J. A. C. Kolk

Spectra of compact locally symmetric manifolds of negative curvature

Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

On the transverse symbol of vectorial distributions and some applications to harmonic analysis

The transverse symbol of a vector-valued distribution supported on a submanifold is introduced and a micro-local vanishing theorem for spaces of such distributions invariant under a Lie group is proved. We give transparent proofs of results of Bruhat and Harish-Chandra on the irreducibility of parabolically or normally induced representations, and of Harish-Chandra in Whittaker theory.

On Euler numbers, Hilbert sums, Lobachevskii integrals, and their asymptotics

Convolution. Denote by x the characteristic function of the interval [ 4,

Convergence of Distributions

1 c R, and by xn = x * . . * x the n-fold convolution product (n E N). Then x,, is a spline function, viz. a CnM2-function on R that is piecewise polynomial of degree n 1, for y1 2 2, having [ 2, 51 as its support. Indeed, let xi’ (0 I i < n) be the i-th derivative of xn. We have the equality ~(‘1 = S-4 6; of distributions on R, where S, denotes the Dirac measure at x E R; whence

Distributions with Compact Support

We now introduce a notion of convergence in the linear space\(\mathcal{D}^\prime(X)\) of distributions on an open set X in R n.

Convolution of Distributions

If u is locally integrable on an open set X in R n and has compact support, the integral \(u(\phi ) = \int_X {u(x)\phi (x)\ dx}\) is absolutely convergent for every \(\phi \in C^\infty (X),\) as follows from a slight adaptation of the proof of Theorem 3.5. More generally, every distribution with compact support can be extended to a continuous linear form on \(C^\infty (X).\)

Multidimensional Real Analysis I: Contents

Convolution involves translation; that makes it difficult to define the former operation for functions or distributions supported by arbitrary open subsets in R n. Therefore we initially consider objects defined on all of R n.

Multidimensional Real Analysis I by J. J. Duistermaat

Mathematical mistakes are indicated by the symbol M; the majority of the corrections are minor textual changes. Example 1.3.10 on page 16. Insert the following words immediately after Example 1.3.10: (Pï uckers conoid). Lemma 2.1.1 on page 40. There exists a slightly less computational proof of this lemma. In fact, apply Lemma 1.1.7.(ii) to Ah = 1≤j≤n h j Ae j , which follows from Formula (1.1), in order to obtain Ah ≤ 1≤j≤n Now the Cauchy–Schwarz inequality from Proposition 1.1.6 immediately yields Ah ≤ h 1≤j≤n Ae j 2 = h A Eucl. Penultimate sentence preceding Proposition 2.2.1 on page 42. Add the following to the sentence: , as will be shown in Definition 2.2.2. Definition 2.2.2 on page 43 Replace the first part of the second sentence in the definition by the following: Then the mapping f is said to be differentiable at a Lemma 2.2.7 on page 45. As additional motivation for the proof of Hadamards Lemma one may offer the following argument. On the one hand, for differentiable f , one requires f (x) − f (a) = φ a (x)(x − a). On the other hand, in view of x − a 2 = (x − a) t (x − a) ∈ R, the reformulation of differentiability in Formula (2.10) implies f (x) − f (a) = Df (a)(x − a) + a (x − a) = Df (a)(x − a) + 1 x − a 2 a (x − a)(x − a) t (x − a). A formal division of the right–hand side by x − a now suggests the formula for φ a (x) as given in the proof.

Multidimensional Real Analysis I: Acknowledgments

Mathematical mistakes are indicated by the symbol M; the majority of the corrections are minor textual changes. Example 1.3.10 on page 16. Insert the following words immediately after Example 1.3.10: (Pï uckers conoid). Lemma 2.1.1 on page 40. There exists a slightly less computational proof of this lemma. In fact, apply Lemma 1.1.7.(ii) to Ah = 1≤j≤n h j Ae j , which follows from Formula (1.1), in order to obtain Ah ≤ 1≤j≤n Now the Cauchy–Schwarz inequality from Proposition 1.1.6 immediately yields Ah ≤ h 1≤j≤n Ae j 2 = h A Eucl. Penultimate sentence preceding Proposition 2.2.1 on page 42. Add the following to the sentence: , as will be shown in Definition 2.2.2. Definition 2.2.2 on page 43 Replace the first part of the second sentence in the definition by the following: Then the mapping f is said to be differentiable at a Lemma 2.2.7 on page 45. As additional motivation for the proof of Hadamards Lemma one may offer the following argument. On the one hand, for differentiable f , one requires f (x) − f (a) = φ a (x)(x − a). On the other hand, in view of x − a 2 = (x − a) t (x − a) ∈ R, the reformulation of differentiability in Formula (2.10) implies f (x) − f (a) = Df (a)(x − a) + a (x − a) = Df (a)(x − a) + 1 x − a 2 a (x − a)(x − a) t (x − a). A formal division of the right–hand side by x − a now suggests the formula for φ a (x) as given in the proof.

Multidimensional Real Analysis I: Frontmatter

Mathematical mistakes are indicated by the symbol M; the majority of the corrections are minor textual changes. Example 1.3.10 on page 16. Insert the following words immediately after Example 1.3.10: (Pï uckers conoid). Lemma 2.1.1 on page 40. There exists a slightly less computational proof of this lemma. In fact, apply Lemma 1.1.7.(ii) to Ah = 1≤j≤n h j Ae j , which follows from Formula (1.1), in order to obtain Ah ≤ 1≤j≤n Now the Cauchy–Schwarz inequality from Proposition 1.1.6 immediately yields Ah ≤ h 1≤j≤n Ae j 2 = h A Eucl. Penultimate sentence preceding Proposition 2.2.1 on page 42. Add the following to the sentence: , as will be shown in Definition 2.2.2. Definition 2.2.2 on page 43 Replace the first part of the second sentence in the definition by the following: Then the mapping f is said to be differentiable at a Lemma 2.2.7 on page 45. As additional motivation for the proof of Hadamards Lemma one may offer the following argument. On the one hand, for differentiable f , one requires f (x) − f (a) = φ a (x)(x − a). On the other hand, in view of x − a 2 = (x − a) t (x − a) ∈ R, the reformulation of differentiability in Formula (2.10) implies f (x) − f (a) = Df (a)(x − a) + a (x − a) = Df (a)(x − a) + 1 x − a 2 a (x − a)(x − a) t (x − a). A formal division of the right–hand side by x − a now suggests the formula for φ a (x) as given in the proof.