2 (a c)^2 \|f\|_\infty^4
\le \bigip{ (I-Q^2) f, f }
$. Apply this inequality to the functions $Q^l f$ for $l = 0, \ldots, k$ and sum the resulting inequalities to obtain (k+1) 2 (a c)\^2 Q\^k f\_\^4 &2 (a c)\^2 \_[l=0]{}\^k Q\^l f\_\^4 \_[l=0]{}\^k &= \_[l=0]{}\^k = 1 for $\ip{f, f} \le 1$. This shows that the norm of $Q^k : \ell^2(V, \pi) \to \ell^\infty(V)$ is bounded by $$\beta_k := [(2k+2) (a c)^2]^{-1/4}
\,.$$ The same bound holds for $Q^k : \ell^1(V, \pi) \to \ell^2(V, \pi)$ by duality. Therefore, considering $Q^{2k} = Q^k \circ Q^k$, we find that the norm of $Q^{2k} : \ell^1(V, \pi) \to \ell^\infty(V)$ is at most $\beta_k^2$, while the norm of $Q^{2k+1} : \ell^1(V, \pi) \to \ell^\infty$ is at most $\beta_k
\beta_{k+1}$. Applying these inequalities to $f := \II x/\pi(x)$ gives \[e\].infinb/.

The case of finite $\pi$ is quite similar. The essential difference is that we work with $\ell^2_0(V, \pi)$, the orthogonal complement of the constants in $\ell^2(V, \pi)$. Note that $\bfo$ is an eigenfunction of $Q$ and that $\ell^2_0(V, \pi)$ is invariant under $Q$. We may still conclude \[e\].1/ for all $f$ that have at least one nonnegative value and at least one nonpositive 