Distributions generated by the boundary values of functions in Privalov spaces

We characterise the distributions generated by the boundary values of functions from Privalov spaces.


Introduction
We use the following notation and preliminaries. stands for the open unit disc in C and is its boundary, i.e.
(Ω) is the space of measurable functions on Ω such that is the space of measurable functions on Ω such that for every compact set ⊂ Ω the following holds ∫ | ( )| < ∞ К .
Privalov spaces on and + and their properties: Privalov  Every function in Nevalina class, ( ), because of Fatou's lemma, has a nontangentional (radial) limit on almost everywere; every function in Privalov class, ( ), has a nontangentional (radial) limit on almost everywere, in both cases we denote the boundary value with * ( ) = lim →1 ( ).
is an element in ′ and it is called the regular distribution generated by the function .

Main results
Theorem ii. lim Theorem 1. Let * ∈ ′ is generated by the boundary value * ( ) of a function ( ) in (Π + ). There exist sequence of polynomials { ( )}, ∈ Π + , and respectivelly { }, ∈ ′ , generated by the boundary values * ( ) of the polynomials ( ), i.e. = * such that: Proof. Let the assumptions of the theorem hold. Since ∈ (Π + ), one has ∈ (Π + ) and there exist a constant > 0 such that In what follows we prove i. and ii. In the previous calculations we use the notation ( ) for the Lebesgue measure of the set K, = max{ ( ): ∈ } and ′ = + [ * ( ) − ( )]. It is obvious that ′ → 0 when → ∞. The Later calculation implies that 〈 , 〉 → 〈 * , 〉 when → ∞ for every, but fixed, , meaning → * weakly in ′. To prove the convergence in the strong topology it sufficies to prove the same convergence for ∈ for an arbitrary bounded set in . Choose ⊂ , arbitrary bounded set. The condition of boundnes implies that there exists a compact set such that ∈ , || || ( ) < , for every ∈ . Note that the calculations at the beginning of the paragraph hold for every ∈ and the new compact set chosen for the boundness condition. Hence, → * in ′.
implies the existence of ∈ (Π + ) such that the sequence of polynomials converges to , uniformly on arbitrary compact subsets of Π + when → ∞.
Firstly we will prove that this function is holomorphic and satisfies the condition for all = + ∈ Π + and arbitrary compact set ⊂ .
Indeed, we use the condition ii., i.e. We will prove ( * ).

Conclusion
We obtain necessary and sufficient condition for a distribution generated from an element of the Privalov class to be boundary value of analytic functions on upper half space. The boundary values are taken in the distributional sense.