We discuss the disjointness of two classes of stable stochastic processes: moving averages and Fourier transforms. Results on the incompatibility of these two representations date back to Urbanik. Here we extend various disjointness results to encompass larger classes of processes.

2000 Mathematics Subject Classification: 35Q55,42B10.In this paper, we study the Schrödinger equation associated with the Dunkl operators, we study the dispersive phenomena and we prove the Strichartz estimates for this equation. Some applications are discussed.

We study the spaces of Lorentz-Zygmund multipliers on compact abelian groups and show that many of these spaces are distinct. This generalizes earlier work on the non-equality of spaces of Lorentz multipliers.

There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous...

We prove good-$\lambda$ inequalities for the area integral, the nontangential maximal function, and the maximal density of the area integral. This answers a question raised by R. F. Gundy. We also prove a Kesten type law of the iterated logarithm for harmonic functions. Our Theorems 1 and 2 are for Lipschitz domains. However, all our results are new even in the case of ${\mathbf{R}}_{+}^2$.

For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, $({(-\Delta )}^{\alpha }u,\varphi )=(u,{(-\Delta )}^{\alpha }\varphi )$, α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean...

We consider sequences of linear operators Uₙ with a localization property. It is proved that for any set E of measure zero there exists a set G for which $U{ₙ}_G\left(x\right)$ diverges at each point x ∈ E. This result is a generalization of analogous theorems known for the Fourier sum operators with respect to different orthogonal systems.

We costruct functions in $H_w^1$ ($w\in A_1$) whose Fourier integral expansions are almost everywhere non-summable with respect to the Bochner-Riesz means of the critical order.

In the present paper we introduce some double sequence spaces defined by a sequence of modulus function $F=\left(f_{k,l}\right)$ over $n$-normed spaces. We also make an effort to study some topological properties and inclusion relations between these spaces.

Denote by $f_{ss}(x,y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that $f_{ss}$ belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Let $-(n+1)<m\le -(n+1)(1-\rho )$ and let $T_a\in {ℒ}_{\rho ,\delta }^m$ be pseudo-differential operators with symbols $a(x,\xi )\in {ℝ}^n\times {ℝ}^n$, where $0<\rho \le 1$, $0\le \delta <1$ and $\delta \le \rho$. Let $\mu$, $\lambda$ be weights in Muckenhoupt classes $A_p$, $\nu ={\left(\mu {\lambda }^{-1}\right)}^{1/p}$ for some $1<p<\infty$. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_a$ with weighted BMO functions $b\in {\mathrm{BMO}}_{\nu }$, namely, the commutator $[b,T_a]$ is bounded from $L^p\left(\mu \right)$ into $L^p\left(\lambda \right)$. Furthermore, the range of $m$ can be extended to the whole $m\le -(n+1)(1-\rho )$.

Let u be a solution to a second order elliptic equation with singular magnetic fields, vanishing continuously on an open subset Γ of the boundary of a Lipschitz domain. An elementary proof of the doubling property for u² over balls centered at some points near Γ is presented. Moreover, we get the unique continuation at the boundary of Dini domains for elliptic operators.

For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space ${\Lambda }_{\phi ,w}$ or the sequence space ${\lambda }_{\phi ,w}$, equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $inf\int \phi ⁎(f*/|g\left|\right)\left|g\right|:g\prec w$, where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular ${\int }_I\phi ⁎((f*)⁰/w)w$,where (f*)⁰ is Halperin’s level function...