Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups

Abstract

However, since it has not yet been published in the general case, we outline the method of obtaining it. Let be the Poisson kernel in H n+1 , where xeU n , £eOT, y>0, <o: = (x,y)eH n+1 , and || -|| denotes the Euclidean norm in R". Then one can show in a standard way that the expressionis the kernel of the operator (-L n + 1 -t(n -t)I)~r in the region where dm is the Lebesgue measure on R" and L r t + 1^0 is the hyperbolic Laplacian on H rt+1 . (See the above-mentioned references.) Now let K(co, 0)') be the kernel of an invariant integral operator on H n+1 . If we define the Selberg function h bywhere do is the invariant measure on H" +1 , y(co) is the y coordinate of co, and we have assumed that JH \K((o, (o')then h is clearly bounded on the spectrum and this implies that the corresponding operator with kernel K((o, co') is bounded on L 2 . Consequently one has the representation /•Jn+joo /.where the integral with respect to s must be regarded initially as lim r _* j the limit being taken in the L 2 sense. However, if we choose the function h in such a way that it satisfies h(s) = h(n -s) and has the right growth inside a strip around Re(s) = \n, then the double integral in (2.1) is absolutely convergent. Next, we need the following identity:Proof. We split this case to the following two cases: (i) a>\(n + l) and (ii) i Case (i). It suffices to show the validity of Relation (b). Relation (a) follows in a similar way, but it is actually easier. We have (u + I ) " " = r(or)" 1 f ,«-i e -'(«+ §) dt. Jo This implies that K n (f ; a) = r(ar)" 1 [ V v^ ; a) dt, J where K n (t; o) is the real heat kernel on H". Now bearing in mind the recurrence relations for the heat kernel we get. LEMMA 4.2. Let f e C C (U) and let df/du be piecewise continuously differentiable. Then the recurrence relations (a) and (b) hold provided that n > 1. Proof. In this case we use the representation s g(s)ds=\ J -00 e-ius where This implies that K n (f;o)=\ K n (l + is;o)g(s)ds J -00 196 E. B. DA VIES AND N. MANDOUVALOSprovided that we can reverse the order of integration. (In the above we have used the recurrence relations for a complex heat kernel.) Here again we have to show that the above double integral is absolutely convergent. For this we use the bound of the heat kernel for a complex time which we obtained in §3. Since Re(l + is) = 1, the same bound \K n+1 (t; p)\ ^ c \t\-l n e -Re(i^+p'/40 (1 + p + ^* e -\n P> w h e r e t = holds for n even or odd. Next, we use the inequalities -( 1 + S 2y } f o r a n y m>0> and find that the original double integral is dominated by [ J -0 + py n e *" p (ch 2 \p -o) 2 sh p dp.The integral with respect to s is absolutely convergent provided that m<\. The integral with respect to p is also absolutely convergent provided that n > 1. If n = 1 then absolute convergence of the second integral requires m > \, which contradicts the above restriction on m. Hence this method fails when n = 1. Relation (a) follows in a similar manner. LEMMA 4.3. Let fdA = (A + iA~a\ p~" u Q(<i\d<i J -00 and V\s\) 2m \g{s)\ds<+«...

Cowling-price theorem and characterization of heat kernel on symmetric spaces

“…(A) As noted earlier in the real hyperbolic spaces H n , α, β do not satisfy the restriction in Theorem 5.1 and from [9] we get the estimate (5.10) only for the bi-invariant heat kernel h t . To get a similar estimate for h δ t , where δ is not the trivial representation, we point out that in SO(n, 1), q δ = 0 for all δ and that for (α, β ) of H n , (α + p δ , β ) are the corresponding parameters for the higher dimensional hyperbolic space H n+2p δ (see [11]).…”

“…That is, h δ t of H n is the same as h t of H n+2p δ . Thus we can again appeal to the result of [9] to show that h δ t satisfies the estimate (5.10) substituting α by α + p δ .…”

“…real hyperbolic space H n = SO(n, 1)/SO(n), where α = n−2 2 and β = − 1 2 , Davies and Mandouvalos [9] already has the estimate of the type (5.10) for the heat kernel h t . For δ ∈ K 0 , h δ t = h (α+p δ ,β +q δ ) t denotes the heat kernel corresponding to the operator L α+p δ ,β +q δ .…”

HEAT KERNEL BOUNDS, POINCARÉ SERIES, AND L2 SPECTRUM FOR LOCALLY SYMMETRIC SPACES

“…In a series of papers sharp upper and lower bounds for the heat kernel K (t, x, y) on X were obtained: Davies and Mandouvalos [6] derived sharp upper and lower bounds in the hyperbolic setting, that is, X = H n . Anker and Ji [1] generalized this result to Riemannian symmetric spaces of nonpositive curvature for those t > 0 and x, y ∈ X such that 1 + t ≥ cd(x, y) for a constant c > 0.…”

“…Our methods are inspired by similar results due to Davies and Mandouvalos for hyperbolic manifolds M = \H n in [6] but, because of nonconstant sectional curvature, the proofs in the more general case of locally symmetric spaces are a little more involved. We also want to emphasize that we make no restrictions concerning the rank of X , that is, the dimension of a maximal flat in X .…”