Regularity Properties of Functional Equations in Several Variables
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This book is about regularity properties of functional equations. In the second part of his fifth problem, Hilbert asked, concerning functional equations, "In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption?” This book contains, in a unified fashion, most of the modern results about regularity of non-composite functional equations with several variables. These results show that "weak” regularity properties, say measurability or continuity, of solutions imply that they are in C[infinity], and hence the equation can be reduced to a differential equation. A long introduction highlights the basic ideas for beginners. Several applications are also included.
Ch. IPreliminaries1Ch. IISteinhaus type theorems53Ch. IIIBoundedness and continuity of solutions73Ch. IVDifferentiability and analyticity109Ch. VRegularity theorems on manifolds157Ch. VIRegularity results with fewer variables169Ch. VIIApplications231
