Spectral properties of the logarithmic Laplacian

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $\lambda_1(\Omega)$ and compare them with previously known inequalities.


INTRODUCTION
In the present paper, we study spectral estimates for the logarithmic Laplacian L ∆ = log(−∆), which is a (weakly) singular integral operator with Fourier symbol 2 log |η| and arises as formal derivative ∂ s s=0 (−∆) s of fractional Laplacians at s = 0. The study of L ∆ has been initiated recently in [CW], where its relevance for the study of asymptotic spectral properties of the family of fractional Laplacians in the limit s → 0 + has been discussed. A further motivation for the study of L ∆ is given in [JSW], where it has been shown that this operator allows to characterize the s-dependence of solution to fractional Poisson problems for the full range of exponents s ∈ (0, 1). The logarithmic Laplacian also arises in the geometric context of the 0-fractional perimeter, which has been studied recently in [DNP].
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 → ϕ 2 * := R d log(e + |ξ|) | ϕ(ξ)| 2 dξ. (1.2) Then (·, ·) log defines a closed, symmetric and semibounded quadratic form with domain H(Ω) ⊂ L 2 (Ω), see Section 2 below. Here and in the following, we identify L 2 (Ω) with the space of functions u ∈ L 2 (R d ) with u ≡ 0 on R d \ Ω. Let be the unique self-adjoint operator associated with the quadratic form. The eigenvalue problem for H then writes as in Ω, We understand (1.3) in weak sense, i.e.
As noted in [CW,Theorem 1.4], there exists a sequence of eigenvalues λ 1 (Ω) < λ 2 (Ω) ≤ . . . , lim k→∞ λ k (Ω) = ∞ and a corresponding complete orthonormal system of eigenfunctions. We note that the discreteness of the spectrum is a consequence of the fact that the embedding H(Ω) → L 2 (Ω) is compact. In the case of bounded open sets, the compactness of this embedding follows easily by Pego's criterion [P]. In the case of unbounded open sets of finite measure, the compactness can be deduced from [JW,Theorem 1.2] and estimates for · * , see Corollary 2.3 below.
In Section 2, using the results from [CW] and [FKV], we discuss properties of functions from D(H). In particular, we show that e ixξ 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 number of eigenvalues N (λ) of the operator H below λ. Here we use technique developed in papers [Bz1], [Bz2], [LY] and [L]. In [Lap] it was noticed that such technique could be applied for a class of pseudo-differential operators with Dirichlet boundary conditions in domains of finite measure without any requirements on the smoothness of the boundary.
We discuss lower bounds for λ 1 (Ω) in Section 4. In Theorem 4.1 we present an estimate that is valid for arbitrary open sets of finite measure. For sets with Lipschitz boundaries, H.Chen and T.Weth [CW] have proved a Faber-Krahn inequality for the operator H that reduces the problem to the estimate of λ 1 (B), where B is a ball satisfying |B| = |Ω|, see Corollary 4.3. In Theorem 4.4 we find an estimate for λ 1 (B d ), where B d is the unit ball, that is better in lower dimensions than the one obtained in Theorem 4.1. We also compare our results with bounds resulting from previously known spectral inequalities obtained in [BK] and [B].
In Section 5 we obtain asymptotic lower bounds using the coherent states transformation approach given in [G]. It allows us to derive, in Section 6, asymptotics for the Riesz means of eigenvalues in Theorem 6.1 and for N (λ) in Corollary 6.2.
Here Ω ⊂ R N is an arbitrary open set of finite measure without any additional restrictions on the boundary.
Finally in Section 7 we obtain uniform bounds on the Riesz means of the eigenvalues using the fact that for bounded open sets with Lipschitz boundaries we have e ixξ x∈Ω ∈ D(H).

PRELIMINARIES AND BASIC PROPERTIES OF EIGENVALUES
As before, let (·, ·) log denote the quadratic form defined in (1.1), and let, for an open set Ω ⊂ R d , H(Ω) denote the closure of C ∞ c (Ω) with respect to the norm · * defined in (1.2).
Lemma 2.2. Let Ω ⊂ R d be an open set of finite measure. Then defines an equivalent norm to the norm · * defined in (1.2) on C ∞ c (Ω).
(2.9) (ii) H(Ω) contains the characteristic function 1 Ω of Ω and also the restrictions Proof. (i) Let, as in the proof of Corollary 2.3,H(Ω) be the space of functions ϕ ∈ L 2 (R d ) with ϕ ≡ 0 on R d \ Ω and with (2.9), endowed with the norm · * * .
Since Ω ⊂ R d be a bounded open set with Lipschitz boundary, it follows from [CW,Theorem 3.1] that C ∞ 0 (Ω) ⊂H(Ω) is dense. Hence the claim follows from Lemma 2.2.
(ii) follows from (i) and a straightforward computation.
Next we note an observation regarding the scaling properties of the eigenvalues λ k (Ω). Then we have as stated in (2.10).

AN 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 2 (Ω) system of eigenfunctions and the eigenvalues of the operator H respectively. In what follows we denote

Then we have
Theorem 3.1. For the eigenvalues of the problem (1.3) and any λ ∈ R we have Proof. Extending the eigenfunction ϕ k by zero outside Ω and using the Fourier transform we find Using that {ϕ k } is an orthonormal basis in L 2 (Ω) and denoting e ξ = e −i(·,ξ) we have We complete the proof by computing the last integral.
Let η > λ and let us consider the function Denote by χ the step function be the number of the eigenvalues below λ of the operator H.
Then by using the previous statement we have Minimising the right hand side w.r.t. η we find η = λ + 1 d and thus obtain the following Corollary 3.2. For the number N (λ) of the eigenvalues of the operator H below λ we have In this section, we focus on lower bounds for the first eigenvalue λ 1 = λ 1 (Ω). From Corollary 3.2, we readily deduce the following bound.
Let Ω ⊂ R d be an open set of finite measure. Then we have In particular, if |Ω| ≤ (2π) d e |B d | , then the operator H does not have negative eigenvalues. (3.2), and therefore N (λ) = 0. Consequently, H does not have eigenvalues below 1 d log (2π) d e|Ω| |B d | . 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 dimensions d for open boundary sets with Lipschitz boundary. We shall use the following Faber-Krahn type inequality.

2)
and equality holds if Ω is a ball.
Proof. The result follows by combining Theorem 4.2 with the identity which follows from the scaling property of λ 1 noted in Lemma 2.5.
Corollary 4.3 gives a sharp lower bound, but it contains the unknown quantity λ 1 (B d ). By Theorem 4.1, we have The following theorem improves this lower bound in low dimensions d ≥ 2.
Proof. Let u ∈ L 2 (B d ) be radial with u L 2 = 1. Then u is also radial, and Consequently, In the case where, in addition, u is a radial eigenfunction of (1.3) corresponding to λ 1 in Ω = B d , it follows that, for every λ ∈ R, ln 2 τ dτ. We now use the following estimate for Bessel functions of the first kind: A proof of this elementary estimate is given in the Appendix. We wish to apply (4.5) with ν = d 2 − 1. This gives Inserting the value λ = log 2 √ d + 2 from (4.6), we deduce that 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 available in the literature. We first mention Beckner's logarithmic estimate of uncertainty [B,Theorem 1], which implies that 1 Here, as before, ψ = Γ Γ denotes the Digamma function. Next we state a further lower bound for (ϕ, ϕ) log which follows from [CW,Proposition 3.2 and Lemma 4.11]. We have where ζ d is given in (2.7), i.e., Inequality (4.8) implies that (4.9) The latter inequality can also be derived from a lower bound of Bañuelos and Kulczycki for the first Dirichlet eigenvalue λ α 1 (B d ) of the fractional Laplacian (−∆) α/2 in B d . In [BK,Corollary 2.2], it is proved that for α ∈ (0, 2).
Combining this inequality with the characterization of λ 1 (B d ) given in [CW, Theorem 1.5], we deduce that as stated in (4.9).
We briefly comment on the quality of the lower bounds obtained here in low and high dimensions. In low dimensions d ≥ 2, (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, the bound (4.3) yields the best value. The following table shows numerical values of the bounds b 1 (d), b 2 (d), b 3 (d) resp. b 4 (d) given by (4.3), (4.4), (4.7), (4.9), respectively. To compare the bounds in high dimensions, we consider the asymptotics as d → ∞. Since log Γ(t) t = log t − 1 + o(t) as t → ∞, the bound (4.3) yields whereas (4.4) obviously gives Moreover, from (4.7) and the fact that we deduce that Finally, (4.8) and (4.12) yield (4.14) So (4.13) provides the best asymptotic bound as d → ∞.
Numerical computations indicate that the bound (4.4) is better than the other bounds for 2 ≤ d ≤ 21, and (4.7) is the best among these bounds for d ≥ 22.

AN ASYMPTOTIC LOWER TRACE BOUND
Throughout this section, we let Ω ⊂ R d denote an open set of finite measure. In this section we prove the following asymptotic lower bound. A similar statement was 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 Proof. Let us fix δ > 0 and consider Since δ is arbitrary it suffices to show the lower bound (5.1), where Ω is replaced by Ω δ . Let g ∈ C ∞ 0 (R d ) be a real-valued even function, g L 2 (R d ) = 1 with support in {x ∈ R d : |x| ≤ δ/2}. For ξ ∈ R d and x ∈ Ω δ we introduce the "coherent state" e ξ,y (x) = e −iξx g(x − y).
Note that e ξ,y L 2 (R d ) = 1. Using the properties of coherent states [LL,Theorem 12.8] we obtain Since t → (λ − t) + is convex then applying Jensen's inequality to the spectral measure of H we obtain Next we consider the quadratic form Therefore from (5.2) we find Let us redefine the spectral parameter λ = ln µ. Then introducing polar coordinates we find The expression in the latter integral is positive if −r ln r > Cµ −1 . The function −r ln r is concave.

WEYL 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 value problem (1.3) satisfy the following asymptotic formula As a corollary we can obtain asymptotics of the number of the eigenvalues of the operator H.

AN EXACT LOWER TRACE BOUND
In this section we prove the following exact lower bound in the case of bounded open sets with Lipschitz boundary.
For any λ ≥ 2C Ω,τ , we have Remark 3. In the definition of C Ω,τ , we need τ < 1, otherwise the integral might not converge. In particular, if Ω = B d is the unit ball in R d , we have Hence the integral defining C Ω,τ converges if τ < 1. A similar conclusion arises for cubes or rectangles, where C Ω,τ < ∞ for τ ∈ (0, 1).
In the proof of Theorem 7.1, we will use the following elementary estimate.
Lemma 7.2. For r ≥ 0, s > 0 and τ ∈ (0, 1), we have and In particular, Remark 4. The obvious bound log(1+ r s ) ≤ r s 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, 1). Since and, for every r > 0, To see (7.4), we fix s > 1, and we note that Moreover, for 0 < r ≤ s − 1, we have d dr so the inequality holds for r ≤ s − 1. If, on the other hand, r ≥ s − 1, we have obviously We may now complete the Proof of Theorem 7.1. (7.5) Here, since where C Ω,τ is defined in (7.1). Here we used Lemma 7.2. Combining (7.5) and (7.6), we get Let us redefine the spectral parameter λ = log µ again. Then we find For the last inequality, we used the fact that 1 µ 1−τ r ≤ 1 r τ for r ≥ 1 µ .
Combining the last estimate with (7.7), we get the asserted lower bound.

APPENDIX: NOTE ON A BOUND FOR BESSEL FUNCTIONS
The following elementary bound might be known but seems hard to find in this form.