Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

Let $(M, {g})$ be a compact, $d$-dimensional Riemannian manifold without boundary. Suppose further that $(M,g)$ is either two dimensional and has no conjugate points or $(M,g)$ has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by B\'erard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.

where |g| is the determinant of the metric g, g ik are entries of the inverse metric. As usual we define the space of square integrable functions, L 2 (M ), as the completion of the space of smooth functions, C ∞ (M ), with respect to the norm induced by the inner product Here dµ denotes the Riemannian volume element of M , which is given in local coordinates by |g|dx 1 dx 2 . . . dx d . The Laplace operator is self-adjoint, non-negative and has compact resolvent. Therefore there is an orthonormal basis {ϕ i ∈ L 2 (M ) : i ∈ N 0 } consisting of eigenfunctions ∆ϕ i = λ 2 i ϕ i , ϕ i ∈ C ∞ (M ) which is ordered such that 0 = λ 0 ≤ λ 1 ≤ . . .. Let e λ (x, y) be defined as the finite sum λ i <λ ϕ i (x)ϕ i (y). The restriction of e λ to the diagonal is called the local counting function and we will denote it by N x (λ) = e λ (x, x). Integration of N x (λ) over M gives the counting function of the Laplace operator N (λ) = #{i | λ i < λ}.

The local Weyl law states that
where ω d is the volume of the unit ball in R d . This result was proved by Levitan in the case of closed Riemannian manifolds. His work was based on the study of the cosine transform of the spectral function of the Laplace operator. In 1968 Hörmander [9] generalised it to the case of pseudo-differential operators of order m. The term O(λ d−1 ) can not be improved in general as the example of S d shows. Obtaining better estimates of this error term under additional geometric assumptions is still an active area of research. Various improvements are known in the case of compact manifolds with negative sectional curvature or under assumptions on the nature of the dynamics of the geodesic flow. For example, it was shown by Duistermaat and Guillemin [3] that the assumption of the set of periodic trajectories in the cosphere bundle having Liouville measure zero implies that the O λ d−1 may be replaced by o λ d−1 . The most significant for the purposes of this paper is the result of Bérard [1], who in 1977 obtained a logarithmic improvement for manifolds with nonpositive sectional curvature: . It is well known that regularized versions of the counting function have better asymptotic expansions. An example of a regularized counting function is the k-th Riesz means It shown by Hörmander in [8] and [9] that the k-th Riesz means admits an asymptotic expansion with an error term of order O λ d−k−1 . Safarov showed in [13] that the assumption that the set of periodic trajectories in the cosphere bundle under the geodesic flow has Liouville measure zero implies that this error term may be replaced by o λ d−k−1 .
Another well known asymptotic expansion is that of the mollified counting function (see e.g. Duistermaat and Guillemin [3]). Namely, let ρ ∈ S(R),ρ ∈ C ∞ 0 (R),ρ(ξ) = 1 for all ξ in a neighbourhood of zero, then whenever the support ofρ is sufficiently small. Here the coefficients are local densities and are related directly to the local heat kernel coefficients of the Laplace operator. Weyl's asymptotic formula is equivalent to the fact that It is known (see for example [13]) that the existence of such a full asymptotic expansion of the mollified counting function (1), independent of the geometric context, is enough to conclude that for k < d we have (see also [5] for the relation of these coefficients to the heat kernel coefficients). For surfaces (d = 2) of constant negative curvature the Selberg trace formula can be used to prove a logarithmic improvement of this formula (see [6]): A direct combination of Bérard's asymptotic formula, the expansion (3), and the method described in [13] yields in case k < d: In the case k = 1 and d = 2 this does however not reduce to Hejhal's estimate (3) but is weaker by a factor of log λ. The purpose of this paper is to improve the above error estimate to cover (3) and thus to generalize this estimate to the case of possibly nonconstant curvature, higher dimension, and higher Riesz means. Our main result is the following.
where the densities a i (x) are the same as in (1) and may be calculated explicitly. For k ≥ d one has Of course integration with respect to x over M yields corresponding estimates for the counting function.
Apart from the theoretical significance estimates of the form given in Theorem 0.1 have practical applications in numerical computations of eigenvalues. Some algorithms, such as the method of particular solutions ( [4,2], or [14] on manifolds) produce a list of eigenvalues of the Laplace operator and one would then like to have a method to check whether or not eigenvalues are missing in this list. Whereas the error estimate in Weyl's law is too large to detect a missing eigenvalue, the error estimates in some of the higher Riesz means will be sensitive to a change in the counting function by a positive integer. Averaged versions of the Weyl law are being used in numerical computations of large sets of eigenvalues, for example of Maass eigenvalues (see e.g. [11] where such Weyl laws are derived from the Selberg trace formula for certain congruence groups and subsequently being used in this context). This method is sometimes referred to as Turing's method.

Acknowledgements
We would like to thank Andreas Strömbergsson and Andrew Booker for pointing out reference [6] to us. We are also grateful to Yuri Safarov for comments on earlier versions of this paper and for pointing out simplifications of the argument.

Proof and main estimates
Throughout the text ρ ∈ S(R) will be be a real valued Schwartz function such that Hereρ denotes the Fourier transform of ρ defined bŷ where ǫ > 0 is a small parameter, which is smaller than the injectivity radius at x. Under the stated assumptions the first term has a full asymptotic expansion i.e for µ → ∞ as it was proved in [3] and [10]. Because N x is supported on the positive semi-axes and ρ ǫ is the Schwartz function N x * ρ ǫ (µ) is rapidly decreasing as µ → −∞.
The k-th Riesz means of the local counting function is then given by is the analytic continuation of r α + /Γ(α + 1) in the parameter α, as described for example in [7]. Let us apply the Riesz means operator to (4): Convolution with the distribution χ k + may be understood as a repeated integral from −∞ to λ, therefore Together with the full asymptotic expansion (5) we obtain for any ε > 0 as λ → ∞.
Our main result is derived from the following estimates of the various terms appearing in (4) and (7). For convenience we use the following notation λ = (1 + |λ|).
We postpone the proof of these two propositions for the moment and show how they imply the result. The splitting (4), the expansion (5), Proposition 1.1 in the case k = 0 together with Proposition 1.2 immediately imply that there exists a c > 0 such that for T ≥ 1: Now we use a proposition derived by Safarov in [13], which we slightly adapt to our situation.
Proposition 1.3. Suppose that ν 1 , ν 2 ∈ R + . Then there exists a constant C > 0 depending only on ρ, ν 1 and ν 2 such that the following statement holds. Suppose N is a function of local bounded variation, N ∈ S ′ (R) and assume that: where a i are constants. Then, for all T ≥ 1: Proof. Our assumptions on ρ imply that ρ(t)dt = 1 and ρ(t)t k dt = 0 for all k ∈ N. Therefore, Using the triangle inequality and the fact that τ + λ ν ≤ τ |ν| λ ν one obtains This shows the proposition with C = |ρ(t)t| t max{ν 1 ,ν 2 } dt. Proposition 1.3 applied to the estimate (9) shows that for any integer k ≥ 0 there exists a c > 0 such that for T ≥ 1: When we substitute this estimate into (7), use the expansion (8), and Proposition 1.1 we get that This estimate is valid for T ≥ 1. If we take T = α log λ for some small α > 0 of obtain Theorem 0.1 for large λ.
The Laplace operator is non-negative and thus the local counting function is supported on the positive half line λ ≥ 0. Let us define We have (11) Let as usual p α,β be the Schwartz space semi-norms defined by p α,β (f ) = sup x |x α ∂ β x f |. Then it easy to check that for all T > 1 we have Thus, since N x is supported in the half line, N x * (χ k−1 + * (ρ T − ρ ǫ ))(λ) = O(e cT λ −∞ ) as λ → −∞ for any c > 0. Hence, N x * (χ k−1 + * (ρ T − ρ ǫ ))(λ) and N odd x * (χ k−1 + * (ρ T − ρ ǫ ))(λ) differ by a function of order O(e cT λ −∞ ) as λ → ∞ for any c > 0. We use the identity to express the function N odd as the restriction to the diagonal of the integral kernel of To estimate this integral we will as usual be exploiting the properties of a suitable parametrix for the operator e −it √ ∆ . In our case we will use the parametrix for cos(t √ ∆) that was constructed by Bérard in [1] and which we describe in the following. Let π : M → M be the universal cover of M and let Γ be its group of automorphisms, so that M ∼ =M /Γ. Denote byC(t, x, y) the integral kernel of cos(t ∆ ), where∆ is the Laplace operator on the non-compact complete manifoldM , theñ where d(x, y) denotes the distance between x and y onM . The regularizations of the distributions x α /Γ( α+1 2 ), x α /Γ(α), x −α are described for example in [7]. Moreover the functions u l and ∆ m y u l all have at most exponential growth as d(x, y) tends to infinity, i.e. for all l, m exists a c > 0 such that (12) |∆ m y u l (x, y)| ≤ c e c d(x,y) , Moreover, for N ≥ ⌊d/2⌋ + 3, the error term is continuous and bounded uniformly in x and y as follows (13) |ǫ N (t, x, y)| ≤ c N e c N |t| .
The distributional kernel of cos(t √ ∆) on M is given by: The integral kernel of cos(t ∆ ) has the finite propagation speed property, i.e. it is supported where d(x, y) ≤ t. This property implies that for fixed x and y the number of terms in (14) is finite for every t. In fact, the assumptions on the curvature for d > 2 and on the absence of conjugate points for d = 2 imply that the number of terms in (14) is O(e c|t| ) for some c > 0 which depends on the geometry of the manifold. The integral kernels of cos(t √ ∆), (ρ T − ρ ǫ )(t) and Re((it) −k−1 e itλ ) are even functions with respect to the independent variable t. Therefore we may restrict our integration to the positive semi-axes and in order to get a bound on (10) we need to estimate terms of the following form: Note that for γ = id and odd dimension d = 2m−1 the distribution t l+α /Γ(l + α + 1) α=−m is supported at the origin for l < m, therefore in this case the pairing above is 0, sincê ρ T −ρ ǫ is supported for |t| > ǫ/2 otherwise for γ = id we have the Fourier transform of a C ∞ 0 (−T, T ) function. In the following we use the notation O T (g(x)) for O(g(x)) in case the implied constant can be chosen independent of T , i.e. f (x, T ) = O T (g(x)) if |f (x, T )| ≤ C|g(x)| with C not dependent on T . The estimates will be based on the following two Lemata. Assume that η is an even real valued Schwartz function such thatη ∈ C ∞ 0 ([−1, 1]) is an even Schwartz function and letη T be the rescaled function η T (t) =η(t/T ). Lemma 1.4. Let ǫ > 0 and k ≥ 0. If k > 0 we suppose furthermore thatη − 1 vanishes of order k at zero. Then, Proof. One checks directly that for any integer m > 0 the L 1 -norm of d m dt m (η T (t)−η ǫ (t))t −k ) is bounded in T for T ≥ 1. The Lemma then follows by integration by parts. Lemma 1.5. For any fixed ǫ > 0, k ≥ 0, and m ∈ R there exists an L > 0 such that Proof. For m ≥ 0 the estimate follows for L = 2m + 1 − k immediately from the support properties ofη T and the fact thatη T is bounded. It remains to show the estimate in case m < 0. Therefore, assume m < 0. Let φ T ∈ C ∞ 0 (R + ) such that φ T = 0 in a neighborhood of zero, φ T = 1 on [R, T ] and φ T = 0 on [2T, ∞), such that φ (β) T ≤ C β uniformly in T and d for T ≥ 1 and T ≥ R > ǫ. Then one shows easily that the Schwartz semi-norms of φ TηT t −k (t + R) m satisfy p α,β φ TηT t −k (t + R) m ≤ C α,β T α for T ≥ 1 and T ≥ R > ǫ, where C α,β is independent of T and R. If ψ T is the Fourier transform of the function φ TηT t −k (t + R) m we therefore have On the other hand the Fourier transform of is the convolution of ψ T with the Fourier transform of can be bounded by a multiple of sup µ µ q |ψ T (µ)| for all q > (|m + 1| + 1). It follows that the convolution is of order O T,R (T 2 λ −m−1 ) and we may therefore choose L = 2.
Proof of Proposition 1.1: For each T ≥ 1 the number of non-zero terms in the sum (14) in finite, the number of terms grows at most exponentially fast with T . Moreover, the estimate (12) implies that there is a constant c l independent on x such that on the support of ρ T . The above Lemmata, applied with η(ξ) =ρ(ξ), together with these growth estimates show that for some c > 0 2 , for T ≥ 1, This implies the statement of the Proposition.

Proof of Proposition 1.2:
We would now like to estimate N x * (δ − ρ T ). This can be done as follows using a Fourier Tauberian theorem. Letρ ∈ S(R) be a non-negative even Schwartz function such that the Fourier transformρ is supported in the interval [−1/2, 1/2]. Letρ T be the rescaled function defined byρ(t) = Tρ(T t). Following Safarov ([12]) we defineρ 1,0 ∈ S(R) bỹ Defineρ T,0 byρ T,0 (t) = Tρ 1,0 (T t). Lemma 1.4 and Lemma 1.5 applied with η(ξ) =ρ 1,0 and k = 0, together with the above growth estimates then imply the bound for all T ≥ 1. The term c 1 λ d−1 appears here because of the contribution of the identity element in Γ. Under these conditions the Fourier Tauberian Theorem 1.3 in [12] states that |N x * (δ −ρ T )(λ)| ≤ C T N ′ x * ρ T,0 (λ), for all T ≥ 1 with a constant C > 0 that does not depend on T . Therefore, there exists c > 0 such that for all T ≥ 1: The support properties of the Fourier transforms of ρ T andρ T imply that ρ T * ρ T =ρ T . Hence, we have Using λ − µ α ≤ λ |α| µ α one derives the bound This implies the proposition.