Perturbation of Burkholder's martingale transform and Monge--Amp\`ere equation

Let $\{d_k\}_{k \geq 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\infty,$ and \{\e_k\}_{k \geq 0} a sequence in $\{\pm 1\}.$ We obtain the following generalization of Burkholder's famous result. If $\tau \in [-\frac 12, \frac 12]$ and $n \in \Z_+$ then |\sum_{k=0}^n{(\{c} \e_k \tau) d_k}|_{L^p([0,1], \C^2)} \leq ((p^*-1)^2 + \tau^2)^{\frac 12}|\sum_{k=0}^n{d_k}|_{L^p([0,1], \C)}, where $((p^*-1)^2 + \tau^2)^{\frac 12}$ is sharp and $p^*-1 = \max\{p-1, \frac 1{p-1}\}.$ For $2\leq p<\infty$ the result is also true with sharp constant for $\tau \in \R.$Comment: 45 pages, 13 figure

Information Disclosure Strategies for Green Industries

Environmental information disclosure strategies, which involve corporate attempts to increase the availability of information on pollution and emissions, can become a basis for a new wave of environmental protection policy that follows and has the potential to complement traditional command and control and market-based approaches. Although a growing body of literature and operational programs suggest that publicly disclosing the information can motivate improved corporate environmental performance, this phenomenon remains poorly understood. This paper reviews the economic and legitimacy theory behind information disclosure and analyzes the current practice and programs adopted in industrialized and industrializing countries. Admittedly few in number, the cases studied reveal the advantages of such voluntary approaches, when the countries of developing Asia must deal with weak institutions, growing markets, and strong communities. Factors that contributed to widespread success of selected programs in the People's Republic of China, India, Indonesia, the Philippines, and the United States are information quality, the dissemination mechanisms, provision of incentives for good performers, and public and private pressure

COVID-19: Is There Evidence for the Use of Herbal Medicines as Adjuvant Symptomatic Therapy?

Background: Current recommendations for the self-management of SARS-Cov-2 disease (COVID-19) include self-isolation, rest, hydration, and the use of NSAID in case of high fever only. It is expected that many patients will add other symptomatic/adjuvant treatments, such as herbal medicines. Aims: To provide a benefits/risks assessment of selected herbal medicines traditionally indicated for “respiratory diseases” within the current frame of the COVID-19 pandemic as an adjuvant treatment. Method: The plant selection was primarily based on species listed by the WHO and EMA, but some other herbal remedies were considered due to their widespread use in respiratory conditions. Preclinical and clinical data on their efficacy and safety were collected from authoritative sources. The target population were adults with early and mild flu symptoms without underlying conditions. These were evaluated according to a modified PrOACT-URL method with paracetamol, ibuprofen, and codeine as reference drugs. The benefits/risks balance of the treatments was classified as positive, promising, negative, and unknown. Results: A total of 39 herbal medicines were identified as very likely to appeal to the COVID-19 patient. According to our method, the benefits/risks assessment of the herbal medicines was found to be positive in 5 cases (Althaea officinalis, Commiphora molmol, Glycyrrhiza glabra, Hedera helix, and Sambucus nigra), promising in 12 cases (Allium sativum, Andrographis paniculata, Echinacea angustifolia, Echinacea purpurea, Eucalyptus globulus essential oil, Justicia pectoralis, Magnolia officinalis, Mikania glomerata, Pelargonium sidoides, Pimpinella anisum, Salix sp, Zingiber officinale), and unknown for the rest. On the same grounds, only ibuprofen resulted promising, but we could not find compelling evidence to endorse the use of paracetamol and/or codeine. Conclusions: Our work suggests that several herbal medicines have safety margins superior to those of reference drugs and enough levels of evidence to start a clinical discussion about their potential use as adjuvants in the treatment of early/mild common flu in otherwise healthy adults within the context of COVID-19. While these herbal medicines will not cure or prevent the flu, they may both improve general patient well-being and offer them an opportunity to personalize the therapeutic approaches

Weak -type estimates for singular integral and maximal operators

In Chapter 3, the weak-type (1,1) constant for the Calderón-Zygmund singular integral operator T defined on L p([special characters omitted]) by [special characters omitted] is shown to be at worst c log n [special characters omitted]. In particular, for the jth-Riesz transform with [special characters omitted], the constant is at worst logarithmic with respect to dimension. In Chapter 4, it is shown that the weak-type inequality attains a limit as [special characters omitted]. In particular for [special characters omitted] with f ≥ 0,[special characters omitted] In Chapter 5, it is shown that the weak-type inequality attains a limit when f is replaced by certain singular measures on linear subspaces and [special characters omitted]. For example, if [special characters omitted] and [special characters omitted], is a singular measure in [special characters omitted], then [special characters omitted