Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Integral Transformations of Volterra Gaussian Processes

We study integral representations of Gaussian processes with a pre-specified law in terms of other Gaussian processes. The dissertation consists of an introduction and of four research articles. In the introduction, we provide an overview about Volterra Gaussian processes in general, and fractional Brownian motion in particular. In the first article, we derive a finite interval integral transformation, which changes fractional Brownian motion with a given Hurst index into fractional Brownian motion with an other Hurst index. Based on this transformation, we construct a prelimit which formally converges to an analogous, infinite interval integral transformation. In the second article, we prove this convergence rigorously and show that the infinite interval transformation is a direct consequence of the finite interval transformation. In the third article, we consider general Volterra Gaussian processes. We derive measure-preserving transformations of these processes and their inherently related bridges. Also, as a related result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale. In the fourth article, we derive a class of ergodic transformations of self-similar Volterra Gaussian processes.

Breath Testing by Fourier Transform Infrared Spectroscopy for Solvent Intoxication Diagnostics

Technical or contaminated ethanol products are sometimes ingested either accidentally or on purpose. Typical misused products are black-market liquor and automotive products, e.g., windshield washer fluids. In addition to less toxic solvents, these liquids may contain the deadly methanol. Symptoms of even lethal solvent poisoning are often non-specific at the early stage. The present series of studies was carried out to develop a method for solvent intoxication breath diagnostics to speed up the diagnosis procedure conventionally based on blood tests. Especially in the case of methanol ingestion, the analysis method should be sufficiently sensitive and accurate to determine the presence of even small amounts of methanol from the mixture of ethanol and other less-toxic components. In addition to the studies on the FT-IR method, the Dräger 7110 evidential breath analyzer was examined to determine its ability to reveal a coexisting toxic solvent. An industrial Fourier transform infrared analyzer was modified for breath testing. The sample cell fittings were widened and the cell size reduced in order to get an alveolar sample directly from a single exhalation. The performance and the feasibility of the Gasmet FT-IR analyzer were tested in clinical settings and in the laboratory. Actual human breath screening studies were carried out with healthy volunteers, inebriated homeless men, emergency room patients and methanol-intoxicated patients. A number of the breath analysis results were compared to blood test results in order to approximate the blood-breath relationship. In the laboratory experiments, the analytical performance of the Gasmet FT-IR analyzer and Dräger 7110 evidential breath analyzer was evaluated by means of artificial samples resembling exhaled breath. The investigations demonstrated that a successful breath ethanol analysis by Dräger 7110 evidential breath analyzer could exclude any significant methanol intoxication. In contrast, the device did not detect very high levels of acetone, 1-propanol and 2-propanol in simulated breath. The Dräger 7110 evidential breath ethanol analyzer was not equipped to recognize the interfering component. According to the studies the Gasmet FT-IR analyzer was adequately sensitive, selective and accurate for solvent intoxication diagnostics. In addition to diagnostics, the fast breath solvent analysis proved feasible for controlling the ethanol and methanol concentration during haemodialysis treatment. Because of the simplicity of the sampling and analysis procedure, non-laboratory personnel, such as police officers or social workers, could also operate the analyzer for screening purposes.