any j = 1 , , · · · , the function φ j /φ has a continuous extension [ φ j /φ ] up to the Martin boundary ∂ M D of D for L − a (see Theorem 6.3 of [38]).We show in this paper that [SSP] also implies the following condition[SIU]. [SIU] λ is an eigenvalue of L D ; and there exist, for any t > K of D , positive constants A and B such that A φ ( x ) φ ( y ) ≤ p ( x, y, t ) ≤ B φ ( x ) φ ( y ) , x ∈ K, y ∈ D. This notion was introduced by Ba˜nuelos-Davis [9], where they called it onehalf IU. Here we should recall that [IU] implies that for any t > c t > c t φ ( x ) φ ( y ) ≤ p ( x, y, t ) , x, y ∈ D.
4e see that the same argument as in the proof of Theorem 3.1 in [25] (orthe argument in the proof of Theorem 1.2 below) shows that [SIU] impliesthe following condition [NUP] (i.e., non-uniqueness for the positive Cauchyproblem). [NUP]
The Cauchy problem( ∂ t + L ) u = 0 in D × (0 , T ) , u ( x,
0) = 0 on D (1.3)admits a solution u with u ( x, t ) > D × (0 , T ).We say that [UP] holds for (1.3) when any nonnegative solution of (1.3) isidentically zero. We note that [UP] implies that the constant function 1 is a”big” perturbation of L − a on D in some sense (see Theorem 2.1 of [32]).Fix a < λ , and suppose that [SSP] holds. Let D ∗ = D ∪ ∂ M D be theMartin compactification of D for L − a , which is a compact metric space.Denote by ∂ m D the minimal Martin boundary of D for L − a , which is aBorel subset of the Martin boundary ∂ M D of D for L − a . Here, we notethat ∂ M D and ∂ m D are independent of a in the following sense: if [SSP]holds, then for any b < λ there is a homeomorphism Φ from the Martincompactification of D for L − a onto that for L − b such that Φ | D = identity ,and Φ maps the Martin boundary and minimal Martin boundary of D for L − a onto those for L − b , respectively (see Theorem 1.4 of [30]).Now, we are ready to state our main results. In the following theoremswe assume that [SSP] holds for some fixed a < λ . Theorem 1.1
The condition [SSP] implies [SIU].
Theorem 1.2
Assume [SSP]. Then, for any ξ ∈ ∂ M D there exists the limitlim D ∋ y → ξ p ( x, y, t ) φ ( y ) ≡ q ( x, ξ, t ) , x ∈ D, t ∈ R . (1.4)Here, as functions of ( x, t ), { p ( x, y, t ) /φ ( y ) } y converges to q ( x, ξ, t ) as y → ξ uniformly on any compact subset of D × R . Furthermore, q ( x, ξ, t ) is acontinuous function on D × ∂ M D × R such that q > D × ∂ M D × (0 , ∞ ) , (1.5) q = 0 on D × ∂ M D × ( −∞ , , (1.6)( ∂ t + L ) q ( · , ξ, · ) = 0 on D × R . (1.7)5 heorem 1.3 Assume [SSP]. Consider the equation (1.1) for I = (0 , T )with 0 < T ≤ ∞ . Then, for any nonnegative solution u of (1.1) there existsa unique pair of Borel measures µ on D and λ on ∂ M D × [0 , T ) such that λ is supported by the set ∂ m D × [0 , T ), and u ( x, t ) = Z D p ( x, y, t ) dµ ( y ) + Z ∂ M D × [0 ,t ) q ( x, ξ, t − s ) dλ ( ξ, s ) (1.8)for any ( x, t ) ∈ D × I .Conversely, for any Borel measures µ on D and λ on ∂ M D × [0 , T ) suchthat λ is supported by ∂ m D × [0 , T ) and Z D p ( x , y, t ) dµ ( y ) < ∞ , < t < T, (1.9) Z ∂ M D × [0 ,t ) q ( x , ξ, t − s ) dλ ( ξ, s ) < ∞ , < t < T, (1.10)where x is a fixed point in D , the right hand side of (1.8) is a nonnegativesolution of (1.1) for I = (0 , T ) with 0 < T ≤ ∞ .The proof of this theorem will be given in Sections 4 and 5. It is basedupon the abstract integral representation theorem established in [34], withoutassuming [IU], via a parabolic Martin representation theorem and Choquet’stheorem (see [18], [21], [35]). Its key step is to identify the parabolic Martinboundary.This theorem is an improvement of Theorem 1.2 of [34]; where the con-dition [IU], which is more stringent than [SSP], is assumed. It is also ananswer to a problem raised in Remark 4.13 of [34]. Note that (1.8) givesexplicit integral representations of nonnegative solutions to (1.1) providedthat the Martin boundary ∂ M D of D for L − a is determined explicitly. Weconsider that [SSP] is one of the weakest possible condition for getting suchexplicit integral representations.Let us recall that when [UP] hods for (1.3), the structure of all non-negative solutions to (1.1) for I = (0 , T ) is extremely simple. Namely, thefollowing theorem holds (see [5]). Fact AT
Assume [UP]. Then, for any nonnegative solution u of (1.1) with I = (0 , T ), there exists a unique Borel measure µ on D such that u ( x, t ) = Z D p ( x, y, t ) dµ ( y ) , ( x, t ) ∈ D × I. (1.11)6onversely, for any Borel measure µ on D satisfying (1.9), the right handside of (1.11) is a nonnegative solution of (1.1) with I = (0 , T ).It is quite interesting that when [UP] holds, the elliptic Martin boundarydisappears in the parabolic representation theorem; while it enters in manycases of [NUP].Finally, we state an integral representation theorem for the case I =( −∞ , Theorem 1.4
Assume [SSP]. Consider the equation (1.1) for I = ( −∞ , u of (1.1) there exists a unique pair of anonnegative constant α and a Borel measure λ on ∂ M D × ( −∞ ,
0) supportedby the set ∂ m D × ( −∞ ,
0) such that u ( x, t ) = αe − λ t φ ( x ) + Z ∂ M D × ( −∞ ,t ) q ( x, ξ, t − s ) dλ ( ξ, s ) (1.12)for any ( x, t ) ∈ D × ( −∞ , α and a Borel measure λ on ∂ M D × ( −∞ ,
0) such that it is supported by ∂ m D × ( −∞ ,
0) and Z ∂ M D × ( −∞ ,t ) q ( x , ξ, t − s ) dλ ( ξ, s ) < ∞ , −∞ < t < , (1.13)the right hand side of (1.12) is a nonnegative solution of (1.1).This theorem is an improvement of Theorem 6.1 of [34], where [IU] isassumed instead of [SSP].Here, in order to illustrate a scope of Theorems 1.3 and 1.4, we give asimple example. Further examples will be given in Section 7. Example 1.5
Let D be a domain in R with finite area. Then, by Theorem6.1 of [33], the constant function 1 is a small perturbation of L = − ∆ on D .Thus Theorems 1.3 and 1.4 hold true for the heat equation( ∂ t − ∆) u = 0 in D × I. Note that there exist many bounded planar domains for which the heatsemigroup is not intrinsically ultracontractive (see Example 1 of [13] andSection 4 of [9]). Thus, the last assertion of this example is new for suchdomains. 7he remainder of this paper is organized as follows. In Section 2 we proveTheorem 1.1, and Theorem 1.2 is proved in Section 3. Sections 4 and 5 aredevoted to the proof of Theorem 1.3. In Section 4 we show it in the case of I = (0 , ∞ ). In Section 5 we show it in the case of I = (0 , T ) with 0 < T < ∞ by making use of results to be given in Section 4. Theorem 1.4 is provedin Section 6. Finally we shall give two more concrete examples in Section 7with emphasis on sharpness of concrete sufficient conditions of [SSP]. In this section we prove Theorem 1.1.
Proof of Theorem 1.1
We may and shall assume that a = 0 < λ . Let G be the Green function of L on D . For any t >
0, put G t ( x, y ) = Z ∞ t p ( x, y, s ) ds,G t ( x, y ) = Z t p ( x, y, s ) ds. Then G = G t + G t . Let us show that for any t > K of D there exists a constant A >
A φ ( x ) φ ( y ) ≤ p ( x, y, t ) , x ∈ K, y ∈ D. (2.1)Fix a compact subset K . We may assume that x ∈ K . Let K ⊂ D bea compact neighborhood of K . Then the same argument as in the proof ofTheorem 1.5 of [30] shows that C − G ( x , z ) ≤ φ ( z ) ≤ C G ( x , z ) , z ∈ D \ K , (2.2)for some constant C >
0. Fix t >
0, and put ǫ t = 12 λ (cid:0) − e − tλ (cid:1) . By [SSP] and (2.2), there exits a compact subset K ⊃ K such that Z D \ K φ ( z ) G ( z, y ) dµ ( z ) ≤ ǫ t φ ( y ) , y ∈ D \ K , (2.3)8here dµ ( z ) = m ( z ) dν ( z ). Since φ ( y ) λ = Z D G ( y, z ) φ ( z ) dµ ( z ) , and G ( y, z ) = G ( z, y ), (2.3) yields φ ( y ) λ ≤ Z K G t ( z, y ) φ ( z ) dµ ( z ) + Z K G t ( z, y ) φ ( z ) dµ ( z )+ ǫ t φ ( y ) (2.4)for any y ∈ D \ K . By Fubini’s theorem, Z D G t ( z, y ) φ ( z ) dµ ( z ) = Z ∞ t ds Z D p ( z, y, s ) φ ( z ) dµ ( z )= Z ∞ t e − λ s φ ( y ) ds = 1 λ e − λ t φ ( y ) . Thus Z K G t ( z, y ) φ ( z ) dµ ( z ) ≤ λ e − λ t φ ( y ) . This together with (2.4) implies ǫ t φ ( y ) ≤ Z K G t ( z, y ) φ ( z ) dµ ( z ) . (2.5)Choose a compact subset K whose interior includes K . By the parabolicHarnack inequality, there exists a constant C depending on t, K , K suchthat p ( z, y, s ) ≤ C p ( x, y, t ) , for any x, z ∈ K , y ∈ D \ K , and 0 < s ≤ t . We have G t ( z, y ) = Z t p ( z, y, s ) ds ≤ C t p ( x , y, t ) , z ∈ K , y ∈ D \ K . (2.6)Thus Z K G t ( z, y ) φ ( z ) dµ ( z ) ≤ (cid:20) C t Z K φ ( z ) dz (cid:21) p ( x , y, t ) . φ ( y ) ≤ C p ( x , y, t ) , y ∈ D \ K , (2.7)where C = 1 ǫ t C t Z K φ ( z ) dµ ( z ) . By the parabolic Harnack inequality, p ( x , y, t ) ≤ C p ( x, y, t ) , x ∈ K, y ∈ D, for some constant C >
0. This together with (2.7) yields the desired inequal-ity (2.1). It remains to show that for any t > K of D there exists a constant B such that p ( x, y, t ) ≤ B φ ( x ) φ ( y ) , x ∈ K, y ∈ D. (2.8)Fix a compact subset K . We may assume that x ∈ K . Let K ⊂ D bea compact neighborhood of K . By the parabolic Harnack inequality thereexists a constant c > c p ( x , y, t ) ≤ p ( z, y, t ) , z ∈ K , y ∈ D. Thus, for any y ∈ D , e − tλ φ ( y ) = Z D φ ( z ) p ( z, y, t ) dµ ( z ) ≥ Z K φ ( z ) p ( z, y, t ) dµ ( z ) ≥ c (cid:20) Z K φ ( z ) dµ ( z ) (cid:21) p ( x , y, t ) . This implies (2.8), since
C p ( x , y, t ) ≥ p ( x, y, t/ , x ∈ K, y ∈ D, for some constant C >
0. (We should note that in proving (2.8) we haveonly used the consequence of [SSP] that φ is a positive eigenfunction.) (cid:3) Remark 2.1
It is an open problem whether [SIU] implies [SSP] or not. Fur-thermore, the problem whether [SSP] implies [SP] or not in the case n >
Parabolic Martin kernels
In this section we prove Theorem 1.2. Throughout the present section weassume [SSP]. We may and shall assume that a = 0 < λ . Let G be theGreen function of L on D . For any 0 < δ < t , put G tδ ( x, y ) = Z tδ p ( x, y, s ) ds. (3.1)We denote by ∂ M D the Martin boundary of D for L . In order to proveTheorem 1.2, we need two lemmas. Lemma 3.1
Let ξ ∈ ∂ M D . Suppose that a sequence { y n } ∞ n =1 ⊂ D convergesto ξ , and there exists the limitlim n →∞ G tδ ( z, y n ) φ ( y n ) = w ( z, t ) , z ∈ D. (3.2)Then lim n →∞ Z D G ( x, z ) G tδ ( z, y n ) φ ( y n ) dµ ( z ) = Z D G ( x, z ) w ( z, t ) dµ ( z ) (3.3)for any x ∈ D , where dµ ( z ) = m ( z ) dν ( z ). Proof
Fix x ∈ D . Let K ⊂ D be a compact neighborhood of x . By [SSP],there exists a constant C > C − φ ( y ) ≤ G ( x, y ) ≤ C φ ( y ) , y ∈ D \ K . (3.4)Let ǫ >
0. Then there exists a compact subset K ⊃ K such that Z D \ K G ( x, z ) G ( z, y ) G ( x, y ) dµ ( z ) < ǫ C , y ∈ D \ K. Thus, for n sufficiently large, Z D \ K G ( x, z ) (cid:20) G tδ ( z, y n ) φ ( y n ) (cid:21) dµ ( z ) ≤ Z D \ K G ( x, z ) (cid:20) C G ( z, y n ) G ( x, y n ) (cid:21) dµ ( z ) < ǫ .
11y Fatou’s lemma, Z D \ K G ( x, z ) w ( z, t ) dµ ( z ) ≤ ǫ . By Theorem 1.1, there exist constants A and A such that A φ ( x ) φ ( y ) ≤ p ( x, y, δ ) ≤ A φ ( x ) φ ( y ) , x ∈ K, y ∈ D. Then, for any t > δ , the semigroup property yields A e − λ ( t − δ ) φ ( x ) φ ( y ) ≤ p ( x, y, t ) ≤ A e − λ ( t − δ ) φ ( x ) φ ( y ) (3.5)for any x ∈ K, y ∈ D . Thus there exists a constant B > n G tδ ( z, y n ) φ ( y n ) ≤ B φ ( z ) , z ∈ K. Then Lebesgue’s dominated convergence theorem yieldslim n →∞ Z K G ( x, z ) (cid:20) G tδ ( z, y n ) φ ( y n ) (cid:21) dµ ( z ) = Z K G ( x, z ) w ( z, t ) dµ ( z ) . Therefore, for n sufficiently large, (cid:12)(cid:12)(cid:12)(cid:12) Z D G ( x, z ) (cid:20) G tδ ( z, y n ) φ ( y n ) (cid:21) dµ ( z ) − Z D G ( x, z ) w ( z, t ) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ. This shows (3.3). (cid:3)
By Lemma 6.1 of [38], it follows from [SSP] that there exists the limitlim D ∋ y → ξ G D ( y, z ) φ ( y ) = h ( ξ, z ) , ( ξ, z ) ∈ ∂ M D × D, (3.6)and h is a positive continuous function on ∂ M D × D . From this we show thefollowing lemma. Lemma 3.2
Under the same assumptions as in Lemma 3.1, one has Z D h ( ξ, z ) G tδ ( z, x ) dµ ( z ) = lim n →∞ Z D G ( y n , z ) φ ( y n ) G tδ ( z, x ) dµ ( z )= Z D G ( x, z ) w ( z, t ) dµ ( z ) (3.7)for any x ∈ D . 12 roof Fix x ∈ D . Let K ⊂ D be a compact neighborhood of x . ByTheorem 1.1, (3.4) and (3.5), there exists a constant C > C G ( z, x ) ≤ G tδ ( z, x ) ≤ G ( z, x ) , z ∈ D \ K . Let ǫ >
0. By [SSP], there exists a compact subset K ⊃ K such that Z D \ K (cid:20) G ( y n , z ) φ ( y n ) (cid:21) G tδ ( z, x ) dµ ( z ) < ǫ , (3.8)for n sufficiently large. By Fatou’s lemma, Z D \ K h ( ξ, z ) G tδ ( z, x ) dµ ( z ) ≤ ǫ . (3.9)On the other hand, for any sufficiently large n (cid:20) G ( y n , z ) φ ( y n ) (cid:21) G tδ ( z, x ) ≤ C , z ∈ K, where C is a positive constant. By Lebesgue’s dominated convergence the-orem, lim n →∞ Z K G ( y n , z ) φ ( y n ) G tδ ( z, x ) dµ ( z ) = Z K h ( ξ, z ) G tδ ( z, x ) dµ ( z ) . (3.10)Combining (3.8), (3.9) and (3.10), we get the first equality. It remains toshow the second equality of (3.7). By Fubini’s theorem and the symmetry p ( x, y, t ) = p ( y, x, t ) , we have Z D G ( y n , z ) G tδ ( z, x ) dµ ( z ) = Z ∞ dr Z tδ ds p ( y n , x, r + s )= Z D G ( x, z ) G tδ ( z, y n ) dµ ( z ) . This together with Lemma 3.1 implies the second equality. (cid:3)
Proof of Theorem 1.2
Let { y j } ∞ j =1 ⊂ D be any sequence converging to ξ ∈ ∂ M D . Put u j ( x, t ) = p ( x, y j , t ) φ ( y j ) for t > , u j ( x, t ) = 0 for t ≤ . (3.11)13ince [SIU] holds, it follows from the parabolic Harnack inequality and locala priori estimates for nonnegative solutions to parabolic equations (see [6]and [16]) that there exists a subsequence { u j k } ∞ k =1 such that u j k converges,as k → ∞ , uniformly on any compact subset of D × R to a solution u of theequation ( ∂ t + L ) u = 0 in D × R satisfying u > D × (0 , ∞ ) and u = 0 on D × ( −∞ , u is independentof { y j k } ∞ k =1 and uniquely determined by ξ . Let { y j } ∞ n =1 and { y ′ j } ∞ n =1 be twosequences in D converging to ξ . Define u j by (3.11), and u ′ j by (3.11) with y j replaced by y ′ j . Suppose that { u j } ∞ j =1 and { u ′ j } ∞ j =1 converge to u and u ′ ,respectively. For any t > δ >
0, put w ( z, t ) = Z tδ u ( z, s ) ds, w ′ ( z, t ) = Z tδ u ′ ( z, s ) ds. Then we havelim n →∞ G tδ ( z, y n ) φ ( y n ) = w ( z, t ) , lim n →∞ G tδ ( z, y ′ n ) φ ( y ′ n ) = w ′ ( z, t ) . By Lemma 3.2, Z D G ( x, z ) w ( z, t ) dµ ( z ) = Z D h ( ξ, z ) G tδ ( z, x ) dµ ( z )= Z D G ( x, z ) w ′ ( z, t ) dµ ( z ) . Thus w ( x, t ) = w ′ ( x, t ), which implies u ( x, t ) = u ′ ( x, t ). This completes theproof of Theorem 1.2. ✷ I = (0 , ∞ ) In this section we prove Theorem 1.3 in the case T = ∞ .We first state an abstract integral representation theorem which holdswithout [SSP]. For x ∈ D and r >
0, we denote by B ( x, r ) the geodesicball in the Riemannian manifold M with center x and radius r . Let x be areference point in D . Choose a nonnegative continuous function a on D such14hat a ( x ) = 1 on B ( x , r ) and a ( x ) = 0 outside B ( x , r ) for some r > B ( x , r ) ⋐ D . Choose a nonnegative continuous function b on R suchthat 0 < b ( t ) < e γt on (1 , ∞ ) for some γ < λ , and b ( t ) = 0 on ( −∞ , β the measure defined by dβ ( x, t ) = a ( x ) b ( t ) m ( x ) dν ( x ) dt . Forany nonnegative measurable function u on Q = D × (0 , ∞ ), we write β ( u ) = Z Z Q u ( x, t ) dβ ( x, t ) . Denote by P ( Q ) the set of all nonnegative solutions of (1.1) with I = (0 , ∞ ),and put P β ( Q ) = { u ∈ P ( Q ); β ( u ) < ∞} . Note that for any u ∈ P ( Q ) there exists a function b as above such that β ( u ) < ∞ ; thus P ( Q ) = S β P β ( Q ). Furthermore, the parabolic Harnackinequality shows that if β ( u ) = 0, then u = 0. Now, let us define the β -Martin boundary ∂ βM Q of Q with respect to ∂ t + L along the line given in [21]and [18]. Put p ( x, t ; y, s ) = p ( x, y, t − s ) , t > s, x, y ∈ D,p ( x, t ; y, s ) = 0 , t ≤ s, x, y ∈ D. Define the β -Martin kernel K β by K β ( x, t ; y, s ) = p ( x, t ; y, s ) β ( p ( · ; y, s )) , ( x, t ) , ( y, s ) ∈ Q, where β ( p ( · ; y, s )) = RR Q p ( z, r ; y, s ) dβ ( z, r ). Note that β ( p ( · ; y, s )) < ∞ for any ( y, s ) ∈ Q , since 0 < b ( t ) < e γt on (1 , ∞ ) for some γ < λ . Let { D j } ∞ j =1 be an exhaustion of D such that each D j is a domain with smoothboundary, D j ⋐ D j +1 ⋐ D , S ∞ j =1 D j = D , and B ( x , r ) ⋐ D . Put Q j = D j × (1 /j, j ). For Y = ( y, s ) , Z = ( z, r ) ∈ Q , let δ β ( Y, Z ) = ∞ X j =1 − j sup X ∈ Q j | K β ( X ; Y ) − K β ( X ; Z ) | | K β ( X ; Y ) − K β ( X ; Z ) | . Then we see that δ β is a metric on Q , and the topology on Q induced by δ β isequivalent to the original topology of Q . Denote by Q β ∗ the completion of Q with respect to the metric δ β . Put ∂ βM Q = Q β ∗ \ Q . A sequence { Y k } ∞ k =1 in Q
15s called a fundamental sequence if { Y k } ∞ k =1 has no point of accumulation in Q and (cid:8) K β ( · ; Y k ) (cid:9) ∞ k =1 converges uniformly on any compact subset of Q to anonnegative solution of (1.1) with I = (0 , ∞ ). By the local a priori estimatesfor solutions of (1.1), for any Ξ ∈ ∂ βM Q there exist a unique nonnegativesolution K β ( · ; Ξ) of (1.1) and a fundamental sequence { Y k } ∞ k =1 in Q suchthat lim k →∞ ∞ X j =1 − j sup X ∈ Q j (cid:12)(cid:12) K β ( X ; Y k ) − K β ( X ; Ξ) (cid:12)(cid:12) | K β ( X ; Y k ) − K β ( X ; Ξ) | = 0 . Thus the metric δ β is canonically extended to Q β ∗ . Furthermore, Q β ∗ be-comes a compact metric space, since by the parabolic Harnack inequality,any sequence { Y k } ∞ k =1 with no point of accumulation in Q has a fundamentalsubsequence. We call K β ( · ; Ξ), ∂ βM Q and Q β ∗ the β -Martin kernel, β -Martinboundary and β -Martin compactification for ( Q, ∂ t + L ), respectively. Notethat β ( K β ( · ; Ξ)) ≤ K β ( · ; Ξ) ∈ P β ( Q ). A non-negative solution u ∈ P β ( Q ) is said to be minimal if for any nonnegativesolution v ≤ u there exists a nonnegative constant C such that v = Cu . Put ∂ βm Q = n Ξ ∈ ∂ βM Q ; K β ( · ; Ξ) is minimal and β ( K β ( · ; Ξ)) = 1 o , which we call the minimal β -Martin boundary for ( Q, ∂ t + L ).Observe that D × [0 , ∞ ) is embedded into Q β ∗ , and D × { } ⊂ ∂ βM Q .Indeed, with y ∈ D fixed, for any sequence { Y k } ∞ k =1 in Q with lim k →∞ Y k =( y,
0) we have lim k →∞ K β ( x, t ; Y k ) = p ( x, t ; y, /β ( p ( · ; y, K β ( · ; y, = K β ( · ; z,
0) if y = z . We also note that any sequence (cid:8) Y k =( y k , s k ) (cid:9) ∞ k =1 in Q with lim k →∞ s k = ∞ is a fundamental sequence, sincelim k →∞ K β ( · ; Y k ) = 0. We denote by ̟ the point in ∂ βM Q corresponding tothe Martin kernel which is identically zero : K β ( · ; ̟ ) = 0. Put L βm Q = ∂ βm Q \ ( D × { } ∪ { ̟ } ) . We obtain the following abstract integral representation theorem in thesame way as in the proof of Theorem 2.1 and Lemma 2.2 of [34].
Theorem 4.1
For any u ∈ P β ( Q ), there exists a unique pair of finite Borelmeasures κ on D and λ on ∂ βM Q \ ( D × { } ) such that λ is supported by theset L βm Q , u ( x, t ) = Z D p ( x, t ; y, β ( p ( · ; y, dκ ( y ) + Z L βm Q K β ( x, t ; Ξ) dλ (Ξ) (4.1)16or any ( x, t ) ∈ Q , and β ( u ) = κ ( D ) + λ ( L βm Q ) . (4.2)Furthermore, the function v ( x, t ) = u ( x, t ) − Z D p ( x, t ; y, β ( p ( · ; y, dκ ( y )is a nonnegative solution of the equation( ∂ t + L ) v = 0 in D × R such that v = 0 on D × ( −∞ , κ on D and λ on ∂ βM Q \ ( D ×{ } )such that λ is supported by the set L βm Q , the right hand side of (4.1) belongsto P β ( Q ).We put P β ( Q ) = (cid:26) v ∈ P β ( Q ); lim t ↓ v ( x, t ) = 0 on D (cid:27) . We show Theorem 1.3 on the basis of Theorem 4.1. To this end it sufficesto show (1.8) for u ∈ P β ( Q ). The key step in the proof is to identify L βm Q .Under the condition [SSP], we shall show that L βm Q = ∂ m D × [0 , ∞ ). In theremainder of this section we assume [SSP]. We may and shall assume that a = 0 < λ . Lemma 4.2
For any domains U and W with U ⋐ W ⋐ D , there existpositive constants C and α such that p ( x, y, t ) ≤ Cf ( t ) φ ( x ) φ ( y ) , x ∈ U, y ∈ D \ W, t > , (4.3)where f ( t ) = e − α/t for 0 < t <
1, and f ( t ) = e − λ t for t ≥
1. Furthermore, q ( x, ξ, t ) ≤ Cf ( t ) φ ( x ) , x ∈ U, ξ ∈ ∂ M D, t > , (4.4) G ( x, y ) ≤ Cφ ( x ) φ ( y ) , x ∈ U, y ∈ D \ W, (4.5)where G is the Green function of L on D .This lemma is shown in the same way as Lemmas 4.2 and 4.4 of [34].Let K ( x, ξ ) be the Martin kernel for L on D with reference point x ∈ D ,i.e., K ( x , ξ ) = 1 , ξ ∈ ∂ M D . The following lemma gives a relation between K and q . 17 emma 4.3 For any ξ ∈ ∂ M D ,lim D ∋ y → ξ G ( x, y ) φ ( y ) = Z ∞ q ( x, ξ, t ) dt, x ∈ D, (4.6) K ( x, ξ ) = R ∞ q ( x, ξ, t ) dt R ∞ q ( x , ξ, t ) dt , x ∈ D. (4.7)This lemma is shown in the same way as Lemma 4.5 of [34] Lemma 4.4
Let ξ, η ∈ ∂ M D , 0 ≤ s, r < ∞ and C >
0. If q ( x, ξ, t − s ) = Cq ( x, η, t − r ) , ( x, t ) ∈ Q, then ξ = η, s = r and C = 1. Proof
Since q ( x, ξ, τ ) > τ > q ( x, ξ, τ ) = 0 for τ ≤
0, we obtainthat s = r . Thus q ( x, ξ, τ ) = q ( x, η, τ ). This together with (4.7) implies that K ( · , ξ ) = K ( · , η ) on D . Hence ξ = η , and so C = 1. ✷ Now, let β be a measure on Q = D × (0 , ∞ ) as described in the beginningof this section: dβ ( x, t ) = a ( x ) b ( t ) m ( x ) dν ( x ) dt . The following propositiondetermines the β -Martin boundary ∂ βM Q , β -Martin compactification Q β ∗ , and β -Martin kernel K β for ( ∂ t + L, Q ). Recall that p ( x, t ; y, s ) = p ( x, y, t − s )and K β ( · ; y, s ) = p ( · ; y, s ) /β ( p ( · ; y, s )). We write q ( x, t ; ξ, s ) = q ( x, ξ, t − s )for ξ ∈ ∂ M D and 0 ≤ s < ∞ . Proposition 4.5 (i) The β -Martin boundary ∂ βM Q of Q for ∂ t + L is equalto the disjoint union of D × { } , ∂ M D × [0 , ∞ ) and the one point set { ̟ } : ∂ βM Q = D × { } ∪ ∂ M D × [0 , ∞ ) ∪ { ̟ } . (4.8)In particular, ∂ βM Q does not depend on β .(ii) The β -Martin compactification Q β ∗ of Q for ∂ t + L is homeomorphic tothe disjoint union of the topological product D ∗ × [0 , ∞ ) and the one pointset { ̟ } , where a fundamental neighborhood system of ̟ is given by thefamily { ̟ } ∪ D ∗ × ( N, ∞ ) , N >
1. In particular, Q β ∗ does not depend on β .18iii) The β -Martin kernel K β is given as follows: For ( x, t ) ∈ Q , K β ( x, t ; y,
0) = p ( x, t ; y, β ( p ( · ; y, , ( y, ∈ D × { } , (4.9) K β ( x, t ; ξ, s ) = q ( x, t ; ξ, s ) β ( q ( · ; ξ, s )) , ( ξ, s ) ∈ ∂ M D × [0 , ∞ ) , (4.10)and K β ( x, t ; ̟ ) = 0.This proposition is shown in the same way as Proposition 4.8 of [34]. Lemma 4.6
Let ( ξ, s ) ∈ ( ∂ M D \ ∂ m D ) × [0 , ∞ ). Then there exists a finiteBorel measure γ on ∂ M D supported by ∂ m D such that q ( · ; ξ, s ) = Z ∂ m D q ( · ; η, s ) dγ ( η ) . (4.11) Proof
For reader’s convenience, we give a sketch of the proof for the case s = 0. (For details, see the proof of Lemma 4.10 of [34].) By the ellipticMartin representation theorem, there exists a unique finite Borel measure µ on ∂ M D supported by ∂ m D such that K ( x, ξ ) = Z ∂ m D K ( x, η ) dµ ( η ) . This together with (4.7) implies Z ∞ q ( x, ξ, t ) dt = Z ∂ m D (cid:18)Z ∞ q ( x, η, t ) dt (cid:19) dγ ( η ) , (4.12)where dγ ( η ) = [ H ( x , ξ ) /H ( x , η )] dµ ( η ) with H ( x, η ) = Z ∞ q ( x, η, t ) dt. For α >
0, denote by G α the Green function of L + α on D . By the resolventequation and [SSP], we then have Z ∞ e − αt q ( x, η, t ) dt (4.13)= Z ∞ q ( x, η, t ) dt − α Z D G α ( x, z ) (cid:18)Z ∞ q ( z, η, t ) dt (cid:19) m ( z ) dν ( z ) , η ∈ ∂ M D . By combining (4.12) and (4.13), we get Z ∞ e − αt (cid:18)Z ∂ m D q ( x, η, t ) dγ ( η ) (cid:19) dt = Z ∞ e − αt q ( x, ξ, t ) dt. Thus the Laplace transforms of q ( x, ξ, t ) and R ∂ m D q ( x, η, t ) dγ ( η ) coincide;and so (4.11) holds. ✷ Lemma 4.7
Let ( ξ, s ) ∈ ( ∂ M D \ ∂ m D ) × [0 , ∞ ). Then q ( · ; ξ, s ) is not min-imal. Proof
For reader’s convenience, we give a proof. We have (4.11). Supposethat q ( · ; ξ, s ) is minimal. Then, along the line given in the proof of Lemma12.12 of [15], we obtain from (4.11) that the support of γ consists of a singlepoint. Thus, for some η ∈ ∂ m D and constant Cq ( · ; ξ, s ) = Cq ( · ; η, s ) . Hence, by Lemma 4.4, ξ = η ; which is a contradiction. ✷ Lemma 4.8
Let ( ξ, s ) ∈ ∂ m D × (0 , ∞ ). Then q ( · ; ξ, s ) is minimal if andonly if q ( · ; ξ,
0) is minimal.
Proof
Assume that q ( · ; ξ,
0) is minimal. Suppose that a nonnegative so-lution u of (1.1) satisfies u ( · ) ≤ q ( · ; ξ, s ) on Q . Put v ( x, t ) = u ( x, t + s ).Then v ( · ) ≤ q ( · ; ξ, v ( · ) = Cq ( · ; ξ,
0) for some constant C . Hence u ( x, t ) = Cq ( x, t ; ξ, s ) for t > s , and u ( x, t ) = 0 = Cq ( x, t ; ξ, s ) for t ≤ s .This shows that q ( · ; ξ, s ) is minimal. Next, assume that q ( · ; ξ, s ) is minimal.Suppose that a nonnegative solution u of (1.1) satisfies u ( · ) ≤ q ( · ; ξ,
0) on Q . Put v ( x, t ) = u ( x, t − s ) for t > s , and v ( x, t ) = 0 for 0 < t ≤ s .Then v ( · ) ≤ q ( · ; ξ, s ). Thus v ( · ) = Cq ( · ; ξ, s ) for some constant C . Hence u ( x, t ) = Cq ( x, t ; ξ, q ( · ; ξ,
0) is minimal. ✷ By Theorem 4.1 and Lemmas 4.7 and 4.8, we have the following proposi-tion.
Proposition 4.9
There exists a Borel subset R of ∂ M D such that R ⊂ ∂ m D, L βm Q = R × [0 , ∞ ) , u ∈ P β ( Q ) there exists a unique Borel measure λ on ∂ M D × [0 , ∞ )which is supported by R × [0 , ∞ ) and satisfies u ( x, t ) = Z R × [0 , ∞ ) q ( x, ξ, t − s ) dλ ( ξ, s ) ( x, t ) ∈ Q. (4.14) Lemma 4.10
Let ( ξ, s ) ∈ ∂ m D × [0 , ∞ ). Then q ( · ; ξ, s ) is minimal. Proof
Suppose that q ( · ; ξ,
0) is not minimal. Then ξ / ∈ R and q ( x, ξ, t ) = Z R × [0 , ∞ ) q ( x, η, t − s ) dλ ( η, s )for some Borel measure λ . We have K ( x, ξ ) Z ∞ q ( x , ξ, t ) dt = Z ∞ q ( x, ξ, t ) dt = Z R × [0 , ∞ ) dλ ( η, s ) K ( x, η ) Z ∞ q ( x , η, t ) dt. Thus K ( x, ξ ) = Z R K ( x, η ) d Λ( η )for some Borel measure Λ. But ξ ∈ ∂ m D \ R and R ⊂ ∂ m D . This contradictsthe uniqueness of a representing measure in the elliptic Martin representationtheorem. Hence q ( · ; ξ,
0) is minimal; which together with Lemma 4.8 showsLemma 4.10. ✷ Completion of the proof of Theorem 1.3 in the case I = (0 , ∞ ) ByLemma 4.10, R = ∂ m D and L βm Q = ∂ m D × [0 , ∞ ) . Thus Proposition 4.9 shows Theorem 1.3. ✷ < T < ∞ In this section we prove Theorem 1.3 in the case 0 < T < ∞ by making useof the results in Section 4. To this end, the following proposition plays acrucial role. 21 roposition 5.1 Let ξ ∈ ∂ M D and 0 ≤ s < r < ∞ . Then Z D p ( x, y, t − r ) q ( y, r ; ξ, s ) dµ ( y ) = q ( x, t ; ξ, s ) , x ∈ D, t > r, (5.1)where dµ ( y ) = m ( y ) dν ( y ) Proof
We first show (5.1) for ξ ∈ ∂ m D . Define u ( x, t ) by u ( x, t ) = q ( x, t ; ξ, s ) , < t ≤ r,u ( x, t ) = Z D p ( x, y, t − r ) q ( y, r ; ξ, s ) dµ ( y ) , r < t < ∞ . (5.2)(We call u the minimal extension of q from t = r .) Then we see that u is anonnegative solution of ( ∂ t + L ) u = 0 in D × (0 , ∞ ) such that u ( · ) ≤ q ( · ; ξ, s )on D × (0 , ∞ ). By Lemma 4.10, u ( · ) = Cq ( · ; ξ, s ) for some constant C . But u ( x, t ) = q ( x, t ; ξ, s ) for 0 < t ≤ r . Thus C = 1, and so u ( · ) = q ( · ; ξ, s ).Next, let ξ / ∈ ∂ m D . By Lemma 4.6, there exists a finite Borel measure γ on ∂ M D supported by ∂ m D such that q ( · ; ξ, s ) = Z ∂ m D q ( · ; η, s ) dγ ( η ) . (5.3)Thus Z D p ( x, y, t − r ) q ( y, r ; ξ, s ) dµ ( y )= Z ∂ m D dγ ( η ) Z D p ( x, y, t − r ) q ( y, r ; η, s ) dµ ( y )= Z ∂ m D q ( x, t ; η, s ) dγ ( η )= q ( x, t ; ξ, s ) . This proves (5.1). ✷ Lemma 5.2
Let ξ, η ∈ ∂ M D , 0 ≤ s, r < T and C >
0. If q ( x, ξ, t − s ) = Cq ( x, η, t − r ) , x ∈ D, < t < T, (5.4)then ξ = η, s = r and C = 1. 22 roof Choose u such that max( r, s ) < u < T , and construct minimalextensions of both sides of (5.4) from t = u . Then, by (5.1) we have q ( x, ξ, t − s ) = Cq ( x, η, t − r ) , x ∈ D, < t < ∞ . By Lemma 4.4, this implies that ξ = η, s = r and C = 1. ✷ Now, let β be a measure on Q = D × (0 , T ) defined by dβ ( x, t ) = a ( x ) b ( t ) m ( x ) dν ( x ) dt. Here a ( x ) is a nonnegative continuous function on D as described in thebeginning of Section 4, and b ( t ) is a nonnegative continuous function on R such that b ( t ) > T / , T ) and b ( t ) = 0 on R \ ( T / , T ). Let K β ( · ; Ξ), ∂ βM Q , ∂ βm Q , and Q β ∗ be the β -Martin kernel, β -Martin boundary, minimal β -Martin boundary, and β -Martin compactification for ( Q, ∂ t + L ) with Q = D × (0 , T ), respectively. The following proposition is an analogue ofProposition 4.5, and is shown in the same way. Proposition 5.3 (i) The β -Martin boundary ∂ βM Q of Q for ∂ t + L is equalto the disjoint union of D × { } , ∂ M D × [0 , T ) and the one point set { ̟ } : ∂ βM Q = D × { } ∪ ∂ M D × [0 , T ) ∪ { ̟ } . (5.5)In particular, ∂ βM Q does not depend on β .(ii) The β -Martin compactification Q β ∗ of Q for ∂ t + L is homeomorphic tothe disjoint union of the topological product D ∗ × [0 , T ) and the one point set { ̟ } , where a fundamental neighborhood system of ̟ is given by the family { ̟ } ∪ D ∗ × ( T − ε, T ) , < ε < T /
2. In particular, Q β ∗ does not depend on β .(iii) The β -Martin kernel K β is given as follows: For ( x, t ) ∈ Q , K β ( x, t ; y,
0) = p ( x, t ; y, β ( p ( · ; y, , ( y, ∈ D × { } , (5.6) K β ( x, t ; ξ, s ) = q ( x, t ; ξ, s ) β ( q ( · ; ξ, s )) , ( ξ, s ) ∈ ∂ M D × [0 , T ) , (5.7)and K β ( x, t ; ̟ ) = 0. 23 emma 5.4 Let ( ξ, s ) ∈ ( ∂ M D \ ∂ m D ) × [0 , T ). Then q ( · ; ξ, s ) is not mini-mal. Proof
Suppose that q ( · ; ξ, s ) is minimal. Then we obtain from (5.3) that q ( x, ξ, t − s ) = Cq ( x, η, t − s ) , x ∈ D, < t < T, for some η ∈ ∂ m D and C >
0. By Lemma 5.2, this is a contradiction. ✷ Lemma 5.5
Let ( ξ, s ) ∈ ∂ m D × [0 , T ). Then q ( · ; ξ, s ) is minimal. Proof
Let u be a nonnegative solution of ( ∂ t + L ) u = 0 in Q such that u ( · ) ≤ q ( · ; ξ, s ) in Q . For r ∈ ( s, T ), let u r be the minimal extension of u from t = r . By Proposition 5.1, u r ( x, t ) ≤ q ( x, t ; ξ, s ) , x ∈ D, t > . By Lemma 4.10, there exists a constant C r such that u r ( x, t ) = C r q ( x, t ; ξ, s )for t >
0. But u r ( x, t ) = u ( x, t ) for 0 < t < r . Thus C r is independent of r ;and so u ( · ) = Cq ( · ; ξ, s ) in Q for some constant C . ✷ Completion of the proof of Theorem 1.3 in the case < T < ∞ Put L βm Q = ∂ βm Q \ ( D × { } ∪ { ̟ } ) . By Proposition 5.3, Lemmas 5.4 and 5.5, we get L βm Q = ∂ m D × [0 , T ) . Thus, Theorem 2.1 of [34] which is an analogue of Theorem 4.1 completesthe proof. ✷ I = ( −∞ , In this section we prove Theorem 1.4. We begin with the following proposi-tion, which can be shown in the same way as in the proof of Theorem 1 of [9](see also [39]). 24 roposition 6.1
Assume [SIU]. Thenlim t →∞ e λ t p ( x, y, t ) φ ( x ) φ ( y ) = 1 uniformly in ( x, y ) ∈ K × D (6.1)for any compact subset K of D .In the rest of this section we assume [SSP]. We may and shall assumethat a = 0 < λ . By Theorem 1.1, we have the following corollary of Propo-sition 6.1. Corollary 6.2
Assume [SSP]. Then, for any compact subset K of D and N > s →−∞ p ( x, y, t − s ) e λ s φ ( y ) = e − λ t φ ( x ) uniformly in ( x, y, t ) ∈ K × D × ( − N, . Lemma 6.3
The solution e − λ t φ ( x ) is minimal. Proof
Suppose that e − λ t φ ( x ) is not minimal. Then, in view of Corol-lary 6.2, the same argument as in the proof of Theorem 1.3 shows that forany nonnegative solution u of the equation( ∂ t + L ) u = 0 in Q = D × ( −∞ , λ on ∂ M D × ( −∞ ,
0) supported by theset ∂ m D × ( −∞ ,
0) such that u ( x, t ) = Z ∂ M D × ( −∞ ,t ) q ( x, ξ, t − s ) dλ ( ξ, s ) , ( x, t ) ∈ Q. Thus e − λ t φ ( x ) = Z ∂ M D × ( −∞ ,t ) q ( x, ξ, t − s ) dλ ( ξ, s ) , ( x, t ) ∈ Q, (6.2)for such a measure λ . Now, fix x . It follows from Theorems 1.1 and 1.2 thatfor any δ > C δ such that C δ − ≤ q ( x, ξ, τ ) e − λ τ φ ( x ) ≤ C δ , τ ≥ δ, ξ ∈ ∂ M D. (6.3)25y (4.4), q ( x, ξ, τ ) ≤ Ce − α/τ φ ( x ) , ξ ∈ ∂ M D, < τ < , (6.4)for some positive constants α and C . By (6.2) and (6.3), e λ φ ( x ) ≥ Z ∂ M D × ( −∞ , − C − e − λ ( − − s ) dλ ( ξ, s ) . Thus Z ∂ M D × ( −∞ , − e λ s dλ ( ξ, s ) ≤ C φ ( x ) . (6.5)For t < − < δ <
1, we have φ ( x ) = Z ∂ M D ×{ ( −∞ ,t − δ ] ∪ ( t − δ,t ) } e λ ( t − s ) q ( x, ξ, t − s ) e λ s dλ ( ξ, s ) . (6.6)In view of (6.4) and (6.5), we choose δ so small that the integral on ∂ M D × ( t − δ, t ) of the right hand side of (6.6) is smaller than φ ( x ) /
3. Then, in viewof (6.3) and (6.5), we choose t < − | t | being so large that the integralon ∂ M D × ( −∞ , t − δ ] of the right hand side of (6.6) is smaller than φ ( x ) / ✷ Completion of the proof of Theorem 1.4
By virtue of Corollary 6.2and Lemma 6.3, the same argument as in the proof of Theorem 1.3 showsTheorem 1.4. ✷ In this section we give two examples in order to illustrate a scope of Theo-rem 1.3. Throughout this section L is a uniformly elliptic operator on R n of the form L u = − n X i,j =1 ∂ i ( a ij ( x ) ∂ j u ) , where a ( x ) = [ a ij ( x ) ] ni,j =1 is a symmetric matrix-valued measurable functionon R n satisfying, for some Λ > − | ξ | ≤ n X i,j =1 a ij ( x ) ξ i ξ j ≤ Λ | ξ | , x, ξ ∈ R n . .1. Let V ( x ) be a measurable function in L ∞ loc ( R n ), and L = L + V ( x )on D = R n . Theorem 7.1
Suppose that there exist a positive constant c < ρ on [0 , ∞ ) such that c [ ρ ( | x | ) ] ≤ V ( x ) ≤ [ ρ ( | x | ) ] , x ∈ R n , (7.1) c ρ (cid:18) r + cρ ( r ) (cid:19) ≤ ρ ( r ) , r ≥ . (7.2)Assume that Z ∞ drρ ( r ) < ∞ . (7.3)Then 1 is a small perturbation of L on R n . Thus Theorem 1.3 holds true. Remark.
Compare this theorem with a non-uniqueness theorem of [26].
Proof
We first note that (7.2) yields cρ ( r ) ≤ cρ r − cρ ( r ) + cρ (cid:16) r − cρ ( r ) (cid:17) ≤ ρ (cid:18) r − cρ ( r ) (cid:19) , r ≥ cρ (0) , since ρ is increasing. We show the theorem by using the same approach asin the proof of Theorem 5.1 of [31]. Put b = c − and ℓ = inf { j ∈ Z ; ρ (0) < b j } . For k ≥ ℓ , put r k = sup { r ≥ ρ ( r ) ≤ b k } . By the continuity of ρ and(7.3), ρ ( r k ) = b k and lim k →∞ r k = ∞ . By (7.2), ρ ( r k + cb − k ) ≤ c − ρ ( r k ) = b / b k < b k +1 = ρ ( r k +1 ) . Thus r k + cb − k < r k +1 for k ≥ ℓ . Define a positive continuously differentiabeincreasing function e ρ on [0 , ∞ ) as follows: Put e ρ ( r ) = b ℓ for r ≤ r ℓ , e ρ ( r ) = b k +1 for r k + cb − k ≤ r ≤ r k +1 ( k ≥ ℓ );and e ρ ( r ) = ρ k ( r ) for r k ≤ r ≤ r k + cb − k ( k ≥ ℓ ) by choosing a continuouslydifferentiabe function ρ k on [ r k , r k + cb − k ] such that ρ k ( r k ) = b k , ρ k ′ ( r k ) = 0 , ρ k ( r k + cb − k ) = b k +1 , ρ k ′ ( r k + cb − k ) = 0 , ≤ ρ k ′ ( r ) ≤ B b k , r k ≤ r ≤ r k + cb − k , for some constant B > k . Then we have C − ≤ e ρ ( r ) ρ ( r ) ≤ C, ≤ e ρ ′ ( r ) ≤ Cρ ( r ) , r ≥ , (7.4)for some positive constant C . Introduce a Riemannian metric g = ( g ij ) ni,j =1 by g ij = e ρ ( | x | ) δ ij . Then M = R n with this metric g becomes a completeRiemannian manifold . Furthermore, by (7.2) and (7.4), M has the boundedgeometry property (1.1) of [4]. The associated gradient ∇ and divergencediv are written as ∇ = e ρ ( | x | ) − ∇ , div = e ρ ( | x | ) − n ◦ div ◦ e ρ ( | x | ) n , where ∇ and div are the standard gradient and divergence on R n . Put L = e ρ ( | x | ) − L,m ( x ) = e ρ ( | x | ) − n , A ( x ) = [ a ij ( x ) ] ni,j =1 , γ ( x ) = e ρ ( | x | ) − V ( x ) . Then L u = − m div ( mA ∇ u ) + γ = − div ( A ∇ u ) − (cid:10) m A ∇ m, ∇ u (cid:11) + γ, where h· , ·i is the standard inner product on R n . Since the inner product h· , ·i associated with the metric g is written as h X, Y i = h e ρ X, Y i , we have L u = − div ( A ∇ u ) − (cid:10)e ρ − A ∇ mm , ∇ u (cid:11) + γ. (7.5)By (7.4), |∇ m ( x ) | ≤ C | n − | e ρ ( | x | ) m ( x ) . From this we have (cid:10)e ρ − A ∇ mm , e ρ − A ∇ mm (cid:11) ≤ e ρ − Λ ( C | n − | e ρ ) ≤ { Λ( C | n − | ) } .
28y (7.1) and (7.4), c C − ≤ γ ( x ) ≤ C . Thus the operator
L − cC − / L belongs to theclass D M ( θ, ∞ , ǫ ) introduced by Ancona [4], where θ = max (cid:0) Λ , Λ( C | n − | ) , C (cid:1) , ǫ = cC − / . Put L = e ρ ( | x | ) − ( L + 1 ) = L + e ρ ( | x | ) − . In order to apply the results of [4], we proceed to estimate e ρ ( | x | ) − . Let d ( x )be the Riemannian distance dist(0 , x ) from the origin 0 to x , and put ψ ( r ) = Z r e ρ ( s ) ds. Then we see that d ( x ) = ψ ( | x | ). Denote by ψ − the inverse function of ψ ,and put Φ( s ) = (cid:2) e ρ (cid:0) ψ − ( s ) (cid:1) (cid:3) − , s ≥ . Then 0 < e ρ ( | x | ) − = Φ ( d ( x ) ) , x ∈ M. Furthermore, Z ∞ Φ( s ) ds = Z ∞ Φ( ψ ( r ) ) e ρ ( r ) dr = Z ∞ dr e ρ ( r ) ≤ C Z ∞ drρ ( r ) dr < ∞ . Hence, by virtue of Corollary 6.1, Theorems 1 and 2 of [4], e ρ ( | x | ) − is a smallperturbation of L on the manifold M . That is, for any ε > K of D = M such that Z D \ K H ( x, z ) e ρ ( | z | ) − H ( z, y ) e ρ ( | z | ) n dz ≤ εH ( x, y ) , x, y ∈ D \ K, where dz is the Lebesgue measure on R n , and H ( x, z ) is the Green function of L on D with respect to the measure e ρ ( | z | ) n dz . Denote by G ( x, z ) the Greenfunction of L on D with respect to the measure dz . Since L = e ρ ( | x | ) − L , wehave H ( x, z ) = G ( x, z ) e ρ ( | z | ) − n Z D \ K G ( x, z ) e ρ ( | z | ) (2 − n ) − G ( z, y ) e ρ ( | y | ) − n e ρ ( | z | ) n dz ≤ εG ( x, y ) e ρ ( | y | ) − n for any x, y ∈ D \ K . Hence 1 is a small perturbation of L on R n . (cid:3) Remark.
A sufficient condition for (7.2) is the following: ρ is a positivedifferentiable function on [0 , ∞ ) satisfying0 ≤ ρ ′ ( r ) ρ ( r ) − ≤ C, r ≥ , (7.6)for some positive constant C . Indeed, from (7.6) we have X ( δ ) ≡ ρ (cid:18) r + δρ ( r ) (cid:19) ρ ( r ) − ≤ exp[ CδX ( δ )] , r ≥ , δ > . Put δ = (2 Ce ) − , and let γ ∈ (1 , e ) be the solution of the equationexp[ X/ e ] = X. Then we get 1 ≤ X ( δ ) ≤ γ . Thus (7.2) holds with c = min( δ, /γ ).The condition (7.3) is sharp, since Theorem 6.2 of [17] yields the followinguniqueness theorem. Theorem 7.2
Suppose that there exists a positive continuous increasingfunction ρ on [0 , ∞ ) such that | V ( x ) | ≤ ρ ( | x | ) , x ∈ R n . (7.7)Assume that Z ∞ drρ ( r ) = ∞ . (7.8)Then [UP] holds. Thus Fact AT holds true. Throughout this subsection we assume that D is a bounded domainof R n . Let L be an elliptic operator on D of the form L = 1 w ( x ) L , where w is a positive measurable function on D such that w, w − ∈ L ∞ loc ( D ).30 heorem 7.3 Let D be a Lipschitz domain. Suppose that there exists apositive function ψ on (0 , ∞ ) such that s ψ ( s ) is increasing and w ( x ) ≤ ψ ( δ D ( x ) ) , x ∈ D, (7.9)where δ D ( x ) = dist ( x, ∂D ). Assume that Z s ψ ( s ) ds < ∞ . (7.10)Then 1 is a small perturbation of L on D . Thus Theorem 1.3 holds true. Remark. (i) The first assertion of this theorem is implicitly shown in [17](see Theorem 7.11 and Remark 7.12 (ii) there).(ii) The Lipschitz regularity of the domain D is assumed only for theHardy inequality to hold for any function in C ∞ ( D ). Thus, for this theoremto hold, it suffices to assume (for example) that D is uniformly ∆-regularJohn domain or a simply connected domain of R (see [3], [4]). Proof of Theorem 7.3
For x ∈ D , put D x = (cid:26) y ∈ D ; | x − y | < δ D ( x )2 (cid:27) . Then 12 δ D ( x ) ≤ δ D ( y ) ≤ δ D ( x ) , y ∈ D x . Thus δ D ( x ) w ( y ) ≤ δ D ( y ) ψ ( δ D ( y ) ) ≤ (cid:18) δ D ( x ) (cid:19) ψ (cid:18) δ D ( x ) (cid:19) . Put Ψ( s ) = 9 s ψ ((3 / s ). Then Ψ( s ) is increasing, and satisfies δ D ( x ) (cid:18) sup y ∈ D x w ( y ) (cid:19) ≤ Ψ ( δ D ( x ) ) , Z Ψ( s ) s ds < ∞ . Hence, by virtue of Proposition 9.2, Theorem 9.1’ and Corollary 6.1 of [4], w is a small perturbation of L on D . This implies that 1 is a small perturbationof L on D . (cid:3) The condition (7.10) is sharp, since Theorem 7.8 and Lemma 7.6 of [17]yield the following uniqueness theorem.31 heorem 7.4
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