Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Effect of training systems and pruning methods on fruit quality in apple

The introduction of dwarfed rootstocks in apple crop has led to a new concept of intensive planting systems with the aim of producing early high yield and with returns of the initial high investment. Although yield is an important aspect to the grower, the consumer has become demanding regards fruit quality and is generally attracted by appearance. To fulfil the consumer’s expectations the grower may need to choose a proper training system along with an ideal pruning technique, which ensure a good light distribution in different parts of the canopy and a marketable fruit quality in terms of size and skin colour. Although these aspects are important, these fruits might not reach the proper ripening stage within the canopy because they are often heterogeneous. To describe the variability present in a tree, a software (PlantToon®), was used to recreate the tree architecture in 3D in the two training systems. The ripening stage of each of the fruits was determined using a non-destructive device (DA-Meter), thus allowing to estimate the fruit ripening variability. This study deals with some of the main parameters that can influence fruit quality and ripening stage within the canopy and orchard management techniques that can ameliorate a ripening fruit homogeneity. Significant differences in fruit quality were found within the canopies due to their position, flowering time and bud wood age. Bi-axis appeared to be suitable for high density planting, even though the fruit quality traits resulted often similar to those obtained with a Slender Spindle, suggesting similar fruit light availability within the canopies. Crop load confirmed to be an important factor that influenced fruit quality as much as the interesting innovative pruning method “Click”, in intensive planting systems.