We prove$L$^{$q$}-inequalities for the gradient of the Green potential ($Gf$) in bounded, connected NTA-domains in$R$^{$n$},$n$≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent$q$for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.

The$H$^{$p$}corona problem is the following: Let$g$_{1}, ...,$g$_{$m$}be bounded holomorphic functions with 0<δ≤Σ‖$g$_{$i$}‖. Can we, for any$H$^{$p$}function ϕ, find$H$^{$p$}functions$u$_{1}, ...,$u$_{$m$}such that Σ$g$_{$i$}$u$_{$i$}=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. The$H$^{$p$}estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

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In this note, we characterize maximal invariant subspaces for a class of operators. Let$T$be a Fredholm operator and $1-TT^{*}\in\mathcal{S}_{p}$ for some$p$≥1. It is shown that if$M$is an invariant subspace for$T$such that dim$M$$⊖$$TM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. As an immediate consequence, we obtain that if$M$is a shift invariant subspace of the Bergman space and dim$M$$⊖$$zM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. We also apply the result to translation operators and their invariant subspaces.

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the$p$-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.