On the boundedness of the mapping \(f\to| f|\) in Besov spaces. (English) Zbl 0766.46018

Oswald, P.

Commentat. Math. Univ. Carol. 33, No. 1, 57-66 (1992).

In this paper the author studies the boundedness of the mapping \((*)\;T:f\to| f|\) in the scale of Besov spaces \(B^ s_{p,q}\) on \(\mathbb{R}^ 1\), where \(1\leq p\), \(q\leq\infty\), and \(s>0\).
The author proves the following main result: Let the parameter \(p\), \(q\), \(s\) be as given above. Then the mapping \(T\) defined by \((*)\) is bounded in \(B^ s_{p,q}\) if and only if \(0<s<1+1/p\).
The result relies on linear spline approximation theory.

Reviewer: S.Wedrychowicz (Rzeszów)

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A15	Spline approximation
35B45	A priori estimates in context of PDEs

Keywords:

scale of Besov spaces; linear spline approximation