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H-measures (also known as microlocal defect measures) of Tartar [1]
and G\'erard [2] obtained for weakly convergent sequences in $L^2$,
and their generalization to $L^p$, called H-distributions [3], are widely
used to determine weather a weakly convergent sequence converge
strongly.We extend the concept of H-distributions to the Sobolev
spaces and give necessary and sufficient condition so that the weak
convergence in $W^{-k,p}$, $p\in(1,\b)$, implies the strong one.

References
[1] Tartar, L.$H$-measures, a new approach for studying homogenisation,
oscillations and concentration effects in partial differential equations.
Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 193--230.
[2] G\'{e}rard, P.
Microlocal defect measures.
Comm. Partial Differential Equations 16 (1991), no. 11, 1761--1794
[3] Antoni\'c, N.; Mitrovi\' c, D.
H-distributions: an extension of H-measures to an $L^p-L^q$
setting. Abstr. Appl. Anal. 2011, Art. ID 901084, 12 pp.

$H$-distribution; $H$-measure; $W^{k,p}$ framework; weak and strong convergence;