Lebesgue Measure Research Articles

Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.

Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions

Let f f be a meromorphic function on the complex plane with Nevanlinna characteristic T ( r , f ) T(r,f) and maximal radial characteristic ln ⁡ M ( t , f ) \ln M(t,f) , where M ( t , f ) M(t,f) is the maximum of the modulus | f | |f| on circles centered at zero and of radius t t . A number of well-known and widely used results make it possible to estimate from above the integrals of ln ⁡ M ( t , f ) \ln M (t,f) over subsets E E on segments [ 0 , r ] [0,r] in terms of T ( r , f ) T(r,f) and the linear Lebesgue measure of E E . In the paper, such estimates are obtained for the Lebesgue–Stieltjes integrals of ln ⁡ M ( t , f ) \ln M(t,f) with respect to an increasing integration function m m , and the sets E E on which the function m m is not constant can have fractal nature. At the same time, it is possible to obtain nontrivial estimates in terms of the h h -content and h h -Hausdorff measure of the set E E , as well as their partial d d -dimensional power versions with d ∈ ( 0 , 1 ] d\in (0,1] . All preceding similar estimates known to the author correspond to the extreme case of d = 1 d=1 and an absolutely continuous integration function m m with density of class L p L^p for p > 1 p>1 . The main part of the presentation is carried out immediately for the differences of subharmonic functions, or δ \delta -subharmonic functions, on circles centered at zero with explicit constants in the estimates. The only restriction in the main theorem is that the modulus of continuity of the function m m should satisfy the Dini condition at zero, and this condition, as a counterexample shows, is essential.

Continuous R-valuations

We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Th\'eron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $\tau$-smooth measure $\mu$ (defined on the Borel $\sigma$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $\mu$, when the support of $\mu$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

Rotated odometers

AbstractWe describe the infinite interval exchange transformations, called the rotated odometers, which are obtained as compositions of finite interval exchange transformations and the von Neumann–Kakutani map. We show that with respect to Lebesgue measure on the unit interval, every such transformation is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends. We describe the dynamics of rotated odometers by means of Bratteli–Vershik systems, derive several of their topological and ergodic properties, and investigate in detail a range of specific examples of rotated odometers.

Mixing properties of erasing interval maps

AbstractWe study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $ , defined by $$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$ We prove that, although the map is singular, its square preserves the Lebesgue measure and is strongly mixing, thus ergodic, with respect to it. We discuss the extension of the results to more general erasing maps.

Nowhere-monotone differentiable functions and set of monstrous shift

Given a strictly increasing differentiable function f:R→R whose derivative vanishes on a dense subset of R, it has been shown by Ciesielski that there is at least a comeager amount of t∈R such that f(x)−f(x−t) is nowhere-monotone. In this paper, we show that the amount of their shifted difference that are nowhere-monotone is not only comeager but also of full Lebesgue measure in R. Interestingly, we also provide a specific pair of strictly increasing differentiable functions φ,ψ:R→R such that φ(x)−ψ(x−t) is nowhere-monotone for every t∈R.

Convexity of complements of limit sets for holomorphic foliations on surfaces

Let $${\mathcal {F}}$$ be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $${\mathcal {F}}$$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representations, answering a question asked by Brunella. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $${\mathcal {F}}$$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set by extending Brunella’s method to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.

On a New Measure on the Levi-Civita Field $$ \mathcal{R} $$

The Levi-Civita field $$ \mathcal{R} $$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on $$ \mathcal{R} $$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over $$ \mathcal{R} $$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on $$ \mathcal{R} $$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on $$ \mathcal{R} $$ that proves to be a better generalization of the Lebesgue measure from $$ \mathbb{R} $$ to $$ \mathcal{R} $$ and that leads to a family of measurable sets in $$ \mathcal{R} $$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in $$ \mathbb{R} $$ hold.

Hybrid Bounds on Two-Parametric Families of Weyl Sums Along Smooth Curves

We obtain a new bound on Weyl sums with degree k≥2 polynomials of the form (τx+c)ω(n)+xn, n=1,2,…, with fixed ω(T)∈Z[T] and τ∈R, which holds for almost all c∈[0,1) and all x∈[0,1). We improve and generalize some recent results of Erdoǧan and Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials xn+yω(n) for all (x,y)∈[0,1)2 with f(x,y)=z for a set of z∈[0,1) of full Lebesgue measure, provided that f is a Hölder function.

ERDŐS PROPERTIES OF SUBSETS OF THE MAHLER SETS

AbstractErdős proved that every real number is the sum of two Liouville numbers. A setWof complex numbers is said to have the Erdős property if every real number is the sum of two members ofW. Mahler divided the set of all transcendental numbers into three disjoint classesS,TandUsuch that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the setUand has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if$m\in [0,\infty )$, then there exist$2^{\mathfrak {c}}$dense subsetsWofSeach of Lebesgue measuremsuch thatWhas the Erdős property and no two of theseWare homeomorphic. It is also proved that there are$2^{\mathfrak {c}}$dense subsetsWofSeach of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are$2^{\mathfrak {c}}$dense subsetsWofSsuch that every complex number is the sum of two members ofWand such that no two of theseWare homeomorphic.

Rayleigh–Faber–Krahn, Lyapunov and Hartmann–Wintner Inequalities for Fractional Elliptic Problems

In this paper, in the cylindrical domain, we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh–Faber–Krahn inequality. Also, we give Lyapunov and Hartmann–Wintner inequalities for the fractional elliptic boundary value problem.

The twisted baker map

As a model to provide a hands-on, elementary understanding of ‘vortex dynamics’, we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalised Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.

Improbability of Collisions of Point-Vortices in Bounded Planar Domains

In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point-vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer-Verlag, 1984] for the disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. The establishment of some useful estimates is discussed and the details of the proof are given in the original article [M. Donati, SIAM J. Math. Anal., 54 (2022), pp. 79--113].

SIGEST

The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, \alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.

Intersections of middle-α Cantor sets with a fixed translation

For let C λ be the middle- Cantor set in . Given , excluding the trivial case we show that Λ(t):=λ∈(0,1/3]:Cλ∩(Cλ+t)≠∅ is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of , which reveals a dimensional variation principle. Furthermore, for any we show that the level set Λβ(t):={λ∈Λ(t):dimH(Cλ∩(Cλ+t))=dimP(Cλ∩(Cλ+t))=βlog2−logλ} has equal Hausdorff and packing dimension . We also show that the set of for which has full Hausdorff dimension.

Constructions of some families of smooth Cauchy transforms

AbstractFor a given Beurling–Carleson subset E of the unit circle $\mathbb {T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from $\mathbb {D}$ to $\mathbb {T}$ . The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

Thick hyperbolic repelling invariant Cantor sets and wild attractors

Let D be the set of such that f β is a symmetric tent map with finite critical orbit. For , by operating Denjoy-like surgery on f β , we constructed a C 1 unimodal map admitting a thick hyperbolic repelling invariant Cantor set which contains a wild attractor. The smoothness of is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. D is dense in , and can not be because the hyperbolic repelling invariant Cantor set of map has Lebesgue measure equal to zero.

Risk-neutral multiobjective optimal control of random Volterra integral equations

We analyze risk-neutral multiobjective optimal control problems, governed by Volterra integral equations with random inputs, and subject to expectation-type final/pure state constraints and mixed pointwise control-state constraints. Moreover, controls are unbounded, and inclusion-form pure state as well as control-state constraints are given by measurable set-valued mappings, whose images are nonempty closed subsets of infinite dimensional spaces. With such constraints, the multiplier corresponding to pure state constraints belongs to the dual space of the Banach space-valued continuous functions. We set up a decomposition result for this dual, and develop a Lagrange multiplier theorem for an abstract multi-criteria optimization model, in which a Robinson-type constraint qualification is applied. Then, making use of the obtained outcomes, exploiting higher integrability and regularity of the random field control/state variables, and employing Banach space-valued integration techniques, we derive Fritz-John necessary optimality conditions, with integrable functions and (countably additive) measures singular with respect to the Lebesgue measure as multipliers, for local weak Pareto solutions of studied problems.

Rigidity for measurable sets

Let Ω⊂Rd, d≥2, be a set with finite Lebesgue measure such that, for a fixed radius r>0, the Lebesgue measure of Ω∩Br(x) is equal to a positive constant when x varies in the essential boundary of Ω. We prove that Ω is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than r which is either open and connected, or of finite perimeter and indecomposable. The proof requires reinventing each step of the moving planes method by Alexandrov in the framework of measurable sets.

Analysis of Finite Solution Spaces of Second-Order ODE with Dirac Delta Periodic Forcing

Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical solutions are often prone to failures in the presence of discontinuities. Algebraic decompositions are employed for analysis in such cases, assuming that regularities exist, operators are present in Banach (solution) spaces, and there is finite measurability. This paper proposes a generalized, finite-dimensional algebraic analysis of the solution spaces of second-order ODEs equipped with periodic Dirac delta forcing. The proposed algebraic analysis establishes the conditions for the convergence of responses within the solution spaces without requiring relative smoothness of the forcing functions. The Lipschitz regularizations and Lebesgue measurability are not considered as preconditions maintaining generality. The analysis shows that smooth and locally finite responses can be admitted in an exponentially stable solution space. The numerical analysis of the solution spaces is computed based on combinatorial changes in coefficients. It exhibits a set of locally uniform responses in the solution spaces. In contrast, the global response profiles show localized as well as oriented instabilities at specific neighborhoods in the solution spaces. Furthermore, the bands of the expansions–contractions of the stable response profiles are observable within the solution spaces depending upon the values of the coefficients and time intervals. The application aspects and distinguishing properties of the proposed approaches are outlined in brief.