Anisotropic Hörmander Spaces on the Lateral Surface of a Cylinder

Theorems on Isomorphisms for Some Parabolic Initial-Boundary-Value Problems in Hörmander Spaces: Limiting Case

“…Relation (25) is obtained from (24) by the well-known method of "straightening" and "gluing" the manifold S. The proof of (25) completely coincides with the proof of Theorem 2 in [29]. To show this, it is necessary to set s := s 0 , use parameter (19) as the interpolation parameter , and apply the interpolation formula (24).…”

“…The right-hand sides g of the boundary conditions (3) and (4) belong to anisotropic Hörmander spaces defined on the lateral surface S = Γ ⇥ (0, ⌧ ) of the cylinder ⌦. We define these spaces by using special local maps on S (see [29], Sec. 1).…”

“…The parabolic problems in Hörmander 2b-anisotropic spaces H s,s/(2b), ' , where 2b is a parabolic weight of the equation and the parameters s and ' are the same as in the mentioned elliptic theory, were investigated in [24][25][26][27][28][29][30].…”

Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces

“…This implies that the mapping (35) extends by continuity to the bounded linear operator (36). Let us build the linear mapping T : (L 2 (Γ)) r → L 2 (S) whose restriction to H s;ϕ (Γ) is a right-inverse of (36). Consider the linear mapping of flattening of Γ L : v → (χ 1 v) • θ 1 , .…”

“…This follows immediately from the boundedness of the operators (51), (52), and (53). The operator (37) is right inverse to (36). Indeed, choosing a vector v = (v 0 , v 1 , .…”