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Eric Sawyer

Publications and source records attributed to Eric Sawyer.

25 records · Page 2Linked to original sources
arXiv2011

Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem

In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.

math.AP↗
arXiv2010

The Dirichlet space: A Survey

In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on certain function theoretic properties of the Dirichlet space and omit covering the interesting connections between this space and operator theory. The results discussed in this survey show what is known about the Dirichlet space and compares it with the related results for the Hardy space.

math.CV↗
arXiv2010

Fibre Positioning Revisited: The use an off-the-shelf assembly robot for OPTIMOS-EVE

The OPTIMOS-EVE instrument proposed for the E-ELT aims to use the maximum field of view available to the E-ELT in the limit of natural or ground-layer-corrected seeing for high multiplex fibre-fed multi-object spectroscopy in the visible and near-IR. At the bare nasmyth focus of the telescope, this field corresponds to a focal plane 2.3m in diameter, with a plate-scale of ~3mm/arcsec. The required positioning accuracy that is implied by seeing limited performance at this plate-scale brings the system into the range of performances of commercial off-the-shelf robots that are commonly used in industrial manufacturing processes. The cost-benefits that may be realized through such an approach must be offset against the robot performance, and the ease with which a useful software system can be implemented. We therefore investigate whether the use of such a system is indeed feasible for OPTIMOS-EVE, and the possibilities of extending this approach to other instruments that are currently in the planning stage.

astro-ph.IM↗
arXiv2008

Function Spaces Related to the Dirichlet Space

We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies.

math.CV↗
arXiv2008

Bilinear Forms on the Dirichlet Space

Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= < fg,b >_{\mathcal{D}} $, where we are looking at the inner product in the space $\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\mathcal{D}\times\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\mathcal{X}$ if the measure $dμ_{b}:=| b^{\prime}(z)| ^{2}dA$ is a Carleson measure for $\mathcal{D}$ and norm $\mathcal{X}$ by $$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\in\mathcal{X}$ and $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$

math.CV↗