L. C. Evans, Partial Differential Equations, Second Edition.
Graduate Studies in Mathematics, volume 19.
American Mathematical Society, Providence, RI, 2010.
Chap. 2: Solution formulas for the transport, Laplace and heat equations.
Chap. 5: Basic theory of Sobolev spaces.
Chap. 6: Applications of Sobolev spaces to elliptic PDEs.
Additionally, the appendices in the book; they cover relevant background material from real analysis, functional analysis, and measure theory.