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In this work we give an explicit construction of parametrices for certain elliptic and degenerate - elliptic boundary value problems. In what follows, t ∈ [0, T) with T > 0, x ∈ X = unit open ball in Rⁿ, and ξ ∈ Rⁿ \ . When the roots λ_j (x, t, ξ), j = l,....,m, of the principal Symbol of the elliptic operator under study are C^∞ with respect to (x, ξ), at t = 0, we present a new construction of a parametrix by means of pseudodifferential operators. Still in the elliptic situation, when the roots are not of constant multiplicity but do coincide when t = 0, we construct a new parametrix built up of Fourier integral operators with complex phase functions. We also construct & parametrix for a class of degenerate - elliptic boundary value problems, once again expressed by pseudodifferential operators. Byfusing these parametrices and especially the pseudolocalípropertY'of pseudodifferential operators, we prove the regularity of the solutions.