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.

Rough Maximal Oscillatory Singular Integral Operators

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25

Sobolev-Morrey Type Inequality for Riesz Potentials, Associated with the Laplace-Bessel Differential Operator

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35

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.

On a Ratio of Functions of Exponential Random Variables and Some Applications

Consider L independent and identically distributed exponential random variables (r.vs) X-1, X-2 ,..., X-L and positive scalars b(1), b(2) ,..., b(L). In this letter, we present the probability density function (pdf), cumulative distribution function and the Laplace transform of the pdf of the composite r.v Z = (Sigma(L)(j=1) X-j)(2) / (Sigma(L)(j=1) b(j)X(j)). We show that the r.v Z appears in various communication systems such as i) maximal ratio combining of signals received over multiple channels with mismatched noise variances, ii)M-ary phase-shift keying with spatial diversity and imperfect channel estimation, and iii) coded multi-carrier code-division multiple access reception affected by an unknown narrow-band interference, and the statistics of the r.v Z derived here enable us to carry out the performance analysis of such systems in closed-form.

Uncovering Dependence Clusters and Linchpin Functions

Dependence clusters are (maximal) collections of mutually dependent source code entities according to some dependence relation. Their presence in software complicates many maintenance activities including testing, refactoring, and feature extraction. Despite several studies finding them common in production code, their formation, identification, and overall structure are not well understood, partly because of challenges in approximating true dependences between program entities. Previous research has considered two approximate dependence relations: a fine-grained statement-level relation using control and data dependences from a program’s System Dependence Graph and a coarser relation based on function-level controlflow reachability. In principal, the first is more expensive and more precise than the second. Using a collection of twenty programs, we present an empirical investigation of the clusters identified by these two approaches. In support of the analysis, we consider hybrid cluster types that works at the coarser function-level but is based on the higher-precision statement-level dependences. The three types of clusters are compared based on their slice sets using two clustering metrics. We also perform extensive analysis of the programs to identify linchpin functions – functions primarily responsible for holding a cluster together. Results include evidence that the less expensive, coarser approaches can often be used as e�ective proxies for the more expensive, finer-grained approaches. Finally, the linchpin analysis shows that linchpin functions can be e�ectively and automatically identified.

Image quality optimization, via application of contextual contrast sensitivity and discrimination functions

What is the best luminance contrast weighting-function for image quality optimization? Traditionally measured contrast sensitivity functions (CSFs), have been often used as weighting-functions in image quality and difference metrics. Such weightings have been shown to result in increased sharpness and perceived quality of test images. We suggest contextual CSFs (cCSFs) and contextual discrimination functions (cVPFs) should provide bases for further improvement, since these are directly measured from pictorial scenes, modeling threshold and suprathreshold sensitivities within the context of complex masking information. Image quality assessment is understood to require detection and discrimination of masked signals, making contextual sensitivity and discrimination functions directly relevant. In this investigation, test images are weighted with a traditional CSF, cCSF, cVPF and a constant function. Controlled mutations of these functions are also applied as weighting-functions, seeking the optimal spatial frequency band weighting for quality optimization. Image quality, sharpness and naturalness are then assessed in two-alternative forced-choice psychophysical tests. We show that maximal quality for our test images, results from cCSFs and cVPFs, mutated to boost contrast in the higher visible frequencies.

Maximal-Element Rationalizability

We examine the maximal-element rationalizability of choice functions with arbitrary do-mains. While rationality formulated in terms of the choice of greatest elements according to a rationalizing relation has been analyzed relatively thoroughly in the earlier litera-ture, this is not the case for maximal-element rationalizability, except when it coincides with greatest-element rationalizability because of properties imposed on the rationalizing relation. We develop necessary and sufficient conditions for maximal-element rationaliz-ability by itself, and for maximal-element rationalizability in conjunction with additional properties of a rationalizing relation such as re exivity, completeness, P-acyclicity, quasi-transitivity, consistency and transitivity.

The C(K, X) spaces for compact metric spaces K and X with a uniformly convex maximal factor

In this paper, we prove that if a Banach space X contains some uniformly convex subspace in certain geometric position, then the C(K, X) spaces of all X-valued continuous functions defined on the compact metric spaces K have exactly the same isomorphism classes that the C(K) spaces. This provides a vector-valued extension of classical results of Bessaga and Pelczynski (1960) [2] and Milutin (1966) [13] on the isomorphic classification of the separable C(K) spaces. As a consequence, we show that if 1 < p < q < infinity then for every infinite countable compact metric spaces K(1), K(2), K(3) and K(4) are equivalent: (a) C(K(1), l(p)) circle plus C(K(2), l(q)) is isomorphic to C(K(3), l(p)) circle plus (K(4), l(q)). (b) C(K(1)) is isomorphic to C(K(3)) and C(K(2)) is isomorphic to C(K(4)). (C) 2011 Elsevier Inc. All rights reserved.

Nutritional regulation of the fatty acid synthase promoter in vivo: Sterol regulatory element binding protein functions through an upstream region containing a sterol regulatory element

The transcription of fatty acid synthase (FAS), a central enzyme in de novo lipogenesis, is dramatically induced by fasting/refeeding and insulin. We reported that upstream stimulatory factor binding to the −65 E-box is required for induction of the FAS transcription by insulin in 3T3-L1 adipocytes. On the other hand, we recently found that two upstream 5′ regions are required for induction in vivo by fasting/refeeding and insulin; one at −278 to −131 albeit at a low level, and the other at −444 to −278 with an E-box at −332 where upstream stimulatory factor functions for maximal induction. Here, we generated double transgenic mice carrying the chloramphenicol acetyltransferase reporter driven by the various 5′ deletions of the FAS promoter region and a truncated active form of the sterol regulatory element (SRE) binding protein (SREBP)-1a. We found that SREBP participates in the nutritional regulation of the FAS promoter and that the region between −278 and −131 bp is required for SREBP function. We demonstrate that SREBP binds the −150 canonical SRE present between −278 and −131, and SREBP can function through the −150 SRE in cultured cells. These in vivo and in vitro results indicate that SREBP is involved in the nutritional induction of the FAS promoter via the −278/−131 region and that the −150 SRE is the target sequence.

Inhibition of G-protein betagamma-subunit functions by phosducin-like protein.

Phosducin is a cytosolic protein predominantly expressed in the retina and the pineal gland that can interact with the betagamma subunits of guanine nucleotide binding proteins (G proteins) and thereby may regulate transmembrane signaling. A cDNA encoding a phosducin-like protein (PhLP) has recently been isolated from rat brain [Miles, M. F., Barhite, S., Sganga, M. & Elliott, M. (1993) Proc. Natl. Acad. Sci. USA 90, 10831-10835. Here we report the expression of PhLP in Escherichia coli and its purification. Recombinant purified PUP inhibited multiple effects of G-protein betagamma subunits. First, it inhibited the betagamma-subunit-dependent ADP-ribosylation of purified alpha(o) by pertussis toxin. Second, it inhibited the GTPase activity of purified G(o). The IC50 value of PhLP in the latter assay was 89 nM, whereas phosducin caused half-maximal inhibition at 17 nM. And finally, PhLP antagonized the enhancement of rhodopsin phosphorylation by purified betagamma subunits. The N terminus of PhLP shows no similarity to the much longer N terminus of phosducin, the region shown to be critical for phosducin-betagamma-subunit interactions. Therefore, PhLP appears to bind to G-protein betagamma subunits by an as yet unknown mode of interaction and may represent an endogenous regulator of G-protein function.

A Recession Notion for a Class of Monotone Bivariate Functions

Using monotone bifunctions, we introduce a recession concept for general equilibrium problems relying on a variational convergence notion. The interesting purpose is to extend some results of P. L. Lions on variational problems. In the process we generalize some results by H. Brezis and H. Attouch relative to the convergence of the resolvents associated with maximal monotone operators.