Excimer laser refractive surgery : corneal wound healing and clinical results

Although the first procedure in a seeing human eye using excimer laser was reported in 1988 (McDonald et al. 1989, O'Connor et al. 2006) just three studies (Kymionis et al. 2007, O'Connor et al. 2006, Rajan et al. 2004) with a follow-up over ten years had been published when this thesis was started. The present thesis aims to investigate 1) the long-term outcomes of excimer laser refractive surgery performed for myopia and/or astigmatism by photorefractive keratectomy (PRK) and laser-in situ- keratomileusis (LASIK), 2) the possible differences in postoperative outcomes and complications when moderate-to-high astigmatism is treated with PRK or LASIK, 3) the presence of irregular astigmatism that depend exclusively on the corneal epithelium, and 4) the role of corneal nerve recovery in corneal wound healing in PRK enhancement. Our results revealed that in long-term the number of eyes that achieved uncorrected visual acuity (UCVA)≤0.0 and ≤0.5 (logMAR) was higher after PRK than after LASIK. Postoperative stability was slightly better after PRK than after LASIK. In LASIK treated eyes the incidence of myopic regression was more pronounced when the intended correction was over >6.0 D and in patients aged <30 years.Yet the intended corrections in our study were higher for LASIK than for PRK eyes. No differences were found in percentages of eyes with best corrected visual acuity (BCVA) or loss of two or more lines of visual acuity between PRK and LASIK in the long-term. The postoperative long-term outcomes of PRK with two different delivery systems broad beam and scanning laser were compared and revealed no differences. Postoperative outcomes of moderate-to-high astigmatism yielded better results in terms of UCVA and less compromise or loss of two more lines of BCVA after LASIK that after PRK.Similar stability for both procedures was revealed. Vector analysis showed that LASIK outcomes tended to be more accurate than PRK outcomes, yet no statistically differences were found. Irregular astigmatism secondary to recurrent corneal erosion due to map-dot-fingerprint was successfully treated with phototherapeutic keratectomy (PTK). Preoperative videokeratographies (VK) showed irregular astigmatism. However, postoperatively, all eyes showed a regular pattern. No correlation was found between pre- and postoperative VK patterns. Postoperative outcomes of late PRK in eyes originally subjected to LASIK showed that all (7/7) eyes achieved UCVA ≤0.5 at last follow-up (range 3 — 11 months), and no eye lost lines of BCVA. Postoperatively all eyes developed and initial mild haze (0.5 — 1) into the first month. Yet, at last follow-up 5/7 eyes showed a haze of 0.5 and this was no longer evident in 2/7 eyes. Based on these results, we demonstrated that the long-term outcomes after PRK and LASIK were safe and efficient, with similar stability for both procedures. The PRK outcomes were similar when treated by broad-beam or scanning slit laser. LASIK was better than PRK to correct moderate-to-high astigmatism, yet both procedures showed a tendency of undercorrection. Irregular astigmatism was proven to be able to depend exclusively from the corneal epithelium. If this kind of astigmatism is present in the cornea and a customized PRK/LASIK correction is done based on wavefront measurements an irregular astigmatism may be produced rather than treated. Corneal sensory nerve recovery should have an important role in the modulation of the corneal wound healing and post-operative anterior stromal scarring. PRK enhancement may be an option in eyes with previous LASIK after a sufficient time interval that in at least 2 years.

Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

The decline of malaria in Finland – the impact of the vector and social variables

Background: Malaria was prevalent in Finland in the 18th century. It declined slowly without deliberate counter-measures and the last indigenous case was reported in 1954. In the present analysis of indigenous malaria in Finland, an effort was made to construct a data set on annual malaria cases of maximum temporal length to be able to evaluate the significance of different factors assumed to affect malaria trends. Methods: To analyse the long-term trend malaria statistics were collected from 1750–2008. During that time, malaria frequency decreased from about 20,000 – 50,000 per 1,000,000 people to less than 1 per 1,000,000 people. To assess the cause of the decline, a correlation analysis was performed between malaria frequency per million people and temperature data, animal husbandry, consolidation of land by redistribution and household size. Results: Anopheles messeae and Anopheles beklemishevi exist only as larvae in June and most of July. The females seek an overwintering place in August. Those that overwinter together with humans may act as vectors. They have to stay in their overwintering place from September to May because of the cold climate. The temperatures between June and July determine the number of malaria cases during the following transmission season. This did not, however, have an impact on the longterm trend of malaria. The change in animal husbandry and reclamation of wetlands may also be excluded as a possible cause for the decline of malaria. The long-term social changes, such as land consolidation and decreasing household size, showed a strong correlation with the decline of Plasmodium. Conclusion: The indigenous malaria in Finland faded out evenly in the whole country during 200 years with limited or no counter-measures or medication. It appears that malaria in Finland was basically a social disease and that malaria trends were strongly linked to changes in human behaviour. Decreasing household size caused fewer interactions between families and accordingly decreasing recolonization possibilities for Plasmodium. The permanent drop of the household size was the precondition for a permanent eradication of malaria.