Duals of variable exponent Hörmander spaces (0<p−≤p+≤10< p^- \le p^+ \le 1) and some applications

In this paper we characterize the dual \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' of the variable exponent H\"or\-man\-der space \B^c_{p(\cdot)} (\Omega) when the exponent $p(\cdot)$ satisfies the conditions $0 < p^- \le p^+ \le 1$, the Hardy-Littlewood maximal operator $M$ is bounded on $L_{p(\cdot)/p_0}$ for some $0 < p_0 < p^-$ and $\Omega$ is an open set in $\R^n$. It is shown that the dual \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' is isomorphic to the H\"ormander space \B^{\mathrm{loc}}_\infty (\Omega) (this is the $p^+ \le 1$ counterpart of the isomorphism \bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' \simeq \B^{\mathrm{loc}}_{\widetilde{p'(\cdot)}} (\Omega), $1 < p^- \le p^+ < \infty$, recently proved by the authors) and hence the representation theorem \bigl( \B^c_{p(\cdot)} (\Omega) \bigr)' \simeq l^{\N}_\infty is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of \B^c_{p(\cdot)} (\Omega) and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $p(\cdot) \equiv p$, $0 < p < 1$, are also given (e.g. \B^c_p (\Omega) does not contain any infinite-dimensional $q$-Banach subspace with $p < q \le 1$ or the quasi-Banach space \B_p \cap \E'(Q) contains a copy of $l_p$ when $Q$ is a cube in $\R^n$). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent H\"ormander spaces is proposed.J. Motos is partially supported by grant MTM2011-23164 from the Spanish Ministry of Science and Innovation. The authors wish to thank the referees for the careful reading of the manuscript and for many helpful suggestions and remarks that improved the exposition. In particular, the remark immediately following Theorem 2.1 and the Question 2 were motivated by the comments of one of them.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2015). Duals of variable exponent Hörmander spaces ($0< p^- \le p^+ \le 1$) and some applications. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 109(2):657-668. https://doi.org/10.1007/s13398-014-0209-zS6576681092Aboulaich, R., Meskine, D., Souissi, A.: New diffussion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008)Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002)Bastero, J.: $$l^q$$ l q -subspaces of stable $$p$$ p -Banach spaces, $$0 < p \le 1$$ 0 < p ≤ 1 . Arch. Math. (Basel) 40, 538–544 (1983)Boas, R.P.: Entire functions. Academic Press, London (1954)Boza, S.: Espacios de Hardy discretos y acotación de operadores. Dissertation, Universitat de Barcelona (1998)Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue spaces, foundations and harmonic analysis. Birkhäuser, Basel (2013)Cruz-Uribe, D.: SFO, A. Fiorenza, J. M. Martell, C. Pérez: The boundedness of classical operators on variable $$L^p$$ L p spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and sobolev spaces with variable exponents. lecture notes in mathematics, vol. 2007. Springer, Berlin, Heidelberg (2011)Hörmander, L.: The analysis of linear partial operators II, Grundlehren 257. Springer, Berlin, Heidelberg (1983)Hörmander, L.: The analysis of linear partial operators I, Grundlehren 256. Springer, Berlin, Heidelberg (1983)Kalton, N.J., Peck, N.T., Roberts, J.W.: An $$F$$ F -space sampler, London Mathematical Society Lecture Notes, vol. 89. Cambridge University Press, Cambridge (1985)Kalton, N.J.: Banach envelopes of non-locally convex spaces. Canad. J. Math. 38(1), 65–86 (1986)Kalton, N.J., Mitrea, M.: Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350(10), 3903–3922 (1998)Kalton, N.J.: Quasi-Banach spaces, Handbook of the Geometry of Banach Spaces, vol. 2. In: Johnson, W.B., Lindenstrauss, J. (eds.), pp. 1099–1130. Elsevier, Amsterdam (2003)Köthe, G.: Topological vector spaces I. Springer, Berlin, Heidelberg (1969)Motos, J., Planells, M.J., Talavera, C.F.: On variable exponent Lebesgue spaces of entire analytic functions. J. Math. Anal. Appl. 388, 775–787 (2012)Motos, J., Planells, M.J., Talavera, C.F.: A note on variable exponent Hörmander spaces. Mediterr. J. Math. 10, 1419–1434 (2013)Stiles, W.J.: Some properties of $$l_p$$ l p , $$0 < p < 1$$ 0 < p < 1 . Studia Math. 42, 109–119 (1972)Triebel, H.: Theory of function spaces. Birkhäuser, Basel (1983)Vogt, D.: Sequence space representations of spaces of test functions and distributions. In: Zapata, G.I. (ed.) Functional analysis, holomorphy and approximation theory, Lecture Notes in Pure and Applied Mathematics, vol. 83, pp. 405–443 (1983

Paternal obesity is associated with IGF2 hypomethylation in newborns: results from a Newborn Epigenetics Study (NEST) cohort

Data from epidemiological and animal model studies suggest that nutrition during pregnancy may affect the health status of subsequent generations. These transgenerational effects are now being explained by disruptions at the level of the epigenetic machinery. Besides in vitro environmental exposures, the possible impact on the reprogramming of methylation profiles at imprinted genes at a much earlier time point, such as during spermatogenesis or oogenesis, has not previously been considered. In this study, our aim was to determine associations between preconceptional obesity and DNA methylation profiles in the offspring, particularly at the differentially methylated regions (DMRs) of the imprinted Insulin-like Growth Factor 2 (IGF2) gene

Concepts of Cardiac Development in Retrospect

Recent research, enabled by powerful molecular techniques, has revolutionized our concepts of cardiac development. It was firmly established that the early heart tube gives rise to the left ventricle only, and that the remainder of the myocardium is recruited from surrounding mesoderm during subsequent development. Also, the cardiac chambers were shown not to be derived from the entire looping heart tube, but only from the myocardium at its outer curvatures. Intriguingly, many years ago, classic experimental embryological studies reached very similar conclusions. However, with the current scientific emphasis on molecular mechanisms, old morphological insights became underexposed. Since cardiac development occurs in an architecturally complex and dynamic fashion, molecular insights can only fully be exploited when placed in a proper morphological context. In this communication we present excerpts of important embryological studies of the pioneers of experimental cardiac embryology of the previous century, to relate insights from the past to current observations

Quantization of Midisuperspace Models

We give a comprehensive review of the quantization of midisuperspace models. Though the main focus of the paper is on quantum aspects, we also provide an introduction to several classical points related to the definition of these models. We cover some important issues, in particular, the use of the principle of symmetric criticality as a very useful tool to obtain the required Hamiltonian formulations. Two main types of reductions are discussed: those involving metrics with two Killing vector fields and spherically symmetric models. We also review the more general models obtained by coupling matter fields to these systems. Throughout the paper we give separate discussions for standard quantizations using geometrodynamical variables and those relying on loop quantum gravity inspired methods.Comment: To appear in Living Review in Relativit

Hybridization and invasive species in a threatened freshwater fish community under environmental pressures: Morphometric and molecular evidence

Mediterranean freshwater systems are under threat owing to increased drought driven by climate change, intensive human land uses and non-native species. This is causing increased fish hybridization in isolated watercourses. The genetic and morphological characteristics of hybrids of sympatric native and non-native fish species were studied in four streams of the Mediterranean Guadalquivir basin (south-west Spain). Fish morphology was analysed using geometric morphometrics, and molecular determination of parenthood was inferred through one mitochondrial gene (cytb) and one nuclear gene (Beta-actin) for all hybrids and a subset of pure parental specimens. Molecular analyses confirmed hybrids between the native Squalius alburnoides and non-native Alburnus alburnus in a stream with continuous flow. Haplotype analyses suggested that they originated from backcrossing of hybrid offspring. Intergeneric crosses between native species S. alburnoides and Pseudochondrostoma willkommii, and S. alburnoides and Iberochondrostoma lemmingii were detected in streams under reduced connectivity scenarios. Morphometrics revealed that hybrid phenotypes were similar to S. alburnoides. In some cases, molecular markers uncovered hybridization events that were neither detected in the field nor by morphometric analyses, potentially supporting a backcrossing/introgression scenario. Hybridization is likely to be increasing in Mediterranean rivers where S. alburnoides are present owing to increased fragmentation caused by summer drought exacerbated by climate change and human land uses and pressures. This can become a problem for these endemic vulnerable species if genetic diversity is lost, morphological homogenization occurs and hybrids cannot be easily detected in the field. The potential risks could be addressed by monitoring and eradication of non-native species and segregation from natives. To avoid native–native crosses, habitat quality and desiccation risk could be tackled by improved water quality and riparian reforestation to provide shade and reduce evapotranspiration. This would need increased coordination and intervention between the institutions that share conservation responsibilities in the area

Evolutionary and Experimental Assessment of Novel Markers for Detection of Xanthomonas euvesicatoria in Plant Samples

BACKGROUND: Bacterial spot-causing xanthomonads (BSX) are quarantine phytopathogenic bacteria responsible for heavy losses in tomato and pepper production. Despite the research on improved plant spraying methods and resistant cultivars, the use of healthy plant material is still considered as the most effective bacterial spot control measure. Therefore, rapid and efficient detection methods are crucial for an early detection of these phytopathogens. METHODOLOGY: In this work, we selected and validated novel DNA markers for reliable detection of the BSX Xanthomonas euvesicatoria (Xeu). Xeu-specific DNA regions were selected using two online applications, CUPID and Insignia. Furthermore, to facilitate the selection of putative DNA markers, a customized C program was designed to retrieve the regions outputted by both databases. The in silico validation was further extended in order to provide an insight on the origin of these Xeu-specific regions by assessing chromosomal location, GC content, codon usage and synteny analyses. Primer-pairs were designed for amplification of those regions and the PCR validation assays showed that most primers allowed for positive amplification with different Xeu strains. The obtained amplicons were labeled and used as probes in dot blot assays, which allowed testing the probes against a collection of 12 non-BSX Xanthomonas and 23 other phytopathogenic bacteria. These assays confirmed the specificity of the selected DNA markers. Finally, we designed and tested a duplex PCR assay and an inverted dot blot platform for culture-independent detection of Xeu in infected plants. SIGNIFICANCE: This study details a selection strategy able to provide a large number of Xeu-specific DNA markers. As demonstrated, the selected markers can detect Xeu in infected plants both by PCR and by hybridization-based assays coupled with automatic data analysis. Furthermore, this work is a contribution to implement more efficient DNA-based methods of bacterial diagnostics