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, \(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(
We prove dispersive and Strichartz estimates for Schrödinger equations on a class of locally symmetric spaces Γ\X, where X = G/K is a symmetric space and Γ is a torsion free discrete subgroup of G. We deal with the cases when either X has rank one or G is complex. We present Strichartz estimates applications to the well-posedness and scattering for nonlinear Schrödinger equations.1991 Mathematics Subject Classification. 35Q55, 43A85, 22E30, 35P25, 47J35, 58D25.
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