2.5. Miscellaneous Comments 115
On the other hand, even continuity of B in the topology of E may be
an excessively strong assumption, especially in view of the applications
where A is a linear differential operator and B a (lower order) perturbation
of A. Several results exist where (Holder) continuity properties of B are
postulated in the graph norm of fractional powers of A (see Sobolevskii
[ 1960: 1], [1960: 2], [1961: 1], [1964: 1], Kato-Fujita [ 1962: 1 ], Henry [ 1978:
11, Pecher [1975: 11, and Webb [1977: 2]. Other type of results where the
assumptions are on S(t) B instead than on B itself have been proved by
Weissler [1979: 1]. A detailed treatment of the equation (2.5.60) can be
found in Martin [1976: 1]. Other references on different types of semilinear
equations are Alirov [ 1972: 1 ], Ahundov [ 1972: 1 ], Ahundov-Jakubov [ 1969:
1], [1970: 1], Ardito-Ricciardi [1974: 11, Akinyele [1976: 1], Bartak [1976: 1],
Biroli [ 1972: 1], [1974: 1 ], Brezis-Strauss [ 1973: 11, Calvert [ 1976: 11, Chafee
[1977: 1], Daleckii-Krein [1970: 1], Datko [1968: 1], [1970: 1], [1970: 2],
[1972: 1], De Blasi-Myjak [1977: 1], Domslak [1962: 1], [1965: 1], [1975: 11,
Enikeeva [ 1968: 1], [1969: 1], [1969: 2], [1972: 1 ], Gel'man-Gerst [ 1968: 11,
Gajewski [1971: 1], [1972:
1], Heinz-Von Wahl [1975:
1], Hughes-Kato-
Marsden [1976: 11, lannelli [1976: 1], boss [1972: 1], Jakubov [1966: 3],
[1967: 2], [1967: 3], [1970: 3], [1970: 4], Jakubov-Aliev [1976: 1], Jakubov-
Ismailova [1974: 1], Kakita [1974/75: 1], Kartsatos [1969:
1], Kielhofer
[1973: 1], [1975: 1], Kirjanen [1974: 11, Kluge-Bruckner [1974: 1], Konishi
[ 1976: 2], Krein [ 1967: 11, Lightbourne [ 1976: 1 ], Lunin [ 1973: 11, [1974: 1 ],
Mamedov [1964: 2], [1965: 11, Mamii [1966:
1],
[1967:
1],
[1967: 2],
Mamii-Mirzov [1971:
1], Martin [1975:
1], [1977:
1], Massey [1976:
1],
[ 1977: 1 ], Monari [ 1971: 1 ], Nikloenko [ 1973: 1 ], Pavel [ 1972: 11, [1974: 11,
[1974: 2], Pazy [1972: 1], [1975: 11, Pao [1972: 1], [1973: 1], [1974/75: 1],
Pogorelenko-Sobolevskii [1967: 2], [1967: 3], [1970:
1], [1972:
1], Raskin
[1973: 1], Raskin-Sobolevskii [1968: 11, [1968: 2], Rautmann [1973:
1],
Reichelt [1975: 1], Ricciardi-Tubaro [1973: 1], Simonenko [1970: 1], [1973:
1], Sobolevskii [1958: 3], [1968: 5], Tsutsumi [1971: 1], [1972: 1], Veliev-
Mamedov [1973: 1], [1974: 11, Vilella-Bressan [1974:
11, Ward [1976:
1],
Zarubin [ 1970: 1], [1970: 2], and Lovelady [ 1973/74: 1 ].
(1) Approximation. Consider the space E = L2 (0, 2 it). Given u E E,
write
(%(t)u)(x)=
e-Inltanernx
where the (an; - oo < n < oo) are the Fourier coefficients of u. We check
that (W(t ); t > 0) is a strongly continuous semigroup in E (called the
Abel-Poisson semigroup); the fact that
W(t) u - u as t -+ 0
simply expresses that the Fourier series of an element u of E converges in
the sense of Abel to u in the L2 norm. A natural question is how closely
does W (t) u approximate u for t small; for instance, which are the u E E