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It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn, Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.
As an application of the boundary parametrization developed in our previous papers, we propose a new method to deduce information on the connected components of the interior of tiles. This gives a systematic way to study the topology of a certain class of self-affine tiles. An example due to Bandt and Gelbrich is examined to prove the efficiency of the method.
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.
Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that
In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.
We prove that triangular configurations are plentiful in large subsets of Cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.
In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.
The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.
We give a new necessary and sufficient condition for an iterated function system to satisfy the deterministic chaos game. As a consequence, we give a new example of an iterated function system which satisfies the deterministic chaos game.
Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.
A set is shy or Haar null (in the sense of Christensen) if there exists a Borel set and a Borel probability measure μ on C[0, 1] such that and for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.
The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy many f ∈ C[0, 1]?
The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.
In an earlier paper, Balka et al. proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦ f–1 are not perfect form a Haar ambivalent set in C[0, 1].
We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f([0, 1]) the level set f–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set Pf ⊂ [0, 1] such that fʹ(x) = ∞ for all x ∈ Pf is Haar ambivalent.
Let μλ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ1/(2k). This is surprising because some spectra for the measure μ1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ1/(2k).
Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.
The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are
$$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$
where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings:
in which the infimum is taken over cubic coverings {Ci} of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D(A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.
We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.
We consider a special kind of structure resolvability and irresolvability for measurable spaces and discuss analogues of the criteria for topological resolvability and irresolvability.
We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^{n}$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in works by Sawyer [Mathematika34(1) (1987), 69–76] and Wisewell [Mathematika51(1–2) (2004), 155–162]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one over $\mathbb{F}_{\ell }\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ presented by Dummit and Hablicsek [Mathematika59(2) (2013), 257–266].
We establish that the intrinsic distance dE associated with an indecomposable plane set E of finite perimeter is infinitesimally Euclidean; namely,
in E. By this result, we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential. We also provide some applications to complex analysis. Moreover, we present a collection of results that would seem to suggest the possibility of developing a De Rham cohomology theory for integral currents.
We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.
We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^{n}$ ($n\geqslant 4$) has upper Minkowski dimension at most $n-4$.