Let M be the Hardy-Littlewood maximal operator defined by:Mf(x) = supx ∈ Q 1/|Q| ∫Q |f| dx, (f ∈ Lloc(Rn)),where the supreme is taken over all cubes Q containing x and |Q| is the Lebesgue measure of Q. In this paper we characterize the Orlicz spaces Lφ*, associated to N-functions φ, such that M is bounded in Lφ*. We prove that this boundedness is equivalent to the complementary N-function ψ of φ satisfying the Δ2-condition in [0,∞), that is, sups>0 ψ(2s) / ψ(s) < ∞.

We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(ℝⁿ), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M, we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.

We prove some time mollification properties and imbedding results in inhomogeneous Orlicz-Sobolev spaces which allow us to solve a second order parabolic equation in Orlicz spaces.

It is proved that the Musielak-Orlicz function space LF(mu,X) of Bochner type is P-convex if and only if both spaces LF(mu,R) and X are P-convex. In particular, the Lebesgue-Bochner space Lp(mu,X) is P-convex iff X is P-convex.

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{-1}{\sum }_{i=0}^{n-1}f\circ {\tau }^i\left(x\right)$ converges almost everywhere to a function f* in $L(p_1,q_1]$, where (pq) and $(p_1,q_1]$ are assumed to be in the set $(r,s):r=s=1,or1<r<\infty and1\le s\le \infty ,orr=s=\infty$. Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized and unified...

We introduce generalized Campanato spaces ${}_{p,\varphi }$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then ${}_{p,\varphi }=BMO$. We give a characterization of the set of all pointwise multipliers on ${}_{p,\varphi }$.

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X\times {ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in L_{loc}^p\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi (a,r)(1/\mu \left(B(a,r)\right){ʃ}_{B(a,r)}|f\left(x\right)-f_{B(a,r)}{{|}^{p}d\mu )}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of...

Let w be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents 0 < α < β. In the present paper, sharp estimates of the compositions of the power means ${}_{\alpha }w\left(x\right):={\left((1/x){\int }_0^x{w}^{\alpha }\left(t\right)dt\right)}^{1/\alpha }$, x > 0, are obtained for various exponents α. As a result, for the function w a property of self-improvement of summability exponents is established.