Weak Type Inequalities Research Articles

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

A Unified Version of Weighted Weak-Type Inequalities for the One-Sided Hardy–Littlewood Maximal Function in Orlicz Classes

Criteria of a Two-Weight, Weak-Type Inequality in Orlicz Classes for Maximal Functions Defined on Homogeneous Spaces

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form φ1(λ)∫{x∈X:Mf(x)>λ}ϱ(x)dμ(x)≤c∫Xφ2c|f(x)|σ(x)dμ(x) are obtained in Orlicz classes, where Mf is a Hardy–Littlewood maximal function defined on homogeneous spaces and ϱ is a weight function.

Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets

We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.

Weighted weak-type inequalities for maximal operators and singular integrals

Weighted weak-type inequalities for maximal operators and singular integrals

Sobolev embeddings in Musielak-Orlicz spaces

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.

Weak and strong type inequalities for Orlicz spaces

We revisit the generalisation of Calderon’s Transfer Principle as espoused in [7]. This principle is used to generate weak type maximal inequalities for ergodic operators in the setting of σ-compact locally compact Hausdorff groups acting measure-preservingly on σ-finite measure spaces. In particular we develop a much more robust protocol for transferring weak and strong type inequalities from Orlicz spaces in the group setting to Orlicz spaces in the measure space setting. This is an important addition to the protocol developed in [7], which to date has only yielded weak type inequalities. The current approach also places fewer restrictions on the underlying Young functions describing the Orlicz spaces involved.

Weighted mixed weak-type inequalities for multilinear fractional operators

Weighted mixed weak-type inequalities for multilinear fractional operators

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

A Weighted Weak-Type Inequality for One-Sided Maximal Operators

Mixed weak type inequalities for the fractional maximal operator

Mixed weak type inequalities for the fractional maximal operator

Almost everywhere convergence of T means with respect to the Vilenkin system of integrable functions

UDC 517.5 We prove and discuss some new weak-type (1,1) inequalities for the maximal operators of T means with respect to the Vilenkin system generated by monotone coefficients. We also apply these results to prove that these T means are almost everywhere convergent. As applications, we present both some well-known and new results.

Weak-type maximal function estimates on the infinite-dimensional torus

We prove necessary and sufficient conditions for the weak-L^p boundedness, for p in (1,infty ), of a maximal operator on the infinite-dimensional torus. In the endpoint case p=1 we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator

Abstract This paper deals with the introduction and study of a new, Orlicz type space associated with the Gegenbauer differential operator and briefly called here a G-Orlicz space. We prove that such spaces are Banach function spaces. The necessary and sufficient conditions are established which ensure the validity of weak and strong type inequalities for fractional maximal functions and fractional integrals of G-Orlicz spaces.

A weighted weak-type inequality for the one-sided maximal operators

UDC 517.5 We obtain some necessary and sufficient conditions for a weighted weak-type inequality of the form ∫ { M g + ( f ) > λ } φ ˜ ( λ ω 3 ( x ) ω 4 ( x ) ) ω 4 ( x ) ⅆ x ≤ C 1 ∫ - ∞ + ∞ φ ˜ ( C 1 | f ( x ) | ω 1 ( x ) ω 2 ( x ) ) ω 2 ( x ) ⅆ x to be true, which generalize some known results.

A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on $${\mathbb {R}}^2$$

A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on $${\mathbb {R}}^2$$

Some weak type inequalities and almost everywhere convergence of Vilenkin–Nörlund means

We prove and discuss some new weak type (1,1 ) inequalities of maximal operators of Vilenkin–Nörlund means generated by monotone coefficients. Moreover, we use these results to prove a.e. convergence of such Vilenkin–Nörlund means. As applications, both some well-known and new inequalities are pointed out.

Some remarks on dimension-free estimates for the discrete Hardy—Littlewood maximal functions

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy—Littlewood averaging operators over convex symmetric bodies acting on ℝd and ℤd. Firstly we show, in the full range of p ∈ [1, ∞], that these optimal constants in Lp (ℝd) are always not larger than their discrete analogues in ℓp(ℤd); and we also show that the equality holds for the cubes in the case of p = 1. This in particular implies that the best constant in the weak type (1, 1) inequality for the discrete Hardy—Littlewood maximal function associated with centered cubes in ℤd grows to infinity as d → ∞, and if d = 1 it is equal to the largest root of the quadratic equation 12C2 − 22C + 5 = 0. Secondly we prove dimension-free estimates for the ℓp(ℤd) norms, p ∈ (1, ∞], of the discrete Hardy—Littlewood maximal operators with the restricted range of scales t ≥ Cqd corresponding to q-balls, q ∈ [2, ∞). Finally, we extend the latter result on ℓ2(ℤd) for the maximal operators restricted to dyadic scales 2n ≥ Cqd1/q.

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

Fractional Besov Trace/Extension-Type Inequalities via the Caffarelli–Silvestre Extension

Let $$u(\cdot ,\cdot )$$ be the Caffarelli–Silvestre extension of f. The first goal of this article is to establish the fractional trace-type inequalities involving the Caffarelli–Silvestre extension $$u(\cdot ,\cdot )$$ of f. In doing so, firstly, we establish the fractional Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$\nabla _{(x,t)}u(x,t).$$ Then, we prove the fractional anisotropic Sobolev/logarithmic Sobolev/Hardy trace inequalities in terms of $$ {\partial _{t} u(x,t)}$$ or $$(-\varDelta )^{-\gamma /2}u(x,t)$$ only. Moreover, based on an estimate of the Fourier transform of the Caffarelli–Silvestre extension kernel and the sharp affine weighted $$L^p$$ Sobolev inequality, we prove that the $${\dot{H}}^{-\beta /2}({\mathbb {R}}^n)$$ norm of f(x) can be controlled by the product of the weighted $$L^p$$ -affine energy and the weighted $$L^p$$ -norm of $${\partial _{t} u(x,t)}.$$ The second goal of this article is to characterize non-negative measures $$\mu $$ on $${\mathbb {R}}^{n+1}_+$$ such that the embeddings $$\begin{aligned} \Vert u(\cdot ,\cdot )\Vert _{L^{q_0,p_0}_{\mu }({\mathbb {R}}^{n+1})}\lesssim \Vert f\Vert _{{\dot{\varLambda }}^{p,q}_\beta ({\mathbb {R}}^n)} \end{aligned}$$ hold for some $$p_0$$ and $$q_0$$ depending on p and q which are classified in three different cases: (1) $$p=q\in (n/(n+\beta ),1];$$ (2) $$(p,q)\in (1,n/\beta )\times (1,\infty );$$ (3) $$(p,q)\in (1,n/\beta )\times \{\infty \}.$$ For case (1), the embeddings can be characterized in terms of an analytic condition of the variational capacity minimizing function, the iso-capacitary inequality of open balls, and other weak-type inequalities. For cases (2) and (3), the embeddings are characterized by the iso-capacitary inequality for fractional Besov capacity of open sets.

Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) −Extension

Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to L q ( ℝ + n + 1 , μ ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp –capacity, including its dual form, the Lp –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

Global Regularity Results for Non-homogeneous Growth Fractional Problems

This article concerns with the global H\"older regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior H\"older regularity for local weak solutions, which need not be assumed bounded. The global H\"older regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.