  We use bosonization and renormalization group methods to determine the ground
state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system
consisting of a one-dimensional spin-1/2 Luttinger liquid coupled by a Kondo
exchange interaction $J_K$ to a frustrated quantum antiferromagnetic Heisenberg
chain, with a nearest-neighbor exchange coupling $J_1$ and a
next-nearest-neighbor (frustrating) exchange interaction $J_2$. We analyze the
interplay of quantum frustration in the antiferromagnetic chain with the Kondo
exchange coupling $J_K$ with the Luttinger liquid. We discuss the structure of
the phase diagram of this system as a function of the ratios $J_K/J_1$,
$J_2/J_1$ and of the parameters of the Luttinger liquid. In particular we
discuss in detail the regimes in which a pair-density-wave state may be
realized and its relation with the spin correlations in the frustrated
antiferromagnetic chain.
