A diagonal embedding theorem for function spaces with dominating mixed smoothness properties. (English) Zbl 0707.46020

Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ. 22, 475-486 (1989).
[For the entire collection see Zbl 0681.00013.]
Spaces with dominating mixed smoothness properties of Sobolev type were introduced by S. M. Nikol’skij [Dokl. Akad. Nauk SSSR 146, 542-545 (1962; Zbl 0196.443)]. The simplest case on the plane \({\mathbb{R}}^ 2\) is characterized by the norm \(\| f\| +\| \frac{\partial f}{\partial x_ 1}\| +\| \frac{\partial f}{\partial x_ 2}\| +\| \frac{\partial^ 2f}{\partial x_ 1\partial x_ 2}\|\) where \(\| \|\) is the \(L_ p({\mathbb{R}}^ 2)\) norm.
This paper deals with spaces \(b^ r_ p({\mathbb{R}}^ 2)\) where \(r=(r_ 1,r_ 2)\), \(-\infty <r_ j<\infty\), \(1\leq p\leq \infty\). The norm is of Besov type and provides dominating mixed smoothness of order \(r_ j\) with respect to \(x_ j\) for \(j=1,2\). The corresponding trace spaces are the Besov spaces \(b^{\rho}_ p({\mathbb{R}})\). It is proved that the diagonal mapping T: f(x\({}_ 1,x_ 2)\to f(x_ 1,x_ 1)\) is a retraction from \(b^ r_ p({\mathbb{R}}^ 2)\) onto \(b^{\rho}_ p({\mathbb{R}})\) where \(\rho =\min (r_ 1+r_ 2-1/p,r_ 1,r_ 2)\).
Reviewer: A.Pryde


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems