## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Citations:

Zbl 1029.46030; Zbl 1028.46050
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