  For the incompressible Euler equations the pressure formally scales as a
quadratic function of velocity. We provide several optimal regularity estimates
on the pressure by using regularity of velocity in various Sobolev, Besov and
Hardy spaces. Our proof exploits the incompressibility condition in an
essential way and is deeply connected with the classic Div-Curl lemma which we
also generalise as a fractional Leibniz rule in Hardy spaces. To showcase the
sharpness of results, we construct a class of counterexamples at several
end-points.
