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...

Let $\Phi \left(t\right)={ʃ}_0^{t}a\left(s\right)ds$ and $\Psi \left(t\right)={ʃ}_0^{t}b\left(s\right)ds$, where a(s) is a positive continuous function such that ${ʃ}_1^{\infty }a\left(s\right)/sds=\infty$ and b(s) is quasi-increasing and $li{m}_{s\to \infty }b\left(s\right)=\infty$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_0$ such that ${ʃ}_1^{s}a\left(t\right)/tdt\ge c_{1}b\left(c_{1}s\right)$ for all $s\ge s_0$; (jj) there exist positive constants $c_2$ and $c_3$ such that ${ʃ}_0^{2\pi }\Psi (\left(c_2\right)/{\left(\right|⨍|}_{}\left)\right|⨍\left(x\right)\left|\right)dx\le c_3+c_3{ʃ}_0^{2\pi }{\Phi (1/(\left|⨍\right|}_{}\left)\right)Mf\left(x\right)dx$ for all $⨍\in L^1\left(\right)$.

We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_{\alpha }$ maps weak weighted Orlicz$-\phi$ spaces into appropriate weighted versions of the spaces $BM{O}_{\psi }$, where $\psi \left(t\right)=t^{\alpha /n}{\phi }^{-1}(1/t)$. This generalizes known results about boundedness of $I_{\alpha }$ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha$ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_{\alpha }$ acting on weak$-L_{\phi }$ for $\phi$ of lower type equal or greater than $n/\alpha$, is the same as the one solving the problem...

Necessary and sufficient conditions are shown in order that the inequalities of the form $\varrho \left(M_{\mu }f>\lambda \right)\Phi \left(\lambda \right)\le C{ʃ}_X\Psi \left(C\right|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$, or $\varrho \left(M_{\mu }f>\lambda \right)\le C{ʃ}_X\Phi (C{\lambda }^{-1}|f\left(x\right)\left|\right)\sigma \left(x\right)d\mu$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_{\mu }$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.

We study the canonical injection from the Hardy-Orlicz space $H^{\Psi }$ into the Bergman-Orlicz space ${}^{\Psi }$.

In this paper we study the relationship between one-sided reverse Hölder classes $R{H}_r^{+}$ and the $A_p^{+}$ classes. We find the best possible range of $R{H}_r^{+}$ to which an $A_1^{+}$ weight belongs, in terms of the $A_1^{+}$ constant. Conversely, we also find the best range of $A_p^{+}$ to which a $R{H}_{\infty }^{+}$ weight belongs, in terms of the $R{H}_{\infty }^{+}$ constant. Similar problems for $A_p^{+}$, $1<p<\infty$ and $R{H}_r^{+}$, $1<r<\infty$ are solved using factorization.

Negative association for a family of random variables $\left(X_i\right)$ means that for any coordinatewise increasing functions f,g we have $(X_{i₁},...,X_{i_k})g(X_{j₁},...,X_{j_l})\le f(X_{i₁},...,X_{i_k})g(X_{j₁},...,X_{j_l})$ for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple...

We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with $L_{\infty }.$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair...

For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space ${\Lambda }_{\phi ,w}$ or the sequence space ${\lambda }_{\phi ,w}$, equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $inf\int \phi ⁎(f*/|g\left|\right)\left|g\right|:g\prec w$, where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular ${\int }_I\phi ⁎((f*)⁰/w)w$,where (f*)⁰ is Halperin’s level...

We consider an embedding of the group of invertible transformations of [0,1] into the algebra of bounded linear operators on an Orlicz space. We show that if this embedding preserves the group action then the Orlicz space is an $L^p$-space for some 1 ≤ p < ∞.

CONTENTSPreface..............................................................................................................................4Introduction........................................................................................................................51. Orlicz spaces..................................................................................................................6 1.1. Orlicz functions...........................................................................................................6 1.2....

CONTENTSIntroduction............................................................................... 51. Definitions and preliminary results......................................... 72. Completeness of $L^{\Phi }(\mu ,\Re )$.............................. 93. Linear functionals on $L^{\Phi }(\mu ,\Re )$....................... 264. Geometry of Fenchel-Orlicz spaces........................................ 41References....................................................................................... 54

Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ${\int }_{ℝ₊}M\left(\omega \left(x\right)\right|u\left(x\right)\left|\right)exp(-\phi \left(x\right))dx\le C{\int }_{ℝ₊}M\left(\right|u^{\text{'}}\left(x\right)\left|\right)exp(-\phi \left(x\right))dx$, where u belongs to some set of locally absolutely continuous functions containing $C{₀}^{\infty }\left(ℝ₊\right)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants. ...

We will present relationships between the modular ρ* and the norm in the dual spaces $\left(L_{\Phi }\right)*$ in the case when a Musielak-Orlicz space $L_{\Phi }$ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space $\left(L{⁰}_{\Phi }\right)*$ will be presented.