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Relative subanalytic sheaves

Given a projection $f$ of a product of real analytic manifolds onto one factor, let us say, $S$, and a subanalytic sheaf $\mathcal{F}$ on the associated subanalytic site, we give a natural construction of the (subanalytic) relative sheaf $\mathcal{F}^S$. Applying our construction to the subanalytic sheaves of tempered distributions, holomorphic functions and Whitney $\mathcal{C}^{\infty}$-functions we obtain their relative versions and study their properties.