AbstractAim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form $$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$
L
u
:
=
∑
i
,
j
=
1
n
a
ij
(
x
)
u
x
i
x
j
=
∇
·
f
,
for almost all
x
∈
Ω
where the coefficients $$a_{ij}$$
a
ij
belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces $$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$
LM
~
{
x
0
}
p
,
φ
. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.
Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces
