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Register a new userComposition of Haar Paraproducts: The Random Case
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When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variants of this question, finding non-classical characterizations, for dyadic paraproducts.
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We establish $L^ptimes L^q$ to $L^r$ estimates for some paraproducts, which arise in the study of the bilinear Hilbert transform along curves.
We prove bounds in the local $ L^2 $ range for exotic paraproducts motivated by bilinear multipliers associated with convex sets. One result assumes an exponential boundary curve. Another one assumes a higher order lacunarity condition.
Some Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $tau$, $p$, $q$ and $n$ of this characterization. Therefore, the authors measure the regularity of the characteristic function $mathcal X$ of the unit cube in $mathbb{R}^n$ via the Besov-type spaces $B^{s,tau}_{p,q}(mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $langle f, mathcal{X} rangle$ generates a continuous linear functional on $B^{s,tau}_{p,q}(mathbb{R}^n)$.
Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. An decomposition of the commutator [H,b] into paraproducts is presented.
We characterize the Schauder and unconditional basis properties for the Haar system in the Triebel-Lizorkin spaces $F^s_{p,q}(Bbb R^d)$, at the endpoint cases $s=1$, $s=d/p-d$ and $p=infty$. Together with the earlier results in [10], [4], this completes the picture for such properties in the Triebel-Lizorkin scale, and complements a similar study for the Besov spaces given in [5].
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