Robert S. Strichartz

Analysis of the Laplacian on the Complete Riemannian Manifold

To what extent can the classical analysis on Euclidean space be carried over to the setting of a complete Riemannian manifold? This is the problem we address in this paper. We are particularly interested in the noncompact case, since analysis on compact manifolds-with or without the Riemannian structure-is quite well understood. Also we strive to avoid making unnecessary hypotheses on the manifold. We will show that much of the classical theory of he Laplacian remains valid for the Laplace-Beltrami operator on a complete Riemannian manifold. This includes the essential self-adjointness, properties of the heat semi-group e IA, the Bessel potentials (I -A)=I*, the Riesz potentials (-,)-,‘*, and the Sobolev spaces based on Bessel potentials. Our results are complementary to those of Aubin [2,3], who studies Sobolev spaces defined by covariant derivatives, and Yau 135) who studies the heat semi-group under the assumption that the Ricci curvature is bounded on both sides. Several other recent papers [6-8, 331 study rncbr’z detailed properties of the heat semi-group under special assumptions on the curvature. The essential self-adjointness has previously been established in [9,23]. We also give some generalizations of an inequality of McKean [ 191, which applies only to simply-connected manifolds with negative curvature (bounded above by a negative constant -k), and which implies that the spectrum of the Laplacian is bounded away from zero. We prove ]lf]], < (p/(n I)@) l/Vf&, for I< p < co and f compactly supported, and we investigate under what circumstances Vf E Lp implies S c e Lp for some

The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations

The Campbell-Baker-Hausdorff-Dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems. Specifically, suppose u(t) takes values in a C∞ manifold and satisfies the initial value problem u′(t) = A(t)(u(t)), u(0) = a, where A(t) is a vector field on the manifold depending continuously on t. Then u(t) = exp z(t)(a) (here exp z(a) means the solution at s = 1 to v′(s) = z(v(s)), v(0) = a) for z(t)∼∑r=1∞∑ωϵPr(−)e(ω)r2r−1e(ω)∫Tr(t) , x[⋯[A(Sω(1))A(Sω(2))]⋯]A(Sω(r))]ds as, t → 0, where Tr(t)={sϵRr:0<S1<S2⋯<Sr<t}, Pr is the set of permutations on {1,…, r}, e(σ) is the number of errors in ordering consecutive terms in {σ(1),…, σ(r)}, and [ ] is the usual commutator of vector fields. Under appropriate analyticity assumptions the series for z(t) is convergent for small t. This formula gives an explicit formulation of results of K.-T. Chen published in 1957. Applications are given to problems in sub-Riemannian geometry, and to improving convergence estimates for the Campbell-Baker-Hausdorff-Dynkin formula in the context of Banach algebras. Our formula can be thought of as a noncommutative generalization of the familiar formula u(t) = a exp(∝0t A(s) ds) in the scalar linear case, in the same way that the Campbell-Baker-Hausdorff-Dynkin formula is a noncommutative generalization of the familiar formula exey = ex + y.

Convolutions with kernels having singularities on a sphere

We prove that convolution with (1xI2)+ and related convolutions are bounded from LI to Lq for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing il/p-l/q, in contrast with fractional integration where il/p-l/q is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolevs inequality.

Fourier asymptotics of fractal measures

Abstract A measure μ on R n will be called locally uniformly α-dimensional if μ(Br(x)) ⩽ crα for all r ⩽ 1 and all x, where Br(x) denotes the ball of radius r about x. For ƒ ϵ L 2 (dμ) , the measure ƒ dμ is in I′ so (ƒ dμ) is well-defined. We show it is locally in L2 and sup r⩾1 r δ−n ∫ B r (y) |(f dμ) ^ (ξ)| 2 dξ ⩽ c ∥f∥ 2 · Under additional hypotheses we show that lim r→∞ r δ−n ∫ B r (y) |(f dμ) ^ (ξ)| 2 (ξ)| 2 dξ is comparable in size to ∥ƒ∥ 2 2 . A number of other related results are established. The special case when α is an integer and μ is the surface measure on a C1 manifold was treated by S. Agmon and L. Hormander (J. Analyse Math. 30, 1976, 1–38).

Function spaces on fractals

Abstract We construct function spaces, analogs of Holder–Zygmund, Besov and Sobolev spaces, on a class of post-critically finite self-similar fractals in general, and the Sierpinski gasket in particular, based on the Laplacian and effective resistance metric of Kigami. This theory is unrelated to the usual embeddings of these fractals in Euclidean space, and so our spaces are distinct from the function spaces of Jonsson and Wallin, although there are some coincidences for small orders of smoothness. We show that the Laplacian acts as one would expect an elliptic pseudodifferential operator of order d+1 on a space of dimension d to act, where d is determined by the growth rate of the measure of metric balls. We establish some Sobolev embedding theorems and some results on complex interpolation on these spaces.

Fractal differential equations on the Sierpinski gasket

Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian. For harmonic functions, biharmonic functions, and Dirichlet eigenfunctions of Δ, we give efficient algorithms to compute the solutions exactly, we display the results of implementing these algorithms, and we prove various properties of the solutions that are suggested by the data. Completing the work of Fukushima and Shima [8] who computed the Dirichlet eigenvalues and their multiplicities, we show how to construct a basis (but not orthonormal) for the eigenspaces, so that we have the analog of Fourier sine series on the unit interval. We also show that certain eigenfunctions have the property that they are a nonzero constant along certain lines contained in SG. For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms. We give strong evidence that the analog of finite propagation for the wave equation does not hold because of inconsistent scaling behavior in space and time.

Lp harmonic analysis and Radon transforms on the Heisenberg group

In this paper, harmonic analysis in the Heisenberg group Heisn = Cn × R with group law (z, t) · (z′, t′) = (z + z′, t + t′ − 12Im z · z′) is taken to mean the joint spectral theory of the operators L (Heisenberg Laplacian) and T = ∂∂t. The spectral decomposition of ƒ e L2 is given explicitly as ƒ = Σλ Σe∝0κƒ ∗ ϑλ, κ, e dλ, where e = ±1, k = 0, 1, 2, … and ϑλ, κ,e(z, t) = λn(2π)n + 1 (n + 2k)n · 1 × exp(−ieλtn + 2k) exp (−λ ¦z¦24(n + 2k)) Lkn − 1(λ¦z¦22(n + 2k)), where Lkn − 1 denotes the Laguerre polynomial. The eigenvalues are iTƒ ∗ ϑλ,κ,e = (eλ(n + 2k)) ƒ ∗ ϑλ,κe and −Lƒ∗ϑλ,k,e·=λƒ∗ϑλ,k,e·. This decomposition is essentially well known, but has not previously been described in this light. It is also the decomposition into irreducibles of the representation of the Heisenberg motion group (semi-direct product of Heisn and U(n)) on L2(Heisn). The main result is a summability in Lp of the decomposition for ƒ e Lp, 1 < p < ∞, ƒ = limr → 1 Σλ Σβ rk∝0∞ ƒ ∗ ϑλ,κ,edλ. For p = 1 only a weaker substitute is available, and for p = ∞ the decomposition is false. There is also a Plancherel formula for ƒ ϵ L2, ‖ƒ2=2φ∑k∑e(n+2k)∫0∞|ƒ∗ϑλ,k,e(z,0)|2dzdλ A number of explicit examples of the decomposition are computed, including part of the Schrodinger propagator eisL. The Heisenberg Radon transform is defined by Rƒ(z, t) = ∝n ƒ((z, t) · (w, 0)) dw. This operator was studied by Geller and Stein. It is shown to be bounded from Lp to Lq if and only if p = (2n + 2)(2n + 1) and q = 2n + 2. Some generalizations of these results to the free step two nilpotent Lie groups are given, but the results are not as complete.

Harmonic analysis on hyperboloids

Abstract The regular representation of O ( n , N ) acting on L 2 ( O(n, N) O(n, N − 1) ) is decomposed into a direct integral of irreducible representations. The homogeneous space O(n, N) O(n, N − 1) is realized as the Hyperboloid H = {(x, t) ϵ R n + N : ¦ t ¦ 2 − ¦ x ¦ 2 = 1} . The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □ h on H , which is the tangential part of the operator □ = Δ x − Δ t on R n + N . The spectrum of □ h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □ u = 0 which vanish in the region ¦ t ¦ , and a continuous part H − . As a representation of O ( n , N ), H − ⊕ H − is unitarily equivalent to the regular representation on L 2 of the cone {(x, t) : ¦ x ¦ 2 = ¦ t ¦ 2 } , and the intertwining operator is obtained by solving the equation □ u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL (2, R ).

Harmonic analysis as spectral theory of Laplacians

By writing the Fourier inversion formula on Euclidean space in polar coordinates, we obtain f(x) = ∝0∞ fgl(x) dx, where Δfgl(x) = −λ2fgl(x), which is the spectral theory of the Laplacian. How do properties of f relate to properties of the family of eigenfunctions fgl? Answers are provided for the following spaces: L2, S, S′, D (in odd dimensions). Analogous results are obtained for harmonic analysis on hyperbolic space, constant curvature semi-Riemannian spaces, the Heisenberg group, and for differential forms on hyperbolic space. For f ϵ L2 there is a “Plancherel Formula” ‖f‖22=πlimt → ∞∫o∞1t∫bt(z) |fκ(x)|2dx dκ where Bt(z) denotes the ball of radius t about an arbitrary point z, which is independent of the dimension, and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted. For certain semi-Riemannian symmetric spaces, we obtain a “Paley-Wiener Theorem” that explains the role of discrete series in producing functions of compact support, and also involve certain non-unitary representations that contain the discrete series representations. For the Heisenberg group the “Plancherel Formula” is of a different nature, requiring that the eigenfunctions be almost periodic in one variable.

Convergence of mock Fourier series

AbstractFor certain Cantor measures μ on ℝn, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse2πiγ·x for λεΛ. a discrete subset of ℝn called aspectrum for μ. For anyL1 functionf, we define coefficientscγ(f)=∝f(y)e−2πiγiydμ(y) and form the Mock Fourier series ∑λ∈Λcλ(f)e2πiλ·x. There is a natural sequence of finite subsets Λn increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by