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

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.


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