Finite Lebesgue Measure Research Articles

Transcendental Julia sets of positive finite Lebesgue measure

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

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.

Exponential decay for the KdV equation on ℝ with new localized dampings

In this paper, we prove several results on the exponential decay in $L^{2}$ norm of the KdV equation on the real line with localized dampings. First, for the linear KdV equation, the exponential decay holds if and only if the averages of the damping coefficient on all intervals of a fixed length have a positive lower bound. Moreover, under the same damping condition, the exponential decay holds for the (nonlinear) KdV equation with small initial data. Finally, with the aid of certain properties of propagation of regularity in Bourgain spaces for solutions of the associated linear system and the unique continuation property, the exponential decay for the KdV equation with large data holds if the damping coefficient has a positive lower bound on $E$, where $E$ is equidistributed over the real line and the complement $E^{c}$ has a finite Lebesgue measure.

Observability inequality at two time points for the KdV equation from measurable sets

We prove two observability inequalities at two time points for the linear KdV equation on the real line. In the two inequalities, the observable region in space is the complement of (i) a measurable set with finite Lebesgue measure, and (ii) of a measurable set contained in a half line with some density conditions, whose Lebesgue measure may be infinite.

Frame Spectral Pairs and Exponential Bases

Given a domain \(\varOmega \subset {\mathbb {R}}^d\) with positive and finite Lebesgue measure and a discrete set \(\varLambda \subset {\mathbb {R}}^d\), we say that \((\varOmega , \varLambda )\) is a frame spectral pair if the set of exponential functions \({\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}\) is a frame for \(L^2(\varOmega )\). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting \({\mathbb {Z}}_N^d\), \(d, N\ge 1\), a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in \({\mathbb {R}}^d\) by “adding” a frame spectral pair in \({\mathbb {R}}^{d}\) to a frame spectral pair in \({\mathbb {Z}}_N^d\). Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.

On Efficiency and Localisation for the Torsion Function

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set {Omega }subset mathbb {R}^{m} of finite Lebesgue measure 0<|{Omega }|<infty with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

AbstractWe give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha&gt;0$ and all z outside a set of finite Lebesgue measure.

On the Torsion Function with Mixed Boundary Conditions

Let D be a non-empty open subset of ℝm,m≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb {R}^{m}, m\\ge 2$\\end{document}, with boundary ∂D, with finite Lebesgue measure |D|, and which satisfies a parabolic Harnack principle. Let K be a compact, non-polar subset of D. We obtain the leading asymptotic behaviour as ε↓ 0 of the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{\\infty }$\\end{document} norm of the torsion function with a Neumann boundary condition on ∂D, and a Dirichlet boundary condition on ∂(εK), in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that D ∖ K is a non-trap domain.

The norm of the characteristic function of a set in the John‐Nirenberg space of exponent p

We find the concrete value of for any measurable set of positive and finite Lebesgue measure, where stands for the John‐Nirenberg space of exponent . In the case we show that for any interval with and any where

Non-separable Lattices, Gabor Orthonormal Bases and Tilings

Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $\Lambda=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(\Lambda)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}\chi_K, \Lambda)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block triangular matrix with diagonal matrices $A$ and $B$, we prove that if $\mathcal G(|K|^{-1/2}\chi_K, \Lambda)$ is an orthonormal basis, then $K$ can be written as a finite union of fundamental domains of $A({\mathbb Z}^d)$ and at the same time, as a finite union of fundamental domains of $B^{-t}({\mathbb Z}^d)$. If $A^tB$ is an integer matrix, then there is only one common fundamental domain, which means $K$ tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domains is also possible. We also provide a constructive way for forming a Gabor window functions for a given upper triangular lattice. Our study is related to a Fuglede's type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

Three Problems on Exponential Bases

AbstractWe consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$. Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$.

Spectral heat content for Lévy processes

AbstractIn this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in , . We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a Lp-Lebesgue space, of a function f in WkLpnn+kp,p(Ω), where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, k,n∈N, k<n and nn−k<p<+∞. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.

On two functionals involving the maximum of the torsion function

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we considerT(Ω)∕(M(Ω)|Ω|) andM(Ω)λ1(Ω), whereΩis a bounded open set of ℝdwith finite Lebesgue measure |Ω|,M(Ω) denotes the maximum of the torsion function,T(Ω) the torsion, andλ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

Some Localization Properties of the Lp Continuous Wavelet Transform

ABSTRACTWe shall prove some simultaneous localization or concentration inequalities for the continuous wavelet transform. We will also show that simultaneous localization in the scale-time(space) is impossible, in the sense that the scale sections of the support of wavelet transform of a nonnull Lp-function can not have finite Lebesgue measure. Finally, some properties of the support of continuous wavelet transform of band-limited functions are studied.

On the number of Courant-sharp Dirichlet eigenvalues

We consider arbitrary open sets \Omega  in Euclidean space with finite Lebesgue measure, and obtain upper bounds for (i) the largest Courant-sharp Dirichlet eigenvalue of \Omega , (ii) the number of Courant-sharp Dirichlet eigenvalues of  \Omega . This extends recent results of P. Bérard and B. Helffer.

On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue

Let \(\Omega \) be an open set in Euclidean space with finite Lebesgue measure \(\vert \Omega \vert \). We obtain some properties of the set function \(F:\Omega \mapsto {\mathbb {R}}^+\) defined by $$\begin{aligned} F(\Omega )=\frac{T(\Omega )\lambda _1(\Omega )}{\vert \Omega \vert } , \end{aligned}$$ where \(T(\Omega )\) and \(\lambda _1(\Omega )\) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Polya bound \(F(\Omega )\le 1,\) and show that $$\begin{aligned} F(\Omega )\le 1- \nu _m T(\Omega )|\Omega |^{-1-\frac{2}{m}}, \end{aligned}$$ where \(\nu _m\) depends only on m. For any \(m=2,3,\ldots \) and \(\epsilon \in (0,1)\) we construct an open set \(\Omega _{\epsilon }\subset {\mathbb {R}}^m\) such that \(F(\Omega _{\epsilon })\ge 1-\epsilon \).

Heat content for convolution semigroups

Let X={Xt}t≥0 be a Lévy process in Rd and Ω be an open subset of Rd with finite Lebesgue measure. In this article we consider the quantity H(t)=∫ΩPx(Xt∈Ωc)dx related to X which is called the heat content. We study its asymptotic behaviour as t goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

On the Beer Index of Convexity and Its Variants

Let S be a subset of mathbb {R}^d with finite positive Lebesgue measure. The Beer index of convexity{text {b}}(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio{text {c}}(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of convexity. We show that every set Ssubseteq mathbb {R}^2 with simply connected components satisfies {text {b}}(S)leqslant alpha {text {c}}(S) for an absolute constant alpha , provided {text {b}}(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. that this estimate holds for simple polygons. We also consider higher-order generalizations of {text {b}}(S). For 1leqslant kleqslant d, the k-index of convexity{text {b}}_k(S) of a set Ssubseteq mathbb {R}^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every dgeqslant 2 there is a constant beta (d)>0 such that every set Ssubseteq mathbb {R}^d satisfies {text {b}}_d(S)leqslant beta {text {c}}(S), provided {text {b}}_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma (d)>0 such that for every varepsilon in (0,1) there is a set Ssubseteq mathbb {R}^d of Lebesgue measure 1 satisfying {text {c}}(S)leqslant varepsilon and {text {b}}_d(S)geqslant gamma frac{varepsilon }{log _2{1/varepsilon }}geqslant gamma frac{{text {c}}(S)}{log _2{1/{text {c}}(S)}}.

Characterization of ellipsoids as K-dense sets

Let K ⊂ ℝN be any convex body containing the origin. A measurable set G ⊂ ℝN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r &gt; 0, the measure of G ⋂ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In a previous work, we proved for the case N = 2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in ℝN: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, which builds upon results obtained in our previous work, relies on an asymptotic formula for the measure of G ⋂ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to Petty.