Fractional Integration and Optimal Estimates for Elliptic Systems

Abstract

In this paper we prove the following optimal Lorentz embedding for the Riesz potentials: Let α ∈ (0, d). There exists a constantWe then show how this result implies optimal Lorentz regularity for a Div-Curl system (which can be applied, for example to obtain new estimates for the magnetic field in Maxwell's equations), as well as for a vector-valued Poisson equation in the divergence free case.

Endpoint $L^1$ estimates for Hodge systems

“…While the argument of (1.1) in [11] for a single loop with a ball growth condition involves only estimates for various maximal functions, in this paper we observe that it can be further simplified by the consideration of a very natural stronger quantity that arises in Stolyarov's estimates:…”

“…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.…”

“…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…”

“…To this end, let us recall the approach to (1.1) in [11]: For the space of divergence free measures one finds appropriate atoms, one demonstrates the sufficiency of an estimate on an atom, and one establishes the estimate for a single atom. The atoms in this case are piecewise-C 1 loops which satisfy the uniform ball-growth condition: To any piecewise-C 1 loops Γ ⊂ R d one associates the measure which is given by integration along the curve…”

“…This approximation/decomposition and the triangle inequality then yields that it suffices to prove the estimate for a single loop which satisfies the ball growth condition (1.5). Finally, the estimate (1.1) for a single loop was argued in [11] by a hands on interpolation that utilizes several pointwise estimates for Riesz potentials and bounds for various maximal functions.…”