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Sharp embeddings of Besov spaces involving only logarithmic smoothness. (English) Zbl 1161.46017
The paper contains sharp embedding assertions for spaces \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) of Besov type into spaces \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) of Lorentz-Zygmund type. The Besov spaces are defined by means of the modulus of continuity \(\omega_1(f,t)_p\), \[ \| f| B^{0,\beta}_{p,r}(\mathbb{R}^n)\| = \| f| L_p(\mathbb{R}^n)\| + \left(\int_0^1 (1+| \ln t| )^{\beta r } \omega_1^r(f,t)_p \frac{dt}{t}\right)^{1/r}, \] whereas the Lorentz-Zygmund spaces \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) collect all \(f\) with finite quasi-norm \[ \left(\int_0^1 t^{q/p} (1+| \ln t| )^{\gamma q } f^\ast(t)^q \frac{dt}{t}\right)^{1/q}, \] where \(1\leq p<\infty\), \(1\leq r,q\leq \infty\), \(\beta, \gamma\in\mathbb{R}\) with \(\beta+1/r>0\), and \(f^\ast\) denotes the non-increasing rearrangement, as usual. The essential point here is that the Besov spaces have zero classical smoothness and differ from their counterparts defined by Fourier analytical methods, say. Thus many of the standard approaches to prove optimality or sharpness of embeddings do not work and the essential benefit of this paper is to close this gap and, moreover, find new phenomena in connection with growth envelopes and limiting embeddings.
The main results, listed in Section 3 of this interesting paper, contain criteria for the above embedding of the form that \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) is embedded into \(L^{\mathrm{loc}}_{p,q;\gamma}(\mathbb{R}^n)\) with \(\gamma=\beta+\frac1r + \frac{1}{\max(p,q)}-\frac1q\) if, and only if, \(q\geq r\) (Theorem 3.1). Furthermore, the growth envelope of the space \(B^{0,\beta}_{p,r}(\mathbb{R}^n)\) is determined by the pair \((t^{-\frac1p} (1+| \ln t| )^{-\beta-\frac1r}, \max(p,r))\).
The proofs, presented in Section 5-9, rely on an inequality by V. I. Kolyada [Math. USSR, Sb. 64, No. 1, 1–21 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 1(5), 3–23 (1988; Zbl 0693.46030)] and some inverse of it, stated as Proposition 3.5 and proved in this paper in Section 4. It connects the modulus of continuity and the non-increasing rearrangement in a tricky way and is certainly of separate interest.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
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