On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

Actin Dynamics in Muscle Cells

In every cell, actin is a key component involved in migration, cytokinesis, endocytosis and generation of contraction. In non-muscle cells, actin filaments are very dynamic and regulated by an array of proteins that interact with actin filaments and/or monomeric actin. Interestingly, in non-muscle cells the barbed ends of the filaments are the predominant assembly place, whereas in muscle cells actin dynamics was reported to predominate at the pointed ends of thin filaments. The actin-based thin filament pointed (slow growing) ends extend towards the middle of the sarcomere's M-line where they interact with the thick filaments to generate contraction. The actin filaments in muscle cells are organized into a nearly crystalline array and are believed to be significantly less dynamic than the ones in other cell types. However, the exact mechanisms of the sarcomere assembly and turnover are largely unknown. Interestingly, although sarcomeric actin structures are believed to be relatively non-dynamic, many proteins promoting actin dynamics are expressed also in muscle cells (e.g ADF/cofilin, cyclase-associated protein and twinfilin). Thus, it is possible that the muscle-specific isoforms of these proteins promote actin dynamics differently from their non-muscle counterparts, or that actin filaments in muscle cells are more dynamic than previously thought. To study protein dynamics in live muscle cells, I used primary cell cultures of rat cardiomyocytes. My studies revealed that a subset of actin filaments in cardiomyocyte sarcomeres displays rapid turnover. Importantly, I discovered that the turnover of actin filaments depends on contractility of the cardiomyocytes and that the contractility-induced actin dynamics plays an important role in sarcomere maturation. Together with previous studies those findings suggest that sarcomeres undergo two types of actin dynamics: (1) contractility-dependent turnover of whole filaments and (2) regulatory pointed end monomer exchange to maintain correct thin filament length. Studies involving an actin polymerization inhibitor suggest that the dynamic actin filament pool identified here is composed of filaments that do not contribute to contractility. Additionally, I provided evidence that ADF/cofilins, together with myosin-induced contractility, are required to disassemble non-productive filaments in developing cardiomyocytes. In addition, during these studies we learned that isoforms of actin monomer binding protein twinfilin, Twf-1 and Twf-2a localise to myofibrils in cardiomyocytes and may thus contribute to actin dynamics in myofibrils. Finally, in collaboration with Roberto Dominguez s laboratory we characterized a new actin nucleator in muscle cells - leiomodin (Lmod). Lmod localises towards actin filament pointed ends and its depletion by siRNA leads to severe sarcomere abnormalities in cardiomyocytes. The actin filament nucleation activity of Lmod is enhanced by interactions with tropomyosin. We also revealed that Lmod expression correlates with the maturation of myofibrils, and that it associates with sarcomeres only at relatively late stages of myofibrillogenesis. Thus, Lmod is unlikely to play an important role in myofibril formation, but rather might be involved in the second step of the filament arrangement and/or maintenance through its ability to promote tropomyosin-induced actin filament nucleation occurring at the filament pointed ends. The results of these studies provide valuable new information about the molecular mechanisms underlying muscle sarcomere assembly and turnover. These data offer important clues to understanding certain physiological and pathological behaviours of muscle cells. Better understanding of the processes occurring in muscles might help to find strategies for determining, diagnosis, prognosis and therapy in heart and skeletal muscles diseases.

Long-term results of operative treatment for Hirschsprung's disease and internal anal sphincter achalasia

The aim of the study was to evaluate long-term results of operative treatment for Hirschsprung's disease(HD) and internal anal sphincter achalasia. Fecal continence and quality of life were evaluated by a questionnaire in 100 adult patients who had undergone surgery for HD, during 1950-75. Fecal continence was evaluated using a numerical scoring described by Holschneider. Fifty-four of the 100 patients underwent clinical examination, rigid sigmoidoscopy and manometric evaluation. In anorectal manometry basal resting pressure(BRP)and maximal squeeze pressure(MSP) were measured and voluntary sphincter force(VSF) was calculated by subtracting the BRP from MSP. The results of operative treatment for adult HD were compared with the results of the patients operated in childhood. In adult HD the symptoms are such mild that the patients attain adolescence or even adulthood. The patients with HD and cartilage-hair-hypoplasia were specifically evaluated. The outcome of the patients with internal anal sphincter achalasia operated on by myectomy was evaluated by a questionnaire and continence was evaluated using a numerical scoring described by Holschneider. Of the 100 patients operated on for HD 38 patients had completely normal bowel habits. A normal or good continence score was found in 91 our of 100 patients. Nine patients had fair continence. One of the patients with fair continence had Down's syndrome and two were mentally retarded for other reasons. Only one patient suffered from constipation. In anorectal manometry the difference in BRP between patients with normal and good continence was statistically significant, whereas the difference between good and fair continence groups was not statistically significant. The differences on MSP and VSF between patient groups with different continence outcome were not statistically significant. The differences between patient groups and normal controls were statistically significant in BRP and MSP. In VSF there was not statistically significant difference between the patients and the normal controls. The VSF reflects the working power of the muscles including external sphincter, levator ani and gluteal muscles. The patients operated at adult age had as good continence as patients operated in childhood. The patients with HD and cartilage-hair-hypoplasia had much more morbidity and mortality than non-cartilage-hair-hypoplasia HD patients. The mortality was as high as 38%. In patients with internal anal sphincter achalasia the constipation was cured or alleviated by myectomy whereas a significant number suffered from soiling-related social problems.

Search for Maximal Flavor Violating Scalars in Same-Charge Lepton Pairs in pp̅ Collisions at √s=1.96 TeV

Models of Maximal Flavor Violation (MxFV) in elementary particle physics may contain at least one new scalar SU$(2)$ doublet field $\Phi_{FV} = (\eta^0,\eta^+)$ that couples the first and third generation quarks ($q_1,q_3$) via a Lagrangian term $\mathcal{L}_{FV} = \xi_{13} \Phi_{FV} q_1 q_3$. These models have a distinctive signature of same-charge top-quark pairs and evade flavor-changing limits from meson mixing measurements. Data corresponding to 2 fb$^{-1}$ collected by the CDF II detector in $p\bar{p}$ collisions at $\sqrt{s} = 1.96$ TeV are analyzed for evidence of the MxFV signature. For a neutral scalar $\eta^0$ with $m_{\eta^0} = 200$ GeV/$c^2$ and coupling $\xi_{13}=1$, $\sim$ 11 signal events are expected over a background of $2.1 \pm 1.8$ events. Three events are observed in the data, consistent with background expectations, and limits are set on the coupling $\xi_{13}$ for $m_{\eta^0} = 180-300$ GeV/$c^2$.

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.