Continuity of extremal elements in uniformly convex spaces

Abstract

Abstract. In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space.Our purpose is to study extremal problems over uniformly convex Banach spaces. For a given linear functional φ, what elements x of the space with norm x = 1 maximize Re φ(x)? Because of the uniform convexity, this problem will always have a unique solution, which is called the extremal element. In this paper, we show that the extremal element depends continuously on the functional φ, and it can be approximated by the solutions of the same extremal problem over subspaces of the original Banach space.Moreover, we show that an extremal element can arise from at most one linear functional of unit norm, and that the functional depends continuously on the extremal element. Using these results, we give a streamlined proof of a theorem of Ryabykh, which says that for a functional defined on the Bergman space A p , with kernel in the Hardy space H q , the extremal element is in H p , where 1 < p < ∞ and 1 p + 1 q = 1. Uniform convexity and extremal problemsLet X be a complex Banach space and let X * be its dual space. For a given linear functional φ ∈ X * with φ = 0, we are interested in all elements x ∈ X with norm x = 1 such that (1.1)Re φ(x) = sup y =1Re φ(y) = φ .Such a problem is referred to as an extremal problem, and x is an extremal element.

Uniform Approximation of Extremal Functions in Weighted Bergman Spaces

“…The argument in [7,Theorem 4.1] shows that, if T n is the best approximate of F of degree n and E p,α n < δ and T n = T n / T n p,α ,…”

“…It is known (see [4]) that the spaces A p α , since they are subspaces of L p spaces, are uniformly convex. In [7], general results are proven about approximating extremal functions in uniformly convex spaces, and a proof is given there of the well known fact that extremal functions are unique in uniformly convex spaces. See [2,3] for more information on extremal problems in spaces of analytic functions.…”

Bergman–Hölder Functions, Area Integral Means and Extremal Problems

“…Let F ∈ A p α be such that F = 1 and Re D F k dA α is as large as possible. There is always a unique function F with this property because L p (dA α ) is uniformly convex, see for example [6]. The next theorem allows us to obtain knowledge about regularity of F from knowledge about the regularity of k. For similar results that give regularity results about k from knowledge of the regularity of F , see [4].…”

“…Several results are known that allow one to deduce regularity properties of F from regularity properties of k, and vice-versa. See for example [5][6][7]12]. Our result is of this type and says that if 0 < β ≤ 2 and k(e it ·) + k(e −it ·) − 2k(·) A q α ≤ C|t| β then F (e it ·) + F (e −it ·) − 2F (·) A p α ≤ C ′ |t| β/p for p > 2 and F (e it ·) + F (e −it ·) − 2F (·) A p α ≤ C ′ |t| β/2 for 1 < p < 2.…”

Extremal problems in Bergman spaces and an extension of Ryabykh's Hardy space regularity theorem for 1 < p < infinity

“…In this paper we study regularity results for the extremal problem of maximizing Re φ(f ) among all functions f ∈ A p of unit norm. An important regularity result is Ryabykh's theorem, which states that if the kernel is actually in the Hardy space H q , then the extremal function must be in the Hardy space H p (see [14] or [6] for a proof). In [7], the following extensions of Ryabykh's theorem are shown in the case where p is an even integer:…”