Boundary behaviour of analytic functions in spaces of Dirichlet type

Abstract

For 0 < p < ∞ and α > −1, we let Ᏸ p α be the space of all analytic functions f in D = {z ∈ C : |z| < 1} such that f belongs to the weighted Bergman space A p α . We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces Ᏸ p α . We also study the size of the exceptional set E( f ) = {e iθ ∈ ∂D : V ( f ,θ) = ∞}, where V ( f ,θ) denotes the radial variation of f along the radius [0,e iθ ), for functions f ∈ Ᏸ p α .

Boundary multipliers of a family of Möbius invariant function spaces

“…In case that p ≤ 2, it is known [12] that one can take t = s in Proposition 3.2. The following construction will be a key for the proof of part (3).…”

Uniformly locally univalent functions and Hardy spaces

“…If we rewrite the above integral by polar coordinates, we obtain It is known that the lemma remains valid for 0 < p 2 (cf. [6]), though we shall not make use of it below. In view of the above lemmas, we recognize importance of the estimation of…”

On weighted Dirichlet spaces of monogenic functions in

“…As is well known, the Dirichlet space  2 1 (D) coincides with H 2 (D), the classical Hardy space in D, while the general Dirichlet space  2 (D), > 1, coincides with the weighted Bergman space H 2 −2 (D) of holomorphic functions in D. For more information on the weighted Bergman and Dirichlet spaces on the disk, we refer to literature. [2][3][4][5][6] Weighted Dirichlet spaces of harmonic functions on the unit ball of R n are also well known, see, eg, previous studies. [7][8][9][10][11] Let B n ∶= {x ∈ R n ∶ |x| < 1} be the open unit ball in R n and h(B n ) be the collection of harmonic functions on B n .…”