We give a generalization of Dorronsoro’s theorem on critical
L
p
\mathrm {L}^p
-Taylor expansions for
B
V
k
\mathrm {BV}^k
-maps on
R
n
\mathbb {R}^n
; i.e., we characterize homogeneous linear differential operators
A
\mathbb {A}
of
k
k
th order such that
D
k
−
j
u
D^{k-j}u
has
j
j
th order
L
n
/
(
n
−
j
)
\mathrm {L}^{n/(n-j)}
-Taylor expansion a.e. for all
u
∈
B
V
loc
A
u\in \mathrm {BV}^\mathbb {A}_{\operatorname {loc}}
(here
j
=
1
,
…
,
k
j=1,\ldots , k
, with an appropriate convention if
j
≥
n
j\geq n
). The space
B
V
loc
A
\mathrm {BV}^\mathbb {A}_{\operatorname {loc}}
, a single framework covering
B
V
\mathrm {BV}
,
B
D
\mathrm {BD}
, and
B
V
k
\mathrm {BV}^k
, consists of those locally integrable maps
u
u
such that
A
u
\mathbb {A} u
is a Radon measure on
R
n
\mathbb {R}^n
.
For
j
=
1
,
…
,
min
{
k
,
n
−
1
}
j=1,\ldots ,\min \{k, n-1\}
, we show that the
L
p
\mathrm {L}^p
-differentiability property above is equivalent to Van Schaftingen’s elliptic and canceling condition for
A
\mathbb {A}
. For
j
=
n
,
…
,
k
j=n,\ldots , k
, ellipticity is necessary, but cancellation is not. To complete the characterization, we determine the class of elliptic operators
A
\mathbb {A}
such that the estimate
(1)
‖
D
k
−
n
u
‖
L
∞
⩽
C
‖
A
u
‖
L
1
\begin{align}\tag {1} \|D^{k-n}u\|_{\mathrm {L}^\infty }\leqslant C\|\mathbb {A} u\|_{\mathrm {L}^1} \end{align}
holds for all vector fields
u
∈
C
c
∞
u\in \mathrm {C}^\infty _c
. Surprisingly, the (computable) condition on
A
\mathbb {A}
such that \eqref{eq:abs} holds is strictly weaker than cancellation.
The results on
L
p
\mathrm {L}^p
-differentiability can be formulated as sharp pointwise regularity results for overdetermined elliptic systems
A
u
=
μ
,
\begin{align*} \mathbb {A} u=\mu , \end{align*}
where
μ
\mu
is a Radon measure, thereby giving a variant for the limit case
p
=
1
p=1
of a theorem of Calderón and Zygmund which was not covered before.