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Maximal Theory of Hardy Spaces

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Book cover Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2142))

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Abstract

The main goal of this chapter is to introduce Hardy spaces in the context of d-Ahlfors-regular quasi-metric spaces by defining H p(X) as a collection of distributions whose maximal belongs to L p(X). This is in the spirit of the pioneering work of C.

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Notes

  1. 1.

    which essentially make it an approximation to the identity

  2. 2.

    The atomic Hardy spaces considered in [Uch80] are of a slightly different variety of than those in [CoWe77]; see [Uch80, p. 581] for details.

  3. 3.

    In this work, the pair \((\mathcal{X},\tau )\) shall be referred to as a topological vector space provided \(\mathcal{X}\) is a vector space and τ is a topology on \(\mathcal{X}\) such that the vector space operations of addition and scalar multiplication are continuous with respect to τ. Under these assumptions, the topological space \((\mathcal{X},\tau )\) may not be Hausdorff. If, in addition to the above considerations, one assumes that the set \(\{x\} \subseteq (\mathcal{X},\tau )\) is closed for each \(x \in \mathcal{X}\) then \((\mathcal{X},\tau )\) is necessarily Hausdorff. In light of this, part of the literature includes the latter condition in the definition of a topological vector space (see, e.g., [Ru91, p. 7]).

  4. 4.

    Given a vector space \(\mathcal{X}\) over \(\mathbb{C}\), call a function \(\|\cdot \|: \mathcal{X} \rightarrow [0,\infty )\) a quasi-norm provided there exists a constant C ∈ (0, ∞) such that for each \(x,y \in \mathcal{X}\) the following three conditions hold (i) \(\|x\| = 0\, \Leftrightarrow \, x = 0\), (ii) \(\|\lambda x\| = \vert \lambda \vert \cdot \| x\|\), \(\forall \,\lambda \in \mathbb{C}\), and (iii) \(\|x + y\| \leq C(\|x\| +\| y\|)\).

  5. 5.

    This variety of Hardy spaces was introduced in [MiMiMiMo13] where the authors considered a slightly less general geometric measure theoretic ambient than the one in this work.

  6. 6.

    This class of Hardy spaces was introduced in [MaSe79ii] in the setting of normal spaces (1-AR spaces) although the notation is due to the authors in [MiMiMiMo13].

  7. 7.

    Given a vector space \(\mathcal{X}\) over \(\mathbb{C}\), call a function \(\|\cdot \|: \mathcal{X} \rightarrow [0,\infty )\) a quasi-semi-norm provided there exists a constant C ∈ (0, ∞) with the property that for each \(x,y \in \mathcal{X}\) the following three conditions hold (i) x = 0 implies that \(\|x\| = 0\), (ii) \(\|\lambda x\| = \vert \lambda \vert \cdot \| x\|\), \(\forall \,\lambda \in \mathbb{C}\), and also (iii) \(\|x + y\| \leq C(\|x\| +\| y\|)\).

  8. 8.

    Given \(z \in \mathbb{C}\) we denote by \(\mathrm{Re}z \in \mathbb{R}\) and \(\mathrm{Im}z \in \mathbb{R}\), respectively, the real and imaginary parts of z.

  9. 9.

    Call a pair \((\mathcal{X},\|\cdot \|)\) (or simply \(\mathcal{X}\)) a quasi-Banach space provided \(\mathcal{X}\) is a vector space (over \(\mathbb{C}\)) and \(\|\cdot \|\) is a quasi-norm on \(\mathcal{X}\) with the property that \(\mathcal{X}\) is complete in the quasi-norm \(\|\cdot \|\), i.e., every sequence of points in \(\mathcal{X}\) which is Cauchy with respect to \(\|\cdot \|\) converges to a point in \(\mathcal{X}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Ryan Alvarado & Marius Mitrea

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  1. Ryan Alvarado
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  2. Marius Mitrea
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Alvarado, R., Mitrea, M. (2015). Maximal Theory of Hardy Spaces. In: Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces. Lecture Notes in Mathematics, vol 2142. Springer, Cham. https://doi.org/10.1007/978-3-319-18132-5_4

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