Some boundary value problems for pseudodifferential equations with degeneration (Q294329)

Following the paper by the first author [Sov. Math., Dokl. 26, 182--185 (1982); translation from Dokl. Akad. Nauk SSSR 265, 1044--1046 (1982; Zbl 0528.35044)], where the special integral transform \(F_\alpha\) was defined the authors introduce a new class of degenerate pseudodifferential operators \(K^\sigma(t,D_x,D_{\alpha,t})\), \(x\in\mathbb{R}^{n-1}\), \(t\geq 0\) with a \(t\)-variable symbol \(\lambda(t,\xi,\eta)\) and the Sobolev type spaces \(H_{s,\alpha}(\mathbb{R}^n_+)\), \(H_{s,\alpha,q}(\mathbb{R}^n_+)\) in which boundary value problems for \(K^{(q)}_{v-}\partial_t v= F\) in \(\mathbb{R}^n_+\) are studied. Eight theorems are formulated dealing with existence, uniqueness and a-priori estimates for the corresponding solutions \(v\) in the above-mentioned functional spaces.