An abstract characterization of Orlicz-Kantorovich lattices constructed by a measure with values in the ring of measurable functions is presented.

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the...

It is well known that if φ(t) ≡ t, then the system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is a basis in some Lebesgue space $L_p$. The aim of this short note is to show that the answer to this question is negative.

Étant donnés un compact $K$ du plan complexe, et une mesure non nulle sur $K$, on étudie $H^{\infty }\left(\mu \right)$, l’adhérence dans $L^{\infty }\left(\mu \right)$, pour la topologie $\sigma \left(L^{\infty }\right(\mu ),L^1(\mu \left)\right)$, de l’algèbre des fractions rationnelles d’une variable complexe, à pôles hors de $K$. Le résultat principal obtenu est qu’il existe un sous-ensemble $E_{\mu }$ de $K$, éventuellement vide, mesurable pour la mesure de Lebesgue plane, et une mesure ${\mu }_s$, éventuellement nulle, absolument continue par rapport à la mesure $\mu$, tels que : $H^{\infty }\left(\mu \right)$ soit isométriquement isomorphe à $H^{\infty }({\lambda }_{E_{\mu }})\oplus L^{\infty }({\mu }_s)$, où ${\lambda }_{E_{\mu }}$ désigne la...

We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^p(\mu ,X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^p(\mu ,X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological space and...

The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,\mu )$ by multiples of projections or unitary operators are examined.

We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.

We study the Orlicz type spaces Hω, defined as a generalization of the Hardy spaces Hp for p ≤ 1. We obtain an atomic decomposition of Hω, which is used to provide another proof of the known fact that BMO(ρ) is the dual space of Hω (see S. Janson, 1980, [J]).

We study the boundary behaviour of the nonnegative solutions of the semilinear elliptic equation in a bounded regular domain Ω of RN (N ≥ 2),⎧   Δu + uq = 0,   in Ω⎨⎩   u = μ,      on ∂Ωwhere 1 &lt; q &lt; (N + 1)/(N - 1) and μ is a Radon measure on ∂Ω. We give a priori estimates and existence results. The lie on the study of superharmonic functions in some weighted Marcinkiewicz spaces.

The purpose of this paper is to prove an embedding theorem for Sobolev type functions whose gradients are in a Lorentz space, in the framework of abstract metric-measure spaces. We then apply this theorem to prove absolute continuity and differentiability of such functions.