Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators

We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k<d$. For these operators we prove a spectral multiplier theorem of Mihlin-H\"ormander type, which is optimal whenever $2k \leq d$, and a corresponding Bochner-Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.


Introduction
In this paper we continue the study of spherical Grushin-type operators started in [CCM1] with the case of the two-dimensional sphere.The focus here is on a family of hypoelliptic operators {L d,k } 1≤k<d , acting on functions defined on the unit sphere S d in R 1+d , i.e., on for some d ≥ 2. As it is well known, the groups SO(1 + r) with 1 ≤ r ≤ d can be naturally identified with a sequence of nested subgroups of SO(1 + d) and correspondingly they act on S d by rotations.We denote by ∆ r the (positive semidefinite) second-order differential operator on S d corresponding through this action to the Casimir operator on SO(1 + r).The operators ∆ r commute pairwise and ∆ d turns out to be the Laplace-Beltrami operator on S d .The operators we are interested in are defined as with k = 1, . . ., d − 1.By introducing a suitable system of "cylindrical coordinates" (ω, ψ) on S d , where ω ∈ S k and ψ = (ψ k+1 , . . ., ψ d ) ∈ (−π/2, π/2) d−k (see Section 3.3 below for details), one can write L d,k more explicitly as 2010 Mathematics Subject Classification.33C55, 42B15, 43A85 (primary); 53C17, 58J50 (secondary).
The first and the second author were partially supported by GNAMPA (Project 2018 "Operatori e disuguaglianze integrali in spazi con simmetrie") and MIUR (PRIN 2016 "Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis").Part of this research was carried out while the third author was visiting the Dicea, Università di Padova, Italy, as a recipient of a "Visiting Scientist 2019" grant; he gratefully thanks the Università di Padova for the support and hospitality.The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
where the Y r and their formal adjoints Y + r (with respect to the standard rotationinvariant measure σ on S d ) are vector fields only depending on ψ, to wit, , (1.4) and V : (−π/2, π/2) d−k → R is given by (1 + tan 2 ψ j ) − 1.
(1.5) Since V(ψ) vanishes only for ψ = 0, the formulae above show that each L d,k is elliptic away from the k-submanifold S k × {0} of S d ; the loss of global ellipticity is anyway compensated by the fact that each L d,k is hypoelliptic and satisfies subelliptic estimates, as shown by an application of Hörmander's theorem for sums of squares of vector fields [Hö1].Indeed the expression (1.3) reveals the analogy of the operators L d,k with certain degenerate elliptic operators G d,k on R d , given by where x, y are the components of a point in R d−k x × R k y and ∆ x , ∆ y denote the corresponding (positive definite) partial Laplacians.
In light of [G1, G2], the operators G d,k are often called Grushin operators; sometimes they are also called Baouendi-Grushin operators, since shortly before the papers by V. V. Grushin appeared, M. S. Baouendi introduced a more general class of operators containing also the G d,k [Ba].In these and other works (see, e.g., [FGW, RoSi, DM]), the coefficient |x| 2 in (1.6) may be replaced by a more general function V (x).As prototypical examples of differential operators with mixed homogeneity, operators of the form (1.6) have attracted increasing interest in the last fifty years; we refer to [CCM1] for a brief list of the main results, focused on the field of harmonic analysis.More recently, the study of Grushin-type operators began to develop also on more general manifolds than R n , from both a geometric and an analytic perspective [BFI1,BFI2,Pe,BoPSe,BoL,GMP1,GMP2].
In this article, we investigate L p boundedness properties of operators of the form F ( L d,k ) in connection with size and smoothness properties of the spectral multiplier F : R → C; here L p spaces on the sphere S d are defined in terms of the spherical measure σ, and the operators F ( L d,k ) are initially defined on L 2 (S d ) via the Borel functional calculus for the self-adjoint operator L d,k .The study of the L p boundedness of functions of Laplace-like operators is a classical and very active area of harmonic analysis, with a number of celebrated results and open questions, already in the case of the classical Laplacian in Euclidean space (think, e.g., of the Bochner-Riesz conjecture).Regarding the spherical Grushin operators L d,k , in the case d = 2 and k = 1 a sharp multiplier theorem of Mihlin-Hörmander type and a Bochner-Riesz summability result for L d,k were obtained in [CCM1].Here we treat the general case d ≥ 2, 1 ≤ k < d, and obtain the following result.
Let η ∈ C ∞ c ((0, ∞)) be any nontrivial cutoff, and denote by L q s (R) the L q Sobolev space of (fractional) order s on R.
(i) For all continuous functions supported in (ii) For all bounded Borel functions F : R → C such that F | (0,∞) is continuous, Hence, whenever the right-hand side is finite, the operator F ( L d,k ) is of weak type (1, 1) and bounded on L p (S d ) for all p ∈ (1, ∞).
Part (i) of the above theorem and a standard interpolation technique imply the following Bochner-Riesz summability result.It is important to point out that weaker versions of the above results, involving more restrictive requirements on the smoothness parameters s and δ, could be readily obtained by standard techniques.Indeed the sphere S d , with the measure σ and the Carnot-Carathéodory distance associated to L d,k , is a doubling metric measure space of "homogeneous dimension" Q = d + k, and the operator L d,k satisfies Gaussian-type heat kernel bounds.As a consequence (see, e.g., [He2,CoSi,DOSi,DzSi]), one would obtain the analogue of Theorem 1.1 with smoothness requirement s > Q/2, measured in terms of an L ∞ Sobolev norm, and the corresponding result for Bochner-Riesz means would give L p boundedness only for δ > Q|1/p − 1/2|.Since Q > D > D − 1, the results in this paper yield an improvement on the standard result for all values of d and k.
As a matter of fact, in the case k ≤ d/2, the above multiplier theorem is sharp, in the sense that the lower bound D/2 to the order of smoothness s required in Theorem 1.1 cannot be replaced by any smaller quantity.Since L d,k is elliptic away from a negligible subset of S d , and D = d is the topological dimension of S d when k ≤ d/2, the sharpness of the above result can be seen by comparison to the Euclidean case via a transplantation technique [Mi, KeStT].
The fact that for subelliptic nonelliptic operators one can often obtain "improved" multiplier theorems, by replacing the relevant homogeneous dimension with the topological dimension in the smoothness requirement, was first noticed in the case of sub-Laplacians on Heisenberg and related groups by D. Müller and E. M. Stein [MüS] and independently by W. Hebisch [He1], and has since been verified in multiple cases.However, despite a flurry of recent progress (see, e.g., [MMü2,CCM1,DM,MMüN] for more detailed accounts and further references), the question whether such an improvement is always possible remains open.The results in the present paper can therefore be considered as part of a wider programme, attempting to gain an understanding of the general problem by tackling particularly significant particular cases.
In these respects, it it relevant to point out that Theorem 1.1 above can be considered as a strengthening of the multiplier theorem for the Grushin operators G d,k on R d proved in [MSi]: indeed a "nonisotropic transplantation" technique (see, e.g., [M2,Theorem 5.2]) allows one to deduce from Theorem 1.1 the analogous result where S d and L d,k are replaced by R d and G d,k .
The structure of the proof of Theorem 1.1 broadly follows that of the analogous result in [CCM1], but additional difficulties need to be overcome here.An especially delicate point is the proof of the "weighted spectral cluster estimates" stated as Propositions 5.1 and 5.2 below, essentially consisting in suitable weighted L 1 → L 2 norm bounds for "weighted spectral projections" associated with bands of unit width of the spectrum of L d,k .These can be thought of as subelliptic analogues of the Agmon-Avakumovič-Hörmander spectral cluster estimates for the elliptic Laplacian ∆ d , which are valid more generally when √ ∆ d is replaced with an elliptic pseudodifferential operator of order one on a compact dmanifold [Hö2], and are the basic building block for a sharp multiplier theorem for elliptic operators on compact manifolds and related restriction-type estimates [So1,So2,SeeSo,FSab]. Thanks to pseudodifferential and Fourier integral operator techniques, estimates of the form (1.8) can be proved for elliptic operators in great generality, but these techniques break down when the ellipticity assumption is weakened.Nevertheless alternative ad-hoc methods may be developed in many cases, based on a detailed analysis of the spectral decomposition of the operator under consideration, often made possible by underlying symmetries.
In the case of the spherical Grushin operator L d,k , as a consequence of its spectral decomposition in terms of joint eigenfunctions of the operators ∆ d , . . ., ∆ k , the integral kernel of the "weighted projection" in (1.7) involves sums of (d − k)-fold tensor products of ultraspherical polynomials.This is a substantial difference from the case considered in [CCM1] (where d − k = 1) and requires new ideas and greater care.Section 5 of this paper is devoted to the proof of these estimates.As in [CCM1], here we make fundamental use of precise estimates for ultraspherical polynomials, which are uniform in suitable ranges of indices.These estimates, which are consequences of the asymptotic approximations of [O1,O2,O3,BoyD], could be of independent interest, and their derivation is presented in an auxiliary paper [CCM2].
In the context of subelliptic operators on compact manifolds, "weighted spectral cluster estimates" were first obtained in the seminal work of Cowling and Sikora [CoSi] for a distinguished sub-Laplacian on SU(2), leading to a sharp multiplier theorem in that case; their technique was then applied to many different frameworks [CoKSi,CCMS,M2,ACMM].However, the general theory developed in [CoSi], based on spectral cluster estimates involving a single weight function, does not seem to be directly applicable to the spherical Grushin operator L d,k (which, differently from the sub-Laplacian of [CoSi], is not invariant under a transitive group of isometries of the underlying manifold).For this reason, here we take the opportunity to establish an "abstract" multiplier theorem, which applies to a rather general setting of self-adjoint operators on bounded metric measure spaces, satisfying the volume doubling property, and extends the analogous result in [CoSi] to the framework of a family of scale-dependent weights.
It would be of great interest to establish whether Theorem 1.1 is sharp when k > d/2 or alternatively improve on it.The corresponding question for the Grushin operators G d,k on R d has been settled in [MMü1]; based on that result, one may expect that Theorem 1.1 and Corollary 1.2 actually hold with D replaced by d.However, when the dimension k of the singular set is larger than the codimension, the approach developed in this paper, which is based on a "weighted Plancherel estimate with weights on the first layer", does not suffice to obtain such result and new methods (inspired, for instance, to those in [MMü1] and involving the "second layer" as well) appear to be necessary.
The paper is organised as follows.In Section 2 we state our abstract multiplier theorem, of which Theorem 1.1 will be a direct consequence; in order not to burden the exposition, we postpone the proof of the abstract theorem to an appendix (Section 7).In Section 3 we introduce the spherical Laplacians and the Grushin operators on S d .A precise estimate for the sub-Riemannian distance ̺ associated with the Grushin operator L d,k is also given.Moreover, we introduce a system of cylindrical coordinates on S d which is key to our approach.In Section 4 we recall the construction of a complete system of joint eigenfunctions of ∆ d , . . ., ∆ k on S d , in terms of which we explicitly write down the spectral decomposition of the Grushin operator L d,k = ∆ d − ∆ k .We also prove some Riesz-type bounds for L d,k , and we state the refined estimates for ultraspherical polynomials, which are the building blocks in the joint spectral decomposition.Section 5 is devoted to the proof of the crucial "weighted spectral cluster estimates" for the Grushin operators L d,k .In Section 6 we use the Riesz-type bounds and the weighted spectral cluster estimates to prove "weighted Plancherel-type estimates" for the Grushin operator L d,k .After this preparatory work, the proof of Theorem 1.1, which boils down to verifying the assumptions of the abstract theorem, concludes the section.
Throughout the paper, for any two nonnegative quantities X and Y , we use X Y or Y X to denote the estimate X ≤ CY for a positive constant C. The symbol X ≃ Y is shorthand for X Y and Y X.We use variants such as k or ≃ k to indicate that the implicit constants may depend on the parameter k.

An abstract multiplier theorem
We state an abstract multiplier theorem, which is a refinement of [CoSi,Theorem 3.6] and [DOSi,Theorem 3.2].The proof of our main result, Theorem 1.1, for the operator L d,k will follow from this result.
As in [CoSi, DOSi], for all q ∈ [2, ∞], N ∈ N \ {0} and F : R → C supported in [0, 1], we define the norm F N,q by Moreover, by K T we denote the integral kernel of an operator T .
Since the subject is replete with technicalities, which could weigh on the discussion, we defer the proof of the abstract theorem to an appendix (Section 7).
Let us just observe that Assumption (b) only requires a polynomial decay in space (of arbitrary large order) for the heat kernel; hence this assumption is weaker than the corresponding ones in [DOSi], where Gaussian-type (i.e., superexponential) decay is required, and in [CoSi], where finite propagation speed for the associated wave equation is required (which, under the "on-diagonal bound" implied by (2.3), is equivalent to "second order" Gaussian-type decay [Si]), and matches instead the assumption in [He2] (see also [M2,Section 6]).
Another important feature of the above result, which is crucial for the applicability to the spherical Grushin operators L d,k considered in this paper, is the use of a family of weight functions, where the weight π r may depend on the scale r in a nontrivial way; this constitutes another important difference to [CoSi], where the weights considered are effectively scalar multiples of a single weight function (compare Assumptions (d) and (e) above with [CoSi,Assumptions 2.2 and 2.5]).
The attentive reader will have noticed that it is actually enough to verify Assumptions (c) and (d) for scales r = 1/N for N ∈ N \ {0} (indeed, one can redefine π r as π 1/⌊1/r⌋ when 1/r / ∈ N); the slightly redundant form of the above assumptions is just due to notational convenience.

Spherical Laplacians and Grushin operators
3.1.The Laplace-Beltrami operator on the unit sphere.For d ∈ N, d ≥ 1, let S d denote the unit sphere in R 1+d , as in (1.1).The Euclidean structure on R 1+d induces a natural, rotation-invariant Riemannian structure on S d .Let σ denote the corresponding Riemannian measure, and ∆ d the Laplace-Beltrami operator on the unit sphere S d in R 1+d .It is possible (see, e.g., [Ge]) to give a more explicit expression for ∆ d , namely, where Indeed the rotation group SO(1 + d) acts naturally on R 1+d and S d ; via this action, the vector fields Z j,r (0 ≤ j < r ≤ d) correspond to the standard basis of the Lie algebra of SO(1 + d), and ∆ d corresponds to the Casimir operator.The commutation relations are easily checked and correspond to those of the Lie algebra of SO(1 + d).

3.2.
A family of commuting Laplacians and spherical Grushin operators.By (3.2), the operator ∆ d commutes with all the vector fields Z j,r (this corresponds to the fact that the Casimir operator is in the centre of the universal enveloping algebra of the Lie algebra of SO(1 + d)); in particular it commutes with each of the "partial Laplacians" for r = 1, . . ., d.
Assume that d ≥ 2. We now observe that, for r = 1, . . ., d − 1, we can identify SO(1 + r) with a subgroup of SO(1 + d), by associating to each A in SO(1 + r) the element A 0 0 I of SO(1 + d).Via this identification, the operator ∆ r corresponds to the Casimir operator of SO(1 + r), and therefore it commutes with all the operators ∆ s for s = 1, . . ., r.
In conclusion, the operators ∆ 1 , . . ., ∆ d commute pairwise, and admit a joint spectral decomposition.In what follows we will be interested in the study of the Grushin-type operator r} is the family of vector fields appearing in the sum (3.4), then it is easily checked that, for all z ∈ S d , (3.5) On the other hand, the commutation relations (3.2) give that for all j, j ′ = 0, . . ., d − 1; in particular the vector fields in Z d,k , together with their Lie brackets, span the tangent space of S d at each point.In other words, the family of vector fields Z d,k satisfies Hörmander's condition and (together with the Riemannian measure σ) determines a (non-equiregular) 2-step sub-Riemannian structure on S d with the horizontal distribution H d,k described in (3.5).The corresponding sub-Riemannian norm on the fibres of H d,k is given, for all p ∈ S d and v ∈ H d,k p , by For more details on sub-Riemannian geometry we refer the reader to [ABB, BeRi, CaCh, Mo].

Cylindrical coordinates.
In order to study the operator L d,k , it is useful to introduce a system of "cylindrical coordinates" on S d that will provide a particularly revealing expression for L d,k in a neighbourhood of the singular set E d,k .For all ω ∈ S d−1 and ψ ∈ [−π/2, π/2], let us define the point ⌊ω, ψ⌉ ∈ S d by ⌊ω, ψ⌉ = ((cos ψ)ω, sin ψ). (3.7) Away from ψ = ±π/2, the map (ω, ψ) → ⌊ω, ψ⌉ is a diffeomorphism onto its image, which is the sphere without the two poles; so (3.7) can be thought of as a "system of coordinates" on S d , up to null sets.In these coordinates, the spherical measure σ on S d is given by where σ d−1 is the spherical measure on S d−1 .Moreover, the Laplace-Beltrami operator may be written in these coordinates as where ∆ d−1 , given by (3.3), corresponds to the Laplace-Beltrami operator on S d−1 (see, e.g., [V,§IX.5]).
We now iterate the previous construction.Let k ∈ N such that 1 ≤ k < d be fixed.Starting from (3.7), we can inductively define the point In these coordinates, the spherical measure σ on S d is given by where σ k is the spherical measure on S k .Moreover, starting from (3.8), we get inductively that where again ∆ k is the operator given by (3.3).
In particular, the Grushin operator L d,k = ∆ d − ∆ k on S d may be written in these coordinates as in (1.3), where the vector fields Y r and the function V : (−π/2, π/2) d−k → R are defined by (1.4) and (1.5) respectively.Note that V(ψ) vanishes only for ψ = 0, corresponding to the singular set E d,k .We also remark that 1 cos The formula (1.3) for the sub-Laplacian corresponds to a somewhat more explicit expression for the sub-Riemannian norm (3.6) on the fibres of the horizontal distribution, which is better written by identifying, via the "coordinates" (3.9), the tangent space T ⌊ω,ψ⌉ and, for all (v, w) ∈ H d,k ⌊ω,ψ⌉ , its sub-Riemannian norm satisfies 3.4.The sub-Riemannian distance.Thanks to (3.13), we can obtain a precise estimate for the sub-Riemannian distance ̺ associated with the Grushin operator L d,k .This is the analogue of [RoSi,Proposition 5.1], that treats the case of "flat" Grushin operators on R n , and [CCM1, Proposition 2.1], that treats the case of L 2,1 on S 2 .In the statement below we represent the points of the sphere in the form ⌊ω, ψ⌉ for ω ∈ S k , ψ ∈ [−π/2, π/2] d−k , as in (3.9).We also denote by ̺ R,S k and ̺ R,S d the Riemannian distances on the spheres S k and S d .
Consequently, the σ-measure V (⌊ω, ψ⌉ , r) of the ̺-ball centred at ⌊ω, ψ⌉ with radius Note now that the expression in the right-hand side of (3.14) defines a continuous function Φ : Hence, in order to prove the equivalence (3.14), it is enough to show that Φ and ̺ are locally equivalent at each point p 0 ∈ Ω d,k , and then apply ).The associated horizontal distribution H G and sub-Riemannian metric are given by . By the equivalence of norms, up to shrinking A, we may assume that , where the norms in (3.19) are those determined by the Riemannian structures of S k and R k ; similarly, we may also assume that for all p, p ′ ∈ U , where the latter equivalence readily follows from (3.18) and (3.20).

A complete system of joint eigenfunctions
Let d, k ∈ N with 1 ≤ k < d.In this section we briefly recall the construction of a complete system of joint eigenfunctions of ∆ d , . . ., ∆ k on S d .This will give in particular the spectral decomposition of the Grushin operator This construction is classical and can be found in several places in the literature (see, e.g., [V, Ch. IX] or [EMOT, Ch. XI]), where explicit formulas for spherical harmonics on spheres of arbitrary dimension are given, in terms of ultraspherical (Gegenbauer) polynomials.The discussion below is essentially meant to fix the notation that will be used later.
4.1.Spectral theory of the Laplace-Beltrami operator.We first recall some well known facts about the spectral theory of ∆ d (see, e.g., [StW,Ch. 4] or [AxBR,Ch. 5]).The operator ∆ d is essentially self-adjoint on L 2 (S d ) and has discrete spectrum: its eigenvalues are given by where ℓ ∈ N d , and (4.4) The corresponding eigenspaces, denoted by H ℓ (S d ), consist of all spherical harmonics of degree ℓ ′ = ℓ − (d − 1)/2, that is, of all restrictions to S d of homogeneous harmonic polynomials on R 1+d of degree ℓ ′ ; they are finite-dimensional spaces of dimension for ℓ ∈ N d (the last identity only makes sense when d > 1), and in particular (this estimate is also valid when d = 1, provided we stipulate that 0 0 = 1).Since ∆ d is self-adjoint, its eigenspaces are mutually orthogonal, i.e., Here the normalization constant c ℓm is chosen so that that is, by means of [Sz,(4.3.3)], Then, for all (ℓ, m) ∈ I d , we obtain an injective linear map ), which is an isometry with respect to the Hilbert space structures of L 2 (S d−1 ) and L 2 (S d ), and a decomposition [V, p. 466, eq.(1)]).The summands in the right-hand side of (4.12) are joint eigenspaces of ∆ d and ∆ d−1 of eigenvalues λ d ℓ and λ k m respectively; hence they are pairwise orthogonal in L 2 (S d ).
4.3.Joint eigenfunctions of ∆ d , . . ., ∆ k .We go back to the general case 1 ≤ k < d and we look for a complete system of joint eigenfunctions of ∆ d , . . ., ∆ k .
We note that the operators of the form (4.17) include those in the functional calculus of the Grushin operator where 4.4.Riesz-type bounds.In this section we prove certain weighted L 2 bounds involving the joint functional calculus of ∆ d , . . ., ∆ k , which, in combination with the weighted spectral cluster estimates in Section 5 below, play a fundamental role in satisfying the assumptions on the weight in the abstract theorem and proving our main result.A somewhat similar estimate was obtained in [CCM1, Lemma 2.5] in the case d = 2 and k = 1.Differently from [CCM1], the estimate in Proposition 4.1 below is proved for arbitrarily large powers of the weight; this prevents us from using the elementary "quadratic form majorization" method exploited in the previous paper, and requires a more careful analysis, based on the explicit eigenfunction expansion developed in the previous sections.
For later use, it is convenient to reparametrise the functions X d ℓ,m defined in (4.8): namely, we introduce the functions where (ℓ, m) ∈ I d and c ℓm is given by (4.10).Let t d,d : In particular, for all (ω, ψ) where ψ = (ψ k+1 , . . ., ψ d ).Finally, we set, for 1 ≤ k < d, (4.23) Proof.By interpolation, it is enough to prove the estimate in the case N ∈ N.

4.5.
Estimates for ultraspherical polynomials.In this section we present a number of estimates for the functions X d ℓ,m (or rather, their reparametrisations X d ℓ,m from (4.21)), which will play a crucial role in the subsequent developments.
We first state some basic uniform bounds that follow from the previous discussion (see (4.7) and (4.12)).In the statement below, we convene that 0 0 = 1.
/2 for all (ℓ, m) ∈ I d .More refined pointwise estimates can be derived from asymptotic approximations of ultraspherical polynomials in terms of Hermite polynomials and Bessel functions, obtained in works of Olver [O3] and Boyd and Dunster [BoyD] in the regimes m ≥ ǫℓ and m ≤ ǫℓ respectively, where ǫ ∈ (0, 1).Here and subsequently, for all ℓ, m ∈ R with ℓ = 0 and 0 ≤ m ≤ ℓ, a ℓ,m and b ℓ,m will denote the numbers in [0, 1] defined by b ℓ,m = m ℓ (4.32) and The points ±a ℓ,m ∈ [−1, 1] play the role of "transition points" for the functions In the case d = 2, the derivation of the estimates in Theorem 4.3 from the asymptotic approximations in [O3,BoyD] is presented in [CCM1, Section 3]; a number of variations and new ideas are required in the general case d ≥ 2, and we refer to [CCM2] for a complete proof (indeed, in [CCM2] a stronger decay is proved in the regime m ≥ ǫℓ for |x| ≥ 2a ℓ,m than the one given in (4.34)).Here we only remark that combining the above estimates yields the following bound. ) Proof.Let ǫ ∈ (0, 1) be a parameter to be fixed later.If m ≤ ǫℓ, the desired estimates immediately follow from (4.35), by taking any c ≤ log 2 (indeed, note that (1 + m) 4/3 ≥ 1 + m).

Weighted spectral cluster estimates
As a consequence of the estimates in Section 4.5, we obtain "weighted spectral cluster estimates" for the Grushin operators where X d r,s has been defined in (4.21).We are interested in bounds for suitable weighted sums of the X d,k ℓ,m for indices ℓ, m such that the eigenvalue λ d,k ℓ,m of L d,k ranges in an interval of unit length (whence the name "spectral cluster").The bounds that we obtain are different in nature according to whether m ≤ ǫℓ or m ≥ ǫℓ for some fixed ǫ ∈ (0, 1), and are presented as separate statements.We remark that, in the case m ≤ ǫℓ, the eigenvalue λ d,k ℓ,m of L d,k is comparable with the eigenvalue λ d ℓ of ∆ d ; consequently, the range m ≤ ǫℓ will be referred to as the "elliptic regime", while the range m ≥ ǫℓ will be called the "subelliptic regime".Proposition 5.1 (subelliptic regime).Let ǫ ∈ (0, 1) and d where X d,k ℓ,m was defined in (5.1).Analogous estimates are proved in [CCM1, Section 4] in the case d = 2 and k = 1; in that case, each of the products in (5.1) reduces to a single factor.Treating the general case, with multiple factors, presents substantial additional difficulties, as one may appreciate from the discussion below.
5.1.The subelliptic regime.Here we prove Proposition 5.1.To this aim, we first present a couple of lemmas that will allow us to perform a particularly useful change of variables in the proof.
Lemma 5.4.Let w ∈ R n and define the matrix M (w) = (m i,j (w)) n i,j=1 by Proof.Observe that m i,j (w) = δ i,j + ρ i,j w j , where Consequently, if S n denotes the group of permutations of the set {1, . . ., n} and ǫ(σ) denotes the signature of the permutation σ, then where S c = {1, . . ., n} \ S. We note that (ρ l,m ) |S| l,m=1 is a skewsymmetric matrix, so its determinant vanishes when |S| is odd; if |S| is even, instead, its determinant is the square of its pfaffian, and using the Laplace-type expansion for pfaffians (see, e.g., [Ar,§III.5,p. 142]) one can see inductively that the determinant is 1.
Lemma 5.5.Let Ω = {v ∈ R n : vj = −1 for all j = 1, . . ., n}, where Let v → w be the map from Ω to R n defined by Moreover, for all ǫ ∈ (0, 1), the map v → w is injective when restricted to Proof.From the definition it is immediate that where M (w) = {m j,s (w)} n j,s=1 is the matrix defined in Lemma 5.4, so and the desired expression for the determinant follows from Lemma 5.4.
Note that, if v ∈ Ω ǫ , 0 ≤ v j , |v j | ≤ j v j ≤ ǫ < 1, so 1 + vj > 0 and Ω ǫ ⊆ Ω.In addition, the equations w j = v j /(1 + vj ) are equivalent to v j − w j vj = w j , that is, Since w j = v j /(1 + vj ) ≥ 0, from Lemma 5.4 it follows that det M (w) ≥ 1, so the matrix M (w) is invertible and the above equation is equivalent to v = M (w) −1 w; in other words, if v ∈ Ω ǫ , then v is uniquely determined by its image w via the map v → w, that is, the map restricted to Ω ǫ is injective.
Proof of Proposition 5.1.We start by observing that, for all (ℓ, k) ∈ I and in particular ℓ j ≃ ℓ 1, for all j ∈ {k, . . ., d}. (5.6) We also note that since it will suffice to apply (5.7) times, with i replaced by i, i + 1, . . ., i + h − 1, respectively.Due to (4.2), we may restrict without loss of generality to x ∈ [0, 1] d−k .In addition, for each fixed i, the sum in the left-hand side of (5.7) is finite, since ℓ d − ℓ k 1 and therefore ) then shows that the estimate (5.7) is trivially true for each fixed i (with a constant dependent on i), and therefore it is enough to prove it for i sufficiently large.
It is convenient to reindex the above sum.Let us set and let us write (5.8) in particular (5.9) The condition ǫℓ d ≤ ℓ k is then equivalent to ∈ (0, 1), and implies, by (5.5), q j ≤ ǭ4 (p + qj ) (5.10) for j = 1, . . ., d− k.As previously discussed, it will be enough to prove the estimate for i sufficiently large; in the following we will assume that Let us first consider the range (5.12) In light of (4.34), the inequalities (5.13) hold for all j ∈ {1, . . ., d − k}.Moreover, for one of the quantities the better bound |x| −1/2 exp(−cp|x| 2 ) holds for some c > 0, thanks to the second estimate in (4.34) and to (5.6).As a consequence, we obtain for arbitrarily large N ∈ N. Note then that the conditions (5.11) and (5.9) imply since i|x| 1 and k − 2α > 0, provided N is large enough.Note that, in estimating the sum in p, we used the fact that the interval Let us now discuss the range (5.14) We first note that (5.14) implies Note that, by (5.9), for all j = 1, . . ., d − k, (5.15)where ϕ(w) = 4w/(1 + w) 2 .Note that the map ϕ : [0, 1] → [0, 1] is an increasing bijection, such that w ≤ ϕ(w) ≤ 4w; its derivative is given by ϕ ′ (w) = 4 1−w (1+w) 3 and vanishes only at w = 1.As a consequence, setting xj = ϕ −1 (x 2 j ), with j ∈ {1, . . ., d − k}, one has xj ≃ |x j |; moreover, in light of (5.15) and (5.10), In particular, in this range, by (4.34), for all j = 1, . . ., d − k, where Ξ is defined as in (5.4).Then where x = (x d , . . ., xk+1 ), q = (q 1 , . . ., q d−k ) and It is easily seen that |∂ p (1/p)|, |∂ p (q j /(p + qj ))| 1/p for all j = 1, . . ., d − k, on the range of summation (note that q j + |q j | ≤ Q ≤ ǭ2 i 2 /Q ≤ ǭ2 p and ǭ < 1, whence p + qj ≃ p q j 1).Thus, by Lemma 5.3, where the change of variables p = u 2 /Q was used, and It is easily checked that for all j = 1, . . ., d − k, on the range of summation (note that |q j |Q ≤ Q 2 ≤ ǭ2 i 2 ≤ ǭ2 u 2 and ǭ < 1, so u 2 + qj Q ≃ u 2 ), and therefore, by Lemma 5.3 and the Leibniz rule, Hence, by [DM,Lemma 5.7], where the change of variables q j = pv j was used, and and the fact that the interval [i/ √ V , (i + 1)/ √ V ] has length V −1/2 was used.We can now use the change of variables observing that w j ≃ v j for all j ∈ {1, . . ., d − k}, and (see Lemma 5.5 below) on the domain of integration (here we use the fact that v j , |v j | ∈ [0, ǭ2 ] for all j ∈ {1, . . ., d − k} and ǭ < 1), so In order to conclude, it is enough to bound the last integral with a multiple of min{i, |x| −1 } k−2α .To do this, it is convenient to split the domain of integration according to whether w j is larger or smaller than 2x 2 d−j+1 for each j = 1, . . ., d − k, and according to which j corresponds to the maximum component w j of w.In other words, where J c = {1, . . ., d − k} \ J.We estimate separately each summand, depending on the choice of j * ∈ {1, . . ., d − k} and J ⊆ {1, . . ., d − k}, noting that, in the respective domain of integration, |x for all j ∈ J. Suppose first that j * ∈ J, and set which is the desired estimate.Here we used that and we are done.
5.2.The elliptic regime.We now discuss the proof of Proposition 5.2.We first observe that a straightforward iteration of Proposition 4.2(i) yields the following estimate.
Proof of Proposition 5.2.Due to the symmetry property of Jacobi polynomials (4.2), we may restrict to d , it suffices to prove the estimate , so (5.3) follows by applying (5.17 We first consider the terms in the sum with ℓ k = 0 (observe that this may happen only for k = 1).The condition ℓ 2 d ∈ [i 2 , (i + 1) 2 ] uniquely determines the value of ℓ d .Using the estimate in Proposition 4.2(ii) to bound X k+1 ℓ k+1 ,0 (x k+1 ) in the left-hand side of (5.17) and then applying Lemma 5.6, we obtain In what follows, we shall therefore assume ℓ k > 0. Define y j := 1 − x 2 j for j = k + 1, . . ., d.Let us first consider the case where for some j * ∈ {k + 1, . . ., d}.
By (4.36), in this case, , where we also extended the sum in ℓ j * −1 to all N j * −1 .As already observed, due to the condition , for a fixed ℓ k i the sum over ℓ d essentially contains only one term and ℓ d ≃ i.Thus, by applying Lemma 5.6 first to the sum over ℓ j * , • • • , ℓ d−1 and then to the sum over ℓ j * − 2, . . ., ℓ k , we get From here on, we shall assume for all j ∈ {k + 1, . . ., d}.
Note that the above inequality implies that ℓ j−1 ≃ ℓ j y j for all j > k * , and moreover, by Corollary 4.4, for k * < j ≤ d, where Ξ was defined in (5.4).Assume first that k * > k.Then whence, by Corollary 4.4, Note now that, for j = k * + 1, . . ., d, in the range of summation; moreover the interval [ i 2 + ℓ 2 k , (i + 1) 2 + ℓ 2 k ] has length ≃ 1 and its endpoints are ≃ i, because ℓ k i.Hence, in view of Lemma 5.3, we can apply [DM,Lemma 5.7] to the inner sum and obtain The change of variables t j−1 = ℓ j−1 /(ℓ j y j ), j = k * + 1, . . ., d, then gives where Lemma 5.6 was applied to the sum in (ℓ k * , . . ., ℓ k ) and the fact that k * ≥ k + 1 ≥ 2 was used.
We now consider the case k * = k.Here, by (5.18), where the last inequality follows from [CCM1,Lemma 4.1] together with Lemma 5.3, the fact that in the range of summation and the fact that (since The change of variables u = ℓ 2 d − ℓ 2 k in the inner integral then gives where ℓ = (ℓ d−1 , . . ., ℓ k ), y = (y d , . . ., y k+1 ), and We now observe that, since u ∈ [i, i + 1], and on the range of summation.Thanks to Lemma 5.3, we can apply [DM,Lemma 5.7] to majorize the inner sum with the corresponding integral and obtain that The change of variables ℓ j = uy j+1 • • • y d τ j , j = k, . . ., d − 1, then gives Finally, the change of variables and we are done.
Proof.Due to the compactness of S d , both (6.7) and (6.8) are obvious for r ≥ 1.
In the following we assume therefore that r < 1.
.16) The implicit constants may depend on ε.Proof.Note that the sub-Riemannian distance ̺ and the Riemannian distance ̺ R,S d are locally equivalent far from the singular set E d,k : since H d,k p = T p M for all p ∈ S d \ E d,k (see (3.5)), and the Riemannian and sub-Riemannian inner products on T p M depend continuously on p, it is enough to apply [CCM1, Lemma 2.3] by choosing as M and N the Riemannian and sub-Riemannian S d respectively, and as F the identity map restricted to any open subset U of S with compact closure not intersecting orthogonal projection π d ℓ d ,...,ℓ k of L 2 (S d ) onto the joint eigenspace of ∆ d , . . ., ∆ k of eigenvalues λ d ℓ d , . . ., λ k ℓ k is given by 4.29) which is (4.23) in the case k = d − 1.Let now 2 ≤ r ≤ d.By the discussion in Section 3, up to null sets we can identify S d with S r × [−π/2, π/2] d−r with coordinates (ω, ψ) and measure cos ψ d−1 d • • • cos ψ r r+1 dψ dω.Consequently the space L 2 (S d ) is the Hilbert tensor product of the spaces L 2 (S r ) and L 2 ([−π/2, π/2] d−r , cos ψ d−1 d • • • cos ψ r r+1 dψ).Hence the inequality (4.29), applied with d = r, yields a corresponding inequality on the sphere S d , namely we assume ǫℓ ≤ m, then, for all (ℓ d , . . ., ℓ k ) ∈ J (k) d with ℓ d = ℓ and ℓ k = m, ǫℓ j+1 ≤ ℓ j , j ∈ {k, . . ., d − 1}, (5.5) and, for j = 1, . . ., d − k,ℓ d−j+1 + ℓ d−j = p + qj , ..,d−k} |x d−j+1 |.