Notions of Convexity by Lars Hörmander

By Lars Hörmander

The first chapters of this booklet are dedicated to convexity within the classical experience, for services of 1 and several other genuine variables respectively. this offers a history for the research within the following chapters of similar notions which take place within the idea of linear partial differential equations and intricate research similar to (pluri-)subharmonic services, pseudoconvex units, and units that are convex for helps or singular helps with appreciate to a differential operator. furthermore, the convexity stipulations that are suitable for neighborhood or international lifestyles of holomorphic differential equations are mentioned, best as much as Trépreau’s theorem on sufficiency of (capital Greek letter Psi) for microlocal solvability within the analytic category.

At the start of the publication, no necessities are assumed past calculus and linear algebra. afterward, easy proof from distribution thought and sensible research are wanted. In a number of areas, a extra wide heritage in differential geometry or pseudodifferential calculus is needed, yet those sections might be bypassed without lack of continuity. the most important a part of the booklet may still hence be available to graduate scholars in order that it will probably function an creation to advanced research in a single and a number of other variables. The final sections, despite the fact that, are written typically for readers conversant in microlocal analysis.

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Until now we have only used differentiation in a classical sense, and this would still suffice here. Thus we have not underlined the fact that for a convex function / in an open interval / , the derivatives /^ and / / both define f in the sense of the theory of distributions. This follows at once since for a non-negative test function if G CQ{I) we have by monotone convergence, for example, [ f^{x)ip{x) dx = lim l{f{x-\-h)-f{x))h-^ip{x)dx = lim / f{x){(p{x — h) — ip(x))h~^ dx = — f{x)(p\x) dx. In analogous discussions of subharmonic functions later on, the language of distribution theory will be much more essential.

CONVEX FUNCTIONS OF ONE VARIABLE equal to that of / . 2 proves that the convexity extends to a larger interval. 2. Quasi-convexity is not a local property, for a locally quasi-convex function can be monotonic in a number of intervals separated by intervals where it is constant. However, if / is locally quasi-convex and not constant in any open interval, then / is quasi-convex. We leave the verification to the reader. 7. Convexity of t h e m i n i m u m of a one parameter family of functions. 3, but the minimum is usually not.

Let T be an affine map V\ —> V2 where Vj are vector spaces, let Xj be convex subsets ofVj, and let fj be a convex function in Xj with finite values. Then Fi{x) = f2iTx), X G T-'X2, F2{y) = inf h{x), y G TXx, Tx=y are convex functions if F2{y) > —00 for some y in the relative interior of TXi. Proof. 2). To prove the one on F2 let yi,y2 G TXi and Ai,A2 > 0, Ai 4- A2 = 1. For arbitrary Xj G Xi with Txj = yj we have F2{Xiyi + A2y2) < /i(AiXi + A2X2) < Ai/i(a:i) + A2/i(x2). /2 G T X i , so F2 would be equal to —00 in the relative interior of T X i , which is against the hypothesis.

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