Two weight inequality for vector-valued positive dyadic operators by parallel stopping cubes

We study the vector-valued positive dyadic operator $$T_\lambda(f\sigma):=\sum_{Q\in\mathcal{D}} \lambda_Q \int_Q f \mathrm{d}\sigma 1_Q,$$ where the coefficients $\{\lambda_Q:C\to D\}_{Q\in\mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^*$ each have the Hardy--Littlewood property. An example of a Banach lattice with the Hardy--Littlewood property is a Lebesgue space. In the two-weight case, we prove that the $L^p_C(\sigma)\to L^q_D(\omega)$ boundedness of the operator $T_\lambda( \cdot \sigma)$ is characterized by the direct and the dual $L^\infty$ testing conditions: $$\lVert 1_Q T_\lambda(1_Q f \sigma)\rVert_{L^q_D(\omega)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\sigma)} \sigma(Q)^{1/p},$$ $$\lVert1_Q T^*_{\lambda}(1_Q g \omega)\rVert_{L^{p'}_{C^*}(\sigma)}\lesssim \lVert g\rVert_{L^\infty_{D^*}(Q,\omega)} \omega(Q)^{1/q'}.$$ Here $L^p_C(\sigma)$ and $L^q_D(\omega)$ denote the Lebesgue--Bochner spaces associated with exponents $1<p\leq q<\infty$, and locally finite Borel measures $\sigma$ and $\omega$. In the unweighted case, we show that the $L^p_C(\mu)\to L^p_D(\mu)$ boundedness of the operator $T_\lambda( \cdot \mu)$ is equivalent to the endpoint direct $L^\infty$ testing condition: $$\lVert1_Q T_\lambda(1_Q f \mu)\rVert_{L^1_D(\mu)}\lesssim \lVert f\rVert_{L^\infty_C(Q,\mu)} \mu(Q).$$ This condition is manifestly independent of the exponent $p$. By specializing this to particular cases, we recover some earlier results in a unified way.Comment: 32 pages. The main changes are: a) Banach lattice-valued functions are considered. It is assumed that the Banach lattices have the Hardy--Littlewood property. b) The unweighted norm inequality is characterized by an endpoint testing condition and some corollaries of this characterization are stated. c) Some questions about the borderline of the vector-valued testing conditions are pose

Pale Europeans and Dark Africans share sun and common health problems:constancy in regional health differences and sunshine

Regional risk of cardiovascular mortality has only recently been added to the group of major risk factors, but its effective sub-factors are not fully understood. The aim of this study was to assess the CHD mortality in Finland and other Europe, its stability and association with capability in vitamin D synthesis. In a worldwide assay several causes besides geographical and climatological factors restrict vitamin D synthesis in skin and several factors affect the activation as also the inactivation of vitamin D. Sunshine anyhow is the constant source of vitamin D, while dietary habits have changed. Effects of vitamin D could be based on mineral, anti-inflammatory and structural factors. Different availability of sunshine has thus associated with the constancy in proportional difference in CHD mortality in Finland as an example country.Conclusion: Sunshine is associated with lower CHD mortality in Europe. Other additive mechanisms are suggested.Keywords: CHD mortality, regional risk, sunshine, inflammation, silico

Mg/Ca ratio in fertilization and agricultural soils, Mg percent of liming agents and human mortality in Finland during 1961-90

Background: Mg is a cofactor in more than 300 enzymatic reactions and its deficiency has been reported to be associated with cardiovascular diseases. Human Mg balance depends on food composition, food processing and Mg variation in foodstuffs, which can be roughly prognostigated by Mg proportion in fertilization and soil. Strong increase of NPK (nitrogen, phosphorus and potassium) in mineral fertilization (fm) included relative delay in Mg supplementation and dilution in plant available silicon (Si) via recycled nutrients (rcl). (Silicon is not included in essential fertilizers in Finland.) Methods: We have assessed old data on Ca and Mg in agricultural soils and approximate data on fm, rcl, as well as Mg % of liming agents (Mg-%.lim) and total (TOT), CHD and non-CHD (nCHD) death-rates of humans by R squares and graphics, in order to clarify their associations and possible causality. Results: Mg/Ca ratio in total fertilization (ft =fm + rcl) was decreasing in 1951-64 and after that mainly increasing. Soil (Mg/Ca) in 1961- 2000 responded on (Mg/Ca).ft with delay of ca 5 years. During 1961-90 (Mg/Ca).fm "explained" CHD by 74-89 %, non-CHD by 87 - 96 % and TOT by 90 - 94 %. (Mg/Ca) fertilization ratios "explained" better female than male CHD, but TOT and non-CHD more similarly. Soil (Mg/Ca) "explained" male CHD by 94 %, but all other death-rates weaker than (Mg/Ca).fm. Different smoking habits could explain this sex difference. All given associations were highly significant (p &lt; 0.001). Conclusion: Mg/Ca changes in fertilization preceded respective changes in soil Ca by five years. They explained in general better than the soil value changes in death-rates, except M.CHD with obvious “tobacco delay”. Effects of silicon and its association with rcl/ft ratio are discussed

Investor Reactions to Corporate Merger and Acquisition Announcements

This dissertation examines investor reactions to corporate press and stock exchange releases on mergers and acquisitions (M&A). Investor reactions to corporate announcements are measured in changes in the corporate stock price. The dissertation focuses on a corporate’s acquisition target and its strategic intention to move within its value network, hypothesizing that different types of acquisitions create different cumulative abnormal return. Acquisition types are extended from traditional horizontal vs. vertical and related vs. unrelated acquisitions to cover all types of acquisitions. More detailed acquisition categories are needed to focus on strategic company moves and their impact on the share price. Investor reactions have traditionally been studied by using event study on day-level analysis. Such analysis does not sufficiently reflect current stock trading, whereas algorithmic trading represents most of the total volume. Recently high-frequency trading and the overall speed of the information flow have underscored the importance of transaction-level analysis, which was adopted for this dissertation. The hypotheses in this dissertation were tested with all stock transactions during 2006-2010 in NASDAQ OMX Helsinki. These publicly listed companies published over 30,000 releases, including 548 M&A actions. Consistent with theory, the findings showed a positive compounded abnormal return (CAR) in all M&A actions. Additionally, transaction level analysis revealed a CAR in unrelated acquisitions representing an upstream change in the center of gravity, whereas day-level analysis produced no CAR. Finally, the multiple regression model of transaction-level analysis improved the coefficient of determination significantly over day-level analysis. Whereas day-level analysis is too ambiguous and therefore allows possible misinterpretation of the event time, transaction-level analysis will give additional research topics such as the speed of response to press release and investors’ pre-announcement reactions

Dyadic analysis of integral operators : median oscillation decomposition and testing conditions

Controlling integral operators by dyadic model operators, and studying the boundedness of dyadic operators on Lebesgue spaces are central themes in dyadic harmonic analysis. This dissertation consists of an introductory part and five articles contributing to these themes. Many operators of harmonic analysis can be dominated by positive dyadic operators by using Lerner's median oscillation decomposition. In the first and fifth article, we extend this decomposition to Banach space valued functions and non-doubling measures. Dyadic shifts and paraproducts are dyadic model operators for Calderón-Zygmund operators. In the second article, we study the boundedness of these operators on unweighted Lebesgue spaces in an abstract operator-valued setting. We prove that operator-valued dyadic shifts are bounded, and we characterize the boundedness of operator-valued dyadic paraproducts. Furthermore, we extend Hytönen's dyadic representation theorem, which states that every Calderón-Zygmund operator can be represented by dyadic shifts and paraproducts, to the operator-valued setting. In the third article, we characterize the boundedness of linear and bilinear positive dyadic operators from a weighted Lebesgue space to another. We consider the case that the Lebesgue exponent of the range side is strictly less than the Lebesgue exponent of the domain side. We show that, in this range of the exponents, the Sawyer testing condition is insufficient for the boundedness. We introduce a sequential testing condition, of which the Sawyer testing condition can be viewed as an endpoint case, and prove that this testing condition is both sufficient and necessary for the boundedness. No characterization in the bilinear case was available until this article. Furthermore, we show that the sequential testing condition is necessary for the boundedness of any positive linear or bilinear operator, and hence it may be helpful in charactering the boundedness of other operators as well. In the fourth article, we characterize the boundedness of positive dyadic operators from a weighted Lebesgue space to another in an abstract operator-valued setting. The purpose is to understand which kind of testing condition is needed in this setting. We prove that an operator-valued positive dyadic operator is bounded if and only if the operator and its adjoint are each bounded on the class of all functions localized on dyadic cubes and taking values on a unit sphere. Furthermore, we show that the boundedness on unweighted Lebesgue spaces is characterized by an endpoint case of this condition. We work directly with Lebesgue spaces, without using interpolation between endpoint spaces. In the second article, we give new (in our opinion simple) proofs for the key tools that we use: decoupling inequality for martingale differences and a variant of Pythagoras' theorem for Lebesgue spaces.Analyysin keskeisiä tutkimuskohteita ovat funktiot ja operaattorit. Funktio on kuvaus lähtöjoukolta maalijoukolle. Se on sääntö, joka liittää jokaiseen lähtöjoukon alkioon täsmälleen yhden maalijoukon alkion. Operaattori on kuvaus funktioilta funktioille. Dyadisella analyysillä tarkoitetaan dyadisten kuutioiden hyödyntämistä analyysin tutkimuksessa. Dyadiset kuutiot ovat kuutioita, joiden olennainen ominaisuus on sisäkkäisyys: Kaksi dyadista kuutiota ovat joko erillisiä tai sisäkkäisiä. Dyadiset kuutiot voivat olla lähtökohtaisesti läsnä tarkastelussa esimerkiksi tutkittaessa dyadisten kuutioiden avulla määriteltyä eli dyadista operaattoria, tai ne voivat tulla mukaan tarkasteluun jonkin käytettävän tekniikan myötä. Harmoninen analyysi tutkii funktioiden ja operaattoreiden määrällisiä ominaisuuksia. Alan tyypillinen ongelma on osoittaa, että jokin tietty operaattori ei voi kasvattaa funktioiden suuruutta rajattomasti ja arvioida tämän rajan kokoa. Funktion suuruutta voidaan mitata monella eri tavoin, joista kukin huomioi funktion tiettyjä piirteitä. Reaaliluvuilta reaalivuille määritellyn funktion suuruutta voidaan mitata esimerkiksi sen kuvaajan alle jäävällä pinta-alalla. Tämän yleistys on Lebesguen normi. Sillä voidaan mitata mitta-avaruudelta vektoriavaruudelle määritellyn funktion suuruutta, ja se riippuu mitta-avaruuden mitasta, vektoriavaruuden normista ja integroituvuuseksponentista. Alan ydinainesta ovat erinäiset tekniikat, kuten funktioiden hajottaminen osiin, joita räätälöidään kuhunkin tilanteeseen sopiviksi. Esimerkki funktion hajotelmasta on reaalilukuarvoisen funktion ilmaiseminen positiivisen ja negatiivisen osansa summana. Esityslauseet ja testiehdot lukeutuvat alan keskeisiin aiheisiin. Esityslauseissa päämäränä on arvioida monimutkaista operaattoria yksinkertaisempien operaattoreiden avulla. Testiehdoissa päämääränä on selvittää raja sille, kuinka paljon jokin tietty operaattori voi kasvattaa funktioiden suuruutta. Lähtökohtaisesti tämän selvittämiseksi täytyy testata, päteekö raja jokaiselle funktiolle. Testiehtojen perusajatuksena on, että operaattorin rakenteen ansiosta riittääkin testata rajan pätevyys vain tietyille riittävän edustaville funktioille. Väitöskirjani käsittee näitä aiheita dyadisen harmonisen analyysin alueella: operaattoreiden esittämistä yksinkertaisempien dyadisten operaattoreiden avulla ja testiehtojen muotoilemista dyadisille operaattoreille. Esityslauseiden aihepiirissä väitöskirjassani muun muassa ulotetaan funktion mediaanihajotelma vektoriarvoisille funktioille. Tämä hajotelma on keskeinen apuväline, kun arvioidaan operaattoreita tietynlaisten positiivisten keskiarvoistavien dyadisten operaattoreiden avulla. Funktion mediaanihajotelmassa funktiota arvioidaan sen paikallisten heilahtelujen avulla. Lerner muotoili ja todisti hajotelman reaalilukuarvoisille funktioille. Todistus pohjautuu sen tarkasteluun, kuinka paljon funktio eroaa mediaanistaan paikallisesti. Reaalilukujen mediaani lasketaan järjestämällä luvut suuruusjärjestykseen ja valitsemalla näistä luvuista keskimmäinen. Koska vektoreita ei voida järjestää suuruusjärjestykseen, tämä mediaanin määritelmä ei ole mielekäs vektoreille. Ratkaisevaa mediaanihajotelman ulottamisessa vektoriarvoisille funktioille on määritellä mediaanin vastine vektoreille. Testiehtojen aihepiirissä väitöskirjassani muun muassa muotoillaan uudenlainen testiehto sille, että tietynlainen positiivinen dyadinen operaattori kasvattaa rajoitetusti funktioiden Lebesguen normia, kun maalipuolen Lebesguen normin integroituvuuseksponentti on aidosti pienempi kuin lähtöpuolen