Mixed Inequalities Research Articles

Corrigendum to “Improvements on Sawyer‐type estimates for generalized maximal functions”

AbstractThe purpose of this note is to correct a mixed estimate involving the generalized maximal function , when Φ is a Young function satisfying certain properties. The family considered include type functions. Although the obtained estimates turn out to be slightly different, they are still good extensions of mixed inequalities for the classical Hardy–Littlewood maximal operators corresponding to the power functions , with . They also allow us to derive the boundedness properties of between Lebesgue spaces by using interpolation techniques related to endpoint estimates of modular type.

Corrigendum to “From $$A_1$$ to $$A_\infty $$: New Mixed Inequalities for Certain Maximal Operators”

Corrigendum to “From $$A_1$$ to $$A_\infty $$: New Mixed Inequalities for Certain Maximal Operators”

A Sweeping Process Control Problem Subject To Mixed Constraints

In this study, we investigate optimal control problems that involve sweeping processes with a drift term and mixed inequality constraints. Our goal is to establish necessary optimality conditions for these problems. We address the challenges that arise due to the combination of sweeping processes and inequality mixed constraints in two contexts: regular and non-regular. This requires working with different types of multipliers, such as finite positive Radon measures for the sweeping term and integrable functions for regular mixed constraints. For non-regular mixed constraints, the multipliers correspond to purely finitely additive set functions.

Mixed Inequalities for Operators Associated to Critical Radius Functions with Applications to Schrödinger Type Operators

We obtain weighted mixed inequalities for operators associated to a critical radius function. We consider Schrödinger Calderón-Zygmund operators of (s,δ) type, for \(1<s\leq \infty \) and 0 < δ ≤ 1. We also give estimates of the same type for the associated maximal operators. As an application, we obtain a wide variety of mixed inequalities for Schrödinger type singular integrals. As far as we know, these results are a first approach of mixed inequalities in the Schrödinger setting.

Locally Random Groups

In this work, we introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood–Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain–Gamburd and Tao.

Mixed inequalities for commutators with multilinear symbol

We prove mixed inequalities for commutators of Calderón–Zygmund operators (CZO) with multilinear symbols. Concretely, let $$m\in {\mathbb {N}}$$ and $${\mathbf {b}}=(b_1,b_2,\ldots , b_m)$$ be a vectorial symbol such that each component $$b_i\in \mathrm {Osc}_{\mathrm {exp}\, L^{r_i}}$$ , with $$r_i\ge 1$$ . If $$u\in A_1$$ and $$v\in A_\infty (u)$$ we prove that the inequality $$\begin{aligned} uv\left( \left\{ x\in {\mathbb {R}}^n: \frac{|T_{\mathbf {b}}(fv)(x)|}{v(x)}>t\right\} \right) \le C\int _{{\mathbb {R}}^n}\Phi \left( \Vert {\mathbf {b}}\Vert \frac{|f(x)|}{t}\right) u(x)v(x)\,dx \end{aligned}$$ holds for every $$t>0$$ , where $$\Phi (t)=t(1+\log ^+t)^r$$ , with $$1/r=\sum _{i=1}^m 1/r_i$$ . We also consider operators of convolution type with kernels satisfying less regularity properties than CZO. In this setting, we give a Coifman type inequality for the associated commutators with multilinear symbol. This result allows us to deduce the $$L^p(w)$$ -boundedness of these operators when $$1<p<\infty $$ and $$w\in A_p$$ . As a consequence, we can obtain the desired mixed inequality in this context.

Joint chance-constrained programs and the intersection of mixing sets through a submodularity lens

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull.

Sawyer-type inequalities for Lorentz spaces

The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate MfvL1,∞(uv)≤Cu,v‖f‖L1(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| \\frac{Mf}{v}\\right\\| _{L^{1,\\infty }(uv)} \\le C_{u,v} \\Vert f \\Vert _{L^{1}(u)}, \\end{aligned}$$\\end{document}where uin A_1 and uvin A_{infty }. We prove a novel extension of this result to the general restricted weak type case. That is, for p>1, uin A_p^{{mathcal {R}}}, and uv^p in A_infty , MfvLp,∞(uvp)≤Cu,v‖f‖Lp,1(u).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| \\frac{Mf}{v}\\right\\| _{L^{p,\\infty }(uv^p)} \\le C_{u,v} \\Vert f \\Vert _{L^{p,1}(u)}. \\end{aligned}$$\\end{document}From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the A_infty extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of A_p^{{mathcal {R}}}. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator {mathcal {M}}, denoted by A_{mathbf {P}}^{{mathcal {R}}}, establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, A_p^{{mathcal {R}}} and A_{mathbf {P}}^{{mathcal {R}}} weights, and Lorentz spaces.

An introduction to mixed hemivariational inequality problems on Hadamard manifolds

In this article, we introduce a category of mixed hemivariational inequality problems on Hadamard manifolds. Using Fan-KKM lemma we initiate the existence of solutions of mixed hemivariational inequality problems (MHVIP) when the underlying vector field is monotone. We also establish the proximal point algorithm for solving mixed hemivariational inequality problems on these nonlinear spaces. Our outcomes and schemes may stimulate further exploration in this fascinating and interesting area of analysis.

From A1 to $A_{\infty }$: New Mixed Inequalities for Certain Maximal Operators

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in Berra (Proc. Am. Math. Soc. 147(10), 4259–4273, 2019). Concretely, given r ≥ 1, u ∈ A1, \(v^{r}\in A_{\infty }\) and a Young function Φ with certain properties, we have that inequality $$ uv^{r}\left( \left\{x\in \mathbb{R}^{n}: \frac{M_{\Phi}(fv)(x)}{ v(x)}>t\right\}\right)\leq C{\int}_{\mathbb{R}^{n}}{\Phi}\left( \frac{|f(x)|}{t}\right)u(x)v^{r}(x) dx $$ holds for every positive t. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator Mγ,Φ, where 0 < γ < n and Φ is a Young function of \(L\log L\) type.

Social inequalities in overweight and obesity in 26 European countries

Abstract Background Previous studies have shown inequalities in overweight and obesity in disfavor of the socially disadvantaged groups. This study examines the extent of these inequalities in 26 European countries. Methods Data from the 2017 EU Statistics on Income and living Conditions (EU-SILC) were used (18 years and older, n = 482,595). A body mass index of 25.0 to 29.9 kg/m2 was classified as overweight and 30.0 and more as obese. Educational level (EL) was used as socioeconomic indicator. Generalized linear models were fitted to compute low-versus high absolute (RD) and relative (RR) inequality. Absolute inequality amplitude (RDA) was calculated as RD/Prevalence. Results Among men, average EU inequalities for overweight were slightly in disfavor of the low educated (RR = 1.05, RDA=5%). A mixed inequality pattern was observed across countries, as the risk of overweight was higher among high educated men in most Eastern countries, in contrast to other parts of Europe (RR from 0.74 to 1.19, RDA from -27% to 20%). Male obesity showed more pronounced inequalities (RR = 1.22, RDA=18%), and a consistent pattern of higher risk among the low educated and wide variation across countries (RR from 1.20 to 2.18, RDA from 16% to 49%). Among women, significant inequalities in overweight were observed (RR = 1.23, RDA=21%), with a consistent pattern of higher risk among the lowest EL, and substantial variation across countries (RR from 1.06 to 1.53, RDA from 7% to 36%). Inequalities were even larger for female obesity, with average RR and RDA reaching 1.49 and 35%, and wider variation (RR from 1.35 to 2.77, RDA from 12% to 88%). Conclusions Social inequalities in weight status are widespread in Europe, but vary substantially between countries. Inequalities are larger among women. For male overweight, a reverse inequality is observed in most Eastern countries. This study allows countries to benchmark the inequalities observed nationally to the situation in other EU countries. Key messages Social inequalities in weight status are widespread in Europe. The pattern of social inequalities in overweight and obesity varies substantially by country and gender.

Improvements on Sawyer type estimates for generalized maximal functions

AbstractIn this paper we prove mixed inequalities for the maximal operator , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy–Littlewood maximal operator proved by E. Sawyer. We show that given , if are weights belonging to the A1‐Muckenhoupt class and Φ is a Young function as above, then the inequality holds for every positive t.A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of . Moreover, it is well‐known that for the particular case with these maximal functions control, in some sense, certain operators in harmonic analysis.

Essential Boundedness and Singularity in Optimal Control

Sufficient optimality conditions for optimal control problems involving isoperimetric and mixed inequality and equality constraints are derived. The main novelty of our approach is the fact that, for such problems, discontinuous and singular solutions can be detected. In other words, our result can deal with solutions for which the proposed optimal control is not continuous, but only essentially bounded, and the classical, crucial strengthened Legendre-Clebsch condition is no longer imposed.

Global error bounds for mixed Quasi-Hemivariational inequality problems on Hadamard manifolds

ABSTRACT In this paper, we introduce and study a class of mixed quasi-hemivariational inequality problems on Hadamard manifolds (in short, (MQHIP)). Some regularized gap functions for (MQHIP) are proposed under suitable conditions. Finally, global error bounds for (MQHIP) in terms of regularized gap functions are derived by employing some strong monotonicity conditions and using the properties of Clarke's generalized directional derivative. The main results presented in this paper improve and generalize corresponding results in literature.

Existence results for mixed hemivariational-like inequalities involving set-valued maps

A hemivariational-like inequality involving a set-valued map and a (nonlinear) bifunction is considered in this paper. The presence of the set-valued map ensures that not one, but two types of solution can be defined, while the presence of the bifunction does not allow the inequality to be written equivalently as an inclusion, making nonsmooth critical point theory unavailable. Using topological methods we are able to prove that our inequality possesses at least one solution (strong solution) provided the set-valued map is upper semicontinuous (lower semicontinuous, respectively). One of the main points of interest in our approach is that we are able to prove the existence of at least one solution even if the KKM technique fails, therefore some of our results are new even in the particular case when the problem reduces to a classical hemivariational, or variational inequality. In the last section we provide a nontrivial example for which our theoretical results are valid. More precisely, we consider a differential inclusion involving the Φ-Laplacian and mixed boundary conditions whose variational formulation leads to a hemivariational-like inequality.

Mixed Norm Inequalities for Lebesgue Spaces

We give necessary and sufficient condition for the two-dimensional mixed norm Hardy inequality to hold with functions that are non-increasing in each variable. The corresponding inequalities involving adjoint operator and the mixed Hardy operators are also discussed.

A sharp reverse Bonnesen-style inequality and generalization

We investigate the isoperimetric deficit of the oval domain in the Euclidean plane. Via the kinematic formulae of Poincaré and Blaschke, and Blaschke’s rolling theorem, we obtain a sharp reverse Bonnesen-style inequality for a plane oval domain, which improves Bottema’s result. Furthermore, we extend the isoperimetric deficit to the symmetric mixed isoperimetric deficit for two plane oval domains, and we obtain two reverse Bonnesen-style symmetric mixed inequalities, which are generalizations of Bottema’s result and its strengthened form.

Social diversity and modal choice strategies in mixed land-use development in South Africa

ABSTRACTThe consensus of the literature is that mixing land uses enhances social diversity within neighbourhoods and increases the diversity of modal choices. South Africa is burdened by the historic legacies and inherited Euclidean (Single-Use) zoning practises, resulting in the unequal access to public services, unequal benefits of economic spillovers and reduced mobility. This compelled Cape Town authorities to promote the mixture of land uses to ensure a more integrated and socially cohesive city, in which the city has been relatively successful. The study examines the relationship between mixed land use development, ethnic diversity, income inequality and modal choice in Cape Town using parametric estimation techniques. The study determined that significant increases in mixed-use development result in increased ethnic diversity, increased income equality indicative of mixed tenure, increased diversity of modal choices, decreased private motorized transportation and increased non-motorized transportation. However, the extremely low R-squared values and large confidence interval ranges indicate that mixed-use development has very little influence on social behaviour or transportation choices.

Facets for single module and multi-module capacitated lot-sizing problems without backlogging

In this paper, we consider the well-known constant-batch lot-sizing problem, which we refer to as the single module capacitated lot-sizing (SMLS) problem, and multi-module capacitated lot-sizing (MMLS) problem. We provide sufficient conditions under which the (k,l,S,I) inequalities of Pochet and Wolsey 1993, the mixed (k,l,S,I) inequalities, derived using mixing procedure, and the paired (k,l,S,I) inequalities, derived using sequential pairing procedure, are facet-defining for the SMLS problem without backlogging. We also provide conditions under which the inequalities derived using the sequential pairing and the n-mixing procedures are facet-defining for the MMLS problem without backlogging. All aforementioned inequalities are special cases of n-step (k,l,S,C) cycle inequalities of Bansal and Kianfar 2015.

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Our main aim is to generalize the classical mixed volumeV(K1,…,Kn)and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.