## 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.

### 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
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