The martingale Hardy space $H_p\left({[0,1)}^2\right)$ and the classical Hardy space $H_p{(}^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...

A strong summability result is proved for the Ciesielski-Fourier series of integrable functions. It is also shown that the strong maximal operator is of weak type (1,1).

We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw2(a, b) spaces.

We study general continuity properties for an increasing family of Banach spaces $S_w^p$ of classes for pseudo-differential symbols, where $S_w^{\infty }=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\mathrm{Op}\left(S_w^p\right)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ and $S_w^p·S_w^q\subset S_w^r$, provided $1/p+1/q=1/r$. If instead $1/p+1/q=1+1/r$, then $S_w^{p}w*S_w^q\subset S_w^r$. By modifying the definition of the $S_w^p$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.

Soit E un espace de Fréchet séparable ne contenant pas $c_0$; soit de plus $\left(X_n\right)$ une suite symétrique de vecteurs aléatoires à valeurs dans E. Alors si la série de Fourier aléatoire $\sum X_{n}exp(i⟨{\lambda }_n,t⟩)$, $t\in R^d$, a p.s. ses sommes partielles localement uniformément bornées dans E, nécessairement elle converge p.s. uniformément sur tout compact de $R^d$ vers une fonction aléatoire à valeurs dans E et à trajectoires continues.

The symbol calculus on the upper half plane is studied from the viewpoint of the Kirillov theory of orbits. The main result is the $L^p$-estimates for Fuchs type pseudodifferential operators.

We study the $L^p$-boundedness of linear and bilinear multipliers for the symmetric Bessel transform.

A number of recent results in Euclidean harmonic analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen...