## 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

### MSC:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

### Citations:

Zbl 0681.00013; Zbl 0196.443