Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For $f={\left(fₙ\right)}_{n\in \mathbb{Z}₊}\in ℳ$, let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and ${\left|\right|Mg\left|\right|}_X{\le \left|\right|Mf\left|\right|}_X$, then ${\left|\right|Sg\left|\right|}_X{\le C\left|\right|Sf\left|\right|}_X$, where C is a constant independent of f and g.

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

Let $M_S$ denote the strong maximal operator. Let $M_x$ and $M_y$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that ${\int }_Q{M}_y{M}_{x}h<\infty$ but ${\int }_Q{M}_x{M}_{y}h=\infty$. It is shown that if f is a function supported on Q such that ${\int }_Q{M}_y{M}_{x}f<\infty$ but ${\int }_Q{M}_x{M}_{y}f=\infty$, then there exists a set A of finite measure in ℝ² such that ${\int }_A{M}_{S}f=\infty$.

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_p\left({ℝ}^d\right)$ (in the case $p>1$), but (in the case when $1/p(·)$ is log-Hölder continuous and $p_{-}=inf\{p\left(x\right):x\in {ℝ}^d\}>1$) on the variable Lebesgue spaces $L_{p(·)}\left({ℝ}^d\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma$-rectangles. We introduce a general maximal operator $M_s^{\gamma ,\delta }$ and with the help of generalized $\Phi$-functions, the strong-...

Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate $\parallel {\sum }_{k=0}^{\infty }d{f}_k{\parallel }_p\le C_p\parallel ({\sum }_{k=0}^{\infty }\left(\right|d{f}_k\left|²\right|{ℱ}_{k-1}{\left)\right)}^{1/2}{\parallel }_p+\parallel ({\sum }_{k=0}^{\infty }|d{f}_k{{|}^p)}^{1/p}{\parallel }_p,$ with $C_p=O(p/lnp)$ as p → ∞.

We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p,q}$, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p,q,r}$ such that for any $\varphi \in L^{p,q}$, ${\left|\right|ℳ\varphi \left|\right|}_{r,\infty }\le C_{p,q,r}{\left|\right|\varphi \left|\right|}_{p,q}$.

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${\left|\right|X\left|\right|}_{BMO}\equiv su{p}_T\left|\right|E\left[\right|X_{\infty }-X_T\left|\right|{ℱ}_T{\left]\right||}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b\left(X\right)=b>0:su{p}_T\left|\right|E\left[exp\left(b²(⟨X{⟩}_{\infty }-⟨X{⟩}_T)\right)\right|{ℱ}_T]{\left|\right|}_{\infty }<\infty$, where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q\left(X\right)ₜ=E\left[⟨X{⟩}_{\infty }\right|ℱₜ]-E\left[⟨X{⟩}_{\infty }\right|ℱ₀]$. We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty }$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1/4d₁(q\left(X\right),L^{\infty })\le b\left(X\right)\le 4/d₁(q\left(X\right),L^{\infty })$ hold for every continuous BMO-martingale X. ...

The main aim of this paper is to prove that the maximal operator ${\sigma }^{}$ is bounded from the Hardy space $H_{1/2}$ to weak-$L_{1/2}$ and is not bounded from $H_{1/2}$ to $L_{1/2}$.

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$. ...

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf\left(x\right)=2^{-n}{\int }_x^{x+2ⁿ}f$. The square function is defined as $Sf\left(x\right)=({\sum }_{n=-\infty }^{\infty }|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$. The local version of this operator, namely the operator $S₁f\left(x\right)=({\sum }_{n=-\infty }^0|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$, is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^p$ into itself (p > 1) and $L^{\infty }$ into BMO. We prove that the operator S not only maps $L^{\infty }$ into BMO but it also maps BMO into BMO. We also prove that the $L^p$ boundedness...

Let Y be a Banach space and let $S\subset L_p$ be a subspace of an $L_p$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S\left(Y\right)\subset L_p\left(Y\right)$. We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{-tB}$ is a positive contraction...

Let ${}_s$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_s$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_s}{\left|R\right|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{\delta }$ (resp. a $G_{\delta \sigma }$).

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^p$ maximal regularity for some $p\in (1,\mathrm{\infty })$, then it has $L^p$ maximal regularity for every $p\in (1,\mathrm{\infty })$.

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}$ whenever the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot )$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_a$, , maps $L^{p(\cdot )}$ into $L^{q(\cdot )}$, where $1/p(\cdot )-1/q(\cdot )=a/n$.

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

We prove a lower bound in a law of the iterated logarithm for sums of the form ${\sum }_{k=1}^N{a}_{k}f(n_{k}x+c_k)$ where f satisfies certain conditions and the $n_k$ satisfy the Hadamard gap condition $n_{k+1}/n_k\ge q>1$.

It is shown that there is no closed convex bounded non-dentable subset K of $C\left({\omega }^{{\omega }^k}\right)$ such that on subsets of K the PCP and the RNP are equivalent properties. Then applying the Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains a non-dentable subset L so that on L the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C\left({\omega }^{{\omega }^k}\right)$.

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators ${ℳ}^{+}$ and ${ℳ}^{-}$. More precisely, we prove that ${ℳ}^{+}$ and ${ℳ}^{-}$ map $W^{1,p}\left(ℝ\right)\to W^{1,p}\left(ℝ\right)$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $\mathrm{BV}\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ boundedly and map $l^1\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$, that is, $$\mathrm{Var}\left(M^{+}\left(f\right)\right)\le \mathrm{Var}\left(f\right)\text{and}\mathrm{Var}\left(M^{-}\left(f\right)\right)\le \mathrm{Var}\left(f\right)$$ if $f\in \mathrm{BV}\left(\mathbb{Z}\right)$, where $\mathrm{Var}\left(f\right)$ is the total variation of $f$ on $\mathbb{Z}$ and $\mathrm{BV}\left(\mathbb{Z}\right)$ is the set of all functions $f:\mathbb{Z}\to ℝ$ satisfying $\mathrm{Var}\left(f\right)<\infty$.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually...

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

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 $m\ge 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants $C_{p,q}$ such that $su{p}_n{\left(\right|fₙ|}^p{)/(1+Sₙ²\left(f\right))}^q\le C_{p,q}$. Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N,q}^{\text{'}}$ such that $su{p}_n\left(f{ₙ}^{2N-1}\right){(1+Sₙ²\left(f\right))}^q\le C_{N,q}^{\text{'}}$. These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if...

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator $T_s$, 0 < s < 2, to be the linear extension of the map $\left(h_I\right)/{\left(\right|I|}^{1/s})\mapsto \left(h_{\tau \left(I\right)}\right)\left(\right|\tau \left(I\right){|}^{1/s})$, where $h_I$ denotes the $L^{\infty }$-normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that $T_{s₀}$ is bounded on $H^{s₀}$, then for all 0 < s < 2 the operator $T_s$ is bounded on $H^s$.

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.