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.

Abstract Let $$\Omega $$ Ω be a bounded, smooth domain of $$\mathbb {R}^N$$ R N , $$N\ge 2$$ N ≥ 2 . In this paper, we prove some inequalities involving the first Robin eigenvalue of the p -laplacian operator. In particular, we prove an upper bound for the first Robin eigenvalue of nonlinear elliptic operators in terms of the first Dirichlet eigenvalue.

Abstract We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg–Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients.

Abstract We give upper bounds for the Poincaré and logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky-reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. To achieve this we use an interpolation approach based on energy interactions between the boundary and the interior of the manifold as well as the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of doubly weighted Brownian motion with pure sticky reflection.

Abstract We characterize simply connected John domains in the plane with the aid of weak tangents of the boundary. Specifically, we prove that a bounded simply connected domain D is a John domain if and only if, for every weak tangent Y of $$\partial D$$ ∂ D , every connected component of the complement of Y that “originates” from D is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (Trans. Amer. Math. Soc. 369 (9), 6511–6536, 2017), who obtains a necessary condition for a John domain in terms of weak tangents but not a sufficient one. We also establish several properties of weak tangents of John domains.