  In this paper we introduce and study the class of weighted multi-Toeplitz
operators associated with noncommutative polydomains ${\bf D_f^m}$, ${\bf
m}:=(m_1,\ldots, m_k)\in {\bf N}^k$, generated by $k$-tuples ${\bf
f}:=(f_1,\ldots, f_k)$ of positive regular free holomorphic functions in a
neighborhood of the origin. These operators are acting on the tensor product
$F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$ of full Fock spaces with
$n_i$ generators or, equivalently, they can be viewed as multi-Toeplitz
operators acting on tensor products of weighted full Fock spaces. For a large
class of polydomains, we show that there are no non-zero compact multi-Toeplitz
operators. We characterize the weighted multi-Toeplitz operators in terms of
bounded free $k$-pluriharmonic functions on the radial part of ${\bf D_f^m}$
and use the result to obtain an analogue of the Dirichlet extension problem for
free $k$-pluriharmonic functions. We show that the weighted multi-Toeplitz
operators have noncommutative Fourier representations which can be viewed as
noncommutative symbols and can be used to recover the associated operators. We
also prove that the weighted multi-Toeplitz operators satisfy a Brown-Halmos
type equation associated with the polydomain ${\bf D_f^m}$.

Functional Analysis;Operator Algebras