Multivariate Lebesgue bounds for R-Leja lower grids

We establish an algebraic upper bound for the Lebesgue constant of multivariate polynomial interpolation on $\mathfrak R$-Leja lower grids. For each fixed dimension $d\geq2$ and every nonempty finite lower set $A\subset\mathbb N_0^d$, let $P_A$ denote the corresponding grid with $N=|A|$ nodes. We prove $Λ(P_A)\leq C_dN^{β_d}$, where $q_d=d/(d-1)$ and $β_d=1+q_d^{-1}\log_2(1+2^{q_d})$. The exponent satisfies $β_2=2.160964\ldots$ and increases to $\log_2 6=2.584962\ldots$ as $d\to\infty$, improving both the cubic estimate and the exponent $3-1/d$ obtained from quadratic prefix bounds. Our proof exploits the binary structure of known univariate estimates to control sums of powers of Lebesgue constants over interpolation prefixes. A weighted Hölder inequality and a disjoint-box packing argument then transfer these prefix-moment bounds to arbitrary lower sets. The resulting estimate is uniform over all lower-set shapes and choices of disk Leja maximizers, with constants depending only on $d$.