General weighted higher-order Hardy inequality with a boundary weight

Let $B_R$ be the ball of radius $R$ centred at the origin in $\mathbb{R}^N$, let $k\geq 2$, let $1<p<N/k$, let $0<\gamma\leq (N-kp)/(kp-1)$, and let $u\in C_c^k(B_R)$. Weighted higher-order Hardy conjecture. The inequality

$$\left(\prod_{j=1}^{k}\frac{jp-1}{p}\gamma\right)^p\int_{B_R}\frac{|u|^p}{|x|^{kp}\left(1-\left(\frac{|x|}{R}\right)^\gamma\right)^{kp}}\,dx\leq\int_{B_R}|\nabla^k u|^p\,dx$$

holds. Moreover, the constant $\left(\prod_{j=1}^{k}(jp-1)\gamma/p\right)^p$ is optimal and is not attained by any nonzero $u$ for which the right-hand side is finite. This conjecture proposes a sharp higher-order Hardy inequality combining a singularity at the origin with a boundary weight; its validity, sharpness, and non-attainment are formulated as part of the general case considered in the paper.