We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best constant in Sobolev inequality. This gives $u$ in terms of its gradient $\nabla u$ for a bounded open set $U$. My question is,

**For an appropriate pointwise relation from a subsequence, does the reproducing kernel formula for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$ ?**

Note that the smooth functions with compact support are dense in $H^1_0(U)$ but not in $H^1(U)$,so the closure in regard of the test space applies on both sides of the kernel formula only for $H^1_0(U)$, while the closure applies to $L^2$ as well.

Note that on a suitable subsequence, pointwise relations hold almost everywhere; in fact, the limits can be considered in $L^p(U)$ when the support vanishes outside $U$, due to integrability of the kernel in $dim.n$ for $n \geq 2$. I also wish to add that the Atiyah-Singer index is zero for $n > 2$. Interestingly, it is awkward to show membership for both the spaces using the same $H^1$ norm,so the equivalence desired produces different norms.

Since Poincare inequality can be uniformly used for all $L^q(U)$, Hedberg estimates (Lars Inge Hedberg, Proc. AMS (1972) on convolution inequalities) appears to give the estimate for the $L^p$ norm of $\nabla u$ (not $u$), which I wish to confirm.

Note also that participation of the kernel is actually the reason for increased Lebesgue index $p*$

My research interests are applied analysis, PDE, Microlocal Analysis, Infinity Laplacian and Pseudo-differential operators.