Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Product Space Singular Integrals with Mild Kernel Regularity

We develop product space theory of singular integrals with mild kernel regularity. We study these kernel regularity questions specifically in situations that are very tied to the T1 type arguments and the corresponding structural theory. In addition, our results are multilinear.

OPERATORS ON BERGMAN SPACES

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.

The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.