Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

Expansion in perfect groups

Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas are explained in more details in the introduction, typos corrected, results and proofs unchange

The fundamental left-right asymmetry in the Germanic verb cluster

Cinque (2005, 2009, 2014a) observes that there is an asymmetry in the possible ordering of dependents of a lexical head before versus after the head. A reflection on some of the concepts needed to develop Cinque’s ideas into a theory of neutral word order reveals that dependents need to be treated separately by class. The resulting system is applied to the problem of word order in the Germanic verb cluster. It is shown that there is an extremely close match between theoretically derived expectations for clusters made up of auxiliaries, modals, causative ‘let’, a main verb, and verbal particles. The facts point to the action of Cinque’s fundamental left-right asymmetry in language in the realm of the verb cluster. At the same time, not all verb clusters fall under Cinque’s generalization, which, therefore, argues against treating all cases of restructuring uniformly

Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Whole Slide Imaging Versus Microscopy for Primary Diagnosis in Surgical Pathology: A Multicenter Blinded Randomized Noninferiority Study of 1992 Cases (Pivotal Study)

Most prior studies of primary diagnosis in surgical pathology using whole slide imaging (WSI) versus microscopy have focused on specific organ systems or included relatively few cases. The objective of this study was to demonstrate that WSI is noninferior to microscopy for primary diagnosis in surgical pathology. A blinded randomized noninferiority study was conducted across the entire range of surgical pathology cases (biopsies and resections, including hematoxylin and eosin, immunohistochemistry, and special stains) from 4 institutions using the original sign-out diagnosis (baseline diagnosis) as the reference standard. Cases were scanned, converted to WSI and randomized. Sixteen pathologists interpreted cases by microscopy or WSI, followed by a wash-out period of ≥4 weeks, after which cases were read by the same observers using the other modality. Major discordances were identified by an adjudication panel, and the differences between major discordance rates for both microscopy (against the reference standard) and WSI (against the reference standard) were calculated. A total of 1992 cases were included, resulting in 15,925 reads. The major discordance rate with the reference standard diagnosis was 4.9% for WSI and 4.6% for microscopy. The difference between major discordance rates for microscopy and WSI was 0.4% (95% confidence interval, -0.30% to 1.01%). The difference in major discordance rates for WSI and microscopy was highest in endocrine pathology (1.8%), neoplastic kidney pathology (1.5%), urinary bladder pathology (1.3%), and gynecologic pathology (1.2%). Detailed analysis of these cases revealed no instances where interpretation by WSI was consistently inaccurate compared with microscopy for multiple observers. We conclude that WSI is noninferior to microscopy for primary diagnosis in surgical pathology, including biopsies and resections stained with hematoxylin and eosin, immunohistochemistry and special stains. This conclusion is valid across a wide variety of organ systems and specimen types