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Tractable embeddings of Besov spaces into small Lebesgue spaces. (English) Zbl 1361.46026
If \(1\leq p<n\), then \[ \Big( \int^1_0 (1 - \log t)^{p/2} f^* (t)^p \, dt \Big)^{1/p} \leq c \big( \| f \, | L^p \| + \| \nabla f\, | L^p \| \big), \] \(f \in W^{1,p}(\mathbb R^n)\), \(\text{supp }f \subset [0,1]^n\), where \(c>0\) is independent of the dimension \(n\). Here, \(f^*\) is the usual decreasing rearrangement of \(f\). There are counterparts with Besov spaces \(B^s_{p,q}\) in place of the Sobolev spaces \(W^{1,p}\). The main point is the control of the related constants \(c\) on the dimension \(n\). Assertions of this type are called tractable (or dimension-controllable). The paper contributes to this topic replacing the left-hand side of the above inequality by the so-called small Lebesgue spaces, normed by \[ \| f \, | L^{(p,b,q} (\mathbb T^n) \| = \Big( \int^1_0 \Big[ (1- \log t)^b \Big( \int^t_0 f^* (s)^p \, ds \Big)^{1/p} \Big]^q \frac{dt}{t} \Big)^{1/q}\,, \] where \(\mathbb T^n = [0,1]^n\) is the \(n\)-torus. The arguments are based on periodic Besov spaces and related approximations, (limiting) interpolations and extrapolations.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42A10 Trigonometric approximation
46B70 Interpolation between normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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