The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients
This volume is an expanded version of Chapters III, IV, V and VII of my 1963 book "Linear partial differential operators". In addition there is an entirely new chapter on convolution equations, one on scattering theory, and one on methods from the theory of analytic functions of several complex variables. The latter is somewhat limited in scope though since it seems superfluous to duplicate the monographs by Eh renpreis and by Palamodov on this subject. The reader is assumed to be familiar with distribution theory as presented in Volume I. Most topics discussed here have in fact been encountered in Volume I in special cases, which should provide the necessary motivation and background for a more systematic and pre cise exposition. The main technical tool in this volume is the Fourier- Laplace transformation. More powerful methods for the study of operators with variable coefficients will be developed in Volume III. However, the constant coefficient theory has given the guidehnes for all that work. Although the field is no longer very active - perhaps because of its advanced state of development - and although it is pos sible to pass directly from Volume I to Volume III, the material pre sented here should not be neglected by the serious student who wants to gain a balanced perspective of the theory of linear partial differen tial equations.
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analytic function apply assume asymptotic boundary bounded Cauchy data Cauchy problem ch sing choose compact set compact subset compact support completes the proof condition cone conic neighborhood constant coefficients constant strength converges convex set Corollary defined Definition denote differential operators distribution elliptic equal equation P(D equivalent estimate exists finite follows from Theorem Fourier transform Fréchet space fundamental solution gives Hence homogeneous Hörmander hyperbolic with respect hypoelliptic hypothesis implies inequality integral Lemma linear Math mixed problem neighborhood norm notation obtain open set P-convex for singular P-convex for supports partition of unity plurisubharmonic functions polynomial positive proof is complete proof of Theorem proves the theorem replaced right-hand side satisfies Section semi-algebraic sequence shows sing supp singular supports subharmonic function supporting function suppu Tº(P topology valid vanishes variables zeros