Spectral Estimates, Contractions and Hypercontractivity
Abstract
Sharp comparison theorems are derived for all eigenvalues of the weighted Laplacian, for various classes of weightedmanifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly logconcave and logconvex densities, extensions to exponential measures, unitballs of , onedimensional spaces and Riemannian submersions. Our main tool is a general Contraction Principle for “eigenvalues” on arbitrary metricmeasure spaces. Motivated by Caffarelli’s Contraction Theorem, we put forth several conjectures pertaining to the existence of contractions from the canonical sphere (and Gaussian space) to weightedmanifolds of appropriate topological type having (generalized) Ricci curvature positively bounded below; these conjectures are consistent with all known isoperimetric, heatkernel and Sobolevtype properties of such spaces, and would imply sharp conjectural spectral estimates. While we do not resolve these conjectures for the individual eigenvalues, we verify their Weyl asymptotic distribution in the compact and noncompact settings, obtain nonasymptotic estimates using the Cwikel–Lieb–Rozenblum inequality, and estimate the trace of the associated heatkernel assuming that the associated heat semigroup is hypercontractive. As a side note, an interesting trichotomy for the heatkernel is obtained.
1 Introduction
A weightedmanifold is a triplet , where is a complete smooth dimensional Riemannian manifold, endowed with a measure having smooth positive density with respect to the Riemannian volume measure . The manifold can be compact or noncompact, but for simplicity we assume it is without boundary. In addition, there is no restriction on the total mass of the measure . The associated weighted Laplacian is defined as:
so that the usual integration by parts formula is satisfied with respect to :
Here denotes the LeviCivita connection, denotes the usual LaplaceBeltrami operator, and we use . One immediately sees that is a symmetric and positive semidefinite linear operator on with dense domain , the space of compactly supported smooth functions on . In fact, it is wellknown (e.g. [8, Proposition 3.2.1]) that the completeness of ensures that is essentially selfadjoint on the latter domain, and so its graphclosure is its unique selfadjoint extension. We continue to denote the resulting positive semidefinite selfadjoint operator by , with corresponding domain . By the spectral theory of selfadjoint operators (see Subsection 3.2), the spectrum is a subset of . When the spectrum is discrete (such as for compact manifolds), it is composed of isolated eigenvalues of finite multiplicity which increase to infinity; we denote these by , and arrange them in nondecreasing order (repeated by multiplicity) . In the discrete case, when is connected and has finite mass, we always have . For the standard definition of when the spectrum is possibly nondiscrete (as the first eigenvalues until the bottom of the essential spectrum), we refer to Subsection 3.2. In this work, we would like to investigate the spectrum of various classes of weightedmanifolds.
Definition.
The weightedmanifold satisfies the CurvatureDimension condition , and , if:
(1.1) 
as symmetric 2tensors on . Here denotes the Ricci curvature tensor, and is called the generalized (infinitedimensional) Ricci tensor. In this work, we restrict the CurvatureDimension condition to connected manifolds.
The generalized Ricci tensor (1.1) was introduced by Lichnerowicz [56, 57], and extended to arbitrary generalized dimension by Bakry [5] (cf. Lott [61]). Note that in the constant density case , the generalized Ricci tensor boils down to the classical one. The CurvatureDimension condition was introduced by Bakry and Émery in equivalent form in [7] (in the more abstract framework of diffusion generators). Its name stems from the fact that the generalized Ricci tensor incorporates information on curvature and dimension from both the geometry of and the measure , and so may be thought of as a generalizedcurvature lower bound, and as a generalizeddimension upper bound. With the exception of this Introduction, we will always assume that whenever referring to the CurvatureDimension condition, so we omit the more general definition involving an arbitrary (cf. [69]). The condition has been an object of extensive study over the last two decades (see e.g. also [73, 51, 31, 32, 85, 11, 91, 70, 69, 47, 8, 45, 67] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [72], and the extension of the CurvatureDimension condition to the metricmeasure space setting by Lott–Sturm–Villani [79, 62].
1.1 Spectrum Comparison for Positively Curved WeightedManifolds
Let denote the dimensional Gaussian probability measure with covariance , namely , where is a normalization constant. When , we simply write for the standard dimensional Gaussian measure. It is well known [52, 8] that the onedimensional Gaussian space serves as a model comparison space for numerous functional inequalities (such as isoperimetric [9], logSobolev [7] and spectralgap [8]), for the class of connected weightedmanifolds satisfying with (“positively curved weightedmanifolds”). The starting point of this work was to explore the possibility that these classical comparison properties also extend to all higherorder eigenvalues of . Contrary to many functional inequalities, which remain invariant under tensorization, thus implying that the comparison space may be chosen to be onedimensional, the spectrum tensorization property naturally forces us to compare to the dimensional space having the same (topological) dimension. Note that positively curved weightedmanifolds always have discrete spectrum, since they necessarily satisfy a logSobolev inequality by the Bakry–Émery criterion [7], and the latter is known to imply (in our finitedimensional setting!) discreteness of spectrum (see e.g. [63, 88, 30]). In addition, if is positively curved then necessarily has finite totalmass [8, Theorem 3.2.7].
Question 1 (Spectral Comparison Question).
Given an dimensional connected weightedmanifold satisfying with , does it hold that:
At first, this question may seem extremely bold and at the same time classical and wellstudied. As for the latter impression, we are not aware of any previous instances of Question 1. The former impression perhaps stems from the extensive body of work in trying to just provide sharp lower and upper bounds on the first eigenvalue gap under various conditions (e.g. [42, 2, 4, 12, 3]), or various other conjectured lower bounds on the entire spectrum, such as Polya’s conjecture (see e.g. [28, 54, 48]).
Unfortunately, one cannot expect to have a positive answer to the above question in general, at least not for the first eigenvalues. The easiest counterexample is given by the canonical sphere, rescaled to have Ricci curvature equal to (times the metric), so that it satisfies ; its th eigenvalue (given by a linear function on the sphere’s canonical embedding in ) is equal to , whereas the corresponding eigenvalue for the dimensional Gaussian space is already equal to (see Subsection 2.2 for more details).
Nevertheless, we can show:
Theorem 1.1 (Spectral Comparison for Positively Curved ).
Question 1 has a positive answer for any Euclidean space satisfying with .
In view of the above counterexample and theorem, and for reasons which will become more apparent later on, it is plausible that some topological restrictions must be enforced to obtain a positive answer to Question 1. The simplest one is to assume that is diffeomorphic to Euclidean space. We tentatively formulate this as:
Conjecture (Spectral Comparison Conjecture for Positively Curved ).
Question 1 has a positive answer for any satisfying with .
See Section 6 for a more refined version of this conjecture. Clearly, the case boils down to Theorem 1.1, so the conjecture pertains to the range . We take this opportunity to also mention the work of Ledoux [50] (cf. Bakry and Bentaleb [6]), who showed how information on higher order iterated carréduchamp operators (so called operators) may be used to obtain higherorder eigenvalue estimates for the generator ; however, here we only assume the condition, which amounts to information on only (see [8] for more on Calculus).
1.2 Spectrum Comparison for Additional Spaces
Our method of proof of Theorem 1.1, described in the next subsection, is very general, and in particular also yields the following additional results:
Theorem 1.2.
Let denote a Euclidean weightedmanifold where is a probability measure satisfying . Then:
Theorem 1.3.
Let denote a probability measure on for . Let denote a second probability measure on , and assume that is convex and unconditional, meaning that . Then:
Theorem 1.4.
Assume , and let denote the unit ball of , rescaled to have volume 1; the uniform measure on is denoted by . Then:
Theorem 1.5.
Given a weightedmanifold with a probability measure, denote its density by , by its cumulative distribution function, and by its onesided flat isoperimetric profile. Let denote two such measures. Then:
Theorem 1.6.
Let denote two weightedmanifolds, and let denote a Riemannian submersion pushing forward onto up to a finite constant. Then:
In particular, this holds for , the corresponding Riemannian volume measures, if both manifolds are connected and the submersion’s fibers are minimal and compact.
In particular, this holds for any finitesheeted Riemannian covering map between two connected manifolds.
1.3 Contracting and Lipschitz Maps
Let denote a Borel map between two metricmeasure spaces. The map is said to pushforward the probability measure onto , denoted , if . To treat the case when may have different or infinite total mass, we will say that pushes forward onto up to a finite constant, if there exists so that pushes forward onto . The map is called Lipschitz () if:
The map is called a contraction if it is Lipschitz with constant .
All of our spectrum comparison theorems are consequences of the following:
Theorem 1.7 (Contraction Principle).
Let denote an Lipschitz map between two (complete) weightedmanifolds pushingforward onto up to a finite constant. Then:
In particular, if has discrete spectrum, then so does .
In fact, an analogous result holds for compact weightedmanifolds with either Dirichlet or Neumann boundary conditions, see Subsection 3.4. We note that even in the classical nonweighted setting, the contraction principle is easily seen to be completely false if we omit the assumption that pushes forward the first volume measure onto the second (up to a finite constant); moreover, in that case, even if is known to be biLipschitz, the resulting spectrum comparison would depend exponentially on the underlying dimension , which is often useless for applications.
While the derivation of Theorem 1.7 is straightforward, we have not encountered an application of contracting maps for spectrum comparison elsewhere. To see Theorem 1.7, it is easy to verify that on we have:
(1.2) 
where and denote the pushforward and pullback maps between and induced by . Theorem 1.7 subsequently follows by the minmax principle and a density argument. A slightly delicate point is that we do not assume injectivity of (which is useful for some of the applications above), and so the minmax argument should be carefully checked. To better appreciate the above stated comparison, the reader may wish to try and explicitly write out and compare the differential operators appearing in (1.2) using the changeofvariables formula relating , and . In Section 3, we develop an abstract argument for spectrum comparison in the general framework of metricmeasure spaces.
A few words are in order regarding previous approaches towards spectrum comparison between differential operators on Riemannian manifolds (and more generally, linear operators on Hilbert spaces). The closest general argument we have found in the literature is the socalled Kato’s inequality and its generalizations (see [41, 78, 15, 16, 14, 24, 22, 23] and the references therein), which under certain conditions permit comparing the trace of the associated heat semigroups, heatkernels, and even the heat semigroup and resolvent operators themselves in the sense of domination of positivity preserving operators. However, these results typically do not involve the individual eigenvalues (cf. [16, III.6]), and in the few cases that do, the conclusion is in the opposite direction to the one appearing in this work (in an attempt to obtain spectral lower bounds on the source manifold by mapping it onto a simpler one). We also mention two additional classical methods of obtaining estimates on the growth and number of negative eigenvalues of a Schrödinger operator – the Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities [60, 33, 58, 75] (see also [48] and the references therein), the latter of which we will in fact employ in this work as well (see Subsection 2.5).
Back to the Contraction Principle. A celebrated contraction property was discovered by L. Caffarelli in [27]:
Theorem (Caffarelli’s Contraction Theorem).
Let satisfy with . Then there exists a map pushing forward onto up to a finite constant which contracts Euclidean distance.
Together with the Contraction Principle, this immediately yields Theorem 1.1. Caffarelli proved the above result for the Brenier OptimalTransport map [84], which uniquely (up to a nullset deformation) minimizes the averaged transport distance among all maps pushing forward onto . Subsequently in [44], YoungHeon Kim and the author gave an alternative proof and extended Caffarelli’s theorem using a (seemingly) different map involving a naturally associated heatflow, which together with the Contraction Principle immediately yields Theorem 1.3. Similarly, the existence of contracting and Lipschitz maps due to Kolesnikov [46], Latała–Wojtaszczyk [49] and Bobkov–Houdré [20] yield Theorems 1.2, 1.4 and 1.5, respectively; details are provided in Section 4.
Contracting, and more generally, Lipschitz maps between metricmeasure spaces, constitute a very powerful tool for transferring isoperimetric, functional and concentration information from to . However, for these traditional applications, there are numerous other tools available, such as Calculus, other parabolic and elliptic methods, OptimalTransport, Localization, etc.. (see e.g. [8, 47, 45]). As shown in this work, contracting maps also yield sharp comparison estimates for the entire spectrum, going well beyond the capability of the above mentioned alternative methods  we believe this to be a noteworthy (albeit simple) observation.
Motivated by Caffarelli’s Contraction Theorem on one hand, and the wellknown comparison results between weightedmanifolds satisfying and the ( or equivalently dimensional) Gaussian measure () on the other, we tentatively put forth the following conjecture, which by the Contraction Principle, would imply Conjecture :
Conjecture (Contraction Conjecture for Positively Curved ).
For any satisfying with , there exists a map:
pushing forward onto up to a finite constant and contracting the corresponding metrics.
See Section 6 for a more refined version of this conjecture. Conjecture is consistent with the BakryLedoux isoperimetric comparison theorem [9] and the BakryÉmery logSobolev inequality [7] for weightedmanifolds. Note that we have restricted the above conjecture to manifolds diffeomorphic to , as the counterexample of the canonical sphere from Subsection 1.1 shows that one cannot hope for such a map unto a general weightedmanifold satisfying . Moreover, there are topological obstructions to the existence of such a map between and , at least if we assume in addition that is onetoone from the source onto the target manifold: indeed, Brouwer’s Invariance of Domain theorem [71] asserts that an injective, surjective and continuous map between two topological manifolds is in fact open, and hence the two manifolds must be homeomorphic.
For a further discussion and refinement of Conjectures and , we refer to Section 6.
1.4 Extensions to Positively Curved ConstantDensity Manifolds
It is of course very natural to attempt extending the previous conjectures to the class of weightedmanifolds satisfying for and finite generalized dimension . Contrary to the situation with the usual functional inequalities (isoperimetric, Sobolev, spectralgap, cf. [10, 69]), it is not so clear what would be the right (topologically dimensional) model space for comparing the entire spectrum. However, when , which corresponds to the classical case of a complete connected Riemannian manifold, endowed with its canonical Riemannian volume measure and having Ricci curvature bounded below by (times the metric), the natural model space is simply the canonical sphere with its metric rescaled to have . For similar topological reasons as in the previous subsection (see also the ensuing discussion), we restrict to the case when is diffeomorphic to a sphere.
Conjecture 3 (Spectral Comparison Conjecture For Positively Curved ).
For any satisfying with , we have:
Conjecture 3 is consistent with:

The Lichnerowicz spectralgap estimate [55].

The Bérard–Gallot estimate on the trace of the heatkernel [13]:
(1.3) 
It is immediate to show that it is compatible with Weyl’s asymptotic law – see Subsection 2.3.

We can actually show that it holds true up to a dimension independent multiplicative constant for – see Subsection 2.5.
Conjecture 3 would follow immediately from the Contraction Principle and the following previously unpublished conjecture of ours [65]:
Conjecture 4 (Contraction Conjecture for Positively Curved ).
For any satisfying with , there exists a map:
pushing forward onto up to a finite constant and contracting the corresponding metrics.
Note that a connected complete Riemannian manifold with , , is necessarily compact and has finite volume. The reader should note the apparent analogy between the latter conjecture and Caffarelli’s Contraction Theorem, in view of the definition of the generalized Ricci tensor (1.1). Conjecture 4 is consistent with:

The Bonnet–Meyers bound on the diameter of such manifolds [36]:

The sharp Sobolev inequality for spaces [8, Theorem 6.8.3].

The Gromov–Lévy isoperimetric inequality [38].

Conjecture 3 on the full spectrum, including all of the known consequences mentioned after its formulation above.
A positive answer to Conjecture 4 would thus yield a single reason to all of these classical facts (albeit only for manifolds which are diffeomorphic to a sphere). It would be very interesting to adapt and extend the OptimalTransport or HeatFlow approaches of Caffarelli [27] and Kim and the author [44] from the scalar setting (involving densities) to the above tensorial setting (involving metrics)  cf. [65].
As before, we have restricted Conjecture 4 to due to potential topological obstructions. Indeed, a map as in Conjecture 4 must be surjective, since is compact as a continuous image of a compact set, while its open complement satisfies , and hence must be empty. Consequently, if we assume in addition that is injective, Brouwer’s Invariance of Domain theorem would imply as before that is open, and hence must be homeomorphic to .
For simplicity, we have chosen not to explicitly formulate the most general possible conjectures in the above spirit. Let us only remark that if we do not insist on finding a topologically dimensional model source space which conjecturally contracts onto dimensional weightedmanifolds (), thereby giving up on obtaining asymptotically sharp eigenvalue estimates (per Weyl’s law) and on injectivity of the contracting map, then a reasonable choice for such a model source space, at least when is an integer, is the rescaled canonical sphere; this would still be consistent with all known generalizations of the above properties (see [8, 79, 69] and the references therein), and contrary to the counterexample of Subsection 1.1, is easily verified for . It is also possible to consider adding the case to the above setting (under suitable modifications, replacing with ), but we do not have a clear sense of how reasonable this might be.
1.5 Comparison on Average
While we were not able to resolve Conjectures nor 3, we would still like to mention some tools for controlling the eigenvalues in some averaged sense. In Subsection 2.3, we recall Weyl’s asymptotic law for the distribution of eigenvalues in the compact case, and develop its analog in the weighted noncompact setting. However, we would like to obtain some concrete nonasymptotic estimates as well.
In Subsection 2.5, we show that Conjecture 3 is satisfied up to a dimension independent multiplicative constant for exponentially large (in the dimension) eigenvalues, by making use of the classical Cwikel–Lieb–Rozenblum inequality together with the sharp Sobolev inequality on weighted manifolds. We did not manage to verify a similar conclusion for Conjecture , perhaps because the condition does not directly feel the dimension . We therefore proceed to obtain some average estimates for the eigenvalues.
When is a probability measure, a very natural function encapsulating the growth of the eigenvalues is given by the trace of the heat semigroup :
where denotes the heatkernel (with respect to ). It is an interesting question to establish conditions on which ensure that is traceclass, i.e. that for . In particular, upper bounds on yield lower bounds on the individual eigenvalues by the trivial estimate:
(1.4) 
However, it may very well happen that the spectrum is discrete (equivalently, that increase to infinity), and yet for all . Note that will inevitably depend on the dimension , e.g. because of Weyl’s law or because of the spectrum’s tensorization property  see Section 2 for concrete examples such as for the dimensional Gaussian space or sphere. This is in contrast to more traditional objects of study on weightedmanifolds (such as the spectralgap or logSobolev constant), which are invariant under tensorization, and thus often dimensionindependent.
In connection to the discussion regarding previously known estimates on the spectrum, we mention the following result of Bérard and Gallot [13, 14] (see also Besson [16, Appendix]). By employing the Gromov–Lévy isoperimetric inequality [38], these authors showed that for any connected with , , one has:
where denotes renormalized to be a probability measure. In particular, this yields (1.3), confirming Conjecture 3 in an averaged (yet strictly weaker) sense. By employing the Bakry–Ledoux isoperimetric inequality [9], it may also be possible to obtain a somewhat similar onaverage confirmation of Conjecture ; this is not immediate and will be explored elsewhere. Here, we are more interested in another direction.
Upper bounds on and moreover lower bounds on under various assumptions on were obtain by F.Y. Wang in [87, 89] (we refer to the excellent book [8] and to Section 5 for subsequent missing references and terminology, which we only mention here in passing). Whenever the space satisfies a Sobolev inequality (or equivalently a Nash inequality, or finitedimensional logSobolev inequality [8, Chapter 6]), and in particular under a condition for and finite , it is wellknown that is ultracontractive [8, Corollary 6.3.3.], i.e. that the heatkernel is bounded, yielding a trivial upperbound on . This ultracontractive case has been extensively studied in the literature, see e.g. [34, 8, 87]. The borderline case when some additional information is needed is precisely when is only hypercontractive, i.e. when satisfies a logSobolev inequality. In that case, and even under a weaker superPoincaré (or Sobolev) inequality, assuming in addition that the space satisfies for some , Wang obtained very general lower bounds on depending on concentration properties of the distance function to a given point .
In Section 5, we expand on the quantitative relation between hypercontractivity of the heat semigroup, the property of being traceclass (i.e. upper estimates on ), and higherorder integrability properties of the associated heatkernel, both in general and under a condition, . Our approach closely follows Wang’s method, based on his dimensionfree Harnack inequality. Our results are weaker and less general than Wang’s, but the proofs are a bit simplified, yielding estimates with concrete dimensiondependence. Finally, a general interesting trichotomy for the heatkernel is deduced. In Section 6, we provide some concluding remarks.
Acknowledgement. I thank Franck Barthe, Mikhail Gromov, Tobias Hartnick and Michel Ledoux for their comments.
2 Eigenvalue Calculation and Asymptotics
We begin with calculating the eigenvalues or their asymptotic distribution for several notable weightedmanifolds.
2.1 Gaussian Space
It is well known that the onedimensional Gaussian Space has simple spectrum at (so that each of the eigenvalues has multiplicity one), with the eigenfunctions of being precisely the Hermite polynomials. By the tensorization property of the spectrum, it follows that the product space has spectrum , where the sum is repeated times and is counted with multiplicity. In other words, the spectrum consists of and the multiplicity of the eigenvalue is given by . It follows that the eigenvalue counting function satisfies for :
and consequently:
(2.1) 
Furthermore, we record that as we have:
(2.2) 
2.2 Canonical Sphere
Let denote the sphere with its canonical metric and volume measure, embedded as the unitsphere in Euclidean space ; its Ricci curvature is equal to . It is well known that the eigenfunctions of the associated Laplacian are given by sphericalharmonics, i.e. the restriction of harmonic homogeneous polynomials in onto . The eigenvalue associated to harmonic polynomials of degree is [83]. Since it is well known that any homogeneous polynomial of degree can be uniquely decomposed into its harmonic components as follows:
we see that the subspace spanned by spherical harmonics of even degree at most or of odd degree at most is of dimension . Consequently, the subspace of all spherical harmonics of degree at most is of dimension , with corresponding eigenvalues being at most .
Now let us rescale the canonical sphere to have radius , so that its Ricci curvature coincides with the metric and therefore satisfies . Since the eigenvalues scale quadratically in the metric, the eigenvalue counting function consequently satisfies for all :
One could hope that the counting function of the rescaled sphere is always dominated by that of the dimensional Gaussian:
However, this is not the case for the first eigenvalues, and is most apparent for , i.e. linear functions on the sphere. Indeed, for all , on the rescaled sphere is equal to the eigenvalue of the last among its linear functionals, i.e. to , whereas on Gaussian space it is already equal to . This show that in general, one cannot hope for a positive answer to Question 1.
For future reference, we record that the unscaled canonical sphere satisfies for all :
(2.3) 
2.3 Weyl’s asymptotic law for weightedmanifolds
When is compact, then as soon as the density of is bounded away from and , the classical Weyl law [28] for the eigenvalue asymptotics of the unweighted Laplacian applies to the weighted one , and we have as :
(2.4) 
Note that by Bishop’s volume comparison theorem [36], for any (connected) with with , and so we see that Weyl’s formula (2.4) confirms Conjecture 3 in an asymptotic sense (as ):
(2.5) 
When is noncompact, the situation is more delicate. As we have not found an explicit reference in the literature, we derive the asymptotics ourselves from the known results for Schrödinger operators, and for simplicity, we restrict to the Euclidean case .
Recall that . Denote by the isometric isomorphism given by the multiplication operator . Conjugating by , we obtain the Schrödinger operator given by:
with domain . This wellknown procedure is a form of Doob’s htransform (e.g. [8, Section 1.15.8],[29, Section IV]). Since and are unitarily equivalent they are both selfadjoint on their respective domains and have the same spectrum. Clearly and hence . Since it is well known (e.g. [43]) that a Schrödinger operator is essentially selfadjoint on as soon as is in and bounded from below, it follows that in such a case its unique selfadjoint extension necessarily coincides with the one described above having domain . Consequently, we may apply the known Weyl formula for eigenvalue asymptotics of selfadjoint Schrödinger operators (e.g. [40, 18, 77]), which asserts that under suitable regularity assumptions:
(2.6) 
where denotes the following phasespace level set of the operator’s symbol:
More precisely (see e.g. [40, Theorem 6]), (2.6) holds under the assumptions that:

is smooth and bounded below.

for some .

For some we have .
Remark 2.1.
It is frequently assumed in the study of Schrödinger operators that in order to obtain a positive semidefinite operator, and this is also the standing assumption in [40]. However, if is only assumed bounded below, we can simply consider where is a constant so that ; the resulting shift in the spectrum is immaterial for the asymptotic distribution of eigenvalues, thereby justifying our slightly extended assumptions above. The assumption that is bounded below also ensures that is essentially selfadjoint on , as explained above.
Note that in typical situations (e.g. as in the next subsection):
(2.7) 
Assuming w.l.o.g. that the minimum of is attained at the origin, it follows that if with , then and hence . In that case, (2.6) implies that as :
in asymptotic accordance with Theorem 1.1. An extension of this reasoning to the manifold setting would similarly asymptotically confirm Conjecture , but we do not pursue the details here.
For future reference, it will be more convenient to rewrite (2.6) as:
(2.8)  
2.4 Asymptotics for the measures
Let us now calculate the asymptotic distribution of eigenvalues for the product measures , , which appear in various places in this work. We exclude the case of the exponential measure since does not have discrete spectrum (this will also be apparent from the ensuing calculations). Fixing , we have:
and:
An application of Hölder’s inequality verifies that outside the compact set when , and so it is clearly bounded below in that case. We now address the unboundedness of from below when , in tandem with the minor nuisance that and are not smooth on the coordinate hyperplanes (for any noneven ). Indeed, we may always approximate by smooth functions so that:
and so that in addition, when , , whereas when , is bounded above uniformly in . The minmax principle (recalled in Section 3) will immediately ensure that this results in a perturbation in the spectrum by a multiplicative factor of at most , which can be made arbitrarily close to . The above properties of ensure that:
uniformly in . It is then easy to see that the function has polynomial growth, and so the regularity and boundedness assumptions described above are satisfied, and the asymptotic distribution (2.6) is valid, uniformly in . As in (2.7), we conclude from (2.8) that:
where denotes the Beta function. Plugging in the known formulae for the volume of the ball and the Beta function, we finally obtain:
(2.9) 
Note that for , as , this is in precise agreement with the calculation carried out for the Gaussian case (2.2). Also observe that as the function on the righthand side explodes, in accordance with the formation of essential (nondiscrete) spectrum in the limiting case . Finally observe that as , converges to the uniform measure on , and the righthand side converges to:
in precise accordance with the classical Weyl estimate (2.4).
2.5 Cwikel–Lieb–Rozenblum inequality
Let be a connected manifold satisfying with . We have already seen in Subsection 2.3 that Weyl’s law confirms Conjecture 3 in an asymptotic sense, but without delving into its proof, it is not possible to extract from it nonasymptotic estimates on the individual eigenvalues.
However, individual (loose) estimates may be obtained from the Bérard–Gallot heatkernel estimate mentioned in the Introduction ([13, 14], [16, Appendix]):
which confirms Conjecture 3 in an averaged sense. In particular, this may be used to prove Conjecture 3 up to a multiplicative constant (in fact, without restricting to manifolds diffeomorphic to the sphere).
In this subsection, we mention yet another method for obtaining (nonsharp but quite good) explicit estimates on the individual eigenvalues, by means of the Cwikel–Lieb–Rozenblum inequality [33, 58, 75] (see also [59, 76, 48]) for the number of negative eigenvalues of the Schrödinger operator , established independently and by different means by these three authors. Lieb’s approach [59] relied on the ultracontractivity of the associated semigroup, whereas Li and Yau provided in [54] yet another proof based on the Sobolev inequality; these two assumptions were shown to be equivalent by Varopoulos [82] (see also [8]). The CLR inequality has since been generalized to a very abstract setting, and we employ the following version by Levin and Solomyak from [53]:
Theorem (Generalized CLR inequality, Levin–Solomyak).
Let denote a finite measure space. Let denote a selfadjoint operator on