By J. T. Schwartz
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15. x/m T D 0 for T 2 D0 . 15 implies that T D a ı for some k 2 N and j 0 jD0 aj 2 C; j D 0; : : : ; k; ak ¤ 0: If k were larger than or equal to m; then 0 D h ; xm Ti D . R/ with D 1 near 0, yields a contradiction. 3 Differentiation 43 (b) More generally, by Schwartz [246], Ch. III, Thm. XXXVI, p. , jD0 h ; Ti D m X1 h.. 1 below. Still more generally, by Schwartz [246], Ch. III, Thm. XXXVII, p. 102, we have 8T 2 D0 . Tj / jD0 if h W ! M/ ! S/ 2 D0 . S/i D h . M/; 2 D. M/ Z 0 ,! M/ W f 7 ! 7! j/ We also emphasize that .
D . C 1/ . C i C 1/A ;i D i X . 1/ kD0 k CiC1 k has the degree i and the zeros 1; 2; : : : ; i; and it fulfills Pi . i we conclude that Pi . / D . 1/i . C 1/ iŠ . C i/; A ;i D ! 1/ D 1: Thus . 1/i ; iŠ. x &0 /x C m 1 X . i/ Á ı ; iŠ. 7) cf. Schwartz [246], (II, 2; 26), p. 5, p. 87. 4) to several dimensions. We suppose that ; ¤ Rn is open, M is a closed C 1 -hypersurface, and 1 f W n M ! f / W M ! Rn W x 7 ! , Trf 2 L1loc . /n 0 D . f // 2 D0 . x/ 2 Rn ; 2 D. /I M cf. Schwartz [245], (II, 2; 43), p.
2 D0 . / for f 2 C 1 U; D0 . / and Prop. 5. Therefore, @f =@ j 2 C m 1 U; D0 . / for f 2 C m U; D0 . / and m 2 N [ f1g; j D 1; : : : ; l/: (3) The space C m Œ0; 1/; D0 . 0; 1/; D0 . t/ converges if t & 0 for all j D 0; : : : ; m: (As before, note that the notion of “weak differentiability” introduced above coincides with that of “strong differentiability” due to the Montel property of the space D0 . /; see Treves [273], Prop. 11, p. ) Let us employ the notion of differentiable distribution-valued functions to give a useful formula for fundamental solutions of powers of differential operators with non-vanishing constant term.