Dirichlet problem for the heat equation in \(h_ c^ 1(d\omega)\) (Q1913393)

In their previous paper [Ann. Pol. Math. 54, No. 1, 29-44 (1991; Zbl 0757.35029)] the same authors studied the initial Neumann problem for the heat equation \((\Delta_x- D_t) u= 0\) in a cylindrical domain \(D\times (D, T)\) with \(h^1_c(dQ ds)\)-data, where \(D\) is a bounded \(C^1\)-domain in \(\mathbb{R}^n\). Here the space \(h^1_c(dQ ds)\) is a predual of a space \(B_T MOC(dQ ds)\subset BMOC(dQ ds)\). In this paper, they study an initial Dirichlet problem with the data in \(h^1_c(d\omega)\) which is a predual of a space \(B_T MOC(d\omega)\subset BMDC(d\omega)\). The measure involved is the caloric measure \(d\omega\) while in the Neumann problem it is the surface measure \(dQ ds\). Therefore the solution is not represented as a double layer potential but using the kernel function \(K\) associated with \(d\omega\).