# Heat kernel estimates and L p-spectral theory of locally by Andreas Weber

By Andreas Weber

During this paintings we derive top Gaussian bounds for the warmth kernel on in the neighborhood symmetric areas of non-compact variety. moreover, we ensure explicitly the Lp-spectrum of in the community symmetric areas M whose common overlaying is a rank one symmetric area of non-compact sort if both the basic crew of M is small (in a undeniable experience) or if the elemental staff is mathematics and M is non-compact.

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L2 -Eigenfunctions of the Laplace-Beltrami operator Proof. We first remind the reader of the heat kernel estimate (cf. 22) k(t, x˜, y˜) ≤ C1 t−n/2 e2µt P (s; x˜, x˜) P (s; y˜, y˜), where s > δ(Γ). If we choose s > 2||ρ||, the Poincar´e series is (for fixed s) bounded (cf. 4). We obtain ˜ k(t, x˜, y˜) ≤ C2 t−n/2 e2µt =: C(t). This yields for any f ∈ L1 (M ) the estimate ˜ ||e−t∆M f ||L∞ ≤ C(t)||f ||L1 . Therefore, e−t∆M : L1 (M ) → L∞ (M ) is bounded and the operator norm is bounded ˜ by C(t). Because of M k(t, x˜, y˜) dvol(˜ y ) ≤ 1, we have for all f ∈ L∞ (M ): ||e−t∆M f ||L∞ ≤ ||f ||L∞ .

The function σ is differentiable, Γ-invariant, and defines therefore a function on the quotient space Γ\X. Furthermore, we have: (a) There is a constant c > 1 such that 1 c P (s; x, x) ≤ σ(x) ≤ c P (s; x, x). (b) There is a constant F with |∆X σ(x)| ≤ F σ(x). In particular, we have − ∆σX σ ≥ −F . Proof. The differentiability of σ and the Γ-invariance are evidently clear. For the proof of the remaining assertions we first remark that the triangle inequality implies P (s; y, y) ≤ e2sd(x,y) P (s; x, x).

Furthermore, we can find a complete orthonormal system of the Hilbert space L2 (M ) consisting of eigenfunctions for ∆M . Elliptic regularity implies that every eigenfunction Φ is differentiable: Φ ∈ C ∞ (M ). 1) j=1 with uniform convergence on subsets of the form [a, ∞) × M × M , a > 0. The next theorem shows that the heat kernel K on a compact Riemannian manifold converges for t → ∞ exponentially in t and uniformly in x, y ∈ M to the limit vol 1(M ) . Because of the physical interpretation of the heat equation and the heat kernel, this is plausible: The heat tries to diminish differences of temperature in M by flowing from regions of high temperature to regions of low temperature.