A GENERIC FUNCTIONAL INEQUALITY AND RICCATI PAIRS: AN ALTERNATIVE APPROACH TO HARDY-TYPE INEQUALITIES

. We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam ( Proc. Natl. Acad. Sci. USA , 2008 & Math. Ann. , 2011). This concept enables us to give very short/elegant proofs of a number of celebrated functional inequalities on Riemannian manifolds with sectional curvature bounded from above by simply solving a Riccati-type ODE. Among others, we provide alternative proofs for Caccioppoli inequalities, Hardy-type inequalities and their improvements, spectral gap estimates, interpolation inequalities, and Ghoussoub-Moradifam-type weighted inequalities. Concerning the multiplicative form, we prove sharp uncertainty principles on Cartan-Hadamard manifolds, i.e., Heisenberg-Pauli-Weyl uncertainty principles, Hydrogen uncertainty principles and Caﬀarelli-Kohn-Nirenberg inequalities. Some sharpness and rigidity phenomena are also discussed

More than one hundred years elapsed since the first form of the celebrated Hardy inequality appeared (sometimes called also as an uncertainty principle), see Hardy [37].It states that the L 2 -norm of the singular term u(x)/|x| is controlled by the L 2 -Dirichlet norm of the function u ∈ C ∞ 0 (R n ).Several extensions and improvements of the Hardy inequality can be found by now in the literature, which became indispensable from the point of view of applications; indeed, solutions of a large class of elliptic problems involving singular terms are based on the validity of Hardy inequalities.Comprehensive discussions concerning these inequalities can be found in the monographs by Balinsky, Evans and Lewis [4], Ghoussoub and Moradifam [34], and Ruzhansky and Suragan [57].
Generally speaking, Hardy-type inequalities can be written in the following forms: where p > 1, p ′ = p p−1 is the conjugate of p, Ω is an open subset of an ambient space M (which could be the Euclidean space R n , any Riemannian/Finsler manifold, or a stratified group), m is a measure on M and V, W, W 1 , W 2 : Ω → (0, ∞) are certain potentials, possibly containing singular terms.
A milestone result -concerning problem (H) -has been provided by Ghoussoub and Moradifam [35,36] (for p = 2 and the Euclidean setting), where the authors proved that (H) holds if and only if (V, W ) is a Bessel pair.The latter notion is based on the solvability of a second order linear ODE containing the potentials V and W .We note that this ODE agrees with the radial equation obtained from the classical approach of supersolutions, which emerges from the early works of Allegretto [3] and Moss and Piepenbrink [51], see Remark 3.4.The concept of Bessel pairs was extended to general p > 1, see Duy, Lam and Lu [24], and also has applications on nonpositively curved Riemannian manifolds, see Flynn, Lam, Lu and Mazumdar [30], where still the usual notion of Bessel pairs and fine comparison arguments are used.
In this paper we use a genuinely different approach to prove Hardy inequalities.First we provide a generic inequality on Riemannian manifolds in both additive and multiplicative forms, that turn out to be equivalent to each other, producing inequalities of type (H) and (UP), respectively.Since these inequalities contain the Laplacian of a given potential, -which implicitly encode curvature information about the manifold -an appropriate comparison argument furnishes in the additive form a Riccati-type ordinary differential inequality that leads us to the notion of Riccati pair for certain potentials.This notion turns out to be extremely efficient to prove inequalities of the type (H).It is worth to be pointed out that Bobkov and Götze [11] already provided some integral estimates on certain domains by using canonical Riccati and Sturm-Liouville equations on the real line.
To present our approach, we put ourselves into the realm of Riemannian manifolds.Let ∇ g , ∆ g , | • | and dv g be the gradient, Laplace-Beltrami operator, norm and standard volume form on a Riemannian manifold (M, g), respectively.Here is our first main result containing the generic inequality (for a weighted version, see Theorem 3.1): Theorem 1.1.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold, with n ≥ 2.
A seemingly surprising fact emerges: it turns out that (1.1) and (1.2) are equivalent in the sense that they can be deduced from each other, see Remark 3.1.Accordingly, we may roughly say that these two classes of functional inequalities have the same 'algebraic' origin.This fact follows from the 'duality' between H(u) and |H ′ (u)| p ′ , which can be viewed as a competition between the two terms, implying the aforementioned equivalence through a simple scaling argument.
On the one hand, both inequalities in Theorem 1.1 are generic, that is for suitable choices of functions ρ, G and H they produce various functional inequalities.While the additive form (1.1) is tailored to produce Hardy-type inequalities in the form of (H), see §4, the multiplicative form (1.2) will provide various uncertainty principles of the type (UP), see §5.
On the other hand, the aforementioned procedure can be reversed: for a number of Hardy-type inequalities one can find suitable choices of functions providing short/elegant proofs for them.To see this, let us focus on the additive form (1.1).Let p > 1 and observe that if This observation and the Laplace comparison (see Theorem 2.1) suggest the following notion (for a weighted form see Definition 3.1); hereafter, H n stands for the n-dimensional Hausdorff measure.Definition 1.1.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold, with n ≥ 2.
An efficient application of Theorem 1.1 can be stated as follows (for a weighted form, see Theorem 3.2): Theorem 1.2.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold, with n ≥ 2.
Let Ω ⊆ M be a domain, p > 1, and ρ ∈ W 1,p loc (Ω) be a positive function with Just to give a little peek to the efficiency of Theorem 1.2, we sketch a short proof of the celebrated McKean's sharp spectral gap estimate (for details, see Theorem 4.9): if the sectional curvature is bounded from above by κ < 0 on an n-dimensional Cartan-Hadamard manifold 1 (M, g), the essential spectrum 1 (M, g) is a Cartan-Hadamard manifold if it is a complete, simply connected Riemannian manifold with nonpositive sectional curvature.
When the domain Ω is embedded into any of the model space forms (i.e., the sphere, the Euclidean space or the hyperbolic space) and ρ is a distance function from a fixed point of the domain, a closer inspection of the proof of Theorem 1.2 (and its weighted version, Theorem 3.2) shows that the sign restriction on the function G from (1.3) can be relaxed.Indeed, in this particular situation, it turns out that there is equality in the Laplace comparison (i.e.∆ g ρ = L(ρ) for a suitable choice of L), thus certain estimates will be independent on the sign of G; for more details see Remark 3.2.
A natural question arises concerning the connection between Riccati and Bessel pairs.In the Euclidean case, the weighted Riccati pair (see Definition 3.1) is a slight extension of the Bessel pair; indeed, if we choose L(t) = (n − 1)/t, require that G ≥ 0 and restrict (1.3) to the equality case, the two notions coincide, see Proposition 3.1.When we move to the Riemannian context, there exists a natural extension of the usual notion of Bessel pairs (using appropriate comparison L) which still preserves this equivalence whenever the sectional curvature is nonpositive, compare Remark 3.3 and the paper by Flynn, Lam, Lu and Mazumdar [30].However, in the generic case, Riccati pairs 'inherit' the geometry of the ambient Riemannian manifold, reflected in the curvature; see e.g. the alternative proof of Cheng's eigenvalue result (Theorem 4.7).In fact, the use of Riccati pairs is twofold.First, Riccati pairs naturally appear from our generic inequality (1.1), thus no redundant change of variables is required.Second, Riccati pairs are flexible from technical point of view as both L and W can be easily adjusted to meet our needs.Such an example is provided by the proof of McKean's sharp spectral gap estimate, where the choice of L ≡ (n − 1) √ −κ significantly reduces the computations comparing to the expected form L(t) = (n − 1) √ −κ coth( √ −κt).Beyond Proposition 3.1, we point out the relationship between the method of supersolutions, also known as the Allegretto-Moss-Piepenbrink approach, and the aforementioned notions, see Remark 3.4.These relations allow us to incorporate in our method various elements of the criticality theory of Hardy inequalities developed in the approach of supersolutions, see the foundational works by Pinchover [55,56], and Devyver, Fraas and Pinchover [23].We refer to Remark 4.7/(b) for an illustration.
In the sequel we roughly describe the content of the paper.In section 2 those notions and results are recalled that will be used in our arguments, as basic elements from Riemannian geometry as well as volume and Laplace comparison principles.Section 3 is devoted to the proof of our main abstract results: Theorems 3.1 & 3.2, that are the weighted versions of Theorems 1.1 & 1.2, respectively.In addition, the relationship between Bessel and Riccati pairs is also discussed, see Proposition 3.1 and Remark 3.3.
Most of the additive functional inequalities in section 4 are formally well known from the Euclidean setting (and some of them also in Riemannian manifolds).However, we shall focus on their proof on Riemannian manifolds -mainly on Cartan-Hadamard manifolds -by showing the efficiency of our main results, mostly based on Riccati pairs.More precisely, we shall consider the following inequalities: • In §4.1 we prove L p -Caccioppoli-type inequalities on Riemannian manifolds, providing alternative, short proofs for the results obtained by D'Ambrosio and Dipierro [20]  It is well known that Bessel pairs can be efficiently used to prove various Hardy-Rellich-type inequalities, see Ghoussoub and Moradifam [35] and Berchio, Ganguly, Roychowdhury [10].We are wondering if such inequalities of higher-order can be elegantly achieved by Riccati pairs, both in Euclidean spaces and Riemannian manifolds.This question will be addressed in a forthcoming paper.
We also notice that our arguments can be easily extended to reversible Finsler manifolds; in the irreversible case, there are certain technical issues which require the presence of the reversibility constant.The details are left to the interested reader.

Preliminaries
In this section we recall those notions and results that are needed for presenting our results.We mainly follow Gallot, Hulin and Lafontaine [33], Hebey [39], and Kristály [41].
Let (M, g) be an n-dimensional complete Riemannian manifold, with n ≥ 2, In particular, one has for every x 0 ∈ M that lim where ω n = π Let p > 1 and fix u ∈ W 1,p (M ).The gradient of u is ∇ g u, whose local components are where g ij are the local components of g −1 = (g ij ) −1 in the local coordinate system (x i ) on a coordinate neighborhood of x ∈ M ; hereafter the standard summation convention is used.The p-Laplace-Beltrami operator on (M, g) is given by ∆ g,p u = div g (|∇ g u| p−2 ∇ g u), see e.g.Hebey [39].Observe that ∆ g,2 is nothing but ∆ g , the usual Laplace-Beltrami operator on (M, g).
If w ∈ C 2 0 (M ), one has the following integration by parts formula where ∇ g u, ∇ g w ∈ T M, and ∇ g u∇ g w is understood as the scalar product with respect to the metric g.The model space form M n κ is an n-dimensional manifold with constant sectional curvature κ, that is ( For further use, let be the volume of the ball of radius R > 0 in the model space form M n κ .Let d x 0 (x) = d g (x 0 , x) be the distance from a given point x 0 ∈ M .The eikonal equation reads as (2.4) Let inj x 0 be the injectivity radius of x 0 ∈ M .We have the following Laplace and Bishop-Gromov volume comparison principles, see e.g.Gallot, Hulin and Lafontaine [33,Theorem 3.101].
(I) If the sectional curvature is bounded from above as K ≤ κ for some κ ∈ R, then: (II) If the Ricci curvature is bounded from below as Ric ≥ κ(n − 1)g for some κ ∈ R, then: If equality holds in any of the above statements, (M, g) is isometric to the model space form M n κ .

Proof of main results
In this section we prove the main abstract results in weighted form.
Let Ω ⊆ M be a domain, p > 1, and ρ ∈ W 1,p loc (Ω) be nonconstant and positive with provided that there exists a neighborhood U ⊂ R of zero such that Proof.(i) By the convexity of ξ → |ξ| p , it turns out that Fix u ∈ C ∞ 0 (Ω) arbitrarily and let ξ := ∇ g u and η := v∇ g ρ, where On the one hand, we have On the other hand, the chain rule Therefore, one has Multiplying by w(ρ) > 0 and using (G) ρ,w we get An integration by parts and the boundary conditions on H yield The above arguments imply inequality (3.1).
(ii) Recall the definitions of I H and J H from (3.4); due to assumption (3.3) we have that Note that p − 1 > 0, thus we can maximize the right hand side of the latter expression with respect to c; the maximum is achieved for the value c Remark 3.1.We notice that the additive and multiplicative forms can be deduced from each other.First, relation (3.2) follows from (3.1), see (ii).Second, we have that J H ; indeed, when I H ≤ 0, the latter inequality is trivial, while for I H > 0, Young's inequality applies.
Clearly, Theorem 1.1 is a simple consequence of Theorem 3.1 by choosing w ≡ 1 and the function ρ such that |∇ g ρ| = 1 dv g -a.e. in Ω.In particular, (G) ρ,w reduces to (G) ρ .In the sequel, we prove a weighted version of Theorem 1.2, which requires the weighted form of the (p, ρ)-Riccati pairs.Definition 3.1.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold, with n ≥ 2.
Let Ω ⊆ M be a domain, p > 1, and ρ ∈ W 1,p loc (Ω) be a positive function with |∇ g ρ| = 1 dv g -a.e. in Ω.Let us fix the continuous functions L, W : (0, sup Ω ρ) → (0, ∞) and the function w A function G satisfying the above conditions is said to be w-admissible for (L, W ).
On the one hand, if ∆ g ρ = L(ρ) dv g -a.e. in Ω, inequality (3.6) follows directly from (3.1) and (3.7).On the other hand, if H n ({x ∈ Ω : ∆ g ρ(x) > L(ρ(x))}) = 0, by (R2) one has that G is nonnegative.Accordingly, we have again by (3.1) and (3.7) the estimates Remark 3.2.For the model space form M n κ and ρ = d x 0 for some x 0 ∈ M n κ fixed, it follows that ∆ g d x 0 = (n − 1)ct κ (d x 0 ) on M n κ except the point x 0 and its cut locus (which is empty for κ ≤ 0, and the antipodal point of x 0 on the κ-sphere S n κ when κ > 0).Therefore, in such cases we choose L(t) = (n − 1)ct κ (t) (with the usual interval restriction when κ > 0) and no sign restriction should be imposed on the function G, see (R2) and the proof of Theorem 3.2.However, on generic Riemannian manifolds, the nonnegativeness of G is indispensable in our arguments.
In the sequel we establish the connection between Riccati and Bessel pairs.According to Duy, Lam and Lu [24, Definition 1.1], if p > 1 and A, B : (0, R) → R are functions with A being of class C 1 , the couple (A, B) is a p-Bessel pair in (0, R) if the ODE has a positive solution in (0, R); for the initial version (p = 2), see Ghoussoub and Moradifam [35,36].
We have the following is a solution of on (0, R), that is precisely (3.5) with equality and L(t) = n−1 t .Proof.Inserting G from (3.9) into (3.10) a simple computation yields (3.8); since all steps can be reversed, the two equations are equivalent.Remark 3.3.Given a Riemannian manifold (M, g) with sectional curvature K ≤ κ for some κ ∈ R, a more appropriate notion for the p-Bessel pair (A, B) instead of (3.8) is Indeed, when κ = 0, equation (3.11) reduces to (3.8), while for κ = 0, the density s κ encodes the curvature and explains the choice of L(t) = (n − 1)ct κ (t).This observation will be crucial in some functional inequalities in the forthcoming sections that will be obtained by means of Riccati pairs, see e.g.Cheng's comparison principle for the first eigenvalue (see Theorem 4.7).
We conclude the section by establishing the relationship between Bessel/Riccati pairs and the classical approach of supersolutions.Remark 3.4.By the theory of supersolutions, if a second order elliptic operator admits a positive supersolution, then the operator is positive in the sense of quadratic forms, see Davies [21,Theorem 4 ) then the operator (−∆ − W ) ≥ 0 is positive; more precisely, one has Observe that if ϕ(x) = y(|x|) for some function y, then (3.12) translates to The equality case of the above relation perfectly agrees with equation (3.8) for A ≡ 1, B = W and p = 2.We note that similar arguments apply in the weighted case and p > 1.In addition, by a suitable Laplace comparison, these arguments can be extended to Riemannian manifolds via relation (3.11).In addition, the above observations extend to Riccati pairs by Proposition 3.1 and Remark 3. Theorem 4.1.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold, with n ≥ 2.
Let Ω ⊆ M be a domain, p > 1, and Proof.Without loss of generality, based on the assumptions |∇ g ρ| p ρ α−p , ρ α ∈ L 1 loc (Ω), we may assume that ρ is positive; indeed, eventually we may replace ρ by ρ + ǫ for some ǫ > 0, and then take the limit with respect to ǫ > 0, by using the above integrability assumptions.A similar reason is developed in [20].
Alternatively, if (M, g) is a complete Riemannian manifold with nonnegative Ricci curvature and Ω ⊂ M is a domain with non-empty piecewise smooth weakly mean convex boundary, then ρ(x) = dist g (x, ∂Ω) is superharmonic a.e. on Ω, see Chen, Leung and Zhao [18, Corollary 2.9].In particular, in the Euclidean setting the above statement can be reversed as well, see Lewis, Li and Li [47].Related study on Finslertype Minkowski spaces can be found in Della Pietra, di Blasio and Gavitone [22].
An immediate consequence of Theorem 4.1 is the estimate of the first Dirichlet eigenvalue of the Riemannian p-Laplacian operator; for simplicity, we consider the unweighted case (α = 0): Corollary 4.1.Let (M, g) be a complete, non-compact n-dimensional Riemannian manifold with n ≥ 2, Ω ⊆ M be a bounded domain, p > 1, and ρ(x) = dist g (x, ∂Ω) for every x ∈ Ω.If −∆ g ρ ≥ 0 in the distributional sense in Ω, then the first Dirichlet eigenvalue of the Riemannian p-Laplacian can be estimated as Using the notation R Ω = sup x∈Ω ρ(x) from Corollary 4.1, in the spirit of Brezis and Marcus [12], and Barbatis, Filippas and Tertikas [5], we provide an improvement of Theorem 4.1 with a suitable reminder term, whenever 1 < p ≤ 2: Theorem 4.2.Under the same assumptions as in Corollary 4.1, if 1 < p ≤ 2, one has for every u ∈ C ∞ 0 (Ω), where In particular, for p = 2, we have Proof.In Theorem 1.1 we choose the functions Note that sup Ω ρ = R Ω and observe that (G) ρ clearly holds, see Remark 4.1/(b).
An elementary computation shows the validity of (4.2).Note that for the reminder term we have R p (t) ≥ 0 for every t ∈ (0, R Ω ) since z := log −1 t eR Ω ∈ (−1, 0) and, due to the fact that p ∈ (1, 2], the function Hardy inequalities and their improvements on Cartan-Hadamard manifolds.In the sequel, as we already noticed, we shall see that our approach perfectly works on Cartan-Hadamard manifolds.Before doing this, the following L p -Hardy inequality is stated on Riemannian manifolds with certain constraints, which has been first established by Kombe and Özaydin [44, Theorem 2.1] for generic p > 1; the initial version for p = 2 is the celebrated result of Carron [14,Theorem 1.4].
A simple consequence of Theorem 4.3 is the following weighted Hardy inequality.
(c) The proof of Theorem 3.1 can be adapted to the compact case as well, where certain Hardy-type inequalities can be produced; such a result will be provided in the sequel.To do this, let (M, g) be an ndimensional compact Riemannian manifold, Ω ⊂ M be a domain and x 0 ∈ Ω.Let w : (0, sup Ω ρ) → (0, ∞) and G : (0, sup Ω ρ) → R be C 1 functions such that (G) x 0 ρ,w : G(ρ)w(ρ) follows by [49].
In the sequel, we provide an alternative approach to establish improved Hardy inequalities on Cartan-Hadamard manifolds; for the simplicity of presentation, we shall consider the case p = 2 and α = 0.The first such result 'interpolates' between Corollary 4.2 and Theorem 4.4.
Theorem 4.5.Let (M, g) be an n-dimensional Cartan-Hadamard manifold, n ≥ 3, and Ω ⊂ M be a bounded domain.Let x 0 ∈ Ω and D Ω = sup x∈Ω d g (x 0 , x).Then for every u ∈ C ∞ 0 (Ω) one has Proof.We are going to apply Theorem 1.2 for the set Ω \ {x 0 }, and the positive function ρ = d x 0 on Ω \ {x 0 } (with Imρ ⊆ (0, D Ω )), as well as and for some C > 0, that will be defined later.First, according to the Laplace comparison principle, see Theorem 2.1/(I)/(i), one has that ∆ g ρ ≥ L(ρ) on Ω \ {x 0 }.Then, we intend to guarantee that (L, W C ) is a (2, ρ)-Riccati pair in (0, D Ω ); to do this, we are looking for a positive function G C which solves the Riccati-type ODE The fundamental solution of the latter equation is given by The function G C is well-defined and positive whenever C ≤ 1 4 .Clearly, we choose the largest possible value for C, i.e., C = 1  4 .With this choice of C, it turns out that (G) ρ is verified on Ω \ {x 0 }.Therefore, (L, W 1/4 ) is a (2, ρ)-Riccati pair in (0, D Ω ), and Theorem 1.2 provides the proof of (4.8) for functions belonging to where j 0,1 ≈ 2.4048 is the first positive root of the Bessel function J 0 , and ω n is the volume of the unit Euclidean ball.Inequality (4.9) has been obtained by Schwarz symmetrization and an ingenious 1-dimensional analysis.
In the sequel, by using our approach based on Riccati pairs, we provide a Riemannian version of the result by Brezis and Vázquez [13], which sheds new light on the appearance of j 0,1 in (4.9); in fact, we prove a kind of interpolation inequality (see also Remark 4.4).Let j ν,k be the k th positive root of the Bessel function J ν of the first kind and degree ν ∈ R.

.11)
A simple reasoning shows that (G) ρ holds on Ω \ {x 0 } for the above choice of G C , thus (L, W C ) is a (2, ρ)-Riccati pair in (0, D Ω ).Now, we are in the position to apply Theorem 1.2 obtaining Ω .This fact is explained by the Schwarz symmetrization argument, rearranging the sets and functions into radially symmetric objects; indeed, under symmetrization, the Dirichlet energy u → Ω |∇u|2 dx non-increases (Pólya-Szegő inequality), the L 2 -norm is preserved (Cavalieri principle), while the Hardy term u → Ω u(x) 2 /|x| 2 dx non-decreases (Hardy-Littlewood-Pólya inequality).In this way, the problem reduces to a ball (with volume Vol(Ω)) and radially symmetric functions; thus a 1-dimensional analysis suffices.
These arguments can be partially repeated in Cartan-Hadamard manifolds: Pólya-Szegő inequality is valid whenever the Cartan-Hadamard conjecture 2 holds (e.g. in dimensions 2, 3 and 4), see Hebey [39], and the L 2 -norm is preserved; however, nothing is known about the Hardy term u → Ω u 2 /d 2 x 0 dv g under symmetrization.
(b) We notice that in the 'complementary' geometric setting, i.e. on Riemannian/Finsler manifolds with nonnegative Ricci curvature, the Hardy term non-decreases under symmetrization, i.e., a Hardy-Littlewood-Pólya-type inequality holds, see Kristály, Mester and Mezei [42] and Kristály and Szakál [43].In this way, Euclidean spaces can be viewed as threshold geometric structures concerning the symmetric rearrangement of Hardy terms.
(c) Note that if in Theorem 4.6 we consider the ball Ω = B x 0 (R) for some R > 0, then by the Bishop-Gromov volume comparison principle, see Theorem 2.1/(I)/(ii), we have that , which is in a perfect concordance with the original inequality (4.9) containing the volume of the set.
(d) Inequality (4.10) trivially interpolates between the inequality of Brezis and Vázquez (ν = 0) and the celebrated Faber-Krahn inequality (ν = n−2 2 ).In the latter case, we obtain a spectral estimate for the first Dirichlet eigenvalue on a domain Ω.Further details are provided in the next section.4.3.Spectral estimates on Riemannian manifolds.Let (M, g) be an n-dimensional Riemannian manifold (n ≥ 2), Ω ⊂ M be a domain, and p > 1.The first Dirichlet eigenvalue of Ω for the p-Laplace-Beltrami operator −∆ g,p on (M, g) is given by In this section we show the applicability of our method to obtain spectral gap estimates on Riemannian manifolds.First, we provide a new proof of Cheng's comparison principle, see Cheng [19] (whose original proof is based on Barta's argument), then the Faber-Krahn inequality on Cartan-Hadamard manifolds, see Chavel [17], as well as McKean's spectral gap estimate are proved.Then, we conclude the section with a short proof of the main result of Carvalho and Cavalcante [15, Theorem 1.1] concerning the lower bound of λ 1,p for the p-Laplacian on generic Riemannian manifolds.Theorem 4.7.(see Cheng [19]) Let (M, g) be an n-dimensional Riemannian manifold with n ≥ 2 and sectional curvature K ≤ κ for some κ ∈ R. Supposing that x 0 ∈ M and 0 < R < min(inj x 0 , π/ √ κ) (with the usual convention, see Theorem 2.1), one has where B κ (R) stands for a ball of radius R in the model space form M n κ .Proof.By standard compactness and symmetrization arguments, we known that λ 1,2 (B κ (R)) is achieved by a positive, radially symmetric, non-increasing function is the non-increasing profile function of v on (0, R), where 0 is the center of the ball B κ (R) ⊂ M n κ .By the Euler-Lagrange equation, it turns out that which can be written equivalently into the form where , ∀t ∈ (0, R).
Note that G κ ≥ 0 on (0, R), due to the fact that v κ is non-increasing and positive in (0, R).
Since (G) ρ also holds, thus G κ verifies all the requirements that (L, W ) is a (2, ρ)-Riccati pair in (0, R).Accordingly, by Theorem 1.2 one has for every On account of Remark 4.2, the latter inequality is valid for every function in C ∞ 0 (B x 0 (R)), which concludes the proof of (4.12).Remark 4.5.We notice that (4.13) is also equivalent to In the case when the domain is not a ball, as in Cheng's result, a more powerful argument is needed; we shall consider only the case κ = 0, which corresponds to the famous Faber-Krahn inequality on Cartan-Hadamard manifolds: Theorem 4.8.Let (M, g) be an n-dimensional Cartan-Hadamard manifold (n ≥ 2) which satisfies the Cartan-Hadamard conjecture, and Ω ⊂ M be a bounded domain.Then we have where Ω * ⊂ R n is a ball with Vol(Ω * ) = Vol g (Ω).
Proof.Let u ∈ C ∞ 0 (Ω) \ {0} be arbitrarily fixed; without loss of generality, we may assume that u ≥ 0. Let u * : Ω * → [0, ∞) be the symmetric rearrangement of u on the Euclidean ball Ω * ⊂ R n (without loss of generality, we may assume that its center is the origin), see Hebey [39].By layer cake representation and the validity of the Cartan-Hadamard conjecture on (M, g), both the Cavalieri principle and Pólya-Szegő inequality hold, thus . Now, for the right hand side, we are falling into the setting of Theorem 4.7 in the case κ = 0. Therefore, let R = (Vol g (Ω)/ω n ) 1/n , Ω * = B 0 (R) and W C ≡ C = λ 1,2 (B 0 (R)).Then then equation (4.14) for κ = 0 has the solution A similar argument as in the proof of Theorem 4.6 shows that the best choice is Clearly, G 0 is positive on (0, R) and (G) ρ holds; therefore, (L, W C ) is a (2, ρ)-Riccati pair in (0, R).The rest similarly follows as in the proof of Theorem 4.7.
following result is McKean's spectral gap estimate, established by McKean [48] for p = 2 by using fine properties of Jacobi fields; our argument is based on Riccati pairs.Theorem 4.9.Let (M, g) be an n-dimensional Cartan-Hadamard manifold (n ≥ 2), with sectional curvature Proof.Let x 0 ∈ M be arbitrarily fixed, and let ρ = d x 0 > 0 on M \ {x 0 }, as well as for some C, c > 0 that will be determined later.Clearly, (G) ρ holds.The Laplace comparison, see Theorem 2.1/(I)/(i), yields Therefore, by choosing C = (n − 1)c √ −κ−(p − 1)c p ′ one has to conclude the proof of (4.16).Another improvement of the McKean's spectral gap estimate will be provided in the next section.
We conclude this section with providing a short proof of the main result of Carvalho and Cavalcante [15, Theorem 1.1], valid on generic Riemannian manifolds: Theorem 4.10.Let (M, g) be a Riemannian manifold, and Ω ⊂ M be a domain.Given p > 1, we assume that there exists a function ρ : Ω → R such that |∇ g ρ| ≤ a and ∆ g,p ρ ≥ b for some a, b > 0. Then 1) . (4.17) Proof.We apply the additive form of Theorem 1.1 with the choice G ≡ c for some c > 0 (which will be determined later) and H(s) = |s| p p for every s ∈ R. Since the assumptions of Theorem 1.1 are trivially verified, by (1.1) and the facts that p > 1, |∇ g ρ| ≤ a and ∆ g,p ρ ≥ b, we obtain for every u ∈ C ∞ 0 (Ω) that Once we maximize the expression cb−(p − 1)c p ′ a p in c > 0, the extremal point is c = ( b pa p ) p−1 , while the maximum is precisely the value b p p p a p(p−1) in (4.17).The proof is complete., their approach also works on Cartan-Hadamard manifolds with sectional curvature K ≤ κ < 0. In the sequel, we provide here a simple proof of the same result by using our approach, based on Riccati pairs.Theorem 4.11.Let (M, g) be an n-dimensional Cartan-Hadamard manifold (n ≥ 3), having sectional curvature K ≤ κ < 0, and x 0 ∈ M .Then, for every λ ∈ [n − 2, (n−1) 2
(c) We notice that Berchio, Ganguly and Grillo [6, Theorem 2.5] provided a more general version of inequality (4.21) under a mild curvature assumption, which can be stated as follows.Let (M, g) be an n-dimensional Riemannian manifold (n ≥ 3), x 0 ∈ M be a point such that Cut x 0 = ∅ (here, Cut x 0 stands for the cut locus of x 0 ), and the sectional curvature in the radial direction satisfies where ψ is a positive, increasing C 2 function with ψ(0) = ψ ′′ (0) = 0, ψ ′ (0) = 1 and Observe that for κ < 0 and the above inequality reduces to (4.21).As expected, inequality (4.25) can be obtained via Riccati pairs as well.Indeed, the curvature assumption (4.23) implies the following refined Laplace comparison principle see Greene and Wu [38].Thus inequality (4.25) follows from Theorem 3.2 by choosing the positivity of G being guaranteed by the boundary conditions on ψ and relation (4.24).This example shows that Riccati pairs can be efficiently used to extend well-known Hardy inequalities to manifolds satisfying -instead of a universal curvature bound -the pointwise curvature assumption (4.23).
We conclude this subsection with a short proof of the following inequality, see Akutagawa and Kumura [2, Theorem 1.3/(5)].
Theorem 4.12.Let (M, g) be an n-dimensional Cartan-Hadamard manifold with n ≥ 2 and sectional curvature K ≤ κ < 0. Let x 0 ∈ M , R > 0 and Ω = M \ B x 0 (R).Then for every and u ∈ C ∞ 0 (Ω) one has Proof.The proof is similar to the one from Theorem 4.11, except the choice of W : (R, ∞) → R which is defined by One can check that the function is positive and verifies both (G) ρ and the ODE where ρ = d x 0 and L(t) = (n − 1)ct κ (t); thus (L, W ) is a (2, ρ)-Riccati pair on (R, ∞), and the claim follows from Theorem 1.2.
Proof.We first prove that (i) and (ii) are equivalent, i.e., they can be deduced from each other.Indeed, let us assume that (i) holds, i.e., by notation's convenience, if α β > 0 and m ≤ n−2 2 , then for every Thus, we may apply (4.28) with the latter choices, obtaining after a simple computation precisely (4.27).The converse can be performed similarly.Thus, it is enough to prove (i).Accordingly, we assume that αβ > 0 and m ≤ n−2 2 .Define , ∀t > 0, and for some C > 0, that will be determined later.Consider the ODE G ′ (t) + w ′ (t) w(t) + L(t) G(t) − G(t) 2 = W C (t), ∀t > 0.
Closely related to the latter observation is the fact that G contains the expression zf ′ (z)/f (z) for f (z) = 2 F 1 (a, b; 1; −z), z > 0, which is used to characterize the order of starlikeness (with respect to zero) of the function f , see e.g.Küstner [46].Although one can find various results in the literature, to the best of our knowledge, there is no full characterization of the order of starlikeness for Gaussian hypergeometric functions with respect to the full spectrum of parameters.However, under specific constraints on the parameters, we provide the following partial result on Cartan-Hadamard manifolds; for simplicity of presentation, we only focus to the inequality of type (4.26); by equivalence, it can be also formulated in terms of its 'dual' (4.27).In particular, since B ≥ A > 0, it is enough to prove that where µ 0 : [0, 1] → [0, 1] is a non-decreasing function satisfying µ 0 (1)−µ 0 (0) = 1, see Küstner [46].Clearly, by the latter representation our claim easily follows.
Remark 4.9.As we already pointed out in Remark 4.8, numerical tests confirm the positiveness of G for every a, b > 0 and α, β, m ∈ R, whose proof requires some specific arguments from the theory of special functions; at this moment, such approach is not available.In particular, we expect to cancel the additional hypothesis αβ + αβ(αβ + 2(n − 2m − 2)) ≤ 2 from Theorem 4.14.
Proof.For simplicity of notations, let γ = 1 + α p−1 > 0. Let u 0 : M → (0, ∞) be defined by u 0 = e We first claim that the latter integrals are well defined.To see this, we observe that κ = 0 implies γ > 0 and V κ (R) = ω n R n ; while κ < 0 implies γ > 1 and where we used the fact that α + γ − 1 = p ′ α.Since the two marginal terms are equal (and finite), we should have equality in each steps.In particular, we have that ∆ g d x 0 = (n − 1)ct κ (d x 0 ) on M \ {x 0 }, thus by Theorem 2.1 it follows that (M, g) is isometric to the model space form M n κ .
Remark 5.2.The proof of Theorem 5.2 is much simpler than the original rigidity argument performed in Kristály [41] and Nguyen [53], where certain ordinary differential inequalities/equations are compared.In the special case when M = H n −1 , the latter inequality reduces to [8,Corollary 3.1].In a similar way, further inequalities can be obtained via the results from §4.4.

2 .
Hardy inequalities and their improvements on Cartan-Hadamard manifolds 12 4.3.Spectral estimates on Riemannian manifolds 16 4.4.Interpolation: Hardy inequality versus McKean spectral gap 19 4.5.Ghoussoub-Moradifam-type inequalities 22 5. Applications II: Multiplicative Hardy-type inequalities 25 and dv g denote the distance function and the canonical measure induced by the metric g on M , respectively.Denote B x 0 (R) = {y ∈ M : d g (x 0 , y) < R} the open metric ball with center x 0 ∈ M and radius R > 0. The volume of a bounded open set S ⊂ M is Vol g (S) = S dv g = H n (S).

4. 4 .
Interpolation: Hardy inequality versus McKean spectral gap.The main result of this section is to prove an interpolation between the Hardy inequality and McKean's spectral gap, established first by Berchio, Ganguly, Grillo and Pinchover [8, Theorem 2.1]; although the authors stated their result on the hyperbolic space H n −1

Remark 5 . 3 . 4 M u 2 dv g 2 .
By using additive-type inequalities we can produce multiplicative inequalities in spirit of Berchio, Ganguly, Grillo and Pinchover [8, Section 3].For example, inequality (4.20) combined with Schwarz inequality imply (in the same geometric context as in Corollary 4.3) for every u ∈ C ∞ 0 (M ) that M |∇ g u| 2 dv g − (n − 2)|κ| M u 2 dv g M d 2 x 0 u 2 dv g ≥ (n − 2) 2 by Remark 4.2 we have the validity of (4.8) for every function in C ∞ 0 (Ω), which concludes the proof.Another improvement of the Hardy inequality in R n is due to Brezis and Vázquez [13, Theorem 4.1]; more precisely .21) We note that inequalities from Theorem 4.11 and Corollary 4.3 are known to be critical on H n −1 , i.e., the right hand sides of the inequalities cannot be improved with weights that are strictly larger somewhere, see Devyver, Fraas and Pinchover [23, Definition 2.1].The criticality proofs are formulated using the supersolutions approach, however by Remarks 3.3 & 3.4, they can be adapted to Riccati pairs.To see this, let us consider (4.21) on H n −1 and recall the definition of W λ from the proof of Theorem 4.11.As we learned from Possible extensions of these inequalities to Cartan-Hadamard manifolds will be also discussed (see Remark 4.8 and Theorem 4.14), where some technical difficulties arise.Theorem 4.13.(see [35, Theorem 2.12]) Let a, b > 0 and α, β, m ∈ R. The following inequalities hold: