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 .