Functional calculus for first order systems of Dirac type and boundary value problems
Functional calculus for first order systems of Dirac type and boundary value problems
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狄拉克型一阶系统的泛函计算和边值问题
DOI：
10.24033/msmf.452
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Sebastian Stahlhut
中科院分区：
文献类型：
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作者：
P. Auscher;Sebastian Stahlhut
——这本回忆录包含两篇文章。 1）在通过一阶系统对边值椭圆问题的先验估计中，我们证明了椭圆方程或上半空间中具有垂直独立系数的系统的弱解的许多先验估计。这些估计旨在应用于各种拓扑中的狄利克雷和诺依曼类型的边值问题。我们致力于解决方案类别，其中包括能源解决方案。对于这些解决方案，我们使用由其共法梯度满足的一阶系统以及与此类系统相关的哈代空间理论进行描述，但该方法还允许我们设计不一定是能量解决方案的解决方案。我们获得了平方函数、非切向极大函数和边界迹范数之间的精确比较。主要论点是，此类结果的指数范围与这些哈代空间（可能是抽象空间）何时被识别为调和分布的具体空间有关。我们考虑一些改编的非切向锐函数并证明与平方函数的比较。我们获得了层势、边界行为（特别是新的强限制）和跳跃关系的有界结果。一种应用是“à la Šnĕıberg”可解性的外推。另一个是在没有进一步假设的情况下扰动 L∞ 中的系数时可解性的稳定性。我们强调，我们的结果不需要德乔治-纳什假设，并且当我们这样做时，我们会改进可用的假设。 2) 在扰动一阶狄拉克算子全纯函数的 L-L 理论中，目的是证明在一定指数范围内 p ≤ q 的某些狄拉克型扰动一阶微分算子的泛函计算中算子的 L-L 非对角估计和 L-L 有界性。我们用相关全纯函数的衰减性质来描述L-L非对角估计和L-L有界性，并给出L-L有界性的必要条件。将给出分数算子的 Hardy-Littlewood-Sobolev 估计的应用。 c © Mémoires de la Société Mathématique de France 144, SMF 2016
— This memoir contains two articles. 1) In A priori estimates for boundary value elliptic problems via first order systems, we prove a number of a priori estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability “à la Šnĕıberg”. Another one is stability of solvability in perturbing the coefficients in L∞ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so. 2) In L-L theory for holomorphic functions of perturbed first order Dirac operators, the aim is to prove L-L off-diagonal estimates and L-L boundedness for operators in the functional calculus of certain perturbed first order differential operators of Dirac type for with p ≤ q in a certain range of exponents. We describe the L-L off-diagonal estimates and the L-L boundedness in terms of the decay properties of the related holomorphic functions and give a necessary condition for L-L boundedness. Applications to Hardy-Littlewood-Sobolev estimates for fractional operators will be given. c © Mémoires de la Société Mathématique de France 144, SMF 2016