By Carlos A Berenstein
ISBN-10: 3540057463
ISBN-13: 9783540057468
Read Online or Download Analytically Uniform Spaces and their Applications to Convolution Equations PDF
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Extra info for Analytically Uniform Spaces and their Applications to Convolution Equations
Sample text
Thus by C k ( ¢ ) e ~c°(¢) g(In[) = sup{Inl-lZog[C21(~(¢) that the In particular, sequence < for some constants [17]) k E~C. H s = h(s) k = k({Hs};{es};~ ) ~ ~(~). the last along 2, let f be a fixed entire for each in ~+, the sequence an arbitrary majorant (38) follows as the first one. e • for all ~ ~ ~n (of. and any > O, (48) with f = $ 2. tf(~)l C = C 2. for some < c e Blnl-~c°({) The Paley-Wiener ~ ~ ~; and~this theorem (cf. completes Remark 2) then implies the proof of Proposition suitable (Cj }j~l Given ¢(g) To complete the proof BAU-structure for ~'.
By the Cauchy-Poincar4 formula, this can be done provided as~(~-t) (30) lim Jtln'l ~ Itl ÷ J~(C-t+i~v) ~M(t-i~v) Jdv = 0 o However, relation (30) follows from obvious estimates: as~(~-t) Itl n-1 S ... <_ C¢,pJtJn-lasm(~_t)exp[-p~(~-t) + Rasm(~-t)] x O x Cd,Mexp[-d~(t) Here we used the inclusion notation p = Ra s + (s+l)a s + (s+l)as~(~-t)] ÷ 0 . 1 supp ~ c {x : Ixl <_ R - T}, + 1 , d n = ~. and the Therefore, A (31) ~M¢ (~) = $ (~-T)~M(T) dT r~ In the last integral , Jd~ I _< (1 + J~-t l)n2+n(l + r as )n, the corresponding constant from Lemma 1.
Applying the ~ > O, ]~(t+in+u) l ! T~lnle Alnl M~(t) max lu I<_1 Therefore, by (67) and (68), (69) f(~-t) ^ +"in) - ~ ( t ) [ a t Ig(t I S ~(~-t)~(t)dt > C2eX(g) _ T l l q l e A ql ; e x p [ x ( ~ - t ) Using again the superadditivity X(g-t) - ~(t) < X(~) - log(l+t2) • lql ! 6 y > for some 6 < e, - 9(t)]dt of X (cf. (55)), we get Then, for lq I sufficiently small, say (69) yields (70) l cz >- T eX(g) Now, let us define m' as (71) m'(E) = C3exp[6AInl - ~ t l ( ~ ) ] ; C 3 = 2CCIC2 le3As ; 57 and, for fixed, we set z ° = Xo+iY o where ~,~ are as above (see (60)).