Boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures

In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis.Comment: 16 page

Lipschitz spaces and Calderón-Zygmund operators associated to non-doubling measures

In the setting of a metric measure space (X, d, µ) with an n-dimensional Radon measure µ, we give a necessary and sufficient condition for the boundedness of Calder'n-Zygmund operators associated to the measure µ on Lipschitz spaces on the support of µ. Also, for the Euclidean space Rd with an arbitrary Radon measure µ, we give several characterizations of Lipschitz spaces on the support of µ, Lip(α, µ), in terms of mean oscillations involving µ. This allows us to view the "regular" BMO space of X. Tolsa as a limit case for α → 0 of the spaces Lip(α, µ)

Single Cell Genome Amplification Accelerates Identification of the Apratoxin Biosynthetic Pathway from a Complex Microbial Assemblage

Filamentous marine cyanobacteria are extraordinarily rich sources of structurally novel, biomedically relevant natural products. To understand their biosynthetic origins as well as produce increased supplies and analog molecules, access to the clustered biosynthetic genes that encode for the assembly enzymes is necessary. Complicating these efforts is the universal presence of heterotrophic bacteria in the cell wall and sheath material of cyanobacteria obtained from the environment and those grown in uni-cyanobacterial culture. Moreover, the high similarity in genetic elements across disparate secondary metabolite biosynthetic pathways renders imprecise current gene cluster targeting strategies and contributes sequence complexity resulting in partial genome coverage. Thus, it was necessary to use a dual-method approach of single-cell genomic sequencing based on multiple displacement amplification (MDA) and metagenomic library screening. Here, we report the identification of the putative apratoxin. A biosynthetic gene cluster, a potent cancer cell cytotoxin with promise for medicinal applications. The roughly 58 kb biosynthetic gene cluster is composed of 12 open reading frames and has a type I modular mixed polyketide synthase/nonribosomal peptide synthetase (PKS/NRPS) organization and features loading and off-loading domain architecture never previously described. Moreover, this work represents the first successful isolation of a complete biosynthetic gene cluster from Lyngbya bouillonii, a tropical marine cyanobacterium renowned for its production of diverse bioactive secondary metabolites