New PDF release: Linear Partial Differential Operators

By Dr. Lars Hörmander (auth.)

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6) for any two coordinate systems" and 'X'. Hence they define a distribution U which has the required properties. This proves the theorem since the uniqueness of U is a trivial consequence of the proof. Remark. P)' (Q) may also be defined as the space of all continuous linear forms on the space of infinitely differentiable densities with compact support. D,,) for which L(cp)=JL"(cpo'X-l)dx if cpEC~(Q,,). For details we refer to DE RHAM [1] where a density is called an odd form of degree n and a distribution is an even current of degree O.

Xn= O. If x ERn. we shall use the notation x = (x'. x"), where x'= (xv . '. • xn). ) If u E[I' we denote the restriction of u to E by Uz;. thus (uz;) (x') = u (x'. 0) . 8. Let Q be an open set containing some point in E. Suppose that the mapping u--'>- Ul. is a continuous mapping of C':' (Q). with the topology defined by the norm Ilullp,ko into 9' (Ra). Then the function k'W) = ((2n)<1-f· Jk W. n-p'dn- l/p '. 13) Chapter II. 7f'. 1, (R,,). Proof. a) Let tp (x') belong to cgo (Q n 1:). k' For let X E cgo (Q) be equal to I f/' in a neighborhood of the support of tp in 1:.

Ri;) . 22) can be extended to 9" (R;:) It is immediately verified that the partial Fourier transform of Diu'is ~iun' ifj < n, and Dnu", ifj = n. The restriction to R;; of a distribution in 9" (Rn) is obviously in 9" (R;t) and conversely it follows from the Hahn-Banach theorem that every element in 9" (R;t) can be extended to an element in 9" (R n ). #' (R;t) the set of distributions Chapter I. Distribution theory 25 in [/" (Rn) with support in Ri;. 9" (Ri;) are dual. Since we shall not use this fact, the proof may be left to the reader.