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For a an arbitrary periodic Borel measure $\mu$, we prove order$O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalarelliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with$\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $\mu^\varepsilon$ isthe measure obtained by $\varepsilon$-scaling of $\mu.$ Our analysis includesboth the case of a measure absolutely continuous with respect to the standardLebesgue measure and the case of "singular" periodic structures (or"multistructures"), when $\mu$ is supported by lower-dimensional manifolds.

Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures