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Annali di Matematica Pura ed Applicata

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

Multiplication properties of besov spaces

  • Hans Triebel
Article

Summary

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].

Keywords

Characteristic Function Multiplication Property Bounded Operator Besov Space Pointwise Multiplication 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    R. P. Boas,Entire functions, Academic Press, Inc. Publishers, New York, 1954.Google Scholar
  2. [2]
    B. Jawerth,The traces of Sobolev and Besov spaces if 0<p<1, Technical report, Lund, 1976.Google Scholar
  3. [3]
    J. L. Lions -E. Magenes,Problèmes aux limites non homogènes - IV, Ann. Scuola Norm. Sup. Pisa,15 (1961), pp. 311–326.MathSciNetGoogle Scholar
  4. [4]
    J. L. Lions -E. Magenes,Problèmes aux limites non homogènes et applications - I, Dunod, Paris, 1968 (English translation:Non-homogeneous boundary value problems and applications - I, Springer, Berlin-Heidelberg-New York, 1972).Google Scholar
  5. [5]
    S. M. Nikol'skij,Approximation of functions of several variables and imbedding theorems, Nauka, Moskva, 1969 (Russian), (English translation: Springer, Berlin-Heidelberg-New York, 1975).Google Scholar
  6. [6]
    J. Peetre,Remarques sur les espaces de Besov. Le cas 0<p<1, C. R. Acad. Sci. Paris, Sér A–B,277 (1973), A947-A950.MathSciNetGoogle Scholar
  7. [7]
    E. Shamir,Une propriété des espaces H s,p, C. R. Acad. Sci. Paris, Sér A–B,255 (1962), A448-A449.MathSciNetGoogle Scholar
  8. [8]
    R. S. Strichartz,Multipliers on fractional Sobolev spaces, J. Math. Mechanics,16 (1967), pp. 1031–1060.zbMATHMathSciNetGoogle Scholar
  9. [9]
    H. Triebel,Interpolation theory, function spaces, differential operators, VEB Deutscher Verl. Wissenschaften, Berlin, 1977.Google Scholar
  10. [10]
    H. Triebel,General function spaces - II: Inequalities of Plancherel-Polya-Nikol'skij type. L p spaces of analytic functions; 0<p ≦∞, J. Approximation Theory,19 (1977).Google Scholar
  11. [11]
    H. Triebel, On spaces of B∞qs type andC s type, Math. Nachr.Google Scholar
  12. [12]
    H. Triebel,Multiplication properties of the spacesBp,qs and Fp,qs. Quasi-Banach algebras of functions, Ann. Mat. Pura Appl.Google Scholar

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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