Finite Lebesgue Measure Research Articles

On support properties of functions and their Jacobi transform

The qualitative uncertainty principle proved by Benedicks asserts that f and its Fourier transform \(\hat f\) cannot both concentrated in subsets of finite Lebesgue measure. In this paper we obtain some uncertainty principles concerning sets of finite measure in the Jacobi setting.

Exponential estimates for solutions of half-linear differential equations

This paper is concerned with estimates, unimprovable in a certain sense, for positive solutions to the half-linear differential equation ( $${|y^{\prime}|^{\alpha - 1} {\rm sgn} y^{\prime})^{\prime} = p(t) |y|^{\alpha - 1} {\rm sgn} y}$$ , where p is a continuous nonnegative function on $${[0, \infty)}$$ and $${\alpha> 1}$$ . It is shown that any positive increasing solution y of the equation satisfies $${y(t) \geqq y(0) exp{h \int_{0}^{t} p^{\frac{1}{\alpha}}(s) {\rm d}s}}$$ , with $${h< (\alpha - 1)^{-\frac{1}{\alpha}}}$$ , for all t on the complement of a set of finite Lebesgue measure. Under an additional assumption, this estimate holds for all t. Further, a condition is established which guarantees that the equation has exponentially increasing solutions and exponentially decreasing solutions.

Uniform bounds for the heat content of open sets in Euclidean space

We obtain (i) lower and upper bounds for the heat content of an open set in Rm with R-smooth boundary and finite Lebesgue measure, (ii) a necessary and sufficient geometric condition for finiteness of the heat content in Rm, and corresponding lower and upper bounds, (iii) lower and upper bounds for the heat loss of an open set in Rm with finite Lebesgue measure.

On restricted weak-type constants of Fourier multipliers

We exhibit a large class of symbols m: Rd ! C for which the corresponding Fourier multipliers Tm satisfy the following restricted weak-type estimates: if A Rd has finite Lebesgue measure, then... In particular, this leads to novel sharp estimates for the real and imaginary part of the Beurling-Ahlfors operator on C. The proof rests on probabilistic methods: we exploit a stochastic representation of the multipliers in terms of Levy processes and appropriate sharp inequalities for differentially subordinated martingales.

Grand Lebesgue spaces with respect to measurable functions

Let 1<p<∞. Given Ω⊂Rn a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces Lp)(Ω) is given by |f|Lp)(Ω)=sup0<ε<p−1ε1p−ε(1|Ω|∫Ω|f|p−εdx)1p−ε. In this paper we consider the norm |f|Lp),δ(Ω) obtained replacing ε1p−ε by a generic nonnegative measurable function δ(ε). We find necessary and sufficient conditions on δ in order to get a functional equivalent to a Banach function norm, and we determine the “interesting” class Bp of functions δ, with the property that every generalized function norm is equivalent to a function norm built with δ∈Bp. We then define the Lp),δ(Ω) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality.

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels Γ ( t ) and noise term ζ ( t ) for the anharmonic coordinates q ( t ) = ( q α ( t ) ) . For zero temperature, i.e. for ζ ( t ) ≡ 0 , we prove that the support of the Fourier transform of Γ ( t ) and of the time averaged velocity–velocity correlation functions K ( t ) of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory q ( t ) with frequency Ω we find that the energy E T transferred to the harmonic system up to time T is proportional to T α . If Ω equals one of the phonon frequencies ω ν , it is α = 2 . We prove that there is a zero measure set L such that for Ω in its full measure complement R ∖ L , it is α = 0 , i.e. there is no energy dissipation. Under certain conditions L contains a subset L ′ such that for Ω ∈ L ′ the dissipation rate is nonzero and may be subdissipative ( 0 ≤ α < 1 ) or superdissipative ( 1 < α ≤ 2 ) , compared to ordinary dissipation ( α = 1 ) . Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency Ω exist for all Ω in a Cantor set C ( k ) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for G ( t ; Ω ) , related to Γ ( t ) . For Ω ∈ C ( k ) the small denominators do not lead to divergencies such that G ( t ; Ω ) is a smooth and bounded function in t .

Weak Cartan-type second main theorem for holomorphic curves

In this paper, a weak Cartan-type second theorem for holomorphic curve f: C → Pn(C) intersecting hypersurfaces Dj, 1 ≤ j ≤ q, in Pn (C) in general position with degree dj is given as follows: For every ɛ > 0, there exists a positive integer M such that \( \left\| {(q - (n + 1) - \varepsilon )T_f (r)} \right. \leqslant \sum\nolimits_{j = 1}^q {\frac{1} {{d_j }}} N_f^M (r,D_j ) + o(T_f (r)) \) where “∥” means the estimate holds for all large r outside a set of finite Lebesgue measure.

$\alpha $-continuity properties of the symmetric $\alpha $-stable process

Let D be a domain of finite Lebesgue measure in Ρ d and let X t D be the symmetric α-stable process killed upon exiting D. Each element of the set {λ i α } i ∞ =1 of eigenvalues associated to X t D , regarded as a function of a ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bounded, then each λ i α is continuous in α and the set of associated eigenfunctions is precompact.

Comparison of potential theoretic properties of rough domains

We discuss the relationships between the notions of intrinsic ultra-contractivity, the parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and the non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain D and compactness of the 1-resolvent of the Neumann Laplacian in D.

Optimal spectral theory of the linear Boltzmann equation

We give optimal compactness results in L p spaces ( 1 < p < ∞ ) related to spectral theory of general neutron transport equations on spatial domains with finite Lebesgue measure.

On the Henstock-Kurzweil Integral for Riesz-Space-Valued Functions Defined on Unbounded Intervals

In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.

Universal spectra, universal tiling sets and the spectral set conjecture

A subset $\Omega$ of $\mathbf{R}^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $\Lambda\subset\mathbf{R}$ such that ${\mathcal E}_\Lambda :=\{e^{i2\pi \langle\lambda, x\rangle}: \lambda\in\Lambda\}$ form an orthogonal basis of $L^2(\Omega)$. The set $\Lambda$ is called a spectrum of the set $\Omega$. The Spectral Set Conjecture states that $\Omega$ is a spectral set if and only if $\Omega$ tiles $\mathbf{R}^d$ by translation. In this paper we prove the Spectral Set Conjecture for a class of sets $\Omega \subset \mathbf{R}$. Specifically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile $\mathbf{R}$ by translates of a strongly periodic set depending only on the spectrum, and vice versa.

Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

A spectral set in R n is a set of finite Lebesgue measure such that L 2 () has an orthogonal basis of exponentials {e 2�ih �,xi : � ∈ �} restricted to . Any such setis called a spectrum for . It is conjectured that every spectral set tiles R n by translations. A tiling set T of translations has a universal spectrumif every set that tiles R n by T is a spectral set with spectrum �. Recently Lagarias and Wang showed that many periodic tiling sets T have universal spectra. Their proofs used properties of factorizations of abelian groups, and were valid for all groups for which a strong form of a conjecture of Tijdeman is valid. However Tijdeman's original conjecture is not true in general, as follows from a construction of Szabo (17), and here we give a counterexample to Tijdeman's conjecture for the cyclic group of order 900. This paper formulates a new sufficient conditionfor a periodic tiling set to have a universal spectrum, and applies it to show that the tiling sets in the given counterexample do possess universal spectra.

New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform

Gabor and wavelet methods are preferred to classical Fourier methods, whenever the time dependence of the analyzed signal is of the same importance as its frequency dependence. However, there exist strict limits to the maximal time-frequency resolution of these both transforms, similar to Heisenberg's uncertainty principle in Fourier analysis. Results of this type are the subject of the following article. Among else, the following will be shown: if \psi is a window function, f\in L^2({\bf R})\setminus\{0\} an arbitrary signal and G_\psi f(\omega,t) the continuous Gabor transform of f with respect to \psi , then the support of G_\psi f(\omega,t) considered as a subset of the time-frequency-plane {\bf R}^2 cannot possess finite Lebesgue measure. The proof of this statement, as well as the proof of its wavelet counterpart, relies heavily on the well known fact that the ranges of the continuous transforms are reproducing kernel Hilbert spaces, showing some kind of shift-invariance. The last point prohibits the extension of results of this type to discrete theory.

Spectral pairs in cartesian coordinates

Let Ω ⊂ℝd have finite positive Lebesgue measure, and let $$\mathcal{L}^2$$ (Ω) be the corresponding Hilbert space of $$\mathcal{L}^2$$ -functions on Ω. We shall consider the exponential functionse λ on Ω given bye λ(x)=e i2πλ·x . If these functions form an orthogonal basis for $$\mathcal{L}^2$$ (Ω), when λ ranges over some subset Λ in ℝ d , then we say that (Ω, Λ) is a spectral pair, and that Λ is a spectrum. We conjecture that (Ω, Λ) is a spectral pair if and only if the translates of some set Ω′ by the vectors of Λ tile ℝd. In the special case of Ω=Id, the d-dimensional unit cube, we prove this conjecture, with Ω′=Id, for d≤3, describing all the tilings by Id, and for all d when Λ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.

Rearrangements and polar factorisation of countably degenerate functions

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.

Arrovian Social Choice Correspondences

A social choice correspondence is Arrovian if it satisfies Kenneth Arrow's choice axiom and independence of infeasible alternatives. For any such correspondence, there is a large fraction of the population whose preferences are irrelevant to the social decision in a large fraction of situations. The authors consider the case of a finite outcome set, and also an outcome set of positive but finite Lebesgue measure (in some Euclidean space). An unrestricted individual preference domain is assumed. The agenda domain assumption allows for a finite lower bound, possibly greater than two, on the size of a feasible set. Copyright 1996 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

The lemma of the logarithmic derivative for subharmonic functions

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r &gt; 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

Tiling the line with translates of one tile

A region \(T\) is a closed subset of the real line of positive finite Lebesgue measure which has a boundary of measure zero. Call a region \(T\) a tile if\({\Bbb R}\) can be tiled by measure-disjoint translates of \(T\). For a bounded tile all tilings of\({\Bbb R}\) with its translates are periodic, and there are finitely many translation-equivalence classes of such tilings. The main result of the paper is that for any tiling of\({\Bbb R}\) by a bounded tile, any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling. From it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups.

Threshold Growth Dynamics

We study the asymptotic of the occupied region for monotone deterministic dynamics in d-dimensional Euclidean space parametrized by a threshold theta, and a Borel set N with positive and finite Lebesgue measure. If A_n denotes the occupied set of the dynamics at integer time n, then A_n+1 is obtained by adjoining any point x for which the volume of overlap between x+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n converges to a unique limiting shape L starting from any bounded initial region that is suitably large. Moreover, L is computed as the polar transform for 1/w, where w is an explicit width function that depends on N and theta. It is further shown that L describes the limiting of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of 2-d box neighborhoods, these limiting shapes are calculated and the dependence of their anisotropy on theta is examined. Other specific two- and three- dimensional examples are also discussed in some detail.