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Mixed norms and Sobolev type inequalities. (English) Zbl 1114.46024

Figiel, Tadeuz (ed.) et al., Approximation and probability. Papers of the conference held on the occasion of the 70th anniversary of Prof. Zbigniew Ciesielski, Bȩdlewo, Poland, September 20–24, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 72, 141-160 (2006).
The author studies mixed norm spaces that arise in connection with embeddings of Sobolev and Besov spaces. Let \(\sigma\in \mathbb R\). Denote by \(\Lambda^\sigma(\mathbb R)\) the space of all measurable functions \(f\) such that \(\| f\| _{\Lambda^\sigma(\mathbb R)}\equiv \sup_{t>0}t^\sigma[f^\ast(t)- f^\ast(2t)]<\infty\). In particular, the author establishes the following Sobolev type inequalities in terms of these mixed norms: Assume that \(1\leq p<\infty\), \(n\geq 2\) \((n\in\mathbb N)\), and that \(\alpha_k\) \((k=1,\dots,n)\) are positive numbers such that \(\alpha\equiv n(\sum_{k=1}^n\frac1{\alpha_k})^{-1}\leq \frac np\). Let \(\sigma_k=\frac1p-\alpha_k\) and \(V_k\equiv L^p_{\widehat{x_k}}({\mathbb R}^{n-1}) [\Lambda^p_{x_k}(\mathbb R)]\). Then \(\bigcap_{k=1}^n V_k\subset L^{q^\ast, p}({\mathbb R}^n)\), where \(q^\ast=\frac{np}{n-\alpha p}\) and \(L^{q^\ast, p}({\mathbb R}^n)\) denotes the Lorentz space. Applying these results, the author obtains optimal constants in embedding theorems for anisotropic Besov spaces. This gives an extension of the estimates proved by Bourgain, Brezis and Mironescu for isotropic Besov spaces.
For the entire collection see [Zbl 1091.47002].

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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