Titlebook: Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure; Pascal Auscher,Moritz Egert Book 2023 The Editor(s) (i

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

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Identification of Adapted Hardy Spaces,This chapter is concerned with identifying three pre-Hardy spaces, ., ., and ., that play a crucial role for Dirichlet and regularity problems, with classical smoothness spaces.

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Riesz Transform Estimates: Part II,We come back to the Riesz transform interval . defined in (.), the endpoints of which we have denoted by .(.). In Chap. . we have characterized the endpoints of the part of ?(.) in (1, .). The identification theorem for adapted Hardy spaces allows us to complete the discussion in the full range of exponents.

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

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Basic Properties of Weak Solutions,At this point in the monograph we begin to slightly change our perspective from Hardy spaces adapted to .?=??.?÷.?. 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.