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4a) signifies that these traces converge in L,(dR) to a function such that 6 l1611q,aR 5 K , Il4llbmh? 6 is called the trace of the function 4 E W;(R) on 8 0 and denoted by 4 Ian. Note that the preceding inequality with I1611q,aR( k . 4a). Note also that 41anis independent of the choice of the sequence {4k}that approximates 4. 6. If n = m = 1 and p following inequality is valid. )a XER for any 4 E Wi(Q),where the constant K is independent of 4 E Wi(Q). 6, it is easy to obtain the following inequalities, which are known as Sobolev inequalities.

Suppose { 4k}is ulso conruined in L J Q ) and is such rhar JJ2kq d x dr for an!. lc dr 4 E L J Q ) . Then 4 E L,. ,(Q) and. 14) for any q E L,, ,(Q), where ( 114) + ( 1 ’4) = 1 und ( I / r ) + ( 1 / i ) = I . Proof‘. ,(Q) and. ,,i(Q). Clearly. it can be shown that for any subsequence of the sequence { 4 k } there exists a further subsequence such that the above conclusion is valid. Thus the conclusion is also valid with respect to the whole sequence, and hence the proof is complete. The next theorem contains a well-known result on lower semicontinuity of certain functions.

In all these cases the assumption on the boundedness of the set Q plays an important role in their proofs and hence is used without further mention. (i) q, r 2 2. Then 4, V I 2. 1 1) implies the conclusion. (ii) 4, r S 2. Then 4, ? 2 2. 11) is valid for any v] E L,,AQ). 12) gives rise to the conclusion. 4. Some Function Spaces (iii) q 2 2, r 5 2. Then tj I 2, f 2 2. 12) are valid for any q E L,(Q). Thus the conclusion follows readily. (iv) 4 5 2. r 2 2. Then (5 2 2. F 5 2. ,(Q) n L,(Q). 12) hold for any ti E L,(Q).