Spectral properties of the logarithmic Laplacian
SSPECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN
ARI LAPTEV AND TOBIAS WETHA
BSTRACT . We obtain spectral inequalities and asymptotic formulae for thediscrete spectrum of the operator log( − ∆) in an open set Ω ∈ R d , d ≥ , of finite measure with Dirichlet boundary conditions. We also derive someresults regarding lower bounds for the eigenvalue λ (Ω) and compare them withpreviously known inequalities.
1. I
NTRODUCTION
In the present paper, we study spectral estimates for the logarithmic Laplacian L ∆ = log( − ∆) , which is a (weakly) singular integral operator with Fourier sym-bol | η | and arises as formal derivative ∂ s (cid:12)(cid:12)(cid:12) s =0 ( − ∆) s of fractional Laplaciansat s = 0 . The study of L ∆ has been initiated recently in [CW], where its rele-vance for the study of asymptotic spectral properties of the family of fractionalLaplacians in the limit s → + has been discussed. A further motivation for thestudy of L ∆ is given in [JSW], where it has been shown that this operator allowsto characterize the s -dependence of solution to fractional Poisson problems for thefull range of exponents s ∈ (0 , . The logarithmic Laplacian also arises in thegeometric context of the -fractional perimeter, which has been studied recently in[DNP].For matters of convenience, we state our results for the operator H = L ∆ whichcorresponds to the quadratic form ϕ (cid:55)→ ( ϕ, ϕ ) log := 1(2 π ) d (cid:90) R d log( | ξ | ) | (cid:98) ϕ ( ξ ) | dξ. (1.1)Here and in the following, we let (cid:98) ϕ denote the Fourier transform ξ (cid:55)→ (cid:98) ϕ ( ξ ) = (cid:90) R d e − ixξ ϕ ( x ) dx of a function ϕ ∈ L ( R d ) . Let Ω ⊂ R d be an open set of finite measure, and let H (Ω) denote the closure of C ∞ c (Ω) with respect to the norm ϕ (cid:55)→ (cid:107) ϕ (cid:107) ∗ := (cid:90) R d log( e + | ξ | ) | (cid:98) ϕ ( ξ ) | dξ. (1.2) Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate Lon-don SW7 2AZ, UK, [email protected] f¨ur Mathematik, Goethe-Universit¨at, Frankfurt, Robert-Mayer-Straße 10, D-60629 Frank-furt, Germany, [email protected]. a r X i v : . [ m a t h . SP ] S e p ARI LAPTEV AND TOBIAS WETH
Then ( · , · ) log defines a closed, symmetric and semibounded quadratic form withdomain H (Ω) ⊂ L (Ω) , see Section 2 below. Here and in the following, weidentify L (Ω) with the space of functions u ∈ L ( R d ) with u ≡ on R d \ Ω . Let H : D ( H ) ⊂ L (Ω) → L (Ω) be the unique self-adjoint operator associated with the quadratic form. The eigen-value problem for H then writes as (cid:40) H ϕ = λϕ, in Ω , ϕ = 0 , on R d \ Ω . (1.3)We understand (1.3) in weak sense, i.e. ϕ ∈ H (Ω) and ( ϕ, ψ ) log = λ (cid:90) Ω ϕ ( x ) ψ ( x ) dx for all ψ ∈ H (Ω) .As noted in [CW, Theorem 1.4], there exists a sequence of eigenvalues λ (Ω) < λ (Ω) ≤ . . . , lim k →∞ λ k (Ω) = ∞ and a corresponding complete orthonormal system of eigenfunctions. We note thatthe discreteness of the spectrum is a consequence of the fact that the embedding H (Ω) (cid:44) → L (Ω) is compact. In the case of bounded open sets, the compactnessof this embedding follows easily by Pego’s criterion [P]. In the case of unboundedopen sets of finite measure, the compactness can be deduced from [JW, Theorem1.2] and estimates for (cid:107) · (cid:107) ∗ , see Corollary 2.3 below.In Section 2, using the results from [CW] and [FKV], we discuss properties offunctions from D ( H ) . In particular, we show that e ixξ (cid:12)(cid:12) x ∈ Ω ∈ D ( H ) , ξ ∈ R d ,provided Ω is an open bounded sets with Lipschitz boundary.In Section 3 we obtain a sharp upper bound for the Riesz means and for the num-ber of eigenvalues N ( λ ) of the operator H below λ . Here we use technique de-veloped in papers [Bz1], [Bz2], [LY] and [L]. In [Lap] it was noticed that suchtechnique could be applied for a class of pseudo-differential operators with Dirich-let boundary conditions in domains of finite measure without any requirements onthe smoothness of the boundary.We discuss lower bounds for λ (Ω) in Section 4. In Theorem 4.1 we present anestimate that is valid for arbitrary open sets of finite measure. For sets with Lips-chitz boundaries, H.Chen and T.Weth [CW] have proved a Faber-Krahn inequalityfor the operator H that reduces the problem to the estimate of λ ( B ) , where B isa ball satisfying | B | = | Ω | , see Corollary 4.3. In Theorem 4.4 we find an estimatefor λ ( B d ) , where B d is the unit ball, that is better in lower dimensions than theone obtained in Theorem 4.1. We also compare our results with bounds resultingfrom previously known spectral inequalities obtained in [BK] and [B].In Section 5 we obtain asymptotic lower bounds using the coherent states trans-formation approach given in [G]. It allows us to derive, in Section 6, asymptoticsfor the Riesz means of eigenvalues in Theorem 6.1 and for N ( λ ) in Corollary 6.2. PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 3
Here Ω ⊂ R N is an arbitrary open set of finite measure without any additionalrestrictions on the boundary.Finally in Section 7 we obtain uniform bounds on the Riesz means of the eigen-values using the fact that for bounded open sets with Lipschitz boundaries we have e ixξ (cid:12)(cid:12) x ∈ Ω ∈ D ( H ) .2. P RELIMINARIES AND BASIC PROPERTIES OF EIGENVALUES
As before, let ( · , · ) log denote the quadratic form defined in (1.1), and let, for anopen set Ω ⊂ R d , H (Ω) denote the closure of C ∞ c (Ω) with respect to the norm (cid:107) · (cid:107) ∗ defined in (1.2). Lemma 2.1.
Let Ω ⊂ R d be an open set of finite measure. Then ( · , · ) log defines aclosed, symmetric and semibounded quadratic form with domain H (Ω) ⊂ L (Ω) .Proof. Obviously, the form ( · , · ) log is symmetric. For functions ϕ ∈ C ∞ c (Ω) , wehave (2 π ) d (cid:107) ϕ (cid:107) = (cid:107) (cid:98) ϕ (cid:107) ≤ (cid:107) ϕ (cid:107) ∗ . (2.1)Moreover, with c := log( e + 2) + sup t ≥ e + t )log t we have (cid:107) ϕ (cid:107) ∗ c ≤ (cid:107) (cid:98) ϕ (cid:107) + (cid:90) | ξ |≥ ln | ξ || (cid:98) ϕ ( ξ ) | dξ ≤ (2 π ) d (cid:0) (cid:107) ϕ (cid:107) + ( ϕ, ϕ ) log (cid:1) − (cid:90) | ξ |≤ ln | ξ || (cid:98) ϕ ( ξ ) | dξ ≤ (2 π ) d (cid:0) (cid:107) ϕ (cid:107) + ( ϕ, ϕ ) log (cid:1) + (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) (cid:107) (cid:98) ϕ (cid:107) ∞ (2.2)while (cid:107) (cid:98) ϕ (cid:107) ∞ ≤ (cid:107) ϕ (cid:107) ≤ | Ω | (cid:107) ϕ (cid:107) . (2.3)Consequently, ( ϕ, ϕ ) log ≥ (cid:107) ϕ (cid:107) ∗ (2 π ) d c − (cid:32) | Ω | (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) (2 π ) d (cid:33) (cid:107) ϕ (cid:107) (2.4) ≥ (cid:32) c − − | Ω | (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) (2 π ) d (cid:33) (cid:107) ϕ (cid:107) . In particular, ( ϕ, ϕ ) log is semibounded. Moreover, it follows from (2.4) and thecompleteness of ( H (Ω) , (cid:107) · (cid:107) ∗ ) that the form ( ϕ, ϕ ) log is closed on H (Ω) . (cid:3) Lemma 2.2.
Let Ω ⊂ R d be an open set of finite measure. Then ϕ (cid:55)→ (cid:107) ϕ (cid:107) ∗∗ := (cid:90) (cid:90) | x − y |≤ ( ϕ ( x ) − ϕ ( y )) | x − y | d dxdy (2.5) defines an equivalent norm to the norm (cid:107) · (cid:107) ∗ defined in (1.2) on C ∞ c (Ω) . ARI LAPTEV AND TOBIAS WETH
Proof.
Let ϕ ∈ C ∞ c (Ω) . By [FKV, Lemma 2.7], we have (cid:107) ϕ (cid:107) ≤ c (cid:107) ϕ (cid:107) ∗∗ with a constant c > independent of ϕ . (2.6)In particular, (cid:107) · (cid:107) ∗∗ defines a norm on C ∞ c (Ω) . Next we note that, by [CW, Theo-rem 1.1(ii) and Eq. (3.1)], ( ϕ, ϕ ) log = 12 (cid:90) R d [ L ∆ ϕ ( x )] ϕ ( x ) dx = κ d (cid:107) ϕ (cid:107) ∗∗ − (cid:90) R d [ j ∗ ϕ ] ϕ dx + ζ d (cid:107) ϕ (cid:107) with κ d := π − d Γ( d/ , ζ d := log 2 + 12 ( ψ ( d/ − γ ) (2.7)and j : R d \ { } → R , j ( z ) = 2 κ d R d \ B d ( z ) | z | − d . Here ψ := Γ (cid:48) Γ is the Digamma function and γ = − Γ (cid:48) (1) is the Euler-Mascheroniconstant. Consequently, we have (cid:12)(cid:12)(cid:12) ( ϕ, ϕ ) log − κ d (cid:107) ϕ (cid:107) ∗∗ (cid:12)(cid:12)(cid:12) ≤ (cid:107) j (cid:107) ∞ (cid:107) ϕ (cid:107) + ζ d (cid:107) ϕ (cid:107) ≤ (cid:16) (cid:107) j (cid:107) ∞ | Ω | + ζ d (cid:17) (cid:107) ϕ (cid:107) . (2.8)As a consequence of (2.1) and (2.8), we find that (cid:107) ϕ (cid:107) ∗∗ ≤ κ d (cid:104) ( ϕ, ϕ ) log + (cid:0) (cid:107) j (cid:107) ∞ | Ω | + ζ d (cid:1) (cid:107) ϕ (cid:107) (cid:105) ≤ π ) d κ d (cid:16) (cid:107) j (cid:107) ∞ | Ω | + ζ d (cid:17) (cid:107) ϕ (cid:107) ∗ . Moreover, by (2.2), (2.3), (2.6) and (2.8) we have (cid:107) ϕ (cid:107) ∗ c ≤ (2 π ) d (cid:0) (cid:107) ϕ (cid:107) + ( ϕ, ϕ ) log (cid:1) + (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) | Ω |(cid:107) ϕ (cid:107) ≤ (2 π ) d (cid:16) κ d (cid:107) ϕ (cid:107) ∗∗ + (cid:0) (cid:107) j (cid:107) ∞ | Ω | + ζ d (cid:1) (cid:107) ϕ (cid:107) (cid:17) + (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) | Ω |(cid:107) ϕ (cid:107) ≤ c (cid:107) ϕ (cid:107) ∗∗ with c = (2 π ) d κ d + c (cid:2) (2 π ) d (cid:0) (cid:107) j (cid:107) ∞ | Ω | + ζ d (cid:1) + (cid:13)(cid:13) ln | · | (cid:13)(cid:13) L ( B (0)) | Ω | (cid:3) . Hencethe norms (cid:107) · (cid:107) ∗ and (cid:107) · (cid:107) ∗∗ are equivalent on C ∞ c (Ω) . (cid:3) Corollary 2.3.
Let Ω ⊂ R d be an open set of finite measure. Then the embedding H (Ω) (cid:44) → L (Ω) is compact.Proof. Let ˜ H (Ω) be defined as the space of functions ϕ ∈ L ( R d ) with ϕ ≡ on R d \ Ω and (cid:90) (cid:90) | x − y |≤ ( ϕ ( x ) − ϕ ( y )) | x − y | d dxdy < ∞ . By [JW, Theorem 1.2], the Hilbert space ( ˜ H (Ω) , (cid:107) · (cid:107) ∗∗ ) is compactly embedded in L (Ω) . Since, by Lemma 2.2, the norms (cid:107) · (cid:107) ∗ and (cid:107) · (cid:107) ∗∗ are equivalent on C ∞ c (Ω) ,the space H (Ω) is embedded in ˜ H (Ω) . Hence the claim follows. (cid:3) PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 5
Corollary 2.4.
Let Ω ⊂ R d be a bounded open set with Lipschitz boundary. (i) The space H (Ω) is equivalently given as the set of functions ϕ ∈ L ( R d ) with ϕ ≡ on R d \ Ω and (cid:90) (cid:90) | x − y |≤ ( ϕ ( x ) − ϕ ( y )) | x − y | d dxdy < ∞ . (2.9)(ii) H (Ω) contains the characteristic function Ω of Ω and also the restrictionsof exponentials x (cid:55)→ Ω ( x ) e ixξ , ξ ∈ R d .Proof. (i) Let, as in the proof of Corollary 2.3, ˜ H (Ω) be the space of functions ϕ ∈ L ( R d ) with ϕ ≡ on R d \ Ω and with (2.9), endowed with the norm (cid:107) · (cid:107) ∗∗ .Since Ω ⊂ R d be a bounded open set with Lipschitz boundary, it follows from[CW, Theorem 3.1] that C ∞ (Ω) ⊂ ˜ H (Ω) is dense. Hence the claim follows fromLemma 2.2.(ii) follows from (i) and a straightforward computation. (cid:3) Next we note an observation regarding the scaling properties of the eigenvalues λ k (Ω) . Lemma 2.5.
Let Ω ⊂ R d be a bounded open set with Lipschitz boundary, and let R Ω := { Rx : x ∈ Ω } . Then we have λ k ( R Ω) = λ k (Ω) − log R for all k ∈ N .Proof. Since C ∞ (Ω) ⊂ H (Ω) is dense, it suffices to note that ( ϕ R , ϕ R ) log = ( ϕ, ϕ ) log − log R (cid:107) ϕ (cid:107) L ( R d ) for ϕ ∈ C ∞ c ( R d ) (2.10)with ϕ R ∈ C ∞ c ( R d ) defined by ϕ R ( x ) = R − d ϕ ( xR ) , whereas (cid:107) ϕ R (cid:107) L ( R d ) = (cid:107) ϕ (cid:107) L ( R d ) . Since (cid:99) ϕ R = R d (cid:98) ϕ ( R · ) we have ( ϕ R , ϕ R ) log = 1(2 π ) d (cid:90) R d log( | ξ | ) | (cid:99) ϕ R ( ξ ) | dξ = R d (2 π ) d (cid:90) R d log( | ξ | ) | (cid:98) ϕ ( Rξ ) | dξ = 1(2 π ) d (cid:90) R d (cid:0) log( | ξ | ) − log R (cid:1) | (cid:98) ϕ ( ξ ) | dξ = ( ϕ, ϕ ) log − log R (cid:107) ϕ (cid:107) L ( R d ) , as stated in (2.10). (cid:3) ARI LAPTEV AND TOBIAS WETH
3. A
N UPPER TRACE BOUND
Throughout this section, we let Ω ⊂ R d denote an open set of finite measure.Let { ϕ k } and { λ k } be the orthonormal in L (Ω) system of eigenfunctions and theeigenvalues of the operator H respectively. In what follows we denote ( λ − t ) + = (cid:40) λ − t, if t < λ, , if t ≥ λ. Then we have
Theorem 3.1.
For the eigenvalues of the problem (1.3) and any λ ∈ R we have (cid:88) k ( λ − λ k ) + ≤ π ) d | Ω | e dλ | B d | d − , (3.1) where | B d | is the measure of the unit ball in R d .Proof. Extending the eigenfunction ϕ k by zero outside Ω and using the Fouriertransform we find (cid:88) k ( λ − λ k ) + = (cid:88) k ( λ ( ϕ k , ϕ k ) − ( H ϕ k , ϕ k )) + = 1(2 π ) d (cid:32)(cid:88) k (cid:90) R d ( λ − log( | ξ | )) | (cid:99) ϕ k ( ξ ) | dξ (cid:33) + ≤ π ) d (cid:90) R d ( λ − log( | ξ | )) + (cid:88) k | (cid:99) ϕ k ( ξ ) | dξ. Using that { ϕ k } is an orthonormal basis in L (Ω) and denoting e ξ = e − i ( · ,ξ ) wehave (cid:88) k | (cid:99) ϕ k ( ξ ) | = (cid:88) k | ( e ξ , ϕ k ) | = (cid:107) e ξ (cid:107) L (Ω) = | Ω | , and finally obtain (cid:88) k ( λ − λ k ) + ≤ π ) d | Ω | (cid:90) R d ( λ − log( | ξ | )) + = 1(2 π ) d | Ω | e dλ (cid:90) | ξ |≤ log( | ξ | − ) dξ. We complete the proof by computing the last integral. (cid:3)
Let η > λ and let us consider the function ψ λ ( t ) = 1 η − λ ( η − t ) + . Denote by χ the step function χ λ ( t ) = (cid:40) , if t < λ, , if t ≥ λ, PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 7 and let N ( λ ) = { k : λ k < λ } , be the number of the eigenvalues below λ of the operator H .Then by using the previous statement we have N ( λ ) ≤ η − λ (cid:88) k ( η − λ k ) + ≤ η − λ π ) d | Ω | e dη | B d | d − . Minimising the right hand side w.r.t. η we find η = λ + d and thus obtain thefollowing Corollary 3.2.
For the number N ( λ ) of the eigenvalues of the operator H below λ we have N ( λ ) ≤ e λd +1 π ) d | Ω | | B d | . (3.2)4. A LOWER BOUND FOR λ (Ω) In this section, we focus on lower bounds for the first eigenvalue λ = λ (Ω) .From Corollary 3.2, we readily deduce the following bound. Theorem 4.1.
Let Ω ⊂ R d be an open set of finite measure. Then we have λ (Ω) ≥ d log (2 π ) d e | Ω | | B d | . (4.1) In particular, if | Ω | ≤ (2 π ) d e | B d | , then the operator H does not have negative eigenval-ues.Proof. If λ < d log (2 π ) d e | Ω | | B d | , then N ( λ ) < by (3.2), and therefore N ( λ ) = 0 .Consequently, H does not have eigenvalues below d log (2 π ) d e | Ω | | B d | . (cid:3) Remark 1.
Note that the inequalities (3.1) , (3.2) and (4.1) hold for any open set Ω of finite measure without any additional conditions on its boundary. In the following, we wish to improve the bound given in Theorem 4.1 in low di-mensions d for open boundary sets with Lipschitz boundary. We shall use thefollowing Faber-Krahn type inequality. Theorem 4.2. ( [CW, Corollary 1.6] )Let ρ > . Among all bounded open sets Ω with Lipschitz boundary and | Ω | = ρ ,the ball B = B r (0) with | B | = ρ minimizes λ (Ω) . Corollary 4.3.
For every open bounded sets Ω with Lipschitz boundary we have λ (Ω) ≥ λ ( B d ) + 1 d log | B d || Ω | , (4.2) and equality holds if Ω is a ball. ARI LAPTEV AND TOBIAS WETH
Proof.
The result follows by combining Theorem 4.2 with the identity λ ( B r (0)) = λ ( B d ) + log 1 r for r > ,which follows from the scaling property of λ noted in Lemma 2.5. (cid:3) Corollary 4.3 gives a sharp lower bound, but it contains the unknown quantity λ ( B d ) . By Theorem 4.1, we have λ ( B d ) ≥ d log (2 π ) d e | B d | = log(2 π ) − d (cid:0) | B d | (cid:1) = 2 d log Γ ( d/
2) + log 2 + 2 d log d − d . (4.3)The following theorem improves this lower bound in low dimensions d ≥ . Theorem 4.4.
For d ≥ , we have λ ( B d ) ≥ log (cid:0) √ d + 2 (cid:1) − d +1 | B d | ( d + 2) d d (2 π ) d . (4.4) Proof.
Let u ∈ L ( B d ) be radial with (cid:107) u (cid:107) L = 1 . Then (cid:98) u is also radial, and | (cid:98) u ( ξ ) | = | (cid:98) u ( s ) | = s − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) u ( r ) J d − ( rs ) r d dr (cid:12)(cid:12)(cid:12)(cid:12) ≤ s − d (cid:18)(cid:90) r d − u ( r ) dr (cid:19) / (cid:18)(cid:90) rJ d − ( sr ) dr (cid:19) / = s − d (cid:112) | S d − | (cid:18) s − (cid:90) s τ J d − ( τ ) dτ (cid:19) / = s − d (cid:112) | S d − | (cid:18)(cid:90) s τ J d − ( τ ) dτ (cid:19) / for ξ ∈ R d with s = | ξ | .Consequently, | S d − || (cid:98) u ( s ) | ≤ s − d (cid:90) s τ J d − ( τ ) dτ. PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 9
In the case where, in addition, u is a radial eigenfunction of (1.3) corresponding to λ in Ω = B d , it follows that, for every λ ∈ R , (2 π ) d [ λ − λ ] = (cid:90) R d ( λ − ln | ξ | ) | (cid:98) u ( ξ ) | dξ ≤ (cid:90) R d ( λ − ln | ξ | ) + | (cid:98) u ( ξ ) | dξ = | S d − | (cid:90) ∞ s d − ( λ − ln s ) + | (cid:98) u ( s ) | ds ≤ (cid:90) ∞ ( λ − ln s ) + s (cid:90) s τ J d − ( τ ) dτ ds = (cid:90) ∞ τ J d − ( τ ) (cid:90) ∞ τ ( λ − ln s ) + s dsdτ = (cid:90) e λ τ J d − ( τ ) (cid:90) e λ τ λ − ln ss dsdτ = (cid:90) e λ τ J d − ( τ ) (cid:90) λ ln τ ( λ − s ) dsdτ = (cid:90) e λ τ J d − ( τ ) (cid:90) λ − ln τ s dsdτ = 12 (cid:90) e λ τ J d − ( τ ) (cid:0) λ − ln τ (cid:1) dτ = e λ (cid:90) τ J d − ( e λ τ ) ln τ dτ. We now use the following estimate for Bessel functions of the first kind: J ν ( x ) ≤ x ν ν Γ( ν + 1) for ν > √ − , ≤ x < (cid:112) ν + 2) . (4.5)A proof of this elementary estimate is given in the Appendix. We wish to apply(4.5) with ν = d − . This gives e λ J d − ( r e λ τ ) ≤ e dλ τ d − d − Γ ( d ) = d | B d | e dλ (2 π ) d τ d − for τ ∈ [0 , if d ≥ and e λ ≤ √ d + 2 , i.e., if d ≥ and λ ≤ log (cid:0) √ d + 2 (cid:1) . (4.6)Here we used that | B d | = d π d Γ( d/ . Consequently, if (4.6) holds, we find that (2 π ) d [ λ − λ ] ≤ d | B d | e dλ (2 π ) d (cid:90) τ d − ln τ dτ, where (cid:90) τ d − ln τ dτ = − d (cid:90) τ d − ln τ dτ = 2 d (cid:90) τ d − dτ = 2 d . Hence (2 π ) d [ λ − λ ] ≤ | B d | d (2 π ) d e dλ , i.e., λ ≥ λ − | B d | d (2 π ) d e dλ . Inserting the value λ = log (cid:0) √ d + 2 (cid:1) from (4.6), we deduce that λ = λ ( B d ) ≥ log (cid:0) √ d + 2 (cid:1) − d +1 | B d | ( d + 2) d d (2 π ) d , as claimed. (cid:3) Remark 2.
It seems instructive to compare the lower bounds given in (4.3) and(4.4) with other bounds obtained from spectral estimates which are already avail-able in the literature. We first mention Beckner’s logarithmic estimate of uncer-tainty [B, Theorem 1], which implies that ( ϕ, ϕ ) log ≥ (cid:90) R d (cid:20) ψ ( d/
4) + log 2 | x | (cid:21) ϕ ( x ) dx ≥ [ ψ ( d/
4) + log 2] (cid:107) ϕ (cid:107) for functions ϕ ∈ C ∞ c ( B d ) and therefore λ ( B d ) ≥ ψ ( d/
4) + log 2 . (4.7)Here, as before, ψ = Γ (cid:48) Γ denotes the Digamma function. Next we state a furtherlower bound for ( ϕ, ϕ ) log which follows from [CW, Proposition 3.2 and Lemma4.11]. We have ( ϕ, ϕ ) log ≥ ζ d (cid:107) ϕ (cid:107) for ϕ ∈ C ∞ c ( B d ) , (4.8)where ζ d is given in (2.7), i.e., ζ d = log 2 + 12 ( ψ ( d/ − γ ) = − γ + d − (cid:88) k =1 k − , d odd, log 2 − γ + d − (cid:88) k =1 k , d even.Inequality (4.8) implies that λ ( B d ) ≥ ζ d . (4.9)The latter inequality can also be derived from a lower bound of Ba ˜n uelos andKulczycki for the first Dirichlet eigenvalue λ α ( B d ) of the fractional Laplacian ( − ∆) α/ in B d . In [BK, Corollary 2.2], it is proved that λ α ( B d ) ≥ α Γ(1 + α )Γ( d + α )Γ( d ) for α ∈ (0 , .Combining this inequality with the characterization of λ ( B d ) given in [CW, The-orem 1.5], we deduce that λ ( B d ) = lim α → + λ α ( B d ) − α ≥ ddα (cid:12)(cid:12)(cid:12) α =0 α Γ(1 + α )Γ( d + α )Γ( d ) = ζ d , as stated in (4.9).We briefly comment on the quality of the lower bounds obtained here in low andhigh dimensions. In low dimensions d ≥ , (4.4) is better than the bounds (4.3),(4.7) and (4.9). In dimension d = 1 where the bound (4.4) is not available, thebound (4.3) yields the best value. The following table shows numerical valuesof the bounds b ( d ) , b ( d ) , b ( d ) resp. b ( d ) given by (4.3), (4.4), (4.7), (4.9),respectively. We note here that a different definition of Fourier transform is used in [B] and therefore theinequality looks slightly different
PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 11 d b ( d ) − ,
55 0 ,
19 0 ,
55 0 ,
79 0 ,
97 1 ,
12 1 ,
25 1 ,
36 1 ,
46 1 , b ( d ) / ,
28 1 ,
48 1 ,
59 1 ,
67 1 ,
73 1 ,
79 1 ,
84 1 ,
89 1 , b ( d ) − . − , − ,
39 0 ,
12 0 ,
47 0 ,
73 0 ,
94 1 ,
12 1 ,
27 1 , b ( d ) − ,
58 0 ,
12 0 ,
42 0 ,
62 0 ,
76 0 ,
87 0 ,
96 1 ,
03 1 ,
10 1 , To compare the bounds in high dimensions, we consider the asymptotics as d →∞ . Since log Γ( t ) t = log t − o ( t ) as t → ∞ , the bound (4.3) yields λ ( B d ) ≥ log d − o (1) as d → ∞ , (4.10)whereas (4.4) obviously gives λ ( B d ) ≥ log √ d + 2 + log 2 + o (1) as d → ∞ , (4.11)Moreover, from (4.7) and the fact that ψ ( t ) = log t + o (1) as t → ∞ , (4.12)we deduce that λ ( B d ) ≥ log d − log 2 + o (1) as d → ∞ , (4.13)Finally, (4.8) and (4.12) yield λ ( B d ) ≥ log √ d + log 2 − γ o (1) as d → ∞ . (4.14)So (4.13) provides the best asymptotic bound as d → ∞ .Numerical computations indicate that the bound (4.4) is better than the otherbounds for ≤ d ≤ , and (4.7) is the best among these bounds for d ≥ .5. A N ASYMPTOTIC LOWER TRACE BOUND
Throughout this section, we let Ω ⊂ R d denote an open set of finite measure. Inthis section we prove the following asymptotic lower bound. A similar statementwas obtained in [G] for the Dirichlet boundary problem for a fractional Laplacian. Theorem 5.1.
For the eigenvalues of the problem (1.3) and any λ ∈ R we have lim inf λ →∞ e − dλ (cid:88) k ( λ − λ k ) + ≥ π ) d | Ω | | B d | d − . (5.1) Proof.
Let us fix δ > and consider Ω δ = { x ∈ Ω : dist( x, R d \ Ω) > δ } . Since δ is arbitrary it suffices to show the lower bound (5.1), where Ω is replaced by Ω δ . Let g ∈ C ∞ ( R d ) be a real-valued even function, (cid:107) g (cid:107) L ( R d ) = 1 with supportin { x ∈ R d : | x | ≤ δ/ } . For ξ ∈ R d and x ∈ Ω δ we introduce the “coherentstate” e ξ,y ( x ) = e − iξx g ( x − y ) . Note that (cid:107) e ξ,y (cid:107) L ( R d ) = 1 . Using the properties of coherent states [LL, Theorem12.8] we obtain (cid:88) k ( λ − λ k ) + ≥ π ) d (cid:90) R d (cid:90) Ω δ ( e ξ,y , ( λ − H ) + e ξ,y ) L (Ω) dydξ. Since t (cid:55)→ ( λ − t ) + is convex then applying Jensen’s inequality to the spectralmeasure of H we obtain (cid:88) k ( λ − λ k ) + ≥ π ) d (cid:90) R d (cid:90) Ω δ (cid:0) λ − ( H e ξ,y , e ξ,y ) L (Ω) (cid:1) + dydξ. (5.2)Next we consider the quadratic form ( H e ξ,y , e ξ,y ) L (Ω) = 1(2 π ) d (cid:90) R d (cid:90) Ω (cid:90) Ω e i ( x − z )( η − ξ ) g ( x − y ) g ( z − y ) log( | η | ) dzdxdη = 1(2 π ) d (cid:90) R d (cid:90) Ω (cid:90) Ω e i ( x − z ) ρ g ( x − y ) g ( z − y ) log( | ξ − ρ | ) dzdxdρ = 1(2 π ) d (cid:90) R d (cid:90) Ω (cid:90) Ω e i ( x − z ) ρ g ( x − y ) g ( z − y ) (log | ξ | + log ( | ξ − ρ | / | ξ | )) dzdxdρ = log | ξ | + R ( y, ξ ) . Since g ∈ C ∞ ( R d ) we have for any M > R ( y, ξ ) =1(2 π ) d (cid:90) R d (cid:90) Ω (cid:90) Ω e i ( x − y ) ρ g ( x − y ) e i ( y − z ) ρ g ( z − y ) log ( | ξ − ρ | / | ξ | ) dzdxdρ = (cid:90) R d | (cid:98) g | log ( | ξ − ρ | / | ξ | ) dρ ≤ C M (cid:90) R d (1 + | ρ | ) − M log ( | ξ − ρ | / | ξ | ) dρ ≤ C | ξ | − . Therefore from (5.2) we find (cid:88) k ( λ − λ k ) + ≥ (2 π ) − d | Ω δ | (cid:90) R d ( λ − log | ξ | − C | ξ | − ) + dξ. (5.3)Let us redefine the spectral parameter λ = ln µ . Then introducing polar coordinateswe find (cid:90) R d ( λ − log | ξ | − C | ξ | − ) + dξ = (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) ∞ (cid:18) ln µr − Cr (cid:19) + r d − dr = µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) ∞ (cid:18) ln 1 r − Cµr (cid:19) + r d − dr (5.4)The expression in the latter integral is positive if − r ln r > Cµ − . The function − r ln r is concave. PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 13
Its maximum is achieved at r = 1 /e at the value /e . The equation − r ln r = Cµ − has two solutions r ( µ ) and r ( µ ) such that r ( µ ) → and r ( µ ) → as µ → ∞ Therefore (cid:90) ∞ (cid:18) ln 1 r − Cµr (cid:19) + r d − dr ≥ (cid:90) r ( µ ) r ( µ ) (cid:18) ln 1 r − Cµr (cid:19) r d − dr = − d r d ln r (cid:12)(cid:12)(cid:12) r ( µ ) r ( µ ) + Cµ ( d + 1) r d +1 (cid:12)(cid:12)(cid:12) r ( µ ) r ( µ ) + 1 d r d (cid:12)(cid:12)(cid:12) r ( µ ) r ( µ ) → d as µ → ∞ . (5.5)Putting together (5.3), (5.4) and (5.5) and using µ = e λ we obtain lim inf λ →∞ e − dλ (cid:88) k ( λ − λ k ) + ≥ π ) d | Ω δ | | B d | d − . Since δ > is arbitrary we complete the proof of Theorem 5.1. (cid:3)
6. W
EYL ASYMPTOTICS
Throughout this section, we let Ω ⊂ R d denote an open set of finite measure.Combining Theorems 3.1 and 5.1 we have Theorem 6.1.
The Riesz means of the eigenvalues of the Dirichlet boundary valueproblem (1.3) satisfy the following asymptotic formula lim λ →∞ e − dλ (cid:88) k ( λ − λ k ) + = 1(2 π ) d | Ω | | B d | d − . (6.1)As a corollary we can obtain asymptotics of the number of the eigenvalues of theoperator H . Corollary 6.2.
The number of the eigenvalues N ( λ ) of the Dirichlet boundaryvalue problem (1.3) below λ satisfies the following asymptotic formula lim λ →∞ e − dλ N ( λ ) = 1(2 π ) d | Ω | | B d | . (6.2) Proof.
In order to prove (6.2) we use two simple inequalities. If h > , then ( λ + h − λ k ) + − ( λ − λ k ) + h ≥ ( −∞ , λ ) ( λ k ) (6.3) and ( λ − λ k ) + − ( λ − h − λ k ) + h ≤ ( −∞ , λ ) ( λ k ) (6.4)The inequality (6.3) implies, together with Theorems 3.1 and 5.1, that lim sup λ →∞ e − dλ N ( λ ) ≤ lim sup λ →∞ e − dλ (cid:88) k ( λ + h − λ k ) + − ( λ − λ k ) + h ≤ h (cid:104) e dh lim sup λ →∞ e − d ( λ + h ) (cid:88) k ( λ + h − λ k ) + − lim inf λ →∞ e − dλ (cid:88) k ( λ − λ k ) + (cid:105) ≤ | Ω || B d | d (2 π ) d e dh − h for every h > and thus lim sup λ →∞ e − dλ N ( λ ) ≤ | Ω || B d | d (2 π ) d lim h → + e dh − h = | Ω | | B d | (2 π ) d . (6.5)Moreover, (6.3) implies, together with Theorems 3.1 and 5.1, that lim inf λ →∞ e − dλ N ( λ ) ≥ lim inf λ →∞ e − dλ (cid:88) k ( λ − λ k ) + − ( λ − h − λ k ) + h ≥ h (cid:104) e dh lim inf λ →∞ e − dλ (cid:88) k ( λ − λ k ) + − e − dh lim sup λ →∞ e − d ( λ − h ) (cid:88) k ( λ − h − λ k ) + (cid:105) ≥ | Ω || B d | d (2 π ) d − e − dh h for every h > and therefore lim inf λ →∞ e − dλ N ( λ ) ≥ | Ω || B d | d (2 π ) d lim h → + − e − dh h = | Ω | | B d | (2 π ) d . (6.6)The claim follows by combining (6.5) and (6.6). (cid:3)
7. A
N EXACT LOWER TRACE BOUND
In this section we prove the following exact lower bound in the case of boundedopen sets with Lipschitz boundary.
Theorem 7.1.
Let Ω ⊂ R d , N ≥ be an open bounded set with Lipschitz bound-ary, let τ ∈ (0 , , and let C Ω ,τ := 1 | Ω | (2 π ) d (cid:90) R d (1 + | ρ | ) τ log(1 + | ρ | ) | (cid:99) Ω ( ρ ) | dρ, (7.1) where Ω denotes the indicator function of Ω .For any λ ≥ C Ω ,τ , we have (cid:88) k ( λ − λ k ) + ≥ | Ω | | B d | (2 π ) d d (cid:104) e dλ − a τ C Ω ,τ e ( d − τ ) λ − b τ C ,τ e ( d − τ ) λ − ( dλ + 1) (cid:105) PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 15 with a τ := d ( d − τ ) − d − τ and b τ := 4 dτ . Remark 3.
In the definition of C Ω ,τ , we need τ < , otherwise the integral mightnot converge. In particular, if Ω = B d is the unit ball in R d , we have (cid:99) Ω ( ρ ) = (2 π ) d | ρ | − d J d ( | ρ | ) where J d ( r ) = O ( √ r ) as r → ∞ . Hence the integral defining C Ω ,τ converges if τ < . A similar conclusion arises for cubes or rectangles, where (cid:99) Ω ( ρ ) = f ( ρ ) · · · · · f d ( ρ d ) and f j ( s ) = O ( s ) as | s | → ∞ , j = 1 , . . . , d .On the other hand, if Ω ⊂ R d is an open bounded set with Lipschitz boundary, wehave C Ω ,τ < ∞ for τ ∈ (0 , . (7.2) Indeed, in this case, Ω has finite perimeter, i.e., Ω ∈ BV ( R d ) . Therefore, as notede.g. in [Lom, Theorem 2.14] , Ω also has finite fractional perimeter P τ (Ω) = (cid:90) Ω (cid:90) R d \ Ω | x − y | − d − τ dxdy = 12 (cid:90) (cid:90) R d (1 Ω ( x ) − Ω ( y )) | x − y | d + τ dxdy for every τ ∈ (0 , . Moreover, P τ (Ω) coincides, up to a constant, with the integral (cid:90) R d | ρ | τ | (cid:99) Ω ( ρ ) | dρ which therefore is also finite for every τ ∈ (0 , . Since moreover Ω and thereforealso (cid:99) Ω are functions in L ( R d ) and for every ε > there exists C ε > with (1 + | ρ | ) τ log(1 + | ρ | ) ≤ C ε (cid:0) | ρ | τ + ε (cid:1) for ρ ∈ R d ,it follows that (7.2) holds. In the proof of Theorem 7.1, we will use the following elementary estimate.
Lemma 7.2.
For r ≥ , s > and τ ∈ (0 , , we have log (cid:16) rs (cid:17) ≤ s log(1 + r ) if s ∈ (0 , (7.3) and log (cid:16) rs (cid:17) ≤ (1 + r ) τ s τ log(1 + r ) if s ≥ . (7.4) In particular, log (cid:16) rs (cid:17) ≤ max (cid:26) s , s τ (cid:27) (1 + r ) τ log(1 + r ) for r, s > . Remark 4.
The obvious bound log(1+ rs ) ≤ rs will not be enough for our purposes.We need an upper bound of the form g ( s ) h ( r ) where h grows less than linearly in r . Proof of Lemma 7.2.
Let first s ∈ (0 , . Since log (cid:16) rs (cid:17) (cid:12)(cid:12)(cid:12) r =0 = 0 = 1 s log(1 + r ) (cid:12)(cid:12)(cid:12) r =0 and, for every r > , ddr log (cid:16) rs (cid:17) = 1 s + r ≤ s + sr = ddr s log(1 + r ) , inequality (7.3) follows. To see (7.4), we fix s > , and we note that log (cid:16) rs (cid:17) (cid:12)(cid:12)(cid:12) r =0 = 0 = (1 + r ) τ s τ log(1 + r ) (cid:12)(cid:12)(cid:12) r =0 . Moreover, for < r ≤ s − , we have ddr (1 + r ) τ s τ log(1 + r ) = (1 + r ) τ − s τ (1 + τ log(1 + r )) ≥ (1 + r ) τ − s τ ≥ s ≥ s + r = ddr log (cid:16) rs (cid:17) , so the inequality holds for r ≤ s − . If, on the other hand, r ≥ s − , we haveobviously log (cid:16) rs (cid:17) ≤ log(1 + r ) ≤ (1 + r ) τ s τ log(1 + r ) . (cid:3) We may now complete the
Proof of Theorem 7.1.
For ξ ∈ R d , we define f ξ ∈ L ( R d ) by f ξ ( x ) = √ | Ω | Ω e − ixξ . Note that (cid:107) f ξ (cid:107) L ( R d ) = 1 for any ξ ∈ R d . We write (cid:88) k ( λ − λ k ) + = (cid:88) k ( λ − λ k ) + (cid:107) ϕ k (cid:107) L (Ω) = 1(2 π ) d (cid:88) k ( λ − λ k ) + (cid:107) (cid:99) ϕ k (cid:107) L ( R d ) = | Ω | (2 π ) d (cid:88) k ( λ − λ k ) + (cid:90) R d |(cid:104) f ξ , ϕ k (cid:105)| dξ = | Ω | (2 π ) d (cid:90) R d (cid:88) k ( λ − λ k ) + |(cid:104) f ξ , ϕ k (cid:105)| dξ. Since (cid:80) k |(cid:104) f ξ , ϕ k (cid:105)| = (cid:107) f ξ (cid:107) L ( R d ) = 1 for ξ ∈ R d , Jensen’s inequality gives (cid:88) k ( λ − λ k ) + ≥ | Ω | (2 π ) d (cid:90) R d (cid:16) λ (cid:88) k |(cid:104) f ξ , ϕ k (cid:105)| − (cid:88) k λ k |(cid:104) f ξ , ϕ k (cid:105)| (cid:17) + dξ = | Ω | (2 π ) d (cid:90) R d (cid:16) λ − (cid:88) k λ k |(cid:104) f ξ , ϕ k (cid:105)| (cid:17) + dξ = | Ω | (2 π ) d (cid:90) R d (cid:16) λ − ( H f ξ , f ξ ) (cid:17) + dξ. (7.5) PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 17
Here, since (cid:112) | Ω | (cid:98) f ξ ( η − ξ ) = (cid:90) Ω e − i ( η − ξ ) x e − ixξ dx = (cid:90) Ω e − iηx dx = (cid:99) Ω ( η ) for η, ξ ∈ R d , we have | Ω | (2 π ) d ( H f ξ , f ξ ) = | Ω | (cid:90) R d log | η || (cid:98) f ξ ( η ) | dη = | Ω | (cid:90) R d log | η − ξ || (cid:98) f ξ ( η − ξ ) | dη = (cid:90) R d log | η − ξ || (cid:99) Ω ( η ) | dη ≤ (cid:90) R d [log | ξ | + log(1 + | η | / | ξ | )] | (cid:99) Ω ( η ) | dη ≤ log | ξ | (cid:90) R d | (cid:99) Ω ( η ) | dη + max (cid:26) | ξ | , | ξ | τ (cid:27) (cid:90) R d (1 + | η | ) τ (log(1 + | η | ) | (cid:99) Ω ( η ) | dη = | Ω | (2 π ) d (cid:16) log | ξ | + max (cid:26) | ξ | , | ξ | τ (cid:27) C Ω ,τ (cid:17) for ξ ∈ R d , (7.6)where C Ω ,τ is defined in (7.1). Here we used Lemma 7.2. Combining (7.5) and(7.6), we get (cid:88) k ( λ − λ k ) + ≥ | Ω | (2 π ) d (cid:90) R d (cid:16) λ − log | ξ | − max (cid:26) | ξ | , | ξ | τ (cid:27) C Ω ,τ (cid:17) + dξ. (7.7)Let us redefine the spectral parameter λ = log µ again. Then we find (cid:90) R d (cid:16) λ − log | ξ | − max (cid:26) | ξ | , | ξ | τ (cid:27) C Ω ,τ (cid:17) + dξ = (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) ∞ (cid:18) log µr − max (cid:26) r , r τ (cid:27) C Ω ,τ (cid:19) + r d − dr = µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) ∞ (cid:18) − log r − max (cid:26) µ − τ r , r τ (cid:27) C Ω ,τ µ τ (cid:19) + r d − dr ≥ µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) ∞ µ (cid:18) − log r − r τ C Ω ,τ µ τ (cid:19) + r d − dr. (7.8)For the last inequality, we used the fact that µ − τ r ≤ r τ for r ≥ µ .Next we note that the function r (cid:55)→ f µ ( r ) = − log r − r τ C Ω ,τ µ τ satisfies f µ ( r ) < for r ≥ and lim r → + f µ ( r ) = −∞ . (7.9)Moreover, this function has two zeros r ( µ ) , r ( µ ) with < r ( µ ) < µ
Consequently, (cid:90) R d (cid:16) λ − log | ξ | − max (cid:8) | ξ | , | ξ | τ (cid:9) C Ω ,τ (cid:17) + dξ ≥ µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:90) r ( µ ) µ (cid:18) − log r − r τ C Ω ,τ µ τ (cid:19) + r d − dr = µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:104) − r d d log r + 1 d r d − C Ω ,τ µ τ ( d − τ ) r d − τ (cid:105) r ( µ ) µ = µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:16)(cid:104) − r ( µ ) d d log r ( µ ) + 1 d r ( µ ) d − C Ω ,τ µ τ ( d − τ ) r ( µ ) d − τ (cid:105) − (cid:104) µ − d d log µ + 1 d µ − d − C Ω ,τ µ τ ( d − τ ) µ τ − d (cid:105)(cid:17) , which implies that (cid:90) R d (cid:16) λ − log | ξ | − max (cid:8) | ξ | , | ξ | τ (cid:9) C Ω ,τ (cid:17) + dξ ≥ µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:16) d r ( µ ) d − C Ω ,τ µ τ ( d − τ ) r ( µ ) d − τ − µ − d d log µ − d µ − d (cid:17) = µ d (cid:12)(cid:12)(cid:12) S d − (cid:12)(cid:12)(cid:12) (cid:16) d e − d (cid:0) C Ω ,τµτ + τC ,τµ τ (cid:1) − C Ω ,τ µ τ ( d − τ ) e − ( d − τ ) (cid:0) C Ω ,τµτ + τC ,τµ τ (cid:1) − µ − d d log µ − d µ − d (cid:17) . Since e − d (cid:0) C Ω ,τµτ + τC ,τµ τ (cid:1) ≥ − d (cid:0) C Ω ,τ µ τ + 4 τ C ,τ µ τ (cid:1) and e − ( d − τ ) (cid:0) C Ω ,τµτ + τC ,τµ τ (cid:1) ≤ , we conclude that (cid:90) R d (cid:16) λ − log | ξ | − max (cid:8) | ξ | , | ξ | τ (cid:9) C Ω ,τ (cid:17) + dξ ≥ µ d (cid:12)(cid:12) S d − (cid:12)(cid:12) d (cid:16) − d (cid:0) C Ω ,τ µ τ + 4 τ C ,τ µ τ (cid:1) − C Ω ,τ µ τ ( d − τ ) − µ − d ( d log µ + 1) (cid:17) = | B d | d (cid:16) µ d − C Ω ,τ ( d − d − τ ) µ d − τ − dτ C ,τ µ d − τ − ( d log µ + 1) (cid:17) = | B d | d (cid:16) e dλ − d ( d − τ ) − d − τ C Ω ,τ e ( d − τ ) λ − dτ C ,τ e ( d − τ ) λ − ( dλ + 1) (cid:17) . Combining the last estimate with (7.7), we get the asserted lower bound. (cid:3)
8. A
PPENDIX : N
OTE ON A BOUND FOR B ESSEL FUNCTIONS
The following elementary bound might be known but seems hard to find in thisform.
Lemma 8.1.
For ν ≥ √ − and ≤ x ≤ (cid:112) ν + 2) we have | J ν ( x ) | ≤ x ν ν Γ( ν + 1) . Proof.
We use the representation J ν ( x ) = (cid:16) x (cid:17) ν ∞ (cid:88) m =0 ( − m m !Γ( m + ν + 1) (cid:16) x (cid:17) m . For ≤ x ≤ (cid:112) ν + 2) and m ≥ , we have (cid:16) x (cid:17) ≤ ( m + 1)( m + ν + 1) = ( m + 1)Γ( m + ν + 2)Γ( m + ν + 1) and therefore Γ( ν + 1)( m + 1)!Γ( m + ν + 2) (cid:16) x (cid:17) m +1) ≤ Γ( ν + 1) m !Γ( m + ν + 1) (cid:16) x (cid:17) m . (8.1)Consequently, J ν ( x ) = x ν ν Γ( ν + 1) (cid:104) ∞ (cid:88) m =1 ( − m Γ( ν + 1) m !Γ( m + ν + 1) (cid:16) x (cid:17) m (cid:105) ≤ x ν ν Γ( ν + 1) . From (8.1) we also deduce that J ν ( x ) ≥ x ν ν Γ( ν + 1) (cid:104) − Γ( ν + 1)Γ( ν + 2) (cid:16) x (cid:17) + Γ( ν + 1)2Γ( ν + 3) (cid:16) x (cid:17) − Γ( ν + 1)6Γ( ν + 4) (cid:16) x (cid:17) (cid:105) = x ν ν Γ( ν + 1) (cid:104) − ν + 1 f (cid:0)(cid:0) x (cid:1) (cid:1)(cid:105) with f : R → R given by f ( t ) = t − t ν +2) + t ν +2)( ν +3) . Since f (cid:48) ( t ) = 1 − tν + 2 + t ν + 2)( ν + 3) , and f (cid:48)(cid:48) ( t ) = 1 ν + 2 (cid:0) tν + 3 − (cid:1) we have f (cid:48) ( t ) ≥ f (cid:48) ( ν +3) = 1 − ν + 3 ν + 2 + ν + 32( ν + 2) = 1 − ν + 3 ν + 2 ≥ for t ∈ R if ν ≥ − and therefore f ( t ) ≤ f (2( ν + 2)) = 2( ν + 2) − [2( ν + 2)] ν + 2) + [2( ν + 2)] ν + 2)( ν + 3) = 4( ν + 2) ν + 3) for t ≤ ν + 2) if ν ≥ − . Since ν +2) ν +3) ≤ ν +1 for ν ≥ √ − , we concludethat J ν ( x ) ≥ x ν ν Γ( ν + 1) (cid:104) − ν + 1 f (cid:0)(cid:0) x (cid:1) (cid:1)(cid:105) ≥ − x ν ν Γ( ν + 1) . PECTRAL PROPERTIES OF THE LOGARITHMIC LAPLACIAN 21 for ν ≥ √ − and ≤ x ≤ (cid:112) ν + 2) . The claim thus follows. (cid:3) Acknowledgements . AL was supported by the RSF grant 19-71-30002.R
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