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On the Riesz means of the solutions of the Schrödinger equation

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On the Riesz means of the solutions of the Schrödinger equation ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze SIGRID SJÖSTRAND On theRieszmeans of the solutions of the Schrödinger equation Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 24, no 2 (1970), p. 331-348. <http://www.numdam.org/item?id=ASNSP_1970_3_24_2_331_0> © Scuola Normale Superiore, Pisa, 1970, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE RIESZ MEANS OF THE SOLUTIONS OF THE SOHRÖDINGER EQUATION by SIGRID SJÖSTRAND 0. Introduction. Consider the solution = G (t) f of the initial value problem At least formally we have where y denotes the Fourier transform. From this it is easily seen that G (t) is a bounded, even unitary, operator in L2 = L2 (Rn~. We also have the group property Thus we have a unitary group of operators. In LP = Lp (Rn), p # 2, G (t) is not bounded. See Hormander [2] and Lanconelli [3]. See also Littman- McOarthy.Riviere [4]. A possible substitute for this, motivated by the theory of distribution (semi) groups, is that at least the Riesz means . of sufficiently large order k are bounded in LP. See Peetre [7]. Pervenuto in redazione il 18 Novembre 1969. 332 More generally we consider the operators G (t), t ~ 0, defined by where H is a positive homogeneous function of degree m &#x3E; 0, and H is infinitely differentiable for ~ =1= 0. We will show that I k (~ (t) is bounded in if k &#x3E;

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