On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries
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Abstract
In this note we prove an analogue of the Rayleigh–Faber–Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere \(\mathbb {S}^{n}\) and on the real hyperbolic space \(\mathbb {H}^{n}\). It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on \(\mathbb {H}^{n}\) and prove the Hong–Krahn–Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.
Keywords
Convolution operators nsphere Real hyperbolic space Rayleigh–Faber–Krahn inequality Hong–Krahn–Szegö inequalityMathematics Subject Classification
35P99 47G40 35S151 Introduction
In this note we establish a similar result also on \(\mathbb S^{n}\) and \(\mathbb H^{n}\) thus completing the picture now for all complete, connected, simply connected Riemannian manifolds of constant sectional curvature. In the case of the real hyperbolic space \(\mathbb H^{n}\) we also establish the analogue of the Hong–Krahn–Szegö inequality on \(\mathbb R^{n}\), namely, the description of \(\Omega \) for which the second eigenvalue is maximised. In fact, our results apply to a more general class of convolution type operators than the Riesz transforms (1.2) that we will describe further.
For the Riesz potential operators on \(\mathbb R^{n}\) results analogous to those of the present note have been obtained by Rozenblum and the authors in [8] (see also [9] for the logarithmic potential operator). Thus, here we restrict our attention to \(\mathbb S^{n}\) and \(\mathbb H^{n}\). So, let M denote \(\mathbb S^{n}\) or \( \mathbb H^{n}\).
In this note we prove the Rayleigh–Faber–Krahn inequality for the integral operator \(\mathcal {K}_{\Omega }\), i.e. it is proved (in Theorem 2.1) that the geodesic ball is a maximiser of the first eigenvalue of the integral operator \(\mathcal {K}_{\Omega }\) among all domains of a given measure in M. The proof is based on the Riesz–Sobolev inequality on M (established in [3]) allowing the use of the symmetrization techniques.

Rayleigh–Faber–Krahn type inequalities: the first eigenvalue of \(\mathcal {K}_{\Omega }\) is maximised on the geodesic ball among all domains of a given measure in \(\mathbb S^{n}\) or \(\mathbb H^{n}\);

Hong–Krahn–Szegö type inequality: the maximum of the second eigenvalue of (positive) \(\mathcal {K}_{\Omega }\) among bounded open sets with a given measure in \(\mathbb H^{n}\) is achieved by the union of two identical geodesic balls with mutual distance going to infinity.
2 Main results
Theorem 2.1
Remark 2.2
In other words Theorem 2.1 says that the operator norm \(\Vert \mathcal {K}_\Omega \Vert _{{\mathscr {L}}(L^{2}(\Omega ))}\) is maximised in a geodesic ball among all domains of a given measure. We also note that since \(\lambda _{1}\) is the eigenvalue with the largest modulus according to the ordering (1.5), we will show in Lemma 3.1 that \(\lambda _{1}\) is actually positive. Therefore, (2.2) is the inequality between positive numbers.
We are also interested in maximising the second eigenvalue of positive operators \(\mathcal {K}_{\Omega }\) on \(\mathbb H^{n}\) among open sets of given measure.
Theorem 2.3
A similar type of results on \(\mathbb R^{n}\) is called the Hong–Krahn–Szegö inequality. See, for example, [5] for further references. We note that in Theorem 2.3 we consider only domains \(\Omega \subset \mathbb H^{n}\) for which \(\mathcal {K}_{\Omega }\) are positive operators. However, this can be relaxed:
Remark 2.4
The statement of Theorem 2.3 and its proof remain valid if we only assume that the second eigenvalues \(\lambda _{2}(\Omega )\) of considered operators \(\mathcal {K}_{\Omega }\) are positive.
Remark 2.5
We note that the proofs in the sequel work equally well also in \(\mathbb R^{n}\) and the statements of Theorems 2.1 and 2.3 are valid with \(\mathbb H^{n}\) replaced by \(\mathbb R^{n}\). See also [10] for the announcement.
In the case of \(\mathbb R^{n}\) and Riesz transforms (1.2), Theorem 2.3 holds without the positivity assumption since the Riesz transforms \(\mathcal {R}_{{\alpha },{\Omega }}\) on \(\mathbb R^{n}\) are positive, see [8].
We do not have a version of Theorem 2.3 on the spheres \(\mathbb S^{n}\) because the assumption (2.3) does not make sense due to compactness of the sphere.
3 Proof of Theorem 2.1
Since the integral kernel of \(\mathcal {K}\) is positive, the following statement, sometimes called Jentsch’s theorem, applies. However, for completeness of this note we restate and give its proof on the symmetric space M (that is, \(\mathbb {S}^{n}\) or \(\mathbb {H}^{n}\)).
Lemma 3.1
The first eigenvalue \(\lambda _{1}\) (with the largest modulus) of the convolution type compact operator \(\mathcal {K}\) is positive and simple; the corresponding eigenfunction \(u_{1}\) can be chosen positive.
Proof of Lemma 3.1
Since \(u_{1}\) is positive it follows that \(\lambda _{1}\) is a simple. In fact, if there were an eigenfunction \(\widetilde{u}_{1}\) linearly independent of \(u_{1}\) and corresponding to \(\lambda _{1}\), then for all real c every linear combination \(u_{1}+c\widetilde{u}_{1}\) would also be an eigenfunction corresponding to \(\lambda _{1}\) and therefore, by what has been proved, it could not become zero in \(\Omega \). As c is arbitrary, this is impossible. Finally, it remains to show \(\lambda _{1}\) is positive. It is trivial since \(u_{1}\) and the kernel are positive. \(\square \)
Proof of Theorem 2.1
4 Proof of Theorem 2.3
Proof of Theorem 2.3 is similar to the case of \(\mathbb {R}^{n}\). However, the proof does not work for \(\mathbb {S}^{n}\) since we use the decay property of the kernel at infinity.
Proof Theorem 2.3
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