Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Composition operators on potential spaces


Authors: David R. Adams and Michael Frazier
Journal: Proc. Amer. Math. Soc. 114 (1992), 155-165
MSC: Primary 46E35; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1992-1076570-5
MathSciNet review: 1076570
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By a result of B. Dahlberg, the composition operators $ {T_H}f = H \circ f$ need not be bounded on some of the Sobolev spaces (or spaces of Bessel potentials) even for very smooth functions $ H = H\left( t \right),H\left( 0 \right) = 0$, unless of course, $ H\left( t \right) = ct$. In this note a natural domain is found for $ {T_H}$ that is, in a sense, maximal and on which the $ \left\{ {{T_H}} \right\}$ form an algebra of bounded operators. Here the functions $ H\left( t \right)$ need not be bounded though they are required to have a sufficient number of bounded derivatives.


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

  • [1] D. R. Adams, On the existence of capacitary strong type estimates in $ {\mathbb{R}^n}$, Ark. Mat. 14 (1976), 125-140. MR 0417774 (54:5822)
  • [2] D. R. Adams and M. Frazier, BMO and smooth truncation in Sobolev spaces, Studia Math. 89 (1988), 241-260. MR 956241 (90a:46069)
  • [3] D. R. Adams and J. Polking, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc. 37 (1973), 529-534. MR 0328109 (48:6451)
  • [4] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., vol. 4, Amer. Math. Soc., Providence, R.I., 1961, pp. 33-49. MR 0143037 (26:603)
  • [5] B. Dahlberg, A note on Sobolev spaces, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, R. I. 1979, pp. 183-185. MR 545257 (81h:46030)
  • [6] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. MR 1070037 (92a:46042)
  • [7] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1959), 115-162. MR 0109940 (22:823)
  • [8] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Ser. 1, Duke Univ., Durham, N.C., 1976. MR 0461123 (57:1108)
  • [9] J. Polking, A Leibniz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1972), 1019-1029. MR 0318868 (47:7414)
  • [10] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR 0215084 (35:5927)
  • [11] H. Triebel, Theory of function spaces, Monographs Math., vol. 78, Birkhäuser, Basel, 1983. MR 781540 (86j:46026)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E35, 47B38

Retrieve articles in all journals with MSC: 46E35, 47B38


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1076570-5
Keywords: Sobolev space, potential space, composition operator
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society