It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p,q}$ to $L_{p,q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_1,L_1)$. As a consequence we show that the dyadic integral of a ∞ function $f\in L_1$ is dyadically differentiable and its derivative is f a.e.

An integral criterion for being an $H^1\left({ℝ}^2\right)$ Fourier multiplier is proved. It is applied in particular to suitable regular functions which depend on the product of variables.

A real-valued Hardy space $H{¹}_{\surd }\left(\right)\subseteq L¹\left(\right)$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H{¹}_{\surd }\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{¹}_{\surd }\left(\right)$, and no Orlicz space...

It is proved that if $m:{ℝ}^d\to ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^1$ space on the product domain ${ℝ}^{d_1}\times ...\times {ℝ}^{d_k}$. This implies an estimate of the norm $N(m,L^p\left({ℝ}^d\right)$ of the multiplier transformation of m on $L^p\left({ℝ}^d\right)$ as p→1. Precisely we get $N(m,L^p\left({ℝ}^d\right))\lesssim {(p-1)}^{-k}$. This bound is the best possible in general.

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and $$div\left(u^{-}{v}^{\to }\right)\in H^1\left(R^d\right)$$ which include as a particular case, the result of [3].

Let ⁿ denote the usual n-torus and let $S{̃}_u^{\delta }\left(f\right)$, u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $li{m}_{R\to \infty }1/logR{\int }_0^R\left(\right||S{̃}_u^{\alpha }\left(f\right)-{f\left|\right|}_{H¹\left(ⁿ\right)})/(u+1)du=0$.

In this article we give a wavelet area integral characterization for weighted Hardy spaces Hp(ω), 0 &lt; p &lt; ∞, with ω ∈ A∞. Our wavelet characterization establishes the identification between Hp(ω) and T2p (ω), the weighted discrete tent space, for 0 &lt; p &lt; ∞ and ω ∈ A∞. This allows us to use all the results of tent spaces for weighted Hardy spaces. In particular, we obtain the isomorphism between Hp(ω) and the dual space of Hp'(ω), where 1&lt; p &lt; ∞ and 1/p +...