We prove that if $\Lambda \in M_p\left({ℝ}^N\right)$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_p\left({ℝ}^N\right)$ is the space of multipliers of $L^p\left({ℝ}^N\right)$.

From the fact that the two-dimensional moment problem is not always solvable, we can deduce that there must be extreme ray generators of the cone of positive definite double sequences which are nor moment sequences. Such an argument does not lead to specific examples. In this paper it is shown how specific examples can be constructed if one is given an example of an N-extremal indeterminate measure in the one-dimensional moment problem (such examples exist in the literature). Konrad Schmüdgen had...

The purpose of this paper is to obtain a discrete version for the Hardy spaces $H^p\left(\mathbb{Z}\right)$ of the weak factorization results obtained for the real Hardy spaces $H^p\left(ℝⁿ\right)$ by Coifman, Rochberg and Weiss for p > n/(n+1), and by Miyachi for p ≤ n/(n+1). It represents an extension, in the one-dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type, but with some restrictions on the measure that exclude the case of points...

We extend the well known factorization theorems on the unit disk to product Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on ℝ²₊ × ℝ²₊ and the characterization of BMO(ℝ²₊ × ℝ²₊) recently obtained by Ferguson, Lacey and Sadosky.

The two-dimensional classical Hardy spaces $H_p\left(ℝ\times ℝ\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p\left(ℝ\times ℝ\right)$ to $L_p\left({ℝ}^2\right)$ (1/2 < p ≤ ∞) and is of weak type $(H_1^{}\left(ℝ\times ℝ\right),L_1\left({ℝ}^2\right))$ where the Hardy space $H_1\left(ℝ\times ℝ\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^{(}ℝ\times ℝ)$ ⊃ $LlogL\left({ℝ}^2\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p\left(ℝ\times ℝ\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p\left(ℝ\times ℝ\right)$, the Fejér means...

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...