Vector-valued Sobolev Spaces and Gagliardo-Nirenberg Inequalities

On some classes of inverse problems for parabolic equations

“…The symbol k k stands for the norm in E. For convenience of the reader, below we present some auxiliary propositions accompanied with proofs in necessary cases. This proposition follows from the known results on interpolation of Sobolev spaces (see [1,5,12,13,22] (under certain conditions on the boundary of G). In particular, the inequality (3.1) for integers s 1 ; s 2 ; s with the space W s p instead of B s p;p on the left-hand side of (3.1) is presented in [22].…”

Anisotropic function spaces and elliptic boundary value problems

“…The study of vector valued function spaces has an entire literature of its own, as new difficulties arise concerning operator-valued multipliers and to what extent properties of classical scalar-valued function spaces continue to hold. For some results and further reading, see e.g., [2,6].…”

On some classes of inverse problems for parabolic and elliptic equations

“…In all auxiliary statements, the conditions on the boundary of G are exposed in accord with the conditions required in this method. The interpolation inequalities are examined in [1,15,21,34]. In particular, the inequality (3.3) for integers s 1 , s 2 , s with the space W s p instead of B s p, p on the left-hand side of (3.3) is presented in [34].…”

“…The interpolation inequalities are examined in [1,15,21,34]. In particular, the inequality (3.3) for integers s 1 , s 2 , s with the space W s p instead of B s p, p on the left-hand side of (3.3) is presented in [34]. Define the space of Bessel's potentials H s p (R n ; E) ( p ∈ (1, ∞)) as the class of func-…”

“…The classes H s p (G; E) (G is a bounded domain) are defined conventionally as restrictions of functions defined on R n to the domain G (see, for instance, [8,32,34]). As is known (see [34]), for ζ -convex spaces, we have H m …”