By J. Bergh, J. Lofstrom
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Theorem . Let A be a given couple of normed spaces and put X l = Ao I ,q 1 , where O � (} o < (} 1 � 1 and 1 � q o � co, 1 � q l � CO . Put A = (} l - (} O . Then K(t, a ; X ) '"" (S� /A (S- O o K(s, a ; A )) qO ds/S) l /q O +t(S� A (S - O l K (s, a ; A )) q 1 ds/S) 1 /q l . / Proof· We first prove � Let a = ao + a1 , ai E Ai, i = O, 1 . 2 " ". and Minkowski's inequality it follows that (Sg/ A (S - O K(s, a ; A)) q O ds/S) l /q O � (Sg/ A (S - OO K(s, ao ; A)) q O ds/S) l /q O + (Sg/ A (S - Oo K(s, a 1 ; A)) qO ds/S) l /q O � I a ll x o + C (S�l / A (sA II a 1 1 1 x ) qO dS/S) l /q O � C( II ao II Xo + t II a l ii x ) .
2 . Theorem . (6) " a ll o , q ; J � C s - o J(s, a ; A), a E L1 (A) where C is independent of e and q . Proof: Obviously, Il a ll o , q J is a norm. Assume that T: Aj� Bj' with norm Mj , j == O, 1. For aE A o, q J ' we have, since T: I'(A) � I'(B) is bounded linear, that Tu(t) is measurable, ; ; Ta == T (J� u(t) dt/t) == J� Tu(t) dt/t (convergence in I'(B)). Th us, with this u, J(t, Tu(t)) == max ( II Tu(t) II Bo ' t II Tu(t) 1 1 B l � M ° max ( 1I u(t) 11 Ao' t M 1 / Mo ll u/t) 11 A ) == Mo J(t M 1 /Mo , u(t)) , and we obtain, by the properties of lP Oq ' lPo ,q (J(t, Tu(t))) � M 6 - 0 M� lP Oq (J(t, u(t))) .
2. Lemma (The fundamental lemma of interpolation theory). Assume that min (l , lit) K(t, a) � O as t�O or as t� oo . Then, for any E > 0 , there is a representation a == L v U v (convergence in I'(A)) of a, such that Here y is a universal constant � 3. 1. 3, we have K(t, a) � CO ,q tO II a ll o, q ; K for any given a E K o , q(A). Thus it follows that min (l, l l t) K(t , a) �O as t � O or t� 00. 1 . 0 Proof of the fundamental lemma : For every integer a == a O , v + a l , v , such that for given E > O Thus it follows that v, there is a decomposition 46 3.