Extension of Haar’s theorem

Haar’s theorem ensures a unique nontrivial regular Borel measure on a locally compact Hausdorff topological group, up to multiplication by a positive constant. In this article, we extend Haar’s theorem to the case of locally compact Hausdorff strongly topological gyrogroups. We simultaneously prove the existence and uniqueness of a Haar measure on a locally compact Hausdorff strongly topological gyrogroup, using the method of Steinlage. We then find a natural relationship between Haar measures on gyrogroups and on their related groups. As an application of this result, we study some properties of a convolution-like operation on the space of Haar integrable functions defined on a locally compact Hausdorff strongly topological gyrogroup

$\mu$-statistical convergence and the space of functions $\mu$-stat continuous on the segment

In this work, the concept of a point $\mu$-statistical density is defined. Basing on this notion, the concept of $\mu$-statistical limit, generated by some Borel measure $\mu\left(\cdot \right)$, is defined at a point. We also introduce the concept of $\mu$-statistical fundamentality at a point, and prove its equivalence to the concept of $\mu$-stat convergence. The classification of discontinuity points is transferred to this case. The appropriate space of $\mu$-stat continuous functions on the segment with sup-norm is defined. It is proved that this space is a Banach space and the relationship between this space and the spaces of continuous and Lebesgue summable functions is considered.

On the Moment Problem and Related Problems

Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

Families of Well Approximable Measures

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1] d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N− 1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most ( log N ) d − 1 2 N − 1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

Extremals in nonlinear potential theory

We consider the PDE - Δ p ⁢ u = ρ -\Delta_{p}u=\rho , where 𝜌 is a signed Borel measure on R n \mathbb{R}^{n} . For each p > n p>n , we characterize solutions as extremals of a generalized Morrey inequality determined by 𝜌.

Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

Uniformly Embedding Topological Manifolds into $L^{2}$ Spaces

With a simple argument, we show as a main note that every locally compact second-countableHausdorff space is topologically embeddable into some $L^{2}$ space with respect to some finite nonzero Borel measure, where the embedding may be chosen so that it is uniform and its range is included in some open proper subset of the $L^{2}$ space.