By K.L. Teo and Z.S. Wu (Eds.)

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Let W,','(Q),p 2 1, denote the Banach space of all those functions L,(Q) for which the norm " n 4 in is finite, where 4, = 8 4 / d t , dX,5 84/dxi,and 4, = d2d/dxiaxj. 4. '(Q) for which the V,(Q) the Banach space of all those functions norm l4lQ= is finite, where 114112,m,Q + 21 ll4xll2,Q and V:,'(Q) the Banach space of all those functions 4 in V,(Q) that are continuous in t (with respect to the norm in L,(R)). Q. , t)l12,n -+ 0 as At 0. Note also that Vl9'((Q) is the completion of W:. All these spaces are complete.

4)]. Let Cp E V2(Q). 15) for any r] E W i *'(Q). Proof. 5) holds for Take E W:*'(Q)such that r](x,t ) = 0 for all ( x , t ) E 2 ! any r] E fl;,'(Q). x [T - E, T ) . 5) holds for such r] and hence for all r] E fl:,'(Q). 3. Three Elementary Lemmas 41 q E @;, ’(Q). 15) is true. For this, let E > 0 be a small number and let ,uEbe the function defined by e E [lo, t - E l , t E cz - E, 71, P&(t,7) = (T - t ) / E , 0, t E [z, TI. Let 4j be an arbitrary element of W i . ‘(Q). ) E W;, ‘(Q) and equal to zero for t 2 z.

3), we obtain /I 5 ( ~ I ) ~ ’ ’ I I ~ ~ I I ~<, Qa. 3. 5, we deduce that ~ ~ c $ ~ ~ A ~ , pP~Z ,MQl l $ I Q < a. 8. This completes the proof of the lemma. 2. 5) arefinite, for any 4 E V2(Q)and for any r] E W:*’(Q). Proof. These two facts will be used throughout the proof without further mention. For the first integral The conclusion follows readily from Holder’s inequality. 5we obtain 5 < a. 10) 11. 11) a. 6) that I l C p O r ] ( ~ ? O>II1,y The proof is now complete. 5 l l C p o l l 2 , * l l v ~ O)ll2,* ~~ < a.