By MAN-WAH WONG

An advent to pseudo-differential operators. This variation keeps the scope and elegance of the unique textual content. A bankruptcy at the interchange of order of differentiation and integration is extra firstly to make the ebook extra self-contained, and a bankruptcy on vulnerable suggestions of pseudo-differential equations is additional on the finish to augment the price of the ebook as a piece on partial differential equations. a number of chapters are supplied with extra routines. The bibliography is a bit accelerated and an index is further.

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**Example text**

10. 11. There exists a function ij; € C7°°(K” ) such that 0 < i ’iO ^ ^ = 0 for 1^1 < 1 and ■^(^) = 1 for lel > 2. o(0 < 1 for ^ G = 1 for |^| < 1 and = 0 for 1^1 > 2. 11. To construct (po, let / be any continuous function on M” such that 0 < /( t) < 1 for t G K” , /( t) = 1 for |t| < I and f{t) = 0 for |i| > Let (p € r ) be a real-valued and nonnegative function such that (p{s) = 0 for |s| > and p{s)ds = 1 . 2. 7, Po G C'^CK"). 7 and the location of the supports of / and p, we see th at po{t) = 0 for \t\ > 2.

2. Let <5 : ~~
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~~The Fourier Transform Also, 21 (M yfm = (27t) - / 2 [ e-^-<{Myf){x)dx = (27t)-"/2 f e-^^-^e'^y-^f{x)dx JR'^ = /(^ - y) = (T -yfm ■ Finally, by another change of variable, we have (D ,/K ? r/(f) = H - ( d j /)({ ). 5. P ro o f: Let (p{x) = e ^ . Then (p{^) = e ^ . 4) j=l J-°o 22 An Introduction to Pseudo-Differential Operators Hence it is sufficient to compute oo /-00 g - itc - t ^ ^ g ( - 00, 00) . 5) =| L where is the contour Im^r in the complex ^:-plane. 6), / OO . /R” = 2~^e~ ^ . 8) 3. 6. (T h e A d jo in t F orm ula) tions in L^(R"). ~~