Alexiewicz Norm Research Articles

Estimates of Henstock-Kurzweil Poisson Integrals

AbstractIf f is a real-valued function on [−π, π] that is Henstock-Kurzweil integrable, let ur(θ) be its Poisson integral. It is shown that ∥ur∥p = o(1/(1 − r)) as r → 1 and this estimate is sharp for 1 ≤ p ≤ ∞. If μ is a finite Borel measure and ur(θ) is its Poisson integral then for each 1 ≤ p ≤ ∞ the estimate ∥ur∥p = O((1−r)1/p−1) as r → 1 is sharp. The Alexiewicz norm estimates ∥ur∥ ≤ ∥f ∥ (0 ≤ r < 1) and ∥ur − f∥ → 0 (r → 1) hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when u is in the harmonic Hardy space associated with the Alexiewicz norm and when f is of bounded variation.

On the Dual Space of BV-Integrable Functions in Euclidean Space

The dual space (with respect to the Alexiewicz norm) of the class of ${\cal BV}$-integrable functions on a compact cell ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i] \subset {\mathbb R}^m$ is shown to be isometrically isomorphic to the space of finite signed Borel measures on ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i)$, and the usual integral representation theorem holds. An example is also given to show that Lipschitz functions are not part of this dual space. This answers a question of Thierry De Pauw.

A Full Characterization of Multipliers for the Strong ?-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong ϱ-integral, introduced by Jarnik and Kurzweil. Let \((S_\varrho (E)|| \cdot ||)\) be the space of all strongly ϱ-integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm \(|| \cdot ||.\) We show that each element in the dual space of \((S_\varrho (E)|| \cdot ||)\) can be represented as a strong ϱ-integral. Consequently, we prove that fg is strongly ϱ-integrable on E for each strongly ϱ-integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.

NORM CONVERGENCE AND UNIFORM INTEGRABILITY FOR THE HENSTOCK-KURZWEIL INTEGRAL

We show that a uniformly integrable, pointwise convergent sequence of Henstock-Kurzweil integrable functions converges in the Alexiewicz norm. In particular, this implies that a sequence satisfying the conditions in the Dominated Convergence Theorem is norm convergent.

RIESZ TYPE THEOREMS FOR GENERAL INTEGRALS

The author gives a general descriptive definition for integration, denoted by \({\mathcal P}\), which has as special cases the Lebesgue integral for bounded measurable functions, the Lebesgue integral, the Denjoy-Perron integral \({\mathcal D}^*\), the wide Denjoy integral \({\mathcal D}\), the Foran integral, the Iseki integral and the \(S{\mathcal F}\)-integral (\cite{Ene1}). This \({\mathcal P}\)-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying \(f\) with \(g\) whenever \(f = g\) \(a.e.\) on \([a,b]\)). The classical Riesz representation theorem for the linear and continuous functionals on \((C([a,b]),\|\cdot\|_\infty)\) is a consequence of Theorem 2. In addition it is shown that the space of \({\mathcal P}\)-integrable functions is of the first category in itself (see Section 5). Also a characterization of weak convergence on this space is given.

ON THE ALEXIEWICZ TOPOLOGY OF THE DENJOY SPACE

The paper deals with the space of all Denjoy-Perron integrable functions on a fixed interval endowed with the Alexiewicz norm and the completion of this space. The relatively weakly compact subsets of each space are characterized.

The space of Henstock integrable functions of two variables

We consider the space of Henstock integrable functions of two variables. Equipped with the Alexiewicz norm the space is proved to be barrelled. We give a partial description of its dual. We show by an example that the dual can′t be described in a manner analogous to the one‐dimensional case, since in two variables there exist functions whose distributional partials are measures and which are not multipliers for Henstock integrable functions.