Characterizations of the lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part

In this paper, we study a class of Borel measures on [Formula: see text] that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the [Formula: see text]-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Reconstruction of the One-Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

Criteria of Removable Sets for Harmonic Functions in the Sobolev Spaces L1p,w

Ahlfors and Beurling [16] proved that set 𝐸 is removable for class 𝐴𝐷2 of analytic functions with the finite Dirichlet integral if and only if 𝐸 does not change extremal distances. Their proof uses the conformal invariance of class 𝐴𝐷2, so it does not immediately generalize to 𝑝 ̸= 2 and to the relevant classes of harmonic functions in the space. In 1974 Hedberg [19] proposed new approaches to the problem of describing removable singularities in the function theory. In particular he gave the exact functional capacitive conditions for a set to be removable for class 𝐻𝐷𝑝(𝐺). Here 𝐻𝐷𝑝(𝐺) is the class of real-valued harmonic functions 𝑢 in a bounded open set 𝐺 ⊂ 𝑅𝑛, 𝑛 ≥ 2, and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑑𝑥 < ∞, 𝑝 > 1. In this paper we extend Hedberg’s results on class 𝐻𝐷𝑝,𝑤(𝐺) of harmonic functions 𝑢 in 𝐺 and such that ∫︁ 𝐺 |∇𝑢|𝑝 𝑤𝑑𝑥 < ∞. Here a locally integrable function 𝑤 : 𝑅𝑛 → (0,+∞) satisfies the Muckenhoupt condition [20] sup 1 |𝑄| ∫︁ 𝑄 𝑤𝑑𝑥 ⎛ ⎝ 1 |𝑄| ∫︁ 𝑄 𝑤1−𝑞𝑑𝑥 ⎞ ⎠ 𝑝−1 < ∞, where the supremum is taking over all coordinate cubes 𝑄 ⊂ 𝑅𝑛, 𝑞 ∈ (1,+∞) and 1 𝑝 + 1 𝑞 = 1; by ℒ𝑛(𝑄) = |𝑄| we denote the 𝑛-dimensional Lebesgue measure of 𝑄. We denote by 𝐿1 𝑞 , ˜ 𝑤(𝐺) the Sobolev space of locally integrable functions 𝐹 on 𝐺, whose generalized gradient in 𝐺 are such that ‖𝑓‖𝐿1 𝑞 , ˜ 𝑤(𝐺) = ⎛ ⎝ ∫︁ 𝐺 |∇𝑓|𝑞 ˜ 𝑤𝑑𝑥 ⎞ ⎠ 1 𝑞 < ∞, where ˜ 𝑤 = 𝑤1−𝑞. The closure of 𝐶∞ 0 (𝐺) in ‖ · ‖𝐿1 𝑞 , ˜ 𝑤(𝐺) is denoted by ∘L 1 𝑞, ˜ 𝑤(𝐺). For compact set 𝐾 ⊂ 𝐺 (𝑞, ˜ 𝑤)-capacity regarding 𝐺 is defined by 𝐶𝑞, ˜ 𝑤(𝐾) = inf 𝑣 ∫︁ 𝐺 |∇𝑣|𝑞 ˜ 𝑤𝑑𝑥, where the infimum is taken over all 𝑣 ∈ 𝐶∞ 0 (𝐺) such that 𝑣 = 1 in some neighbourhood of 𝐾. Note that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 is independent from the choice of bounded set 𝐺 ⊂ 𝑅𝑛. We set 𝐶𝑞, ˜ 𝑤(𝐹) = 0 for arbitrary 𝐹 ⊂ 𝑅𝑛 if for every compact 𝐾 ⊂ 𝐹 there exists a bounded open set 𝐺 such that 𝐶𝑞, ˜ 𝑤(𝐾) = 0 regarding 𝐺. To conclude, we formulate the main results. Theorem 1. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶∞ 0 (𝐺 ∖ 𝐸) is dense in ∘L 1 𝑞, ˜ 𝑤(𝐺). Theorem 2. Compact 𝐸 ⊂ 𝐺 is removable for 𝐻𝐷𝑝,𝑤(𝐺) if and only if 𝐶𝑞, ˜ 𝑤(𝐸) = 0. Corollary. The property of being removable for 𝐻𝐷𝑝,𝑤(𝐺) is local, i.e. compact 𝐸 ⊂ 𝐺 is removable if and only if every 𝑥 ∈ 𝐸 has a compact neighbourhood, whose intersection with 𝐺 is removable. Theorem 3. If 𝐺 is an open set in 𝑅𝑛 and 𝐶𝑞, ˜ 𝑤(𝑅𝑛 ∖𝐺) = 0. Then 𝐶∞ 0 (𝐺) is dense in ∘L 1 𝑞, ˜ 𝑤(𝑅𝑛).

Isoperimetric and Functional Inequalities

We establish lower estimates for an integral functional$$\int\limits_\Omega f(u(x), \nabla u(x)) \, dx ,$$where \(\Omega\) -- a bounded domain in \(\mathbb{R}^n \; (n \geqslant 2)\), an integrand \(f(t,p) \, (t \in [0, \infty),\; p \in \mathbb{R}^n)\) -- a function that is \(B\)-measurable with respect to a variable \(t\) and is convex and even in the variable \(p\), \(\nabla u(x)\) -- a gradient (in the sense of Sobolev) of the function \(u \colon \Omega \rightarrow \mathbb{R}\). In the first and the second sections we utilize properties of permutations of differentiable functions and an isoperimetric inequality \(H^{n-1}( \partial A) \geqslant \lambda(m_n A)\), that connects \((n-1)\)-dimensional Hausdorff measure \(H^{n-1}(\partial A )\) of relative boundary \(\partial A\) of the set \(A \subset \Omega\) with its \(n\)-dimensional Lebesgue measure \(m_n A\). The integrand \(f\) is assumed to be isotropic, i.e. \(f(t,p) = f(t,q)\) if \(|p| = |q|\).Applications of the established results to multidimensional variational problems are outlined. For functions \( u \) that vanish on the boundary of the domain \(\Omega\), the assumption of the isotropy of the integrand \( f \) can be omitted. In this case, an important role is played by the Steiner and Schwartz symmetrization operations of the integrand \( f \) and of the function \( u \). The corresponding variants of the lower estimates are discussed in the third section. What is fundamentally new here is that the symmetrization operation is applied not only to the function \(u\), but also to the integrand \(f\). The geometric basis of the results of the third section is the Brunn-Minkowski inequality, as well as the symmetrization properties of the algebraic sum of sets.

The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.

Elementary volume and measurability properties of additive functions

The paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications

We study, for some subsets I of N*, the Banach space E of bounded real sequences {xn}n∈I. For any integer k, we introduce a measure over (E,B(E)) that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration.

Invariant Measure Under the Affine Group Over

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.