1,720,956 research outputs found
Uniform rectifiability and ε-approximability of harmonic functions in Lp
No full textSuppose that E⊂Rn+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in Lp(Ω) for every p∈(1,∞), where Ω:=Rn+1∖E. Together with results of many authors this shows that pointwise, L∞ and Lp type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension 1 Ahlfors–David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.peerReviewe
Uniform rectifiability implies Varopoulos extensions
No full textWe construct extensions of Varopolous type for functions f∈BMO(E), for any uniformly rectifiable set E of codimension one. More precisely, let Ω⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let ≔σ≔Hn⌊∂Ω denote the surface measure on ∂Ω. We show that if f∈BMO(∂Ω,dσ) with compact support on ∂Ω, then there exists a smooth function V in Ω such that |∇V(Y)|dY is a Carleson measure with Carleson norm controlled by the BMO norm of f, and such that V converges in some non-tangential sense to f almost everywhere with respect to σ. Our results should be compared to recent geometric characterizations of Lp-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all f∈Cc(∂Ω), a harmonic extension u, with |∇u(Y)|2dist(Y,∂Ω)dY a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition.peerReviewe
Random and non-random dyadic systems in doubling metric spaces
No full textThe standard system of dyadic cubes in the Euclidean space R^n is a collection of half-open cubes of different sizes such that the cubes of every size partition the space and every cube is a finite union of smaller cubes. The construction of this system is very simple but it does rely strongly on the geometrical properties of the space R^n. Hence, if we give up most of the geometrical properties of the space R^n, the construction of sets with similar properties becomes more complicated. In this paper, we show that there exist dyadic cubes in general doubling metric spaces. We do this using certain maximal sets of points and a carefully defined partial order of those points. We look at several different dyadic systems first in R^n and then in general doubling metric spaces.
We start by proving some basic results related to doubling metric spaces and other related topics and continue by introducing the standard, randomized and adjacent systems of dyadic cubes in R^n. Then, in a general doubling metric space, we construct a system of sets that has similar properties as the standard system of dyadic cubes in R^n. We call also these sets cubes although they are not cubes in the usual sense of the word. After this, we add a probabilistic angle to the constructed system by randomizing them and look at two different random systems. Lastly, we look at some applications of the random dyadic systems.
Our goal is to introduce a new simpler way of randomizing dyadic systems in doubling metric spaces and show that this is an effective way of randomizing the systems. We show this by proving that every point in the space has only a small probability of ending up near the boundary of a cube of given size. This property has an interesting application since we can use it to construct systems of Hölder-continuous spline functions in doubling metric spaces. It is still an open problem to prove whether there exist systems of Lipschitz-continuous spline functions in every doubling metric space. We do not know the answer to this problem but we show that at least there exist systems of Hölder-continuous spline functions of every exponent η ∈ (0,1) in every doubling metric space
Connectivity conditions and boundary Poincaré inequalities
Get PDFInspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets , with codimension Ahlfors--David regular boundaries. First, we prove that if satisfies both the local John condition and the exterior corkscrew condition, then also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if is a -sided chord-arc domain, then the boundary supports a Heinonen--Koskela type weak -Poincaré inequality. We also construct an example of a set such that the boundary is Ahlfors--David regular and supports a weak boundary -Poincaré inequality but is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.40 pages, 5 figures. v3: accepted version; updated grant information and picture formats. To appear in Analysis & PD
Adjacent and random dyadic systems and their applications to metric, Euclidean and vector-valued analysis
No full textDifferent dyadic techniques are an inseparable part of modern-day harmonic analysis both in the Euclidean space and in metric spaces. In this dissertation, we improve adjacent and random dyadic techniques in metric spaces and apply these and previously known techniques for questions related to metric, Euclidean and vector-valued analysis.
The dissertation consists of an introductory part and four research articles. In the first article, we present a general randomization procedure for dyadic systems in metric spaces which can be used for constructing both random and adjacent dyadic systems. As an application of the new random systems, we improve the continuity properties of metric wavelets of P. Auscher and T. Hytönen by exploiting the improved ``smallness of boundary'' property of our random cubes. In the third article, we prove some additional properties for our adjacent dyadic systems to prove a decomposition result for dyadic systems in metric spaces. With its help, we give an alternative proof for the quantitative bound of the Lp norm of shift operators acting on vector-valued functions in metric spaces.
In the second article, we explore certain properties of the Muckenhoupt weight classes, the class of Reverse Hölder weights and their weakened versions in spaces of homogeneous type. In the Euclidean setting, the Muckenhoupt weight classes have numerous different equivalent definitions but in spaces of homogeneous type some of those equivalences break down. We show that although certain definitions are no longer equivalent in this context, their weakened versions still define the same weight classes. We also show that every weak Reverse Hölder weight has a self-improving property. In the literature, these types of weak weights appear especially in the theory of partial differential equations.
In the fourth article, we prove quantitative weighted bounds for so called rough homogeneous singular integrals by combining older techniques with a quantitative version of M. Lacey's recent extension of the A2 theorem. The proof of this extension is based on a domination technique which provides a way to dominate Calderón-Zygmund operators pointwise with the help of a finite number of simple sparse operators associated with adjacent dyadic systems.Tässä väitöskirjassa käsitellään harmonisessa analyysissa tärkeiksi muodostuneita dyadisia tekniikoita ja niiden sovelluksia sekä metrisissä avaruuksissa että euklidisessa avaruudessa. Työssä tarkastellaan erityisesti niin sanottuja rinnakkaisia ja satunnaistettuja dyadisia systeemejä, joilla on ollut tärkeä rooli monissa alan viime vuosien merkittävissä tuloksissa.
Väitöskirja koostuu johdannosta ja neljästä tutkimusartikkelista. Ensimmäisessä artikkelissa esitellään uudenlainen satunnaistustekniikka metristen avaruuksien dyadisille kuutioille ja hyödynnetään tätä tekniikkaa satunnaisten ja rinnakkaisten systeemien rakentamiseen. Näitä satunnaistettuja systeemejä sovelletaan artikkelin lopussa tietynlaisten aallokefunktioiden jatkuvuusominaisuuksien parantamiseen. Kolmannessa artikkelissa osoitetaan, että rinnakkaisia systeemejä voidaan käyttää tehokkaasti hajottamaan dyadisia kuutiokokoelmia äärellisen moneen pienempään hyvin käyttäytyvään osaan. Tällaisen tekniikan avulla on suoraviivaista antaa uusi todistus siirto-operaattoreiden rajoittuneisuudelle vektoriarvoisilla Lebesgue-avaruuksilla. Sekä ensimmäinen että kolmas artikkeli hyödyntävät aktiivisesti todennäköisyyslaskentaan pohjautuvia tekniikoita.
Toisessa artikkelissa käsitellään Muckenhouptin painoluokkia ja niiden heikennettyjä versioita metrisissä avaruuksissa. Metrinen Muckenhouptin painojen teoria on haastavampi kuin vastaava euklidinen teoria, sillä monet euklidisen avaruuden ekvivalentit määritelmät eivät määrittele samoja funktioluokkia metrisissä avaruuksissa. Artikkelissa osoitetaan, että monet näistä määritelmistä muuttuvat jälleen yhtäpitäviksi, mikäli määritelmiä heikennetään. Tämän lisäksi artikkelissa todistetaan heikkojen painojen itseparantuvuusominaisuus. Heikkoja painoja esiintyy erityisesti osittaisdifferentiaaliyhtälöiden teoriassa.
Viimeisessä artikkelissa todistetaan kvantitatiivisia painotettuja epäyhtälöitä niin sanotuille karkeille singulaarisille integraalioperaattoreille. Artikkelissa vanhempia tekniikoita yhdistetään M. Laceyn hiljattain todistamaan A2-lauseen laajennokseen, josta artikkelissa todistetaan aluksi kvantitatiivinen versio. Tämä laajennos perustuu monimutkaisten Calderón-Zygmund-operaattoreiden dominointiin äärellisen monella yksinkertaisemmalla operaattorilla, jotka voidaan rakentaa kätevästi rinnakkaisten dyadisten systeemien avulla.ei saavutettav
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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