  We study a superconformal index for ${\cal N}=4$ super Yang-Mills on $S^1
\times S^3$ with a half BPS duality domain wall inserted at the great
two-sphere in $S^3$. The index is obtained by coupling the 3d generalized
superconformal index on the duality domain wall with 4d half-indices. We
further consider insertions of line operators to the configuration and propose
integral equations which express that the 3d index on duality domain wall is a
duality kernel relating half indices of two line operators related by the
duality map. We explicitly check the proposed integral equations for various
duality domain walls and line operators in the ${\cal N}=4$ SU(2) theory. We
also briefly comment on a generalization to $\mathcal{N}=2$ $A_1$ Gaiotto
theories with a simple example, ${\cal N}=2$ SU(2) SYM with four flavors.
