By Tripathi M. M., Kim J., Kim S.

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**Extra resources for A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds**

**Example text**

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. 1. 2 are valid for the case when Cp E V:*"(Q)or Cp E 8 : - O ( Q ) . The next lemma will be needed in Chapter V1. 3. 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).

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. THEOREM 4. IS. Lct R , c R‘ und R, c R shri such that R , is opcw h,irh compiicr closure trnd R2 is compact and concr’x. rion defined on R, x R,. x on 51, fiw euch s E R , , und { f } he a I. Mathematical Background 26 sequence of measurable functions d&ed on Q, with values in 0,. Zf yk 3' yo in L,(R,, R") z { y = (yl, .

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). Thus we obtain the conclusion. 14. 13 iind { 4 k }he u bourded sequence in L,. ,( Q ) , 1 _< 4 . r 5 x . 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,.