Let $P(\mathbf{x})$ be a bounded multivariate polynomial of degree at most $d$ (for my purposes it can be either coordinate degree or total degree) over $[-1,1]^n$, and assume $|P(\mathbf{x})| \leq 1$. In A bounded polynomial having bounded coefficients: several variables it is discussed that the coefficients of such a polynomial can be very large (more or less exponential in d and n).

My question is if it is possible to have a bound that grows at most linearly in d and n by adding some basic assumptions on $P$, e.g. a bounded lipschitz constant, or bounded 1st and 2nd order partial derivatives?