A boundary point p of a domain D in C^n is a peak point for a class of functions (holomorphic on D, or continuous up to the closure, or holomorphic in a neighbourhood) when some function in that class attains its maximum modulus at p and only at p; a strong support function is the weaker local analogue whose zero set touches the boundary only at p. This area studies when such functions exist on smoothly bounded pseudoconvex domains, the known positive answers (strictly pseudoconvex boundaries, finite type in C^2, convex and lineally convex domains, h-extendible or semiregular points, real-analytic boundaries), the constructions used to produce them (the Bedford-Fornaess sector method with a multiplicative Cousin problem on a Riemann surface over the boundary, Hormander weighted L2 estimates for d-bar, bumping of local defining functions), and the sharp Holder exponents obtained, together with the central open question of whether every boundary point of finite D'Angelo type on a smoothly bounded pseudoconvex domain in C^n for n greater than two is a peak point.
A point p on the smooth boundary of a domain D in complex n-space is a peak point for the algebra A^alpha(D) of functions holomorphic on D and of class C^alpha on its closure if some f in A^alpha(D) …