Finite Positive Borel Measure Research Articles

The Poisson Integral of a Singular Measure

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y > 0, its Poisson integral is defined by the formulawhere CN is chosen so thatSince σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.

The mean Modulus of a Blaschke Product with Zeroes in a Nontangential Region

In this paper we investigate the behavior of a certain class of inner functions, i.e., functions which are bounded and analytic in the unit disc U and whose radial limits have modulus 1 almost everywhere. Recall that every inner function I can be written as where and σ is a finite positive Borel measure on [0,2π] singular with respect to Lebesgue measure. The first factor is called a Blaschke product, and the second is called a singular inner function; see [4] for details. We will deal with Blaschke products whose zeroes {zk } lie in a fixed nontangential region R(α) = {z ∊ U: (1 − |z|)/|1 − z|) > α} for some α > 0. Let C(α) be the class of all infinite Blaschke products with {zk } ⊂ R(α). We shall be concerned with the quantity Now ▵(r,φ ) → 0 as r → 1 for any inner function, and we shall determine bounds from above and below on ▵(r, B) for B ∊ C(alpha;), bounds which can be expressed explicitly in terms of the behavior of {zn }; this is done in part I. In part II we use the results of part I to get information on the derivatives of B ∊ C(α), their Taylor coefficients, and ▵(r, B) for {zn } with specific growth conditions. In what follows, Hp denotes the usual Hardy space of analytic functions on the unit diskLp [0, 1] the usual Lebesgue class on [0,1]. In addition, for 0 <p < 1Bp denotes the class of functions analytic in the unit disk for which Finally, the notation f(x) ≐ g(x) will be used to express the existence of constants C 0, C1 > 0 such that C 0 g(x) ≤ f(x) ≤ C 1, g(x) for all x in some domain which will be obvious from the context; C 0 and C 1 will be independent of x, though they may depend on other quantities. This paper is based on the author's thesis [6], supervised by Patrick Ahem at the University of Wisconsin, Madison.

Specification of the Munsell book of color

We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D(μ) where μ is a finite positive Borel measure on the unit circle. First, we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of D(μ). It allows us to give an estimate of the capacity of points and arcs of the unit circle. We also provide new conditions on closed sets to be polar. Our method is based on sharp estimates of norms of some outer functions, which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.