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We study three-point correlation functions of scalar operators in conformalfield theories with boundaries and interfaces. We focus on two cases wherethere are one bulk and two boundary operators (B$\partial\partial$), or twobulk and one boundary operators (BB$\partial$). We perform a detailed analysisof the conformal blocks in different OPE channels. In particular, we obtain thebulk channel conformal blocks of the BB$\partial$ three-point functions forarbitrary exchanged spins in a series expansion with respect to the radialcoordinates. We also study examples of such three-point functions in thesimplest holographic dual where the $AdS_{d+1}$ space contains a brane fillingan $AdS_d$ subspace. Such a setup arises in top-down models with probe branesand is also relevant for the functional approach to boundary and interface CFTcorrelators. We systematically study the Witten diagrams in this setup both inposition space and in Mellin space. We also discuss in detail how to decomposethese Witten diagrams into conformal blocks.

Aspects of higher-point functions in BCFT$_d$