Norm of the point evaluation functionals on the Zygmund-type spaces

Abstract

Norms of the point evaluation functionals δ<inf>z</inf>(f) = f(z) on the Zygmund-type spaces Z^α(D), where α > 0, with suitably chosen standard weights on the open unit disk D in the complex plane, are calculated. More precisely, when α = 1 we calculate the norm of the functional δ<inf>z</inf> on the quotient space Z/P<inf>1</inf> for the weight (1 − |z|²)^α, whereas when α ∈ (0, +∞) we calculate the norm of the functional on the quotient space Z^α/P<inf>1</inf> for the weight (1 − |z|)^α, where P<inf>1</inf> is the space of linear holomorphic functions (a two-dimensional linear space). The choice of the weight functions is caused by the fact that for such chosen weight functions the norms of the point evaluation functionals can be calculated due to the possibility of calculating some integrals that appear during the calculation of the norms of the functionals.