From the Cauchy-Schwarz inequality, we can prove that $\lVert w(x)\rVert^2_{L^2_{[0,1]}}=\int_0^1 w(x)^2\, dx \leq \sqrt{\int_0^1 w(x) \,dx}\cdot \sqrt{\int_0^1 w(x)^3\, dx}.$
Is it possible to prove another inequality with the other direction, that is, $\int_0^1 w(x)^3\, dx \leq C_1\cdot \left(\int_0^1 w(x)^2 dx\right)^{1/n}\;?$
Thanks a lot!!!