Abstract
We use the notation introduced in the previous sections. In particular, let ℝn be again euclidean n-space where n ∈ ℕ. The Schwartz space S (ℝn), its dual S′ (ℝn) and the spaces L p (ℝn) with 0 < p ≤ ∞ have the same meaning as in 2.1, the latter quasi-normed by (2.1). Let L loc1 (ℝn) be the collection of all complex-valued locally Lebesgue-integrable functions in ℝn. Any f ∈ L loc1 (ℝn) is interpreted in the usual way as a regular distribution. Conversely, as usual, a distribution on ℝn is called regular if, and only if, it can be identified (as a distribution) with a locally integrable function on ℝn. If A(ℝn) is a collection of distributions on ℝn, then
simply means that any element f of A(ℝn) is a regular distribution f ∈ L loc1 (ℝn). Then, in particular, the distribution function μf(λ), the rearrangement f *(t) and its maximal function f **(t) in (10.2)–(10.4) make sense accepting that they might be infinite. If A 1(ℝn) and A 2(ℝn) are two quasi-normed spaces, continuously embedded in S′(ℝn), then
always means that there is a constant c > 0 such that
(continuous embedding). On the other hand we do not use the word embedding in connection with inequalities of type (10.12).
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© 2001 Springer Basel AG
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Triebel, H. (2001). Classical inequalities. In: The Structure of Functions. Monographs in Mathematics, vol 97. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8257-6_11
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DOI: https://doi.org/10.1007/978-3-0348-8257-6_11
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9494-4
Online ISBN: 978-3-0348-8257-6
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