By Alfred Renyi
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Now the Ilk are defined via the k Gk: Qk = U m=-oo G," and ok = 2-k-3. 12) follows from the equality 10i D°V,k = 0k (D°W ek) * XGk on Q \ (Ilk_1)ek and the analogous equality on (Dk)ek-i . 0 Remark 3 Sometimes it is more convenient to suppose that the functions &k in Lemmas 4 and 5 are defined on R" and supp'k C Il. (We shall use the same notation ikk E Co (Il) in this case also). 11)). 11) takes the form 00 00 E V)k=EOk=1 on k=-oo Q. 13) k=ko For Il = R" we shall apply the following analogue of Lemma 5.
2 Nonlinear mollifiers with variable step We start by presenting four variants of smooth partitions of unity, which will be constructed by mollifying discontinuous ones. , s, be open sets and 9 K C U Stk. , s, exist such that 0 < ok < 1 and a t tpk = 1 K. 5) k=1 Idea of the proof. Without loss of generality we may assume that the 1k are bounded. There exists b > 0 such that K C G =_ 6 (S1k)a. Set Gk = (1k)6 \ k=1 k-1 a U (Slm)6 and consider the discontinuous partition of unity: E co,, = Xc on k=1 m=1 R".
1(11) and m < I - 1. ,1 - 1 we have IIBaf - f G E '(6k, fk)w,TM,(n) :5 E J(bk,''kfo)W,m(R") k=1 k=1 By Lemma 11 of Chapter 1 Okfo E W;O(R") and as in the proof of Lemma 1, applying, in addition, footnote 11 of Chapter 1, we establish that w(4,0kf0)W;(R^) :5 M1 is independent of f and k. ,1- 1. 3. (&"! (D°-flakDwf) flX k=1 k=1 Le(n) `(A6, 00 00 D",f)-D°-6,iIjk Dwf II A6. '-"l (R") as in the proof of Lemma 1 we establish that w(bk, D°-p k L'w0 f0)Loo(R°) < M2bkIID°-1 Vk DwfolI W:181(R") , where M2 is independent of f and k.