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By Dorothee D. Haroske

In the past, no e-book has systematically awarded the lately constructed notion of envelopes in functionality areas. Envelopes are rather easy instruments for the examine of classical and extra complex areas, corresponding to Besov and Triebel-Lizorkin forms, in restricting occasions. This conception originates from the classical results of the Sobolev embedding theorem, ubiquitous in all parts of sensible research. Self-contained and obtainable, Envelopes and Sharp Embeddings of functionality areas offers the 1st exact account of the hot idea of development and continuity envelopes in functionality areas. The booklet is definitely dependent into components, first supplying a finished creation after which interpreting extra complex subject matters. a few of the classical functionality areas mentioned within the first half comprise Lebesgue, Lorentz, Lipschitz, and Sobolev. the writer defines progress and continuity envelopes and examines their homes. partially II, the booklet explores the implications for functionality areas of Besov and Triebel-Lizorkin forms. the writer then provides a number of functions of the consequences, together with Hardy-type inequalities, asymptotic estimates for entropy, and approximation numbers of compact embeddings. As one of many key researchers during this progressing box, the writer deals a coherent presentation of the new advancements in functionality areas, delivering priceless details for graduate scholars and researchers in useful research.

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Here α = (α1 , . . , αn ) ∈ Nn0 stands for some multi-index, |α| = α1 + · · · + αn , α ∈ Nn0 , Dα are classical derivatives, Dα = 1 ∂xα 1 ∂ |α| , n . . ∂xα n α ∈ Nn0 , and C m (Rn ) is endowed with the norm f |C m (Rn ) = Dα f |C(Rn ) , |α|≤m where · |C(Rn ) can obviously be replaced by · |L∞ (Rn ) in this case. The set of all compactly supported, infinitely often differentiable functions is denoted by C0∞ (Rn ), as usual. 20) where x, h ∈ Rn . For convenience we may write ∆h instead of ∆1h .

3, Rem. 4]. Let 1 ≤ p, q ≤ ∞ such that 0 ≤ 1r = p1 + 1q ≤ 1. Let α, β ≥ 0. Then max(α,β)) Lip(1,−α) · Lip(1,−β) → Lip(1,− → Lip(1,−(α+β)) . 29) The following extrapolation type result for spaces Lip(1,−α) was obtained p, q in [EH00, Prop. 2(i)], [Har00b, Prop. 7]; for details about extrapolation techniques we refer to [Mil94]. 21 Let 1 ≤ p ≤ ∞. (i) Let q = ∞, α > 0. Then f ∈ Lip(1,−α) if, and only if, f belongs to p, ∞ Lp and there is some c > 0 such that for all λ, 0 < λ < 1, ω(f, t)p ≤ c λ−α .

Let Lexp,a [0, 1] be the exponential space with respect to [R, µ] = [0, 1], |·| . 19 Let a ≥ 0. Then L EG exp,a (·) ∈ Lexp,a [0, 1] . 4(i). 3 Connection with the fundamental function In rearrangement-invariant function spaces X one has the concept of the “fundamental function” ϕX ; we now investigate its connection with the growth envelope function EGX as defined above. , Rn equipped with the (n-dimensional) Lebesgue measure n . We closely follow the presentation in [BS88, Ch. 2, §5]. 11. , if µf (λ) = µg (λ) for all λ ≥ 0 implies (f ) = (g).

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