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 prove unconditionality of general Franklin systems in $L^p\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

Assume that $X$, $Y$ are continuous-path martingales taking values in ${ℝ}^{\nu }$, $\nu \ge 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$\parallel \underset{t\ge 0}{sup}|Y_t{|\parallel }_1\le 2\parallel \underset{t\ge 0}{sup}|X_t{|\parallel }_1.$$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.

The proof that H¹(δ) and H¹(δ²) are not isomorphic is simplified. This is done by giving a new and simple proof to a martingale inequality of J. Bourgain.

Let $\left(X_j\right)$ be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of $X_n...X_2{X}_1$ to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

We study some operators originating from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G* and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on $L^p$. Moreover, we generalize a classical multiplier theorem by weakening its...

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 space with...