Extremal problems in Bergman spaces and an extension of Ryabykh’s theorem

Abstract

We study linear extremal problems in the Bergman space A p of the unit disc for p an even integer. Given a functional on the dual space of A p with representing kernel k ∈ A q , where 1/p + 1/q = 1, we show that if the Taylor coefficients of k are sufficiently small, then the extremal function F ∈ H ∞ . We also show that if q ≤ q 1 < ∞, then F ∈ H (p−1)q 1 if and only if k ∈ H q 1 . These results extend and provide a partial converse to a theorem of Ryabykh.

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

“…• For q ≤ q 1 < ∞, if the extremal function F ∈ H (p−1)q1 , then the kernel k ∈ H q1 . (In fact, the proof in [7] shows that this result holds if 1 < q 1 < ∞). We show that the first two results above hold for all p such that 1 < p < ∞.…”

“…in [4], [8], [9], [12], [13] and [16]. Regularity results for solutions to this and similar problems can be found in [6], [7], [11] and [14]. See also the survey [1].…”

“…In [7], it is shown that if p is an even integer, then for q ≤ q 1 < ∞ the extremal function F ∈ H (p−1)q1 if and only if the kernel k ∈ H q1 . It is also shown that if the Taylor coefficients of k satisfy a certain bound then F ∈ H ∞ , and that the map sending a kernel k ∈ H q to its extremal function F ∈ A p is a continuous map from H q \ {0} into H p .…”

“…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:…”

Bergman–Hölder Functions, Area Integral Means and Extremal Problems

“…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.…”