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On limiting embeddings of Besov spaces. (English) Zbl 1090.46026

Let \(0<s<1\) and \(1\leq p,\,\theta<\infty\). The Besov space \(B_{p,\theta}^{s}({\mathbb R}^n)\) consists of all functions \(f\in L^{p}( {\mathbb R}^n)\) such that \[ \| f\| _{ b_{p,\theta}^{s}}\equiv \left(\int_{0}^{\infty}\left(t^{-\theta}\omega_p(f,t)\right)^\theta {dt\over t}\right)^{1/\theta} <\infty, \] where \(\omega_p(f,t)\) is the \(L^p\)-modulus of continuity of \(f\). The main result of the paper is the following:
Theorem 1.1. Let \(0<s<1\) and \(p<q<\infty\). Let also \(\gamma\equiv n(1/p-1/q)\). Assume that \(\gamma<s\) and \(1\leq \theta<\infty\). If either \(p>1\), \(n\geq 1\) or \(p\geq 1\), \(n\geq 2\), then for any \(f\in B_{p,\theta}^{s}( {\mathbb R}^n)\) we have \[ \| f\| _{ b_{q,\theta}^{s-\gamma}}\leq A{(1-s)^{1/\theta^*}\over {(s-\gamma)^{1/\theta}}}\| f\| _{ b_{p,\theta}^{s}}, \] where \(\theta ^*=\max(p,\theta)\) and the constant \(A\) does not depend on \(s\) and \(f\). It is shown that the main result yields to obtain the theorem of J. Bourgain, H. Brezis and P. Mironescu [J. Anal. Math. 87, 77–101 (2002; Zbl 1029.46030)]. A different elementary proof of a more general result of V. Maz’ya and T. Shaposhnikova [J. Funct. Anal. 195, No. 2, 230–238 (2002; Zbl 1028.46050)] is given by using the rearrangement estimates. It is shown that a stronger inequality holds: for any \(0<s<1\) and \(1\leq p< n/s\), \[ \| f\| _{ L^{q^*,p}({\mathbb R}^n)}\leq c_{p,n}{{s(1-s)}\over {(n-sp)^{p}}}\| f\| _{ b_{p}^{s}({\mathbb R}^n)}\qquad \left(q^*= {{np}\over{n-sp}}\right). \]

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.)
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