The Analysis of Linear Partial Differential Operators I: by Lars Hörmander

oo, cf>EC~(1R). 13. If ur(x)=teitX, x>O and Ur(x) =0, x~O, then Ut(cf» = Jteitxcf>(x)dx=icf>(O)+i J eitxcf>'(x)dx 00 00 o 0 00 =icf>(O)-cf>'(O)/t- if cf>EC~(1R).

Let K be a compact set in JR" which is not the union of finitely many compact connected sets. 1) is not valid for any C and k. (K1u ... uKj) is compact. Choose xjEK j , let Xo be a limit point of {xJ and set u(¢)= I mi¢(x)-¢(xo)) where mj is a positive sequence such that Imjlxj-xol=1, Imj=oo. Such a sequence exists since liminflxj-xol=O. Then lu(¢)1 ~sup WI so u is a distribution. 1) is valid and we choose ¢ E COO equal to 1 in a neighborhood of K 1 u ... u K j and 0 near K '-. (K 1 u ... u K j ), hence at x o, then we obtain I mi~C i~j which is a contradiction when j -+ 00.

10) Ivex) - u(x, y)1 = o(lx - Yl k) uniformly in y. If XEK this follows from the continuity of Uo(x, y). 11) when cPj(xHO. When we have lo~(u(x, Yl) - u(x'Y2))1 =0((lx-y11+IYl -Y 21)k-lYl) for Iyl~k o~(u(x, Yl) - u(x, Y2)) if Yl' Y2 EK, is a polynomial in x of degree k -Iyl and lo~o~(u(x, Yl) -u(x, Y2))x=y,1 = Up+iYl' Y2) IYl - Y2Ik-lfJl-lyl =O(IYl - Y2Ik-lfJ+YI). 10); the general case will be useful later on. (x) = ua:v(x), xEK, where tx. =(0, ... ,1,0, ... ) with 1 just in the v-th place. 2 (in n variables) 50 II.

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