Lebesgue Points Research Articles

Lebesgue points via the Poincaré inequality

We show that in a Q-doubling space (X, d, µ), Q > 1, which satisfies a chain condition, if we have a Q-Poincaré inequality for a pair of functions (u, g) where g ∈ L Q (X), then u has Lebesgue points $\mathcal{H}^h $ -a.e. for $h(t) = \log ^{1 - Q - \varepsilon } (1/t)$ . We also discuss how the existence of Lebesgue points follows for u ∈ W 1,Q (X) where (X, d, µ) is a complete Q-doubling space supporting a Q-Poincaré inequality for Q > 1.

Lebesgue Points of Two-Dimensional Fourier Transforms and Strong Summability

We introduce the concept of modified strong Lebesgue points and show that almost every point is a modified strong Lebesgue point of \(f\) from the Wiener amalgam space \(W(L_1,\ell _\infty )({\mathbb {R}}^2)\). A general summability method of 2D Fourier transforms is given with the help of an integrable function \(\theta \). Under some conditions on \(\theta \) we show that the Marcinkiewicz-\(\theta \)-means of a function \(f\in W(L_1,\ell _\infty )({\mathbb {R}}^2)\) converge to \(f\) at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of \(f\in W(L_p,\ell _\infty )({\mathbb {R}}^2)\), whenever \(1<p<\infty \). As an application we generalize the classical 1D strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for \(f\in W(L_1,\ell _\infty )({\mathbb {R}})\) and for strong \(\theta \)-summability. Some special cases of the \(\theta \)-summation are considered, such as the Weierstrass, Abel, Picar, Bessel, Fejer, de La Vallee-Poussin, Rogosinski and Riesz summations.

Perturbing Misiurewicz Parameters in the Exponential Family

In one-dimensional real and complex dynamics, a map whose post-singular (or post-critical) set is bounded and uniformly repelling is often called a Misiurewicz map. In results hitherto, perturbing a Misiurewicz map is likely to give a non-hyperbolic map, as per Jakobson's Theorem for unimodal interval maps. This is despite genericity of hyperbolic parameters (at least in the interval setting). We show the contrary holds in the complex exponential family $z \mapsto \lambda \exp(z)$: Misiurewicz maps are Lebesgue density points for hyperbolic parameters. As a by-product, we also show that Lyapunov exponents almost never exist for exponential Misiurewicz maps. The lower Lyapunov exponent is negative infinity almost everywhere. The upper Lyapunov exponent is non-negative and depends on the choice of metric.

Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces

Abstract The inversion formula for the continuous wavelet transform is usually considered in the weak sense. With the help of summability methods of Fourier transforms we obtain norm convergence and convergence at Lebesgue points of the inverse wavelet transform for functions from the Lp and Wiener amalgam spaces.

Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, iffis in the Wiener amalgam spaceW(L1,lq)(R)andfis almost everywhere locally bounded, orf∈W(Lp,lq)(R) (1&lt;p&lt;∞,1≤q&lt;∞), then strongθ-summability holds at each Lebesgue point off. The analogous results are given for Fourier series, too.

Поведение экспоненциальных средних рядов Фурье и сопряженных рядов Фурье в точках Лебега

Поведение экспоненциальных средних рядов Фурье и сопряженных рядов Фурье в точках Лебега

Fine properties of the subdifferential for a class of one-homogeneous functionals

Abstract We collect some known results on the subdifferentials of a class of one-homogeneous functionals, which consist in anisotropic and nonhomogeneous variants of the total variation. It is known that the subdifferential at a point is the divergence of some “calibrating field”. We establish new relationships between Lebesgue points of a calibrating field and regular points of the level surfaces of the corresponding calibrated function.

The variable exponent Sobolev capacity and quasi-fine properties of Sobolev functions in the case [formula omitted

In this article we extend the known results concerning the subadditivity of capacity and the Lebesgue points of functions of the variable exponent Sobolev spaces to cover also the case p−=1. We show that the variable exponent Sobolev capacity is subadditive for variable exponents satisfying 1⩽p<∞. Furthermore, we show that if the exponent is log-Hölder continuous, then the functions of the variable exponent Sobolev spaces have Lebesgue points quasieverywhere and they have quasicontinuous representatives also in the case p−=1. To gain these results we develop methods that are not reliant on reflexivity or maximal function arguments.

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \)-summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \)-means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\). Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\), where \(1<r\le \infty \). Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\).

Maximal function and Riesz potential on p-adic linear spaces

For maximal function and Riesz potential on p-adic linear space ℚpn we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential Iα(f) exists at point b, then b is Lq Lebesgue point for some q.

Example of a Monotonic, Everywhere Differentiable Function on ℝ Whose Derivative Is not Continuous

We construct an example of a monotonic function which is differentiable everywhere, but the derivative is not continuous. This is done using a nonnegative discontinuous integrable function whose every point is a Lebesgue point.

Sobolev spaces, Lebesgue points and maximal functions

In this note, we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such maximal operators.

Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $\Delta u -u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of continuity (still in $L^p$ average) for $D^2 u$ and $\dd_t u$. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the $L^p$ average). Moreover, we get a quantitative estimate of the error in this Taylor expansion. Our method is based on decay estimates obtained by contradiction, using blow-up arguments and Liouville type theorems. As a by-product of our approach, we deduce that the regular points of the free boundary are locally contained in a $C^1$ hypersurface for the parabolic distance $\sqrt{x^2 +|t|}$.

Bulk universality holds pointwise in the mean for compactly supported measures

Let μ be a measure with compact support, with orthonormal polynomials {pn}, and associated reproducing kernels {Kn}. We show that without any global assumptions on the measure, a weak local condition leads to the bulk universality limit in the mean. For example, if μ ≥ C > 0 in some open interval J , then at each Lebesgue point ξ of J , and for each r > 0,

B2-convexity implies strong and weak lower semicontinuity of partitions of ℝn

We prove that B2-convexity is sufficient for lower semicontinuity of surface energy of partitions of R n , for any n 2. We establish lower semicontinuity in the usual strong topology, assuming the regions converge in volume. We also establish lower semicontinuity in the more general situation in which we suppose integral currents associated with individual regions converge to some integral current in the weak topology of integral currents. B2-convexity, formulated by F. Morgan in 1995, is a powerful condition since it is easy to work with and since many other conditions from the literature imply it. Our results therefore imply that each of those conditions is sufficient for strong and weak lower semicontinuity of surface energy. We establish other results of independent interest, including a Lebesgue point theorem for partitions and a localization theorem, which shows that if lower semicontinuity holds locally then it holds globally.

Pointwise Convergence of the Calderón Reproducing Formula

In this paper, we study the pointwise convergence of the Calderon reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every \(f\in L_{w}^{p}(\mathbb {R}^{d})\) with an \(\mathcal{A}_{p}\) weight w, 1≤p<∞, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.

Computing Fekete and Lebesgue points: Simplex, square, disk

We have computed point sets with maximal absolute value of the Vandermonde determinant (Fekete points) or minimal Lebesgue constant (Lebesgue points) on three basic bidimensional compact sets: the simplex, the square, and the disk. Using routines of the Matlab Optimization Toolbox, we have obtained some of the best bivariate interpolation sets known so far.

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T n,A,B f(⋅)−f(⋅) in terms of moduli of continuity \(\bar{w}_{\cdot}f\) and w ⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.

Universality in the bulk holds close to given points

Let μ be a measure with compact support. Assume that ξ is a Lebesgue point of μ and that μ ′ is positive and continuous at ξ . Let { A n } be a sequence of positive numbers with limit ∞ . We show that one can choose ξ n ∈ [ ξ − A n n , ξ + A n n ] such that lim n → ∞ K n ( ξ n , ξ n + a K ̃ n ( ξ n , ξ n ) ) K n ( ξ n , ξ n ) = sin π a π a , uniformly for a in compact subsets of the plane. Here K n is the n th reproducing kernel for μ , and K ̃ n is its normalized cousin. Thus universality in the bulk holds on a sequence close to ξ , without having to assume that μ is a regular measure. Similar results are established for sequences of measures.

Hausdorff measures and Lebesgue points for the Sobolev classes W α p , α &gt; 0, on spaces of homogeneous type

Suppose that (X, µ, d) is a space of homogeneous type, where d is themetric and µ is the measure related by the doubling condition with exponent γ > 0, Wαp(X), p > 1, are the generalized Sobolev classes, α > 0, and dimH is the Hausdorff dimension. We prove that, for any function u ∈ Wαp(X), p > 1, 0 < α < γ/p, there exists a set E ⊂ X such that dimH(E) ≤ γ − αp and, for any x ∈ X ∖ E, the limit $$ \mathop {\lim }\limits_{r \to + 0} \frac{1} {{\mu (B(x,r))}}\int_{B(x,r)} {ud\mu = u^ * (x)} $$ exists; moreover, $$ \mathop {\lim }\limits_{r \to + 0} \frac{1} {{\mu (B(x,r))}}\int_{B(x,r)} {|u - u^ * (x)|^q d\mu = 0,} \frac{1} {q} = \frac{1} {p} - \frac{\alpha } {\gamma }. $$ . For α = 1, a similar result was obtained earlier by Hajlasz and Kinnunen in 1998. The case 0 < α ≤ 1 was studied by the author in 2007; in the proof, the structures of the corresponding capacities were significantly used.