Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure

This chapter contains all necessary background on function spaces that will be used later on.

In this chapter, we introduce the elliptic operators used in this monograph and recall their main properties in the Ł² setting. We also recall material on (bi)sectorial operators and their holomorphic functional calculus.

In this chapter we discuss general principles for \(\operatorname {H}^p - \operatorname {H}^q\)-bounded operator families. We provide a toolbox that will allow us to manipulate resolvent families associated with our first- and second-order operators efficiently on an abstract level.

In order to extend the operator theory for L to Hardy spaces, we need to guarantee that certain operators f(L) preserve vanishing zeroth moments or have the conservation property f(L)c = c whenever c is a constant. In absence of integral kernels, the action of such operators on constants is explained via off-diagonal estimates. We discuss several conservation properties, in particular for resolvents and Poisson semigroups.

In this chapter, we define the four numbers that rule the functional calculus properties of our elliptic operators and that will help us to describe the ranges of well-posedness of our boundary value problems. We study intrinsic relations between these numbers, using the machinery developed in Chap. 4.

In this chapter, we characterize the range of exponents p for \(\operatorname {L}^p\)-boundedness of the Riesz transform ∇_xL^−1∕2.

Operator-adapted Hardy–Sobolev spaces are our main tool in this monograph and will be essential for understanding most of the following chapters. They have been developed in various references starting with semigroup generators in Auscher et al. (J Geom Anal 18(1):192–248, 2008), Duong and Li (J Funct Anal 264(6), 1409–1437, 2013), Hofmann and Mayboroda (Math Ann 344(1): 37–116, 2009) and Hofmann et al. (Ann Sci Éc Norm Supér (4) 44(5): 723–800, 2011) up to the recent monographs focusing on bisectorial operators (Amenta and Auscher, Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018; Auscher and Stahlhut, Mém Soc Math Fr (N.S.)(144):vii+164, 2016). Still we need some unrevealed features and we take this opportunity to correct some inexact arguments from the literature. For general properties of adapted Hardy spaces we closely follow Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018, Sec.3), where the authors develop an abstract framework of two-parameter operator families that provides a unified approach to sectorial and bisectorial operators. The application to bisectorial operators with firstorder scaling has been detailed in Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018, Sect. 4) and we review their results in Sect. 8.1. Section 8.2 provides all necessary details in order to apply the framework to sectorial operators with second-order scaling, and we summarize the results that are relevant to us. This will justify using parts of Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018) for sectorial operators in the further course. The abstract framework allows us to treat operator-adapted Besov spaces simultaneously without any additional effort.

This chapter is concerned with identifying three pre-Hardy spaces, \(\mathbb {H}_L^p\), \(\mathbb {H}_L^{1,p}\), and \(\mathbb {H}_{DB}^p\), that play a crucial role for Dirichlet and regularity problems, with classical smoothness spaces.

In this short chapter we present two consequences of the identification theorem for operatoradapted Hardy spaces that are of independent interest. One concerns analyticity, the other one concerns the \(\operatorname {H}^\infty \)-calculus for L.

We come back to the Riesz transform interval defined in (7.​1), the endpoints of which we have denoted by r_±(L). In Chap. 7 we have characterized the endpoints of the part of ℐ(L) in (1, ∞). The identification theorem for adapted Hardy spaces allows us to complete the discussion in the full range of exponents.

In this chapter, we show that the critical numbers are intrinsic in the sense that we could have equivalently defined them through other families of functions of L than resolvents. We focus on the Poisson semigroup and, when a = 1, the heat semigroup.

In this chapter, we discuss Ł^p-boundedness of the Hodge projector associated to L₀ (that is, L in the case when a = 1). We obtain a characterization of the range for p in terms of critical numbers.

In this chapter, we work out a precise relation between kernel bounds and critical numbers p₋(L) strictly below 1. Except for Sect. 14.5 this is an intermezzo not needed for the application to boundary value problems. However, it nicely illustrates the usefulness of our choice for the interval J (L) compared to Auscher (Mem Am Math Soc 186(871):xviii+75, 2007) and connects with the theory of Gaussian estimates in the first chapter of Auscher and Tchamitchian (Astérisque (249):viii+172, 1998). In particular, we obtain resolvent kernels from those of high powers of the resolvent without using heat semigroups (which exist only if ω_L < π∕2).

The identification of adapted Hardy spaces as a key tool to treating boundary value problems has appeared first in the work of Auscher and Stahlhut (Mém Soc Math Fr (N.S.) (144):vii+164, 2016). Although we argue independently of this reference concerning this particular issue, in this chapter, we make the bridge and characterize their admissible range of exponents in terms of our critical numbers.

At this point in the monograph we begin to slightly change our perspective from Hardy spaces adapted to L = −a⁻¹ ÷_xd∇_x to weak solutions to the associated elliptic system in the upper half-space. In this chapter, we gather well-known properties of weak solutions that will frequently be used in the further course.

In this chapter we establish the existence part in our main results on the Dirichlet and Regularity problems with \({\operatorname {H}}^p\)-data, Theorems 1.​1 and 1.​2. When the data f additionally belongs to \({\operatorname {L}}^2\), the (eventually unique) solution is given by the Poisson semigroup. Hence, we proceed in two steps: First, we establish the required semigroup estimates for data \(f \in a^{-1}({\operatorname {H}}^p \cap {\operatorname {L}}^2)\) and \(f \in \dot {\operatorname {H}}^{1,p} \cap \operatorname {W}^{1,2}\), respectively. Second, we obtain existence of a solution by a density argument for the full class of data.

In this chapter, we establish the existence part of Theorem 1.​3, our main result on the Dirichlet problems with boundary data in Hölder spaces and BMO.

In this chapter, we prove the compatible existence on Dirichlet problems with data in homogeneous Hardy–Sobolev and Besov spaces of fractional smoothness that have been announced in Sect. 1.​6. We also compare them to what can be obtained by the general first-order approach (Amenta and Auscher, Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018) when specialized to elliptic systems in block form.

This chapter is needed to prepare the next chapter on uniqueness. We consider the divergence form operator \(\mathcal {L} u = -\div A \nabla u = -\partial _t (a \partial _t u) - \div _x d \nabla _x u\) on \({\mathbb {R}^{1+n}}\). It is of the same class as Λ in (3.​5) but in one dimension higher. Hence, \(\mathcal {L} \) is defined on \({\dot {\operatorname {W}}}^{1,2}({\mathbb {R}}^{1+n})\) via the Lax-Milgram lemma and invertible onto \({\dot {\operatorname {W}}}^{-1,2}({\mathbb {R}}^{1+n})\). It turns out that the inverse \(\mathcal {L}^{-1}\) on particular test functions can explicitly be constructed using abstract single layer operators \( \mathcal {S}_{t}^{\mathcal {L}}\). All this relies on the fundamental observation of Rosén (Publ Mat 57(2):429–454, 2013) that what is called single layer potential in the classical context of elliptic operators with real coefficients can abstractly be defined using the \(\operatorname {H}^\infty \)-calculus for the perturbed Dirac operator DB. We provide new estimates for the abstract single layer operators by incorporating the full strength of \( \operatorname {L}^p-\operatorname {L}^q\)-bounds for the Poisson semigroup associated with the boundary operator L.

This chapter complements Chaps. 17, 18 and 19. We shall prove the uniqueness parts in Theorems 1.​1, 1.​2, 1.​3, and 1.​4.

In this final chapter we are concerned with the Neumann problem. In particular, we shall give the proof of Theorem 1.​5. All relies heavily on our earlier observation that our critical numbers give a description of the range of admissible exponents in the work of Auscher and Mourgoglou (Rev Mat Iberoam 35(1):241–315, 2019) and Auscher and Stahlhut (Mém. Soc. Math. Fr. (N.S.) (144), vii+164, 2016).