We prove that for every p ≥ 1 there exists a bounded function in the analytic Besov space B p whose derivative is "badly integrable" along every radius. We apply this result to study multipliers and weighted superposition operators acting on the spaces B p .

We use properties of the sequences of zeros of certain spaces of analytic functions in the unit disc D to study the question of characterizing the weighted superposition operators which map one of these spaces into another. We also prove that for a large class of Banach spaces of analytic functions in D, Y , we have that if the superposition operator S ϕ associated to the entire function ϕ is a bounded operator from X, a certain Banach space of analytic functions in D, into Y , then the superposition operator S ϕ ′ maps X into Y .2000 Mathematics Subject Classification. Primary 30H30, 30H20; Secondary 46E15, 47H99.

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For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ 𝔺 : |z| < 1} and satisfy ∫𝔻(1 − |z|2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.