Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Besov spaces on domains in $ {\bf R}\sp d$


Authors: Ronald A. DeVore and Robert C. Sharpley
Journal: Trans. Amer. Math. Soc. 335 (1993), 843-864
MSC: Primary 46E35
DOI: https://doi.org/10.1090/S0002-9947-1993-1152321-6
MathSciNet review: 1152321
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study Besov spaces $ B_q^\alpha ({L_p}(\Omega ))$, $ 0 < p,q,\alpha < \infty $, on domains $ \Omega $ in $ {\mathbb{R}^d}$ . We show that there is an extension operator $ \mathcal{E}$ which is a bounded mapping from $ B_q^\alpha ({L_p}(\Omega ))$ onto $ B_q^\alpha ({L_p}({\mathbb{R}^d}))$. This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of $ B_q^\alpha ({L_p}(\Omega ))$, atomic decompositions for the elements of $ B_q^\alpha ({L_p}(\Omega ))$, and a description of the Besov spaces by means of spline approximation.


References [Enhancements On Off] (What's this?)

  • [1] M. Christ, The extension problem for cnrtain function spaces involving fractional orders of differentiability, Ark. Mat. 22 (1984), 63-81. MR 735878 (85k:46032)
  • [2] R. DeVore and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), 397-414. MR 920166 (89h:46044)
  • [3] -, Free multivariate splines, Constr. Approx. 3 (1987), 239-248. MR 889558 (88e:41029)
  • [4] R. DeVore and R. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., No. 293, 1983. MR 727820 (85g:46039)
  • [5] H. Johnen and K. Scherer, On the equivalence of the $ K$-functional and moduli of continuity and some applications, Lecture Notes in Math., vol. 571, Springer-Verlag, Berlin, 1976, 119-140. MR 0487423 (58:7060)
  • [6] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88. MR 631089 (83i:30014)
  • [7] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [8] E. A. Storozhenko and P. Oswald, Jackson's theorem in the spaces $ {L_p}({\mathbb{R}^k}), 0 < p < 1$ , Siberian Math. J. 19 (1978), 630-639.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E35

Retrieve articles in all journals with MSC: 46E35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1152321-6
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society