  For a complex algebraic variety $X$, we introduce higher $p$-Du Bois
singularity by imposing canonical isomorphisms between the sheaves of K\"ahler
differential forms $\Omega_X^q$ and the shifted graded pieces of the Du Bois
complex $\underline{\Omega}_X^q$ for $q\le p$. If $X$ is a reduced
hypersurface, we show that higher $p$-Du~Bois singularity coincides with higher
$p$-log canonical singularity, generalizing a well-known theorem for $p=0$. The
assertion that $p$-log canonicity implies $p$-Du Bois has been proved by
Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two
theorems asserting that the sheaves of reflexive differential forms
$\Omega_X^{[q]}$ ($q\le p$) coincide with $\Omega_X^q$ and
$\underline{\Omega}_X^q$ respectively, and these are shown by calculating the
depth of the latter two sheaves. We construct explicit isomorphisms between
$\Omega_X^q$ and $\underline{\Omega}_X^q$ applying the acyclicity of a Koszul
complex in a certain range. We also improve some non-vanishing assertion shown
by them using mixed Hodge modules and the Tjurina subspectrum in the isolated
singularity case. This is useful for instance to estimate the lower bound of
the maximal root of the reduced Bernstein-Sato polynomial in the case where a
quotient singularity is a hypersurface and its singular locus has codimension
at most 4.
