Contents Introduction.............................................................................................................................3 1. Definitions, notations and some auxiliary lemmas...................................................4 2. The definition of the spaces $H_{p,q;Y}(Ω, ℬ)$..........................................................7 3. Some properties of the spaces $H_{p,q;Y}(Ω, ℬ)$...................................................10 4. Some examples of the spaces $H_{p,q;Y}(Ω, ℬ)$....................................................15 5. Some special cases of the spaces $H_{p,q;μ}(Ω, ℬ)$.............................................20 6. Estimates for elliptic operators depending on a parameter....................................23 7. The density theorems for elliptic operators depending on a parameter...............35 8. Differential equations in the spaces $H_{p,q;μ}(Ω, ℬ)$...........................................38 9. Local theorems.................................................................................................................46 10.Some counter-examples...............................................................................................49 References.............................................................................................................................53