For 0 < q ≤ 1, the author introduces a new Hardy space $L²H_{ℝ}^q\left(ℝ{²}_{+}\times ℝ{²}_{+}\right)$ on the product domain, and gives its generalized Lusin-area characterization. From this characterization, a φ-transform characterization in M. Frazier and B. Jawerth’s sense is deduced.

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...

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f\left(s\right)={\sum }_{n=1}^{\infty }aₙn^{-s}$ with ${\left|\right|f\left|\right|}_{\infty }:=su{p}_{\Re s>0}\left|f\left(s\right)\right|<\infty$, then ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-2}{\le \left|\right|f\left|\right|}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-1/2}{\le C\left|\right|f\left|\right|}_{\infty }$, C being an absolute constant.

In this paper, it is proved that the Fourier integral operators of order $m$, with $-n<m\le -(n-1)/2$, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.

We deal with the Hardy-Lorentz spaces $H^{p,q}\left(ℝⁿ\right)$ where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^p\left(ℝ\right)$, 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^p\left(ℝ\right)$, 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^p\left(ℝ\right)$.

In this paper we study the Hilbert transform and maximal function related to a curve in R2.

We give a Littlewood-Paley function characterization of a new Hardy space HK₂ and its φ-transform characterizations in M. Frazier & B. Jawerth's sense.

Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ which are the local versions of $H^p\left({ℝ}^n\right)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ and discuss the $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous...