L 1 -estimates for eigenfunctions and heat kernel estimates for semigroups dominated by the free heat semigroup
aa r X i v : . [ m a t h . A P ] J un L -estimates for eigenfunctions and heat kernel estimatesfor semigroups dominated by the free heat semigroup Hendrik Vogt ∗ Abstract
We investigate selfadjoint positivity preserving C -semigroups that aredominated by the free heat semigroup on R d . Major examples are semi-groups generated by Dirichlet Laplacians on open subsets or by Schr¨odingeroperators with absorption potentials. We show explicit global Gaussian up-per bounds for the kernel that correctly reflect the exponential decay of thesemigroup. For eigenfunctions of the generator that correspond to eigen-values below the essential spectrum we prove estimates of their L -norm interms of the L -norm and the eigenvalue counting function. This estimateis applied to a comparison of the heat content with the heat trace of thesemigroup.MSC 2010: 35P99, 35K08, 35J10, 47A10Keywords: Schr¨odinger operators, eigenfunctions, L -estimates,heat kernel estimates, heat content, heat trace In the recent paper [BHV13], the authors studied Dirichlet Laplacians on opensubsets Ω of R d . They proved an estimate for the L -norm of eigenfunctions interms of their L -norm and spectral data, and they used this to estimate the heatcontent of Ω by its heat trace. The aim of the present paper is to provide sharperestimates in the following more general setting.Let Ω ⊆ R d be measurable, where d ∈ N , and let T be a selfadjoint positivitypreserving C -semigroup on L (Ω) that is dominated by the free heat semigroup,i.e., 0 T ( t ) f e t ∆ f (cid:0) t > , f ∈ L (Ω) (cid:1) . Let − H denote the generator of T . ∗ Fachbereich 3 – Mathematik, Universit¨at Bremen, 28359 Bremen, Germany, +49 421 218-63702, hendrik.vohugo@[email protected]
1n important example for the operator − H is the Dirichlet Laplacian with alocally integrable absorption potential on an open set Ω ⊆ R d . For more generalabsorption potentials the space of strong continuity of the semigroup will be L (Ω ′ )for some measurable Ω ′ ⊆ Ω.In our first main result we estimate the L -norm of eigenfunctions of H interms of their L -norm and the eigenvalue counting function N t ( H ), which for t < inf σ ess ( H ) denotes the number of eigenvalues of H that are t , counted withmultiplicity. Let ϕ be an eigenfunction of H with eigenvalue λ < inf σ ess ( H ) .Then || ϕ || c d ( t − λ ) − d/ (cid:0) ln tt − λ (cid:1) d N t ( H ) || ϕ || (cid:0) λ < t < inf σ ess ( H ) (cid:1) , with c d = 35 d +1 d d/ . (a) We point out that as in [BHV13; Thm. 1.6] one has the lowerbound || ϕ || > (cid:0) πde (cid:1) d/ λ − d/ || ϕ || . Thus, the factor d d/ in the constant c d is of the correct order. The factor ( t − λ ) − d/ matches the factor λ − d/ ; cf. Corollary 1.3 below. See [BHV13; Example 1.8(3)]for an explanation why one should expect the factor N t ( H ) with some t > λ in theestimate of Theorem 1.1.(b) In [BHV13; Thm. 1.3], in the framework of Dirichlet Laplacians on opensubsets of R d , the estimate || ϕ || C d E − d/ (cid:18) λE λt − λ (cid:19) d (cid:0) ln N t ( H ) (cid:1) d N t ( H ) + (cid:18) λE (cid:19) d − (cid:18) λt − λ (cid:19) d ! || ϕ || was shown under the additional assumption t λ , where E = inf σ ( H ). Ourestimate || ϕ || c d λ − d/ (cid:0) λt − λ (cid:1) d/ (cid:0) ln tt − λ (cid:1) d N t ( H ) || ϕ || improves on this in severalregards; most notably, the factors λE , (cid:0) ln N t ( H ) (cid:1) d and the second summand areremoved altogether.(c) In [BHV13], a partition of R d into cubes was used in the proof. We willwork with a “continuous partition” into balls instead; see the proof of Lemma 4.2.Working with balls leads to a better constant c d in the estimate.(d) In the case d = 1 and H the Dirichlet Laplacian on an open subset of R , animproved estimate is given in [BHV13; Rem. 1.5]. For that estimate it is crucialthat H is a direct sum of Dirichlet Laplacians on intervals. The improvement isnot possible for general H in dimension d = 1; this can be seen similarly as in[BHV13; Example 1.8(3)]. 2f H has compact resolvent, then one can apply Theorem 1.1 with t = (1 + ε ) λ for any ε > λ − d/ as the lower bound of Remark 1.2(a). Assume that H has compact resolvent. Let ϕ be an eigenfunctionof H with eigenvalue λ . Then || ϕ || c d C dε λ − d/ N (1+ ε ) λ ( H ) || ϕ || ( ε > , with c d as in Theorem 1.1 and C ε = ε − / ln(3 + ε ) . The assumption that H has compact resolvent is in particular sat-isfied if Ω has finite volume. Note that then the trivial estimate || ϕ || vol(Ω) || ϕ || holds. We point out that, up to a dimension dependent constant, the estimate ofCorollary 1.3 is never worse since one has the bound N t ( H ) K d vol(Ω) t d/ for all t >
0. (To obtain this bound, apply [LiYa83; Cor. 1] to open sets ˜Ω ⊇ Ω and notethat e − tH e − t ∆ ˜Ω , where ∆ ˜Ω denotes the Dirichlet Laplacian on ˜Ω.)Our second main result is the following heat kernel estimate for semigroupsdominated by the free heat semigroup. This estimate is obtained as a by-productof the preparations for the proof of Theorem 1.1. For all t > the semigroup operator e − tH has an integral kernel p t .If E := inf σ ( H ) > then p t ( x, y ) (cid:18) eE πd (cid:19) d/ exp (cid:18) − E t − | x − y | t (cid:19) (cid:0) t > d E , x, y ∈ R d (cid:1) . (a) For 0 < t < d E one just has the estimate with respect to thefree heat kernel, 0 p t ( x, y ) (4 πt ) − d/ exp (cid:18) − | x − y | t (cid:19) . In combination with Theorem 1.5 this gives0 p t ( x, y ) (4 πt ) − d/ (cid:18) ed E t (cid:19) d/ exp (cid:18) − E t − | x − y | t (cid:19) ( t > . (In the case E = 0 this estimate is true but inconsequential.)(b) In [Ouh06; formula (22)], the following estimate was proved in the frame-work of Dirichlet Laplacians with absorption potentials on open subsets of R d : p t ( x, y ) c ε (4 πt ) − d/ (cid:18) E t + ε | x − y | t (cid:19) d/ exp (cid:18) − E t − | x − y | t (cid:19) , ε > c ε = e (1 + ε ) d/ . Part (a) shows that the summand ε | x − y | t isactually not needed, which may come as a surprise.(c) In the generality of our setting, the estimate provided in Theorem 1.5 isprobably the best one can hope for. Suppose, for example, that the semigroup T is irreducible and that E is an isolated eigenvalue of H . Then the large timebehaviour of p t is known: e E t p t ( x, y ) → ϕ ( x ) ϕ ( y ) ( t → ∞ ) , where ϕ is the non-negative normalized ground state of H ; see, e.g., [KLVW13;Thm. 3.1]. Moreover, if inf σ ( H ) = 1 then E d/ ϕ ( E / · ) is the ground state of anappropriately scaled operator H E with inf σ ( H E ) = E . This explains the factor E d/ in our estimate.Note, however, that better estimates are known for Dirichlet Laplacians un-der suitable geometric assumptions on the domain Ω. Then a boundary termlike ϕ ( x ) ϕ ( y ) can be included in the estimate. This can be shown via intrinsicultracontractivity as in [OuWa07].An important application of Corollary 1.3 is that it allows us to compare the“heat content” of H with its “heat trace”. We assume that H has compact re-solvent, with ( λ k ) the increasing sequence of all the eigenvalues of H , repeatedaccording to their multiplicity. For t > Q H ( t ) := || e − tH Ω || the heat content , by Z H ( t ) := P ∞ k =1 e − tλ k the heat trace of H .Note that Q H , Z H are decreasing functions. It may well occur that Q H ( t ) = ∞ and/or Z H ( t ) = ∞ for some but not all t > Assume that H has compact resolvent and that Z H ( t ) < ∞ forsome t > . Then Q H ( t ) < ∞ for all t > t , Q H ( t ) c ε,d λ − d/ Z H (cid:0) t ε (cid:1) (cid:0) < ε < tt − (cid:1) , with c ε,d = c d C dε as in Corollary 1.3. The proof is rather short, so we give it right here. We will use the followingsimple estimate. (cf. [BHV13; Lemma 5.2]) For
T, λ > one has N λ ( H ) Z H ( T ) e T λ .Proof. If k ∈ N is such that λ k λ , then k e T λ k P kj =1 e − T λ j e T λ Z H ( T ). Thus, N λ ( H ) = { k ; λ k λ } e T λ Z H ( T ). 4 roof of Theorem 1.7. Let T := t ε . Let ( ϕ k ) be an orthonormal basis of L (Ω)such that Hϕ k = λ k ϕ k for all k ∈ N . By Corollary 1.3 and Lemma 1.8 we obtain || ϕ k || c ε,d λ − d/ k N (1+ ε ) λ k ( H ) || ϕ k || c ε,d λ − d/ Z H ( T ) e T (1+ ε ) λ k for all k ∈ N . For f ∈ L (Ω) ∩ L ∞ (Ω) one has e − tH f = P ∞ k =1 h f, ϕ k i e − tλ k ϕ k andhence || e − tH f || ∞ X k =1 || f || ∞ e − tλ k || ϕ k || . Using a sequence ( f k ) in L (Ω) with 0 f k ↑ Ω and recalling T (1 + ε ) = t − T ,we conclude that || e − tH Ω || ∞ X k =1 e − tλ k || ϕ k || c ε,d λ − d/ Z H ( T ) ∞ X k =1 e − T λ k = c ε,d λ − d/ Z H ( T ) . The paper is organized as follows. In Section 2 we investigate properties ofselfadjoint positivity preserving semigroups dominated by the free heat semigroup.In Section 3 we prove Theorem 1.5, and we show off-diagonal resolvent estimatesneeded in the proof of Theorem 1.1, which in turn is given in Section 4.
Throughout this section let Ω ⊆ R d be measurable, and let T be a selfadjointpositivity preserving C -semigroup on L (Ω) that is dominated by the free heatsemigroup, with generator − H . Let τ be the closed symmetric form associatedwith H . The purpose of this section is to collect some basic properties of τ and H .It is crucial that D ( τ ) is a subset of H ( R d ) (in fact an ideal; see, e.g., [MVV05;Cor. 4.3]). Thus we can define a symmetric form σ by σ ( u, v ) := τ ( u, v ) − h∇ u, ∇ v i (cid:0) u, v ∈ D ( σ ) := D ( τ ) (cid:1) . (2.1)This gives a decomposition of the form τ as the standard Dirichlet form plus aform σ that is positive and local in the sense of the following lemma. If − H is theDirichlet Laplacian with an absorption potential V > ⊆ R d ,then σ ( u, v ) = R V uv . In this case the next three results are trivial.
Let u, v ∈ D ( τ ) . Then σ ( u, v ) > , and σ ( u, v ) = 0 if u ∧ v = 0 .Proof. By [MVV05; Cor. 4.3], the first assertion follows from the assumption that T is a positive semigroup dominated by the free heat semigroup. For the secondassertion let w := u − v . Then τ ( u, v ) = τ ( w + , w − ) T is a positive semi-group (see, e.g., [MVV05; Cor. 2.6]). Since h∇ u, ∇ v i = 0, this implies σ ( u, v ) σ ( u, v ) = 0. 5 .2 Lemma. If ξ ∈ W ∞ ( R d ) and u ∈ D ( τ ) , then ξu ∈ D ( τ ) . Moreover, f : R d → D ( τ ) , f ( x ) := ξ ( · − x ) u is continuous.Proof. By [MVV05; Cor. 4.3], D ( τ ) is an ideal of H ( R d ). This implies the firstassertion ξu ∈ D ( τ ) since ξu ∈ H ( R d ) and | ξu | || ξ || ∞ | u | ∈ D ( τ ).For the second assertion it suffices to show continuity at 0, and we can assumewithout loss of generality that ξ , u are real-valued. From the identity f ( x ) − f (0) = ξ ( · − x ) (cid:0) u − u ( · − x ) (cid:1) + ( ξu )( · − x ) − ξu one deduces that f : R d → H ( R d ) is continuous at 0. By Lemma 2.1 we obtain σ (cid:0) f ( x ) − f (0) (cid:1) = σ (cid:0) | f ( x ) − f (0) | (cid:1) σ (cid:0) || ξ ( · − x ) − ξ || ∞ | u | (cid:1) ||∇ ξ || ∞ | x | σ ( | u | ) . Due to the decomposition (2.1) this yields continuity of f : R d → D ( τ ) at 0. Let u, v ∈ D ( τ ) . Then σ ( ξu, v ) = σ ( u, ξv ) for all ξ ∈ W ∞ ( R d ) .Proof. Since D ( τ ) is a lattice, it suffices to show the assertion for u, v > ξ . Throughout the proof we consider only real-valued function spaces.We define a bilinear form b by b ( ϕ, ψ ) := σ ( ϕu, ψv ) (cid:0) ϕ, ψ ∈ D ( b ) := W ∞ , ( R d ) (cid:1) . Then b ( ϕ, ψ ) > ϕ, ψ > σ ( ϕu, ψv ) = Z ϕψ dµ (cid:0) ϕ, ψ ∈ W ∞ , ( R d ) (cid:1) (2.2)for some finite positive Borel measure µ on R d (depending of course on u, v ). Weonly sketch the argument: first one can extend b to a continuous bilinear formon C ( R d ), by positivity. Then one uses the linearisation of b in C ( R d × R d ) ′ toobtain a finite Borel measure ν on R d × R d such that b ( ϕ, ψ ) = R ϕ ( x ) ψ ( y ) dν ( x, y )for all ϕ, ψ ∈ W ∞ , ( R d ). Finally, spt ν ⊆ (cid:8) ( x, x ); x ∈ R d (cid:9) since b ( ϕ, ψ ) = 0 in thecase spt ϕ ∩ spt ψ = ∅ , by Lemma 2.1, and this leads to the asserted measure µ .To complete the proof, we show that the representation (2.2) is valid for all ϕ, ψ ∈ W ∞ ( R d ). Let χ ∈ C ( R d ) such that 0 χ χ | B (0 , = 1. Then u n := χ ( · n ) u → u in H ( R d ) as n → ∞ , and σ ( u n ) σ ( u ) for all n ∈ N byLemma 2.1. Therefore, lim sup τ ( u n ) τ ( u ), and this implies u n → u in D ( τ ).Applying (2.2) to σ (cid:0) χ ( · n ) ϕu, χ ( · n ) ψv (cid:1) and letting n → ∞ we derive (2.2) for any ϕ, ψ ∈ W ∞ ( R d ). For real-valued ξ ∈ W ∞ ( R d ) we now obtain σ ( ξu, v ) = Z ξ dµ = σ ( u, ξv ) .
6n the proof of Theorem 1.1 we will work with operators that are subordinatedto H as follows. For an open set U ⊆ R d let H U denote the selfadjoint operator in L (Ω ∩ U ) associated with the form τ restricted to D ( τ ) ∩ H ( U ). (Observe thatthis form domain is dense in L (Ω ∩ U ).) Let ϕ be an eigenfunction of H with eigenvalue λ . Let U be an opensubset of R d , and let ξ ∈ W ∞ ( R d ) , ξ = 0 on R d \ U . Then ξϕ ∈ D ( H U ) and ( H U − λ )( ξϕ ) = − ∇ ξ · ∇ ϕ − (∆ ξ ) ϕ. Proof.
By Lemma 2.2 we have ξϕ ∈ D ( τ ). Moreover, ξϕ ∈ H ( U ) due to theassumption ξ = 0 on R d \ U . For v ∈ D ( τ ) ∩ H ( U ) we have ξv ∈ D ( τ ) ∩ H ( U )and ( τ − λ )( ϕ, ξv ) = h ( H − λ ) ϕ, ξv i = 0 . Since σ ( ξϕ, v ) = σ ( ϕ, ξv ) by Lemma 2.3, the decomposition (2.1) yields( τ − λ )( ξϕ, v ) = ( τ − λ )( ξϕ, v ) − ( τ − λ )( ϕ, ξv )= h∇ ( ξϕ ) , ∇ v i − h∇ ϕ, ∇ ( ξv ) i = h ϕ ∇ ξ, ∇ v i − h∇ ϕ, v ∇ ξ i . Now ϕ ∇ ξ is in H ( R d ) d and ∇ · ( ϕ ∇ ξ ) = ∇ ϕ · ∇ ξ + ϕ ∆ ξ , so we conclude that( τ − λ )( ξϕ, v ) = −h ∇ ϕ · ∇ ξ + ϕ ∆ ξ, v i for all v ∈ D ( τ ) ∩ H ( U ), which proves the assertion. In this section we prove Theorem 1.5, and we provide resolvent estimates neededin the proof of Theorem 1.1. Throughout we denote C + := { z ∈ C ; Re z > } . We point out that in the following result T is not required to be a semigroup. Let (Ω , µ ) be a measure space, and let ρ : Ω → R be measurable.Let λ ∈ R , and let T : C + → L ( L ( µ )) be analytic, || T ( z ) || e − λ Re z for all z ∈ C + .Assume that there exists C > such that || e αρ T ( t ) e − αρ || Ce α t ( α, t > . Then || e αρ T ( z ) e − αρ || exp (cid:0) α / Re z − λ Re z (cid:1) ( α > , z ∈ C + ) , in particular, || e αρ T ( t ) e − αρ || e α t − λt for all α, t > . || wBw − || := sup (cid:8) || wBw − f || ; f ∈ L ( µ ) , || f || , w − f ∈ L ( µ ) (cid:9) for an operator B ∈ L ( L ( µ )) and a measurable function w : Ω → (0 , ∞ ). Proof of Proposition 3.1.
Observe that M := (cid:8) f ∈ L ( µ ); ρ bounded on [ f = 0] (cid:9) is dense in L ( µ ). Let α >
0, and let f, g ∈ M with || f || = || g || = 1. Define theanalytic function F : C + → C by F ( z ) := e λz − α /z h e αρ/z T ( z ) e − αρ/z f, g i . Let c > | ρ | c on [ f = 0] ∪ [ g = 0]. Then | F ( z ) | || e λz T ( z ) |||| e − αρ/z f || || e αρ/ ¯ z g || exp(2 αc Re z ) ( z ∈ C + ) , in particular | F ( t ) | e αc for all t >
1. Moreover, | F ( t ) | e λt − α /t || e αρ/t T ( t ) e − αρ/t || e λt − α /t · Ce ( α/t ) t Ce | λ | for all 0 < t <
1. Thus, | F ( z ) | z ∈ C + by the next lemma, and thisyields || e αρ/z T ( z ) e − αρ/z || exp (cid:0) α Re z − λ Re z (cid:1) ( α > , z ∈ C + ) . The assertion follows by replacing α with α/ Re z .The following Phragm´en-Lindel¨of type result is similar to [CoSi08; Prop. 2.2]. Let F : C + → C be analytic. Assume that there exist c , c > suchthat | F ( z ) | exp( c Re z ) ( z ∈ C + ) , | F ( t ) | c ( t > . Then | F ( z ) | for all z ∈ C + .Proof. Note that lim sup z → iy | F ( z ) | y ∈ R \ { } . Thus, | F ( z ) | c ∨ z ∈ C + by the Phragm´en-Lindel¨of principle applied to the sectors (cid:8) z ∈ C ;Re z > , Im z > (cid:9) and (cid:8) z ∈ C ; Re z > , Im z < (cid:9) . Then an application of thePhragm´en-Lindel¨of principle to the sector C + implies | F | C + .In the next lemma we state a version of the well-known Davies’ trick; cf. [Dav95;proof of Lemma 19]. For the proof note that inf ξ ∈ R d exp (cid:0) | ξ | t − ξ · x (cid:1) = exp (cid:0) − | x | t (cid:1) for all t > x ∈ R d . 8 .3 Lemma. Let Ω ⊆ R d be measurable, and let B be a positive operator on L (Ω) .For ξ, x ∈ R d let ρ ξ ( x ) := e ξx . Then for t > the following are equivalent: (i) B e t ∆ , (ii) || ρ ξ Bρ − ξ || →∞ (4 πt ) − d/ e | ξ | t for all ξ ∈ R d . In (i), the inequality B e t ∆ is meant in the sense of positivity preservingoperators, i.e., Bf e t ∆ f (cid:0) f ∈ L (Ω) (cid:1) . The following result provides an estimate of the resolvent of H by the freeresolvent. Together with Proposition 3.5 below this will be an important steppingstone in the proof of Theorem 1.1. Let Ω ⊆ R d be measurable, and let T be a selfadjoint positive C -semigroup on L (Ω) that is dominated by the free heat semigroup. Let − H be thegenerator of T , and let E := inf σ ( H ) . Then for all ε ∈ (0 , one has T ( t ) ε − d/ e − (1 − ε ) E t e t ∆ ( t > , ( H − λ ) − ε − d/ (cid:0) (1 − ε ) E − λ − ∆ (cid:1) − ( λ < (1 − ε ) E ) . Proof.
As above let ρ ξ ( x ) := e ξx . The assumptions imply || T ( z ) || → e − E Re z forall z ∈ C + and || ρ ξ T ( t ) ρ − ξ || → || ρ ξ e t ∆ ρ − ξ || → = e t | ξ | ( ξ ∈ R d , t > . By Proposition 3.1 it follows that || ρ ξ T ( t ) ρ − ξ || → e t | ξ | − E t ( ξ ∈ R d , t > . (3.1)Let t >
0, and let k t be the convolution kernel of e t ∆ . Then for ξ ∈ R d thekernel of e − t | ξ | ρ ξ e t ∆ ρ − ξ is given by e − t | ξ | + ξ · ( x − y ) k t ( x − y ) = k t ( x − tξ − y ) ( x, y ∈ R d )since − t | ξ | + ξ · ( x − y ) − | x − y | t = − | x − y − tξ | t . (The above identity is the keypoint in the proof; this is why we need unbounded weights in Proposition 3.1.)Therefore, e − t | ξ | || ρ ξ T ( t ) ρ − ξ || →∞ || e t ∆ || →∞ = || e t ∆ || / →∞ = (8 πt ) − d/ . By duality we also have e − t | ξ | || ρ ξ T ( t ) ρ − ξ || → (8 πt ) − d/ . Using the semigroupproperty and (3.1), we conclude for ε ∈ (0 ,
1] that || ρ ξ T ( t ) ρ − ξ || →∞ || ρ ξ T ( ε t ) ρ − ξ || →∞ || ρ ξ T ((1 − ε ) t ) ρ − ξ || → || ρ ξ T ( ε t ) ρ − ξ || → e t | ξ | (8 π ε t ) − d/ e − E (1 − ε ) t (8 π ε t ) − d/ = (4 πεt ) − d/ e t | ξ | − E (1 − ε ) t . Now the first assertion follows from Lemma 3.3, and this gives the second assertionby the resolvent formula. 9 roof of Theorem 1.5.
The existence of the kernel p t follows from the Dunford-Pettis theorem, and Theorem 3.4 implies p t ( x, y ) (4 πεt ) − d/ e εE t exp (cid:18) − E t − | x − y | t (cid:19) for all t >
0. Then for t > d E the assertion follows by setting ε := d E t .We conclude this section with an off-diagonal L -estimate for the free resolvent. Let
A, B ⊆ R d be measurable, and let d ( A, B ) denote the dis-tance between A and B . Then || A ( µ − ∆) − B || → (1 − θ ) − d/ µ exp (cid:0) − θ √ µ d ( A, B ) (cid:1) for all µ > , < θ < .Proof. Let r := d ( A, B ). By duality we have to show || B ( µ − ∆) − A || ∞→∞ (1 − θ ) − d/ µ exp (cid:0) − θr √ µ (cid:1) =: C, or equivalently, ( µ − ∆) − A C on B . Let x ∈ B . By the resolvent formula weobtain( µ − ∆) − A ( x ) = Z ∞ e − µt Z R d k t ( y ) A ( x − y ) dy dt Z ∞ e − µt Z | y | > r k t ( y ) dy dt, where k t ( y ) = (4 πt ) − d/ exp (cid:0) − | y | t (cid:1) . We substitute y = (4 t ) / z and note that | y | > r if and only if t > (cid:0) r | z | (cid:1) ; then by Fubini’s theorem we infer that( µ − ∆) − A ( x ) π − d/ Z R d Z ∞ ( r | z | ) e − µt dt e −| z | dz = π − d/ µ Z R d exp (cid:18) − µr | z | − | z | (cid:19) dz. Note that θr √ µ µr | z | + θ | z | and hence exp (cid:0) − µr | z | − | z | (cid:1) e − θr √ µ e − (1 − θ ) | z | for all z ∈ R d . We conclude that( µ − ∆) − A ( x ) µ e − θr √ µ π − d/ Z R d e − (1 − θ ) | z | dz = µ e − θr √ µ (1 − θ ) − d/ , which proves the assertion. For µ > (cid:0) dr (cid:1) (where r = d ( A, B )), optimizing the estimate ofProposition 3.5 with respect to θ leads to the choice θ = (cid:0) − dr √ µ (cid:1) / . For µ > (cid:0) d r (cid:1) ,the choice θ = 1 − d r √ µ yields || A ( µ − ∆) − B || → (cid:0) ed + r √ µ (cid:1) d/ µ e − r √ µ . Proof of Theorem 1.1
Throughout this section we assume the setting of Section 2, i.e., Ω ⊆ R d is mea-surable, T a selfadjoint positivity preserving C -semigroup on L (Ω) dominatedby the free heat semigroup, with generator − H , and τ the closed symmetric formassociated with H . We denote E ( H ) := inf σ ( H ) . Recall that, for an open set U ⊆ R d , H U is the selfadjoint operator in L (Ω ∩ U )associated with the form τ restricted to D ( τ ) ∩ H ( U ).For A ⊆ R d we denote by U ε ( A ) = S x ∈ A B ( x, ε ) the ε -neighborhood of A . If A is measurable, then we write | A | for the Lebesgue measure of A . For r > E ( H ) < t < inf σ ess ( H ) we define the sets F r ( t ) := (cid:8) x ∈ R d ; E ( H B ( x,r ) ) < t (cid:9) ,G r ( t ) := R d \ U r ( F r ( t )) . (4.1)For the proof of Theorem 1.1 the following two facts will be crucial. On the onehand, the set F r ( t ) is “small” in the sense that the Lebesgue measure of U r ( F r ( t ))is not too large, as is expressed in the next lemma. On the other hand, the set G r ( t ) is “spectrally small” in the sense that the ground state energy of H G r ( t ) isnot much smaller than t ; see Lemma 4.2 below. Let r > and E ( H ) < t < inf σ ess ( H ) . Then | U s ( F r ( t )) | ω d (2 r + s ) d N t ( H ) ( s > , where ω d := | B (0 , | .Proof. Let M ⊆ F r ( t ) be a maximal subset with the property that the balls B ( x, r ), x ∈ M are pairwise disjoint. Then by the min-max principle and thedefinition of F r ( t ) one sees that M has at most N t ( H ) elements. Moreover, F r ( t ) ⊆ S x ∈ M B ( x, r ) by the maximality of M . Therefore, | U s ( F r ( t )) | X x ∈ M | B ( x, r + s ) | N t ( H ) · ω d (2 r + s ) d . Let E ,d denote the ground state energy of the Dirichlet Laplacianon B (0 , . Then E ,d ( d + 1)( d + 2) ( d + 1) , and E ( H G r ( t ) ) > t − E ,d /r (cid:0) r > , E ( H ) < t < inf σ ess ( H ) (cid:1) . roof. For ψ ∈ W , (cid:0) B (0 , (cid:1) defined by ψ ( x ) = 1 − | x | one easily computes ||∇ ψ || / || ψ || = ( d + 1)( d + 2), thus proving the first assertion. Let now ψ de-note the normalized ground state of the Dirichlet Laplacian on B (0 , r > ψ r := r − d/ ψ ( · r ); note that || ψ r || = 1 and ψ r ∈ W ∞ ( R d ).To prove the second assertion, we need to show that τ ( u ) > (cid:0) t − E ,d /r (cid:1) || u || (4.2)for all u ∈ D ( τ ) ∩ H ( G r ( t )), without loss of generality u real-valued. We will use (cid:0) ψ r ( · − x ) (cid:1) x ∈ R d as a continuous partition of the identity. By Lemma 2.2 we have ψ r ( · − x ) u ∈ D ( τ ) for all x ∈ R d . Using (2.1) and Lemma 2.3 we obtain τ (cid:0) ψ r ( · − x ) u (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12) ψ r ( · − x ) ∇ u + u ∇ ψ r ( · − x ) (cid:12)(cid:12)(cid:12)(cid:12) + σ (cid:0) ψ r ( · − x ) u (cid:1) = Z (cid:16) ∇ (cid:0) ψ r ( · − x ) u (cid:1) · ∇ u + u |∇ ψ r ( · − x ) | (cid:17) + σ (cid:0) ψ r ( · − x ) u, u (cid:1) = τ (cid:0) ψ r ( · − x ) u, u (cid:1) + Z u |∇ ψ r ( · − x ) | . Note that R ψ r ( y − x ) dx = || ψ r || = 1 and Z |∇ ψ r ( y − x ) | dx = ||∇ ψ r || = ||∇ ψ || /r = E ,d /r for all y ∈ R d . Taking into account Lemma 2.2 (with ξ = ψ r ) we thus obtain R τ (cid:0) ψ r ( · − x ) u, u (cid:1) dx = τ ( u, u ) and hence Z τ (cid:0) ψ r ( · − x ) u (cid:1) dx = τ ( u ) + || u || · E ,d /r . To conclude the proof of (4.2), we show that the left hand side of this identity isgreater or equal t || u || : note that ψ r ( · − x ) u ∈ H ( B ( x, r )). For x ∈ R d \ F r ( t ) wehave τ (cid:0) ψ r ( · − x ) u (cid:1) > t || ψ r ( · − x ) u || by the definition of F r ( t ); for x ∈ F r ( t ) wehave ψ r ( · − x ) u = 0 since u ∈ H ( G r ( t )). Therefore, Z τ (cid:0) ψ r ( · − x ) u (cid:1) dx > t Z || ψ r ( · − x ) u || dx = t || u || . It is known that E ,d behaves like d for large d . For d = 3,however, the estimate E ,d ( d + 1)( d + 2) = 10 from Lemma 4.2 is quite sharpsince E , = π > . There exists ρ ∈ C ( R d ) such that spt ρ ⊆ B (0 , , R ρ = 1 and ||∇ ρ || d + 1 , || ∆ ρ || d + 1) . (4.3)12 roof. Let ρ ∈ W ( R d ), ρ ( x ) := d ( d +2)2 σ d − (1 − | x | ) B (0 , ( x ), where σ d − denotesthe surface measure of the unit sphere ∂B (0 , Z ρ = 1 , ||∇ ρ || = d ( d + 2) d + 1 < d + 1and ∆ ρ = d ( d + 2) σ d − (cid:0) − d B (0 , + δ ∂B (0 , (cid:1) in the distributional sense, so ∆ ρ is a measure with || ∆ ρ || = 2 d ( d + 2) < d + 1) .Using a suitable mollifier and scaling, one obtains ρ as asserted. Proof of Theorem 1.1. (i) Let r > (cid:0) E ,d t − λ (cid:1) / , and let F r := F r ( t ), G r := G r ( t ) beas in (4.1). Then E ( H G r ) > λ by Lemma 4.2. We define ξ ∈ C ( R d ) satisfyingspt ξ ⊆ G r , spt( R d − ξ ) ⊆ U r ( F r )as follows: let ρ r := r − d ρ ( · r ), where ρ is as in Lemma 4.4. Then ξ := R d − ρ r/ ∗ U r/ ( F r ) = R d + ρ r/ ∗ ( R d − U r/ ( F r ) )has the above properties, and ||∇ ξ || ∞ ||∇ ρ r/ || = r ||∇ ρ || , || ∆ ξ || ∞ || ∆ ρ r/ || = r || ∆ ρ || . (4.4)By Lemma 2.4 we obtain ξϕ ∈ D ( H G r ) and f r := ( H G r − λ )( ξϕ ) = − ∇ ξ · ∇ ϕ − (∆ ξ ) ϕ, spt f r ⊆ spt ∇ ξ ⊆ U r ( F r ) . Then ξϕ = ( H G r − λ ) − f r = ( H G r − λ ) − U r ( F r ) f r . Since ξ = 1 on Ω \ U r ( F r ), wecan now estimate || ϕ || = || U r ( F r ) ϕ || + || Ω \ U r ( F r ) ξϕ || || U r ( F r ) ϕ || + || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → || f r || . (4.5)The remainder of the proof consists of estimating the terms in this pivotal inequal-ity. Lemma 4.1 implies || U r ( F r ) ϕ || | U r ( F r ) | / || ϕ || (cid:0) ω d (5 r ) d N t ( H ) (cid:1) / || ϕ || (4.6)and || f r || | U r ( F r ) | / || f r || (cid:0) ω d (4 r ) d N t ( H ) (cid:1) / || f r || , (4.7) || f r || ||∇ ξ || ∞ ||∇ ϕ || + || ∆ ξ || ∞ || ϕ || r ||∇ ρ || √ λ || ϕ || + 2 r || ∆ ρ || || ϕ || , (4.8)where in (4.8) we used (4.4) and ||∇ ϕ || = λ || ϕ || .13ii) Next we estimate || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → . Let δ, θ ∈ (0 ,
1) and ε := δ t − λt , µ := (1 − ε ) E ( H G r ) − λ. Then ( H G r − λ ) − ε − d/ ( µ − ∆) − by Theorem 3.4, and hence Proposition 3.5implies || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → ε − d/ (1 − θ ) − d/ µ e − θr √ µ . (4.9)By Lemma 4.2 and the definition of ε we have µ > (1 − ε )( t − E ,d /r ) − λ > t − εt − E ,d /r − λ = (1 − δ )( t − λ ) − E ,d /r . We now choose r such that r = c + E ,d (1 − δ )( t − λ ) , with c > d + 1 to be determined later.Then r / − δ )( t − λ ) c (4.10)since E ,d ( d + 1) c by Lemma 4.2, and µr > (1 − δ )( t − λ ) r − E ,d = c . By (4.9) we thus obtain || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → ε − d/ (1 − θ ) − d/ r c e − θc . (4.11)(iii) In this step we incorporate an estimate for || f r || into (4.11). By (4.3) wehave ||∇ ρ || c and || ∆ ρ || c . Thus, using (4.8), (4.10) and λ < t we obtain r || f r || ||∇ ρ || r √ λ || ϕ || + || ∆ ρ || || ϕ || c r / − δ · r λt − λ || ϕ || + 2 c || ϕ || c C δ r tt − λ || ϕ || , with C δ = p / (1 − δ ) + 2. Recalling ε = δ t − λt , we infer by (4.11) that || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → || f r || ε − d/ (1 − θ ) − d/ e − θc r c || f r || δ − d/ (cid:0) tt − λ (cid:1) ( d +1) / (1 − θ ) − d/ e − θc · C δ || ϕ || . (4.12)Now we set K δ,θ := δ (1 − θ ) and choose c := d + 12 θ ln (cid:18) K δ,θ · tt − λ (cid:19) . δ − d/ (cid:0) tt − λ (cid:1) ( d +1) / (1 − θ ) − d/ e − θc = (cid:0) (cid:1) d/ K / δ,θ , so by (4.7) and (4.12) we obtain || Ω \ U r ( F r ) ( H G r − λ ) − U r ( F r ) || → || f r || (cid:0) ω d (5 r ) d N t ( H ) (cid:1) / · K / δ,θ · C δ || ϕ || . (4.13)(iv) We set θ := and δ := , so that K δ,θ = and hence c = ( d + 1) ln tt − λ > d + 1 (4.14)as required above. Moreover, one easily verifies that K / δ,θ · C δ . By (4.5),(4.6) and (4.13) we conclude that || ϕ || (cid:16) (cid:0) ω d (5 r ) d N t ( H ) (cid:1) / || ϕ || (cid:17) = (cid:0) (cid:1) ω d (5 r ) d N t ( H ) || ϕ || . (4.15)Stirling’s formula yields ω d = π d/ Γ( d + 1) π d/ p πd/ (cid:0) d e (cid:1) d/ = ( πd ) − / (2 πe ) d/ d − d/ , so by (4.10) we obtain ω d (5 r ) d ( πd ) − / (cid:0) πe · / − δ (cid:1) d/ d − d/ · d c d ( t − λ ) − d/ . Using 2 πe · / − δ , ( d + 1) d d d +1 / and (4.14) we finally derive ω d (5 r ) d π − / (7 · d · d d/ (cid:0) ln tt − λ (cid:1) d ( t − λ ) − d/ . Together with (4.15) this proves the assertion since (cid:0) (cid:1) π − / · References [ArWa03]
W. Arendt and M. Warma , Dirichlet and Neumann boundary con-ditions: What is in between?
J. Evol. Equ. (2003), no. 1, 119–135.[BHV13] M. van den Berg, R. Hempel and J. Voigt , L -estimates foreigenfunctions of the Dirichlet Laplacian, to appear in J. Spectr. The-ory .[BeDa89]
M. van den Berg and E. B. Davies , Heat flow out of regions in R m , Math. Z. (1989), no. 4, 463–482.15CoSi08]
T. Coulhon and A. Sikora , Gaussian heat kernel upper boundsvia the Phragm´en-Lindel¨of theorem,
Proc. Lond. Math. Soc. (2008),no. 2, 507–544.[Dav95] E. B. Davies , Uniformly elliptic operators with measurable coeffi-cients,
J. Funct. Anal. (1995), no. 1, 141–169.[KLVW13]
M. Keller, D. Lenz, H. Vogt and R. Wojciechowski , Note onbasic features of large time behaviour of heat kernels, to appear in
J.Reine angew. Math. , DOI: .[LiYa83]
P. Li and S. T. Yau , On the Schr¨odinger equation and the eigenvalueproblem,
Comm. Math. Phys. (1983), no. 3, 309–318.[MVV05] A. Manavi, H. Vogt and J. Voigt , Domination of semigroups as-sociated with sectorial forms,
J. Operator Theory (2005), no. 1,9–25.[Ouh06] E. M. Ouhabaz , Comportement des noyaux de la chaleur desop´erateurs de Schr¨odinger et applications `a certaines ´equationsparaboliques semi-lin´eaires,
J. Funct. Anal. (2006), no. 1, 278–297.[OuWa07]
E. M. Ouhabaz and F.-Y. Wang , Sharp estimates for intrinsicultracontractivity on C ,α -domains. Manuscripta Math.122