A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure $R$ and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition:$$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}\<+\infty \mathrm{forsome}0\<{\alpha }^{*}\<+\infty .$$In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result, we extend...

A Large Deviation Principle (LDP) is proved for the family $\frac{1}{n}{\sum }_1^n𝐟\left(x_i^n\right)·Z_i^n$ where the deterministic probability measure $\frac{1}{n}{\sum }_1^n{\delta }_{x_i^n}$ converges weakly to a probability measure R and ${\left(Z_i^n\right)}_{i\in \mathbb{N}}$ are ${ℝ}^d$-valued independent random variables whose distribution depends on $x_i^n$ and satisfies the following exponential moments condition: $$\underset{i,n}{sup}𝔼{\mathrm{e}}^{{\alpha }^{*}\left|Z_i^n\right|}<+\infty \mathrm{forsome}0<{\alpha }^{*}<+\infty .$$ In this context, the identification of the rate function is non-trivial due to the absence of equidistribution. We rely on fine convex analysis to address this issue. Among the applications of this result,...

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction ${f|}_U$ is not in $L^q\left(U\right)$. When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p’s but nowhere q-integrable for some other q’s (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González,...

It is shown that the same probabilistic metric as used by Schweizer and Sklar to obtain all Lp space metrics can be used to derive the metrics of Orlicz spaces.

We collect and extend results on the limit of ${\sigma }^{1-k}{(1-\sigma )}^k{\left|v\right|}_{l+\sigma ,p,\Omega }^p$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and ${\left|·\right|}_{l+\sigma ,p,\Omega }$ is the intrinsic seminorm of order l+σ in the Sobolev space $W^{l+\sigma ,p}\left(\Omega \right)$. In general, the above limit is equal to $c{\left[v\right]}^p$, where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

We study limiting K- and J-methods for arbitrary Banach couples. They are related by duality and they extend the methods already known in the ordered case. We investigate the behaviour of compact operators and we also discuss the representation of the methods by means of the corresponding dual functional. Finally, some examples of limiting function spaces are given.

We study linear operators from a non-locally convex Orlicz space $L^{\Phi }$ to a Banach space $(X,|\left|·\right|{|}_X)$. Recall that a linear operator $T:L^{\Phi }\to X$ is said to be σ-smooth whenever $uₙ{⟶}^{\left(o\right)}0$ in $L^{\Phi }$ implies ${\left|\right|T\left(uₙ\right)\left|\right|}_X\to 0$. It is shown that every σ-smooth operator $T:L^{\Phi }\to X$ factors through the inclusion map $j:L^{\Phi }\to L^{\Phi ̅}$, where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T:L^{\Phi }\to X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex...

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_p$-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...