Inner factors of analytic functions of variable smoothness in the closed disk
Let p(ζ) be a positive function defined on the unit circle T and satisfying the condition|p(ζ2)−p(ζ1)|≤c0loge|ζ2−ζ1|,ζ1,ζ2∈T,p−=minζ∈Tp(ζ). Futhermore, let 0<α<1, r≥0, r∈Z, and assume that p−>1α. Define a class of analytic functions in the unit disk D as follows: f∈Hr+αp(⋅) ifsup0<ρ<1sup0<|θ|<π∫02π|f(r)(ρei(λ+θ))−f(r)(ρeiλ)|θ|α|p(eiλ)dλ<∞.The following main results are proved.
Theorem 1. Let f∈Hr+αp(⋅), and let I be an inner function, f/I∈H1. Then f/I∈Hr+αp(⋅).
Theorem 2. Let f∈Hr+αp(⋅), and let I be an inner function, f/I∈H∞. Assume that the multiplicity of every zero of f in D is at least r+1. Then fI∈Hr+αp(⋅).