Abstract We consider the simple random walk on the infinite cluster of a general class of percolation models on $$\mathbb {Z}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain “porous interfaces” surrounding the discrete blow-up of a compact set A . These controls substantially generalize previous results obtained in Nitzschner and Sznitman (J. Eur.Math. Soc. (JEMS) 22 (8), 2629–2672, 2020) for Brownian motion in $$\mathbb {R}^d$$ R d and in Chiarini and Nitzschner (Comm. Math. Phys. 386 (3), 1685–1745, 2021) for random walks on $$\mathbb {Z}^d$$ Z d equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

Abstract A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights is established. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. In particular, a Littlewood-Paley estimate recently proved by Peláez and the second author (Peláez and Rättyä Adv. Math. , 391 , 70, 2021) is improved in part. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map f , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under f of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.