×
Appell, Jürgen; Hardy, G.

On products of Sobolev-Orlicz spaces. (English) Zbl 0707.46023

Bull. Aust. Math. Soc. 42, No. 3, 427-436 (1990).
This paper asks when the operation of multiplication by a fixed function is a bounded linear operator on a k-th order Orlicz-Sobolev space, and uses the answer to obtain a sufficient condition for a nonlinear Nemytskii operator to map such a space continuously into itself. Generalisations and extensions of results of Marcus, Mizel and Valent are obtained.
Reviewer: J.F.Toland

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Keywords:

superposition operator; operation of multiplication by a fixed function; Orlicz-Sobolev space; nonlinear Nemytskii operator
Cite Review PDF
Full Text: DOI

References:

[1] DOI: 10.1016/0022-1236(80)90059-2 · Zbl 0444.47048 · doi:10.1016/0022-1236(80)90059-2
[2] Maz’ya, Theory of multipliers in spaces of differentiable functions (1985)
[3] DOI: 10.1002/mana.19800990119 · Zbl 0474.46017 · doi:10.1002/mana.19800990119
[4] Hardy, Bull. Austral. Math. Soc. 30 pp 251– (1984)
[5] Hardy, Bull. Austral. Math. Soc. 23 pp 121– (1981)
[6] DOI: 10.2307/1996957 · Zbl 0239.35045 · doi:10.2307/1996957
[7] DOI: 10.1016/0022-1236(71)90018-8 · Zbl 0216.15702 · doi:10.1016/0022-1236(71)90018-8
[8] Appell, Nonlinear superposition operators (1989)
[9] Appell, Analysis 7 pp 305– (1987) · Zbl 0627.47033 · doi:10.1524/anly.1987.7.34.305
[10] Appell, Untersuchungen zur Theorie nichtlinearer Operatoren und Operatorgleichungen (1985)
[11] DOI: 10.1007/BF01361143 · Zbl 0091.27804 · doi:10.1007/BF01361143
[12] Adams, Sobolev spaces (1976)
[13] Zabrejko, Integral equations (1968)
[14] DOI: 10.1016/0022-1236(79)90113-7 · Zbl 0418.46024 · doi:10.1016/0022-1236(79)90113-7
[15] DOI: 10.1016/0362-546X(78)90072-X · Zbl 0387.46035 · doi:10.1016/0362-546X(78)90072-X
[16] DOI: 10.2307/1998518 · Zbl 0351.46021 · doi:10.2307/1998518
[17] DOI: 10.1007/BF02786819 · Zbl 0328.46028 · doi:10.1007/BF02786819
[18] DOI: 10.1007/BF00263040 · Zbl 0266.46029 · doi:10.1007/BF00263040
[19] DOI: 10.1007/BF00251378 · Zbl 0236.46033 · doi:10.1007/BF00251378
[20] Kufner, Function spaces (1977)
[21] Krein, Interpolation of linear operators (1978) · Zbl 0438.46054
[22] Krasnosel’ski Convex functions and Orlicz spaces (1958)
[23] Hardy, Bull. Austral. Math. Soc. 37 pp 29– (1988)
[24] Zabrejko, Voran. Gas. Univ. Trudy Sem. Funk. Anal. 8 pp 1– (1966)
[25] Wang, Bull. Acad. Polon. Sci. 11 pp 19– (1963)
[26] Valent, Boundary value problems of finite elasticity - local theorems on existence, uniqueness and analytic dependence on data (1987) · Zbl 0648.73019
[27] Valent, Rend. Sem. Mat. Univ. Padova 74 pp 63– (1985)
[28] Moseenkov, Ukrain. Mat. Zh. 34 pp 384– (1982) · doi:10.1007/BF01682129
[29] DOI: 10.2307/1998689 · Zbl 0417.46035 · doi:10.2307/1998689
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.