Let denote the standard unit disk in
. The famous Moser–Trudinger inequality says that
holds. There is another important inequality in analysis, the Hardy inequality which claims that
holds. The one is usuall called the Hardy functional. One can immediately see that
for any . Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant
such that the following
where is the unit ball in
,
and
is the complement of
with respect to the following norm
.
Let us go back to the case . They then defined
where is a regular, bounded and convex domain sitting in
. They then conjectured that the following
still holds for some constant where
denotes the completion of
with the corresponding norm associated with
. Apparently, the conjecture holds true for
.