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.

Predictive Ability of NGAL as a Marker of Renal Damage: evaluation of Multiple Clinical Settings

Introduction. Neutrophil Gelatinase-Associated Lipocalin (NGAL) belongs to the family of lipocalins and it is produced by several cell types, including renal tubular epithelium. In the kidney its production increases during acute damage and this is reflected by the increase in serum and urine levels. In animal studies and clinical trials, NGAL was found to be a sensitive and specific indicator of acute kidney injury (AKI). Purpose. The aim of this work was to investigate, in a prospective manner, whether urine NGAL can be used as a marker in preeclampsia, kidney transplantation, VLBI and diabetic nephropathy. Materials and methods. The study involved 44 consecutive patients who received renal transplantation; 18 women affected by preeclampsia (PE); a total of 55 infants weighing ≤1500 g and 80 patients with Type 1 diabetes. Results. A positive correlation was found between urinary NGAL and 24 hours proteinuria within the PE group. The detection of higher uNGAL values in case of severe PE, even in absence of statistical significance, confirms that these women suffer from an initial renal damage. In our population of VLBW infants, we found a positive correlation of uNGAL values at birth with differences in sCreat and eGFR values from birth to day 21, but no correlation was found between uNGAL values at birth and sCreat and eGFR at day 7. systolic an diastolic blood pressure decreased with increasing levels of uNGAL. The patients with uNGAL <25 ng/ml had significantly higher levels of systolic blood pressure compared with the patients with uNGAL >50 ng/ml ( p<0.005). Our results indicate the ability of NGAL to predict the delay in functional recovery of the graft. Conclusions. In acute renal pathology, urinary NGAL confirms to be a valuable predictive marker of the progress and status of acute injury.

Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.