On the uniqueness of solutions of Duhamel equations

Abstract

We consider the Duhamel equation
' ~ f = g
in the subspace
C1xy = ff 2 C1 ([0; 1] [0; 1]) : f (x; y) = F (xy) for some F 2 C1 [0; 1]g
of the space C1 ([0; 1] [0; 1]) and prove that if ' pxy=06= 0; then this equation
is uniquely solvable in C1xy: The commutant of the restricted double integration
operator Wxyf (xy) :=
R x
0
R y
0 f (t ) ddt on C1xy is also described. Some other
related questions are also discussed.