Riesz Theorem Research Articles

On the functor of comonotonically maxitive functionals

We introduce a functor of functionals that preserve the maximum of comonotone functions and the addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and includes the idempotent measure functor as a subfunctor. The main aim of this paper is to demonstrate that this functor is isomorphic to the capacity functor. We establish this isomorphism using the fuzzy max-plus integral. In essence, this result can be viewed as an idempotent analogue of the Riesz Theorem, which establishes a correspondence between the set of σ-additive regular Borel measures and the set of positive linear functionals.

Bounded Pluriharmonic Functions and Holomorphic Functions on Teichmüller Space

Abstract In this paper, we discuss the boundary behavior of bounded pluriharmonic functions and bounded holomorphic functions on the Teichmüller space. We will show a version of the Fatou theorem that every bounded pluriharmonic function admits the radial limits along the Teichmüller geodesic rays, and a version of the F. and M. Riesz theorem that the radial limit of a non-constant bounded holomorphic function is not constant on any non-null measurable set on the Bers boundary in terms of the pluriharmonic measure. As a corollary, we obtain the non-ergodicity of the action of the Torelli group for a closed surface of genus $g\ge 2$ on the space of projective measured foliations.

Riesz Inequality for Harmonic Quasiregular Mappings

In this paper, we generalize the Riesz theorem for harmonic quasiregular mappings for a special case (when p = 2) in the unit disc. Our results improve similar results in this field and are proved with milder conditions. Moreover, we prove another variant forms of Riesz inequality for harmonic quasiregular functions.

Compactness Criteria for Stieltjes Function Spaces and Applications

In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli–Arzelà and Kolmogorov–Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.

Local vector measures

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean space. In more general curved contexts, the same objects can be perfectly understood via charts. However, charts are often unavailable in the less regular setting of metric geometry, where still the questions make sense.In this paper we propose a way to deal with this sort of problems and, more generally, to give a meaning to a concept of ‘measure acting in duality with sections of a given bundle’, loosely speaking. Despite the generality, several classical results in measure theory like Riesz's and Alexandrov's theorems have a natural counterpart in this setting. Moreover, as we are going to discuss, the notions introduced here provide a unified framework for several key concepts in nonsmooth analysis that have been introduced more than two decades ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions and Miranda's BV functions.Not surprisingly, the understanding of the structure of these objects improves with the regularity of the underlying space. We are particularly interested in the case of RCD spaces where, as we will argue, the regularity of several key measures of the type we study nicely matches the known regularity theory for vector fields, resulting in a very effective theory.We expect that the notions developed here will help creating stronger links between differential calculus in Alexandrov spaces (based on Perelman's DC charts) and in RCD ones (based on intrinsic tensor calculus).

On t-normed integrals with respect to possibility capacities on compacta

Riesz Theorem establishes a correspondence between the set of σ-additive regular Borel measures and the set of linear positively defined functionals. We consider an idempotent analogue of this correspondence between possibility capacities and functionals preserving the maximum operation and t-norm operation using t-normed integrals. Additionally, we establish that this correspondence possesses a categorical sense, representing a morphism of monads. Furthermore, we delve into the investigation of the associated idempotent convexity.

Green functions and smooth distances.

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.

On some properties of functions of the grand Hardy classes

Grand Hardy class H+p), p > 1, is defined and some properties of functions belonging to this class are studied in this work. Namely, the analogs of the Riesz and Smirnov theorems as well as the Cauchy’s formula for representation of function are proved. Necessary and sufficient condition for the validity of Riesz theorem in grand Hardy spaces H+p), p > 1, is found. Subspace GH+p)of the grand Hardy space H+p)generated by this condition is defined.

Some remarks on characterization of t-normed integrals on compacta

A characterization of t-normed integrals was obtained by Luis M. de Campos, María T. Lamata and Serafín Moral (1991) for finite compacta and by T. Radul (2022) for the general case. Such characterization establishes a correspondence between the space of capacities and homogeneous respect t-norm monotone normalized functionals preserving the maximum operation of comonotone functions. In fact, we can consider these theorems as non-additive and non-linear analogues of well-known Riesz Theorem about a correspondence between the set of σ-additive regular Borel measures and the set of linear positively defined functionals. We discuss optimality of such characterization.

Global dynamics of diffusive Hindmarsh–Rose equations with memristors

Global dynamics of the diffusive Hindmarsh–Rose equations with memristors as a new proposed model for neuron dynamics are investigated in this paper. We prove the existence and regularity of a global attractor for the solution semiflow through uniform analytic estimates showing the higher-order dissipative property and the asymptotically compact characteristics of the solutions by the approach of Kolmogorov–Riesz theorem. The quantitative bounds of the regions containing this global attractor respectively in the state space and in the regular space are explicitly expressed by the model parameters.

The Cyclical Compactness in Banach C∞(Q)-Modules

In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete Amodules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X) = Γ(Y). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+i ·C∞(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module.

Boundary values in spaces spanned by rational functions and the index of invariant subspaces

Let μ $\mu$ be a positive compactly supported measure in the complex plane C $\mathbb {C}$ , and for each p , 1 ⩽ p < ∞ $p,1\leqslant p<\infty$ , let H p ( μ ) $H^p(\mu )$ be the closed subspace of L p ( μ ) $L^p(\mu )$ spanned by the polynomials. In 1991, Thomson gave a complete description of its structure, expressing H p ( μ ) $H^p(\mu )$ as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non-trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to R p ( μ ) $R^p(\mu )$ , the closed subspace of L p ( μ ) $L^p(\mu )$ spanned by the rational functions having no poles on the support of μ $\mu$ , by first establishing the existence of boundary values in these spaces. Our results all depend on the semiadditivity of analytic capacity, and ultimately on some form of the F. and M. Riesz theorem.

Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost

This paper are divided into two mains parts with different affairs to discuss. The first part are some history of how the modern strict mathmaticala analysis built up. And there’s a simple counterexample for Rolle theorem. As we all known Rolle theorem has three requirement to work out. So, there’s some counterexamples for which there’s only two requirements exist, and what is going to happen. The second part gives two counterexamples on Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Then there’s a proof for the equivalent of the Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Some recent research are given at the last, which means it’s still a vigor field nowadays.

Synthetic topology in Homotopy Type Theory for probabilistic programming

AbstractThe ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial σ-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.

An F. and M. Riesz Theorem for strong H2-functions on the unit circle

An F. and M. Riesz Theorem states that if f,g∈H2 are such that fg=0 a.e. on the unit circle T then either f=0 a.e. on T or g=0 a.e. on T. In this paper we establish an extension of the F. and M. Riesz Theorem for (operator-valued) strong H2-functions on the unit circle.

A quantitative version of the Kolmogorov–Riesz theorem

In the present paper, we prove a generalization of the celebrated Kolmogorov–Riesz theorem in the $$L^{p}([0,1],X)$$ spaces, with $$1\le p <\infty$$ and X a Banach space. Specifically, our result is a quantitative version of such theorem. Our main tool is the so-called degree of nondensifiability, which measures (in the specified sense), the distance from a given convex subset of a Banach space to the class of its Peano Continua.

On the Solvability of Riemann Problems in Grand Hardy Classes

The grand Hardy classes $$H_{p)}^{+}$$ and $${}_{m}H_{p)}^{-}$$ , $$p>1$$ , of functions analytic inside and outside the unit disk, which are generated by the norms of the grand Lebesgue spaces, are defined. Riemann problems of the theory of analytic functions with piecewise continuous coefficient are considered in these spaces. For these problems in grand Hardy classes, a sufficient solvability condition on the coefficient of the problem is found and a general solution is constructed. It should be noted that grand Lebesgue spaces are nonseparable and, therefore, certain classical facts (for example, part of the Riesz theorem) do not hold in these spaces, as well as in the Hardy spaces generated by them. Therefore, one must find a suitable subspace associated with differential equations and study the problems in these subspaces.

On Parseval’s Identity and Plancherel Relation in 2-Inner Product Spaces

This paper investigates the foundational concepts of linear 2-normed spaces and 2-inner product spaces, presenting essential definitions and theorems that elucidate their structure. We establish Parseval’s identity for finite orthonormal sets within these spaces and examine its relationship with the Plancherel relation. Our findings include theorems related to orthonormal sets and Bessel's inequality, highlighting their implications for countably infinite orthonormal sets. By contributing to the theoretical framework of 2-inner product spaces, we enhance the understanding of orthogonality and summability in this context. Introduced by Diminnie et al. (1973), the 2-inner product extends traditional inner product notions into a two-dimensional framework, offering valuable insights into unique mathematical structures. Recent contributions, such as the formalization of orthonormal sets by Elumalai and Patricia (2000) and the rigorous proofs of the Riesz theorems by Harikrishnan et al. (2011), further deepen our understanding of these spaces. This paper aims to develop important inequalities based on orthonormal sets and present equivalent statements related to them, fostering continued research in various mathematical contexts.

Generalized convergence theorems for monotone measures

In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

Strong uniqueness results for first-order planar equations

We discuss strong uniqueness results in the forward Cauchy Problem for a class of complex fist-order equations with Hölder coefficients defined in the plane. We use an appropriate variant of the similarity principle in order to reduce the original question to a local version of Riesz's uniqueness theorem for holomorphic functions.