Annali di Matematica Pura ed Applicata

, Volume 114, Issue 1, pp 87–102 | Cite as

Multiplication properties of besov spaces

  • Hans Triebel


The paper deals with the question whether f → gf (pointwise multiplication) is a bounded operator from B p,q s =B p,q s (Rn) into itself. Here B p,q s are the general (non-homogeneous isotropic) Besov spaces; −<s<∞; 0<p ≦∞; 0<q ≦. A special case of the main result (formulated in the theorem in3.1) is the following. If g is the characteristic function of a half-space, then f → gf is a bounded operator in B p,q s if
$$s \in \left\{ \begin{gathered} \left( {\frac{1}{p} - 1,\frac{1}{p}} \right)for 1 \leqq p \leqq \infty , \hfill \\ \left( {n\left( {\frac{1}{p} - 1} \right),\frac{1}{p}} \right)for\frac{{n - 1}}{n}< p \leqq 1, \hfill \\ \end{gathered} \right.$$
and it is not a bounded operator in B p,q s if either0<p<n −1 / n or
$$s \notin \left\{ \begin{gathered} \left[ {\frac{1}{p} - 1,\frac{1}{p}} \right]for 1 \leqq p \leqq \infty , \hfill \\ \left[ {n\left( {\frac{1}{p} - 1} \right),\frac{1}{p}} \right]for\frac{{n - 1}}{n}< p \leqq 1. \hfill \\ \end{gathered} \right.$$
(For1<p< these results are known). The paper is the continuation of[12].


Characteristic Function Multiplication Property Bounded Operator Besov Space Pointwise Multiplication 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1977

Authors and Affiliations

  • Hans Triebel
    • 1
  1. 1.JenaD.D.R.

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