This article deals with weighted Fréchet spaces of holomorphic functions which are defined as countable intersections of weighted Banach spaces of type $H^{\infty }$. We characterize when these Fréchet spaces are Schwartz, Montel or reflexive. The quasinormability is also analyzed. In the latter case more restrictive assumptions are needed to obtain a full characterization.

We prove some weighted endpoint estimates for some multilinear operators related to certain singular integral operators on Herz and Herz type Hardy spaces.

Let $M_g^{+}$ be the maximal operator defined by $M_g^{+}⨍\left(x\right)=su{p}_{h>0}({ʃ}_x^{x+h}\left|⨍\right|g)/\left({ʃ}_x^{x+h}g\right)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy ${\Delta }_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u\left(x\in ℝ|M_g^{+}⨍\left(x\right)>\lambda \right)\le C/\left(\Phi \left(\lambda \right)\right){ʃ}_{ℝ}\Phi \left(\right|⨍\left|\right)\omega$ holds for every ⨍ in the Orlicz space $L_{\Phi }\left(\omega \right)$. We also characterize the positive functions ω such that the integral inequality ${ʃ}_{ℝ}\Phi \left(\right|M_g^{+}⨍\left|\right)\omega \le {ʃ}_{ℝ}\Phi \left(\right|⨍\left|\right)\omega$ holds for every $⨍\in L_{\Phi }\left(\omega \right)$. Our results include some already obtained for functions in $L^p$ and yield as consequences...

We consider Bergman projections and some new generalizations of them on weighted $L^{\infty }\left(\right)$-spaces. A new reproducing formula is obtained. We show the boundedness of these projections for a large family of weights v which tend to 0 at the boundary with a polynomial speed. These weights may even be nonradial. For logarithmically decreasing weights bounded projections do not exist. In this case we instead consider the projective description problem for holomorphic inductive limits.

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for ${\Phi }_2^{-1}(ʃ{\Phi }_2\left(w\left(x\right)\right|Tf\left(x\right)\left|\right)t\left(x\right)dx)\le {\Phi }_1^{-1}(ʃ{\Phi }_1\left(Cu\left(x\right)\right|f\left(x\right)\left|\right)v\left(x\right)dx)$ to hold when ${\Phi }_1$ and ${\Phi }_2$ are N-functions with ${\Phi }_2\circ {\Phi }_1^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

Consideramos límites inductivos ponderados de espacios de funciones holomorfas que están definidos como la unión numerable de espacios ponderados de Banach de tipo H∞. Estudiamos el problema de la descripción proyectiva y analizamos cuando estos espacios tienen la condición de densidad dual de Bierstedt y Bonet.

Let ω be a weight on an LCA group G. Let M(G,ω) consist of the Radon measures μ on G such that ωμ is a regular complex Borel measure on G. It is proved that: (i) M(G,ω) is regular iff M(G,ω) has unique uniform norm property (UUNP) iff L¹(G,ω) has UUNP and G is discrete; (ii) M(G,ω) has a minimum uniform norm iff L¹(G,ω) has UUNP; (iii) M₀₀(G,ω) is regular iff M₀₀(G,ω) has UUNP iff L¹(G,ω) has UUNP, where M₀₀(G,ω) := {μ ∈ M(G,ω) : μ̂ = 0 on Δ(M(G,ω))∖Δ(L¹(G,ω))}.