Matts Essén

On minimal thinness, reduced functions and Green potentials

where £ t = E n Qk and c(£t) is the capacity of Ek. We prove that the set E is minimally thin at T e T in (7 if and only if W(x)<oo. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjdgren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsai] [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.

Fix-points and a normal family of analytic functions

Let G be a family of functions analytic in a domain D in the complex plane. We prove that G is a normal family, provided that for each f ∈ GE 9, there exists k = k( f ) > 1 such that the k th iterate f k has no fix-point in D

On beurling's theorem for harmonic measure and the rings of saturn

A classical estimate of harmonic measure for a closed subset of the unit disk is due to A. Beurling. We give a higher-dimensional version of this result.

Best constant inequalities for conjugate functions

A survey is given of sharp forms of some classical inequalities for the conjugate function.

On Estimating Eigenvalues of a Second Order Linear Differential Operator

For an eigenvalue problem on I0 = (0,1), we determine the maxima and infima of all eigenvalues when the coefficient p in the operator –y″ + py is allowed to vary in the class of all integrable functions with ∫ P+ = B or ∫ P+ = B where we integrate over the interval I0.

On Wiener Conditions for minimally thin and rarefied Sets

Let D = {x ∈ ℝ p : x 1 > 0}, where x = (x 1,…, x p ), p ≥ 2 and ∂D is the euclidean boundary of D. If u is subharmonic in D and y ∈ ∂D, we define u(y) = lim sup u(x), x → y, x ∈ D. If u is non-positive on ∂D and sup D u(x)/x 1 < ∞, it is known that

Near-integrability of the conjugate funtion

\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {u\left( x \right)/{x_1} \to \alpha ,}&{x \to \infty ,}&{x \in D\backslash E,} \end{array}} \\ {\begin{array}{*{20}{c}} {\left( {u\left( x \right) - \alpha {x_1}} \right)/\left| x \right| \to 0,}&{x \to \infty ,}&{x \in D\backslash F,} \end{array}} \end{array}

Tauberian Theorems, Convolutions and Some Results of D.C. Russell

where the exceptional set E is minimally thin at infinity in D (cf. [5]) and the exceptional set F is rarefied at infinity in D (cf. [3]).