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 $⨍\in L^1\left(\right)$.

In connection with the $A_{\varphi }$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R{H}_{\varphi }$. We prove that when ϕ is ${\Delta }_2$ and has lower index greater than one, the class $R{H}_{\varphi }$ coincides with some reverse-Hölder class $R{H}_q,q>1$. For more general ϕ we still get $R{H}_{\varphi }\subset A_{\infty }={\bigcup }_{q>1}R{H}_q$ although the intersection of all these $R{H}_{\varphi }$ gives a proper subset of ${\bigcap }_{q>1}R{H}_q$.

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 collect known and prove new necessary and sufficient conditions for the weighted weak type maximal inequality of the form $${\Phi }_1\left(\lambda \right)\varrho \left(\{x\in X:M_{\mu }f\left(x\right)>\lambda \}\right)\le c{\int }_X{\Phi }_2\left(c\right|f\left(x\right)\left|\right)\sigma \left(x\right)\mathrm{d}\mu \left(x\right),$$ which extends some known results.

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. ...

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.

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

Il lavoro presenta diverse caratterizzazioni degli spazi Lorentz-Zygmund generalizzati (GLZ) $L_{p,q;\alpha }\left(R\right)$, con $p,q\in \left(0,+\mathrm{\infty }\right]$, $m\in \mathbb{N}$, $\alpha \in {\mathbb{R}}^m$ e $\left(R,\mu \right)$ spazio misurato con misura $\mu \left(R\right)$ finita. Dato uno spazio misurato $\left(R,\mu \right)$ e $\alpha \in {\mathcal{R}}_{-}^m$ , otteniamo representazioni equivalenti per la (quasi-) norma dello spazio GLZ $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$. Inoltre, se $\left(R,\mu \right)$ è uno spazio misurato con misura finita e $\alpha \in {\mathcal{R}}_{+}^m$, viene presentata in termini di decomposizioni una norma equivalente per lo spazio $L_{1,1;\alpha }\left(R\right)$. Si dimostra che le norme equivalenti considerate per $L_{\mathrm{\infty },\mathrm{\infty };\alpha }\left(R\right)$, con $\left(R,\mu \right)$ uno spazio a misura...

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 characterize the pairs of weights on ℝ for which the operators $M_{h,k}^{+}f\left(x\right)=su{p}_{c>x}h(x,c){ʃ}_x^{c}f\left(s\right)k(x,s,c)ds$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x,c):x<c$, while k is defined on $(x,s,c):x<s<c$. If $h(x,c)={(c-x)}^{-\beta }$, $k(x,s,c)={(c-s)}^{\alpha -1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{\alpha ,\beta }^{+}f=su{p}_{c>x}1/{(c-x)}^{\beta }{ʃ}_x^{c}f\left(s\right)/{(c-s)}^{1-\alpha }ds$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal...

Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies ${\Delta }_2$. Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.

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 < ∞.

We study the holomorphic Hardy-Orlicz spaces ${}^{\Phi }\left(\Omega \right)$, where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in ℂⁿ. The function Φ is in particular such that $¹\left(\Omega \right){\subset }^{\Phi }\left(\Omega \right){\subset }^p\left(\Omega \right)$ for some p > 0. We develop maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from ${}^{\Phi }\left(\Omega \right)$ into ¹(Ω).

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...

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

Suppose that a real nonatomic function space on $[0,1]$ is equipped with two rearrangement-invariant norms $\parallel ·\parallel$ and $\left|\right||·|\left|\right|$. We study the question whether or not the fact that $(X,\parallel ·\parallel )$ is isometric to $(X,|\left|\right|·\left|\right|\left|\right)$ implies that $\parallel f\parallel =\left|\right|\left|f\right|\left|\right|$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify...