Hardy Inequality Research Articles

Global existence for an isotropic modification of the Boltzmann equation

Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau.Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities.

One-Dimensional Discrete Hardy and Rellich Inequalities on Integers

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form nα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^\\alpha $$\\end{document}. We prove the inequality when α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities(with weights nα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^\\alpha $$\\end{document}) which are asymptotically sharp as α→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow \\infty $$\\end{document}. As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.

Some new weighted weak-type iterated and bilinear modified Hardy inequalities

We characterize the good weights for some weighted weak-type iterated and bilinear modified Hardy inequalities to hold.

Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains

Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains

A regularity result for the free boundary compressible Euler equations of a liquid

We derive a priori estimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density.

The sharp Hardy–Moser–Trudinger inequality in dimension 𝑛

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ sup u ∈ W 0 1 , n ( B n ) , ∫ B n | ∇ u | n d x − ( 2 ( n − 1 ) n ) n ∫ B n | u | n ( 1 − | x | 2 ) n d x ≤ 1 ∫ B n e ( 1 − β n ) α n | u | n n − 1 | x | − β d x ≤ C \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right )^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any β ∈ [ 0 , n ) \beta \in [0,n) , n ≥ 3 n\geq 3 where α n = n ω n − 1 1 n − 1 \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and ω n − 1 \omega _{n-1} is the surface area of the n − 1 n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green’s functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324].

Hardy inequalities for magnetic p-Laplacians

We establish improved Hardy inequalities for the magnetic p-Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov–Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free p-Laplacian. We also post some remarks with open problems.

Multidimensional Frank–Laptev–Weidl improvement of the hardy inequality

AbstractWe establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.

Inductive methods for Hardy inequality on trees

We study the Hardy inequality on at most countable rooted tree. The main known criteria in the lower - triangle case for this inequality are two Arcozzi - Rochberg - Sawyer criteria and the capacity criterion. In the survey we show that these two criteria are connected with the criteria for the Hardy inequalities for the sequences, for the Hardy inequality on an interval of the real axis and for the trace inequalities with the Riesz potentials. We provide the examples from the literature, when the trace inequality or another statement is characterized in terms of the validity of the Hardy inequality on the tree. We simplify two known proofs of the Arcozzi - Rochberg - Sawyer criteria, which are based on the Marcinkiewicz interpolation theorem and on the capacity criterion. We provide new proofs for Arcozzi - Rochberg - Sawyer criteria, which are based on the induction in the tree, the inductive formula for the capacity and the formula of integration by parts. The latter of the proofs is written for the Hardy inequality on the tree with a boundary and for the Hardy inequality over the family of all binary cubes. In the diagonal case this proof provides an optimal constant $p$, which coincides with the Bennett constant in the Hardy inequality for the sequences. In the general case we provide a few new inductive criteria for the validity of the Hardy inequality in terms of the existence of a family of functions satisfying an inductive relation. One of these criteria is applied in the proof of a theorem containing additional equivalent conditions for the validity of the Hardy inequality on the trees in the diagonal case.

Hardy type identities and inequalities with divergence type operators on smooth metric measure spaces

<abstract><p>We gave the Hardy type identities and inequalities for the divergence type operator $ L_{f, V} $ on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.</p></abstract>

Local well-posedness of 1D degenerate drift diffusion equation

<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

Some Hardy's inequalities on conformable fractional calculus

Some Hardy's inequalities on conformable fractional calculus

Generalized weighted Hardy's inequalities with compact perturbations

Generalized weighted Hardy's inequalities with compact perturbations

Sharp weighted fractional Hardy inequalities

Sharp weighted fractional Hardy inequalities

Existence and multiplicity results for parameter Kirchhoff double phase problem with Hardy–Sobolev exponents

The solvability of a class of parameter Kirchhoff double phase Dirichlet problems with Hardy–Sobolev terms is considered. We focus on the existence of at least one solution, two solutions, three solutions, and infinitely many solutions to the problem, as the nonlinear terms satisfy different growth conditions, respectively. Our tools are mainly based on variational methods and critical point theory. In particular, in order to establish the relationship between singular terms and the norm of the Musielak–Orlicz–Sobolev space, we extend the Sobolev–Hardy inequality from W01,p to W01,H.

Iterated discrete Hardy-type inequalities with three weights for a class of matrix operators

Iterated Hardy-type inequalities are one of the main objects of current research on the theory of Hardy inequalities. These inequalities have become well-known after study boundedness properties of the multidimensional Hardy operator acting from the weighted Lebesgue space to the local Morrie-type space. In addition, the results of quasilinear inequalities can be applied to study bilinear Hardy inequalities. In the paper, we discussed weighted discrete Hardy-type inequalities containing some quasilinear operators with a matrix kernel where matrix entries satisfy discrete Oinarov condition. The research of weighted Hardy-type inequalities depends on the relations between parameters p, q and θ, so we considered the cases 1 < p ≤ q < θ < ∞ and p ≤ q < θ < ∞, 0 < p ≤1, criteria for the fulfillment of iterated discrete Hardy-type inequalities are obtained. Moreover, an alternative method of proof was shown in the work.

Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

Caffarelli-Kohn-Nirenberg identities, inequalities and their stabilities

We set up a one-parameter family of inequalities that contains both the Hardy inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg inequalities (when the parameter is optimal). Moreover, we study these results with the exact remainders to provide direct understandings to the sharp constants, as well as the existence and non-existence of the optimizers of the Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an application of our identities, we establish some sharp versions with optimal constants and theirs attainability of the stability of the Heisenberg Uncertainty Principle and several stability results of the Caffarelli-Kohn-Nirenberg inequalities (see Theorem 1.1). In particular, for the stability of the Heisenberg uncertainty principle, we introduced a new deficit function δ1(u) and established the best constant for the stability inequality (see Theorem 1.2) which also leads to the stability inequality with sharp constant with respect to the deficit function δ2(u) and improved the known stability result for δ2(u) in the literature. (see Theorem 1.3). A sharp stability inequality for the non-scale invariant Heisenberg uncertainty principle with optimal constant is also obtained in Theorem 1.4.

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research.

A Note on Best Constants for Weighted Integral Hardy Inequalities on Homogeneous Groups

A Note on Best Constants for Weighted Integral Hardy Inequalities on Homogeneous Groups