Consider the sequence ${\left(Cₙ\right)}_{n\ge 1}$ of positive numbers defined by C₁ = 1 and $C_{n+1}=1+Cₙ²/4$, n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

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. ...

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega ,ℱ,\mathbb{P})$ be a probability space and $p(·):\Omega \to (0,\infty )$ be a $ℱ$-measurable function such that $0<{inf}_{x\in \Omega }p\left(x\right)\le {sup}_{x\in \Omega }p\left(x\right)<\infty$. It is proved that a predictable martingale Hardy space ${𝒫}_{p(·)}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy...

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.

This work is concerned with the theory of initial and progressive enlargements of a reference filtration $𝔽$ F with a random time. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an $𝔽$ F -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as...

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 → ∞.

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...

This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^p$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^p$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale...

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...

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 ${}_{p,\varphi }$.

We prove a law of the iterated logarithm for sums of the form ${\sum }_{k=1}^N{a}_{k}f\left(n_{k}x\right)$ where the $n_k$ satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.

We determine the optimal constants $C_{p,q}$ in the moment inequalities ${\left|\right|g\left|\right|}_p\le C_{p,q}{\left|\right|f\left|\right|}_q$, 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

The main aim of this paper is to prove that there exists a martingale $f\in H_{1/2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak-$L_{1/2}$.

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 $C\left({\omega }^{{\omega }^k}\right)$.

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.

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 $n_{k+1}/n_k\ge q>1$.

We define a new type of multiplier operators on $L^p{(}^N)$, where ${}^N$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^p{(}^N)$, to which the theorem applies as a particular example.

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Let $K$ be a non-trivial knot in the $3$-sphere, $E_K$ its exterior, $G_K={\pi }_1\left(E_K\right)$ its group, and $P_K={\pi }_1\left(\partial E_K\right)\subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $g{P}_K{g}^{-1}\cap P_K=\left\{e\right\}$ for any $g\in G_K$ with $g\notin P_K$, unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental...

We consider an SPDE in a Hilbert space $H$ of the form $dX(t)=(AX(t)+b(X(t)))dt+\sigma (X(t))dW(t)$, $X(0)=x\in H$ and the corresponding transition semigroup $P_{t}f(x)=𝔼[f(X(t,x))]$. We define the infinitesimal generator $\overline{L}$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\phi =\frac{1}{2}\mathrm{Tr}[\sigma (x){\sigma }^{*}(x)D^2\phi ]+⟨b(x),D\phi ⟩$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\overline{L}$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular...

The purpose of this work is a study of the following insurance reserve model: $R\left(t\right)=\eta +{\int }_0^{t}p(s,R\left(s\right))ds+{\int }_0^t\sigma (s,R\left(s\right))d{W}_s-Z\left(t\right)$, t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $in{f}_{0\le t\le T}PR\left(t\right)\ge c\ge \gamma$ is considered.

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...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical...

By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^p[0,1]$, 1 < p < ∞.

To each set of knots $t_i=i/2n$ for i = 0,...,2ν and $t_i=(i-\nu )/n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space ${}_{\nu ,n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_i$ and the orthogonal projection $P_{\nu ,n}$ of L²(I) onto ${}_{\nu ,n}$. The main result is $li{m}_{(n-\nu )\wedge \nu \to \infty }\left|\right|P_{\nu ,n}\left|\right|₁=su{p}_{\nu ,n:1\le \nu \le n}\left|\right|P_{\nu ,n}\left|\right|₁=2+(2-\surd 3)²$. This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^2$-approximation of a probability density function $f$. If $f$ itself is a step-function the number of jumps may be unknown.

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf\left(x\right)=1/\left|B\right(0,\left|x\right|\left)\right|{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$, $Tf\left(x\right)=1/\left|B\right(x,\left|x\right|\left)\right|{\int }_{B(x,|x\left|\right)}f\left(t\right)dt$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

Let N ≥ 2 be a given integer. Suppose that $df={\left(dfₙ\right)}_{n\ge 0}$ is a martingale difference sequence with values in $\ell {₁}^N$ and let ${\left(\epsilon ₙ\right)}_{n\ge 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $\mathbb{P}(su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^n{\epsilon }_{k}d{f}_k{\left|\right|}_{\ell {₁}^N}\ge 1)\le (lnN+ln\left(3lnN\right))/(1-{\left(2lnN\right)}^{-1})su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^{n}d{f}_k{\left|\right|}_{\ell {₁}^N}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_N$ in the above estimate satisfies $li{m}_{N\to \infty }{C}_N/lnN=1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $\ell {₁}^N$.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths,...

Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ to a quasi-Banach space ℬ if and only if sup${\left|\right|T\left(a\right)\left|\right|}_{ℬ}$: a is an infinitely differentiable (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ < ∞, where the (p,q,s)-atom of ${Ḟ}_{p,q}^s\left(ℝⁿ\right)$ is as defined by Han, Paluszyński and Weiss.