Hardy--Littlewood--Sobolev inequality for $p=1$

Stolyarov, Dmitriy

doi:10.48550/arxiv.2010.05297

Cited by 4 publications

(15 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that the operator PA ≡ (Div A, Curl(A ⊤ )) is both elliptic and canceling whenever A ∈ Sym n , in which case Proposition 8.1 follows from the work of Van Schaftingen [76] and Stolyarov [64]. However, for non-symmetric matrices, this operator is not elliptic: it is easy to verify that P(ξ)(ξ ⊗ ξ ⊥ ) = 0, where ξ ⊥ is any vector orthogonal to ξ.…”

Section: Other Examplesmentioning

confidence: 95%

Compensation phenomena for concentration effects via nonlinear elliptic estimates

Guerra¹,

Raiță²,

Schrecker³

2021

Preprint

View full text Add to dashboard Cite

We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. The compensation effect takes the form of nonlinear elliptic estimates, where constraining the values of the field to lie in a cone compensates for the lack of ellipticity of the differential operator. We give a series of new examples of this phenomenon, focusing on the case where the cone is a subset of the space of symmetric matrices and the differential operator is the divergence or the curl. One of our main findings is that the maximal gain of integrability is tied to both the differential operator and the cone, contradicting in particular a recent conjecture from arXiv:2106.03077. This appends the classical compensated compactness framework for oscillations with a variant designed for concentrations, and also extends the recent theory of compensated integrability due to D. Serre. In particular, we find a new family of integrands that are Div-quasiconcave under convex constraints. k 3.2. A quantitative characterisation of null Lagrangians 3.3. k-Hessian equations 4. Elliptic estimates in two dimensions 4.1. Elliptic estimates for the divergence 4.2. Elliptic estimates for the curl 4.3. Optimality 4.4. Density of smooth functions and separation 5. New examples of div-quasiconcave integrands 6. Elliptic estimates for the divergence 6.1. Elliptic estimates 6.2. The case of exact constraints: optimality and higher integrability 7. Elliptic estimates for the curl 7.1. Elliptic estimates 7.2. The case of exact constraints: optimality and higher integrability 8. Other examples 8.1. A degenerate example 8.2. A second order example 8.3. A first order example Appendix A.

show abstract

Section: Other Examplesmentioning

confidence: 95%

Compensation phenomena for concentration effects via nonlinear elliptic estimates

Guerra¹,

Raiță²,

Schrecker³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A program in this direction was pioneered in the seminal work of J. Bourgain and H. Brezis [5] (see also [6,25]) and received remarkable contributions from L. Lanzani and E. Stein [14] and J. Van Schaftingen [26][27][28], while endpoint fine parameter improvements on the Lorentz [11,23] and Besov-Lorentz [24] scales have only recently been obtained.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to give a simple proof of the Besov-Lorentz estimates obtained in [24] for a restricted class of operators and to show how this estimate can be used to resolve several open questions in the theory, in particular estimates for Hodge systems [27, Open Problems 1 & 2] and the endpoint extension of [28,Propositions 8.8 & 8.10] in the case of first order operators. Our starting place is an estimate the first and third named authors proved in [11], that for any α ∈ (0, d) there exists a constant C > 0 for which one has the inequality…”

Section: Introductionmentioning

confidence: 99%

“…The estimate (1.1) is a partial replacement for the failure of the Hardy-Littlewood-Sobolev embedding in the L 1 endpoint, cf. [19, p. 119], while a comprehensive resolution of the question of a replacement has been given by D. Stolyarov, who in [24] (see also [2,Conjecture 2] where such an inequality was conjectured to hold) establishes the sharper inequality…”

Section: Introductionmentioning

confidence: 99%

“…Van Schaftingen's class of cocancelling operators [28] (see Definition 2.1 below where we recall this class). The argument in [24] is quite involved, and it is there commented by D. Stolyarov that whether the inequality (1.3) admits a simpler proof if one only seeks its validity for the more restrictive class of divergence free measures is unknown. We will shortly give such a proof, which benefited from several insights from his paper and the series of lectures 1 he gave on the topic.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Endpoint $L^1$ estimates for Hodge systems

Hernandez,

Raita,

Spector

2021

Preprint

View full text Add to dashboard Cite

show abstract

Endpoint $$L^1$$ estimates for Hodge systems

2022

View full text Add to dashboard Cite

Let $$d\ge 2$$ d ≥ 2 . In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$\begin{aligned} \Vert I_\alpha F\Vert _{{\dot{B}}^{0,1}_{d/(d-\alpha ),1}(\mathbb {R}^d;\mathbb {R}^k)} \le C \Vert F \Vert _{L^1(\mathbb {R}^d;\mathbb {R}^k)} \end{aligned}$$ ‖ I α F ‖ B ˙ d / ( d - α ) , 1 0 , 1 ( R d ; R k ) ≤ C ‖ F ‖ L 1 ( R d ; R k ) for all $$F \in L^1(\mathbb {R}^d;\mathbb {R}^k)$$ F ∈ L 1 ( R d ; R k ) which satisfy a first order cocancelling differential constraint, where $$\alpha \in (0,d)$$ α ∈ ( 0 , d ) and $$I_\alpha $$ I α is a Riesz potential. We show how this implies endpoint Besov–Lorentz estimates for Hodge systems with $$L^1$$ L 1 data via fractional integration for exterior derivatives.

show abstract

Hardy--Littlewood--Sobolev inequality for $p=1$

Cited by 4 publications

References 19 publications

Compensation phenomena for concentration effects via nonlinear elliptic estimates

Compensation phenomena for concentration effects via nonlinear elliptic estimates

Endpoint $L^1$ estimates for Hodge systems

Endpoint $$L^1$$ estimates for Hodge systems

Contact Info

Product

Resources

About