Before going, let us recall the so-called Hardy inequality from this note below
Theorem (Hardy’s inequality). Let with . Then
The constant is the best possible constant.
The purpose of this note is to show that the Hardy integral cannot be improved in the usual sense, that is, there are no nontrivial potential and no exponent such that, for any function ,
for some positive constant .
Indeed, let us construct the following family of test functions as follows
Now, making use the co-area formula gives us
Next, for almost everywhere, it holds
Hence, in order to complete the proof, we need to show that
Fortunately, this is trivial. We fix some large enough such that
Then within the ball , it is easy to see that
The proof is now complete.
For more details, we refer the reader to a paper due to de Pino et al. published in J. Funct. Anal. in 2010 [here].