  This work introduces microlocal compactness forms (MCFs) as a new tool to
study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of
functions. Decisively, MCFs retain information about the location, value
distribution, and direction of oscillations and concentrations, thus extending
at the same time the theories of (generalized) Young measures and H-measures.
In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate
between weak and strong compactness, and thus MCFs allow to quantify the
difference in compactness. The definition of MCFs involves a Fourier variable,
whereby also differential constraints on the functions in the sequence can be
investigated easily - a distinct advantage over Young measure theory.
Furthermore, pointwise restrictions are reflected in the MCF as well, paving
the way for applications to Tartar's framework of compensated compactness;
consequently, we establish a new weak-to-strong compactness theorem in a
"geometric" way. After developing several aspects of the abstract theory, we
consider three applications: For lamination microstructures, the hierarchy of
oscillations is reflected in the MCF. The directional information retained in
an MCF is harnessed in the relaxation theory for anisotropic integral
functionals. Finally, we indicate how the theory pertains to the study of
propagation of singularities in certain systems of PDEs. The proofs combine
measure theory, Young measures, and harmonic analysis.
