Lebesgue Points Research Articles

On the Mellin-Gauss-Weierstrass operators in the Mellin-Lebesgue spaces

In this paper, we present the modulus of smoothness of a function $f∈X_{c}^{p}$, which the Mellin-Lebesgue space, and later we state some properties of it. In this way, the rate of convergence is gained. Moreover, we elucidate some pointwise convergence results for the Mellin-Gauss-Weierstrass operators. Especially, we acquire the pointwise convergence of them at any Lebesgue point of a function $f$.

Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function,MCf(x):=supr>0⁡1C(B(x,r))∫B(x,r)|f|dC, for C= the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type (p,p) bound for 1<p≤+∞ on the capacitary integration spaces Lp(C) and a weak-type (1,1) bound on the capacitary integration space L1(C). We show how these estimates clarify and improve the existing literature concerning maximal function estimates on capacitary integration spaces. As a consequence, we deduce correspondingly stronger differentiation theorems of Lebesgue-type, which in turn, by classical capacitary inequalities, yield more precise estimates concerning Lebesgue points for functions in Sobolev spaces.

Lebesgue points of functions in the complex Sobolev space

Let [Formula: see text] be a function in the complex Sobolev space [Formula: see text], where [Formula: see text] is an open subset in [Formula: see text]. We show that the complement of the set of Lebesgue points of [Formula: see text] is pluripolar. The key ingredient in our approach is to show that [Formula: see text] for [Formula: see text] is locally bounded from above by a plurisubharmonic function.

Slowly recurrent Collet–Eckmann maps on the Riemann sphere

In this paper we study perturbations of rational Collet–Eckmann maps for which the Julia set is the whole sphere, and for which the critical set is allowed to be slowly recurrent. Generically, if each critical point is simple, we show that each such Collet–Eckmann map is a Lebesgue point of Collet–Eckmann maps in the space of rational maps of the same degree d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. The same result holds in each subspace, where we fix the multiplicities of the critical points.

Slowly recurrent Collet–Eckmann maps with non‐empty Fatou set

AbstractIn this paper, we study rational Collet–Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, these maps are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree .

Boundary behavior of positive solutions of the heat equation on a stratified Lie group

In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution u at a point on the boundary is the existence of the strong derivative of the boundary measure of u at that point. Moreover, the parabolic limit and the strong derivative are equal. We also construct an example of a positive measure on the Heisenberg group to show that the set of all points where strong derivative exists is strictly larger than the set of Lebesgue points of the measure.

On Lebesgue points of entropy solutions to the eikonal equation

We consider entropy solutions to the eikonal equation$|\nabla u|=1$in two-space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless, they share several of their fine properties with BV functions: we show in particular that the set of non-Lebesgue points has at least one co-dimension.

Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincaré inequality.In particular, when restricted to Euclidean spaces, a closed set E \subset \mathbb{R}^n with zero Lebesgue measure is shown to be removable for W^{1,p}(\mathbb{R}^n \setminus E) if and only if \mathbb{R}^n \setminus E supports a p -Poincaré inequality as a metric space. When p&gt;1 , this recovers Koskela’s result ( Ark. Mat. \mathbf{37} (1999), 291–304), but for p=1 , as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces L^{1,p} . To be able to include p=1 , we first study extensions of Newtonian Sobolev functions in the case p=1 from a noncomplete space X to its completion \hat X . In these results, p -path almost open sets play an important role, and we provide a characterization of them by means of p -path open, p -quasiopen and p -finely open sets. We also show that there are nonmeasurable p -path almost open subsets of \mathbb{R}^n , n \ge 2 , provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with L^p -integrable upper gradients, about p -quasiopen, p -path open and p -finely open sets, and about Lebesgue points for N^{1,1} -functions, to spaces that only satisfy local assumptions.

Generalization of Hardy–Littlewood maximal inequality with variable exponent

AbstractLet be a measurable function defined on and . In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone‐like set defined by a given function ψ. Moreover, instead of the integral means, we consider the ‐means. Let and satisfy the log‐Hülder condition and . Then, we prove that the maximal operator is bounded on if and is bounded from to the weak if . We generalize also the theorem about the Lebesgue points.

A note on maximal Fourier restriction for spheres in all dimensions

We prove a maximal Fourier restriction theorem for hypersurfaces in \(\mathbb{R}^{d}\) for any dimension \(d\geq 3\) in a restricted range of exponents given by the Tomas-Stein theorem (spheres being the most canonical example). The proof consists of a simple observation. When \(d=3\) the range corresponds exactly to the full Tomas-Stein one, but is otherwise a proper subset when \(d&gt;3\). We also present an application regarding the Lebesgue points of functions in \(\mathcal{F}(L^p)\) when \(p\) is sufficiently close to 1.

Remarks on Martio's conjecture

We introduce a certain integrability condition for the reciprocal of the Jacobian determinant which guarantees the local homeomorphism property of quasiregular mappings with a small inner dilatation. This condition turns out to be sharp in the planar case. We also show that every branch point of a quasiregular mapping with a small inner dilatation is a Lebesgue point of the differential matrix of the mapping.

Vilenkin–Lebesgue Points and Almost Everywhere Convergence for Some Classical Summability Methods

The concept of Vilenkin–Lebesgue points was introduced in [12], where the almost everywhere convergence of Fejer means of Vilenkin–Fourier series was proved. In this paper, we present a different (and simpler) approach to prove a similar result, which can be used to prove that the corresponding result holds also in a more general context, namely for regular Norlund and T-means.

Large sets without Fourier restriction theorems

We construct a function that lies in L p ( R d ) L^p(\mathbb {R}^d) for every p ∈ ( 1 , ∞ ] p \in (1,\infty ] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.

Approximation of integrable functions by general linear matrix operators of their Fourier series

Abstract The pointwise estimates of the deviation T n , A f ( ⋅ ) − f ( ⋅ ) {T}_{n,A}f(\cdot )-f\left(\cdot ) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of f f , with application of the rth differences of the entries of A A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.

Convergence of 𝑇 means with respect to Vilenkin systems of integrable functions

Abstract In this paper, we derive the convergence of 𝑇 means of Vilenkin–Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilenkin–Lebesgue points. Moreover, we discuss the pointwise and norm convergence in L p L_{p} norms of such 𝑇 means.

The precise representative for the gradient of the Riesz potential of a finite measure

Given a finite nonnegative Borel measure $m$ in $\mathbb{R}^{d}$, we identify the Lebesgue set $\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}$ of the vector-valued function $$V_{s}(x) = \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y), $$ for any order $0 < s < d$. We prove that $a \in \mathcal{L}(V_{s})$ if and only if the integral above has a principal value at $a$ and $$\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.$$ In that case, the precise representative of $V_{s}$ at $a$ coincides with the principal value of the integral. We also study the existence of Lebesgue points for the Cauchy integral of the intrinsic probability measure associated with planar Cantor sets, which leads to challenging new questions.

CONVERGENCE OF URYSOHN-TYPE NONLINEAR INTEGRAL OPERATORS AT p–LEBESGUE POINT

In this paper, a general form of the family of Urysohn-type nonlinear integral operators with kernel $K_{\lambda }\left( x,t,g\right) $ is discussed and theorems about the point convergence of this family at ${p}$--Lebesgue points of a function in $L_{p}$ are given. Here, $\lambda$ is the accumulation point and is a positive parameter that changes in the real numbers. Kernel function $K_{\lambda }\left( x,t,g\left( t\right) \right)$ is an entire analytic function with respect to its third variable.

Some new results and inequalities for subsequences of N\xf6rlund logarithmic means of Walsh\u2013Fourier series

We prove that there exists a martingale fin H_{p} such that the subsequence {L_{2^{n}}f } of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces H_{p} to the space weak-L_{p} for 0< p<1 . We also prove that for any fin L_{p}, pgeq 1 , L_{2^{n}}f converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.

On the structure of weak solutions to scalar conservation laws with finite entropy production

We consider weak solutions with finite entropy production to the scalar conservation law $$\begin{aligned} \partial _t u+\mathrm {div}_x F(u)=0 \quad \text{ in } (0,T)\times \mathbb {R}^d. \end{aligned}$$ Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.

Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier series

Triangular Cesàro summability and Lebesgue points of two-dimensional Fourier series