Composition operators on potential spaces

Adams, David R.; Frazier, Michael

doi:10.1090/S0002-9939-1992-1076570-5

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.

[1] David R. Adams, On the existence of capacitary strong type estimates in 𝑅ⁿ, Ark. Mat. 14 (1976), no. 1, 125–140. MR 417774, https://doi.org/10.1007/BF02385830
[2] David R. Adams and Michael Frazier, BMO and smooth truncation in Sobolev spaces, Studia Math. 89 (1988), no. 3, 241–260. MR 956241, https://doi.org/10.4064/sm-89-3-241-260
[3] David R. Adams and John C. Polking, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc. 37 (1973), 529–534. MR 328109, https://doi.org/10.1090/S0002-9939-1973-0328109-5
[4] A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037
[5] Björn E. J. Dahlberg, A note on Sobolev spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 183–185. MR 545257
[6] Michael Frazier and Björn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR 1070037, https://doi.org/10.1016/0022-1236(90)90137-A
[7] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
[8] Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR 0461123
[9] John C. Polking, A Leibniz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1971/72), 1019–1029. MR 318868, https://doi.org/10.1512/iumj.1972.21.21082
[10] Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060. MR 0215084
[11] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540