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.

Analisi dei cantieri di Cippatura in merito ad aspetti operativi e di salvaguardia degli operatori

La cippatura è un processo produttivo fondamentale nella trasformazione della materia prima forestale in biomassa combustibile che coinvolgerà un numero sempre più crescente di operatori. Scopo dello studio è stato quantificare la produttività e il consumo di combustibile in 16 cantieri di cippatura e determinare i livelli di esposizione alla polvere di legno degli addetti alla cippatura, in funzione di condizioni operative differenti. Sono state identificate due tipologie di cantiere: uno industriale, con cippatrici di grossa taglia (300-400kW) dotate di cabina, e uno semi-industriale con cippatrici di piccola-media taglia (100-150kW) prive di cabina. In tutti i cantieri sono stati misurati i tempi di lavoro, i consumi di combustibile, l’esposizione alla polvere di legno e sono stati raccolti dei campioni di cippato per l’analisi qualitativa. Il cantiere industriale ha raggiunto una produttività media oraria di 25 Mg tal quali, ed è risultato 5 volte più produttivo di quello semi-industriale, che ha raggiunto una produttività media oraria di 5 Mg. Ipotizzando un utilizzo massimo annuo di 1500 ore, il cantiere semi-industriale raggiunge una produzione annua di 7.410 Mg, mentre quello industriale di 37.605 Mg. Il consumo specifico di gasolio (L per Mg di cippato) è risultato molto minore per il cantiere industriale, che consuma in media quasi la metà di quello semi-industriale. Riguardo all’esposizione degli operatori alla polvere di legno, tutti i campioni hanno riportato valori di esposizione inferiori a 5 mg/m3 (limite di legge previsto dal D.Lgs. 81/08). Nei cantieri semi-industriali il valore medio di esposizione è risultato di 1,35 mg/m3, con un valore massimo di 3,66 mg/m3. Nei cantieri industriali si è riscontrato che la cabina riduce drasticamente l’esposizione alle polveri di legno. I valori medi misurati all’esterno della cabina sono stati di 0,90 mg/m3 mentre quelli all’interno della cabina sono risultati pari a 0,20 mg/m3.

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.

Estimation of Quasi-Rational DSGE Models

Small-scale dynamic stochastic general equilibrium have been treated as the benchmark of much of the monetary policy literature, given their ability to explain the impact of monetary policy on output, inflation and financial markets. One cause of the empirical failure of New Keynesian models is partially due to the Rational Expectations (RE) paradigm, which entails a tight structure on the dynamics of the system. Under this hypothesis, the agents are assumed to know the data genereting process. In this paper, we propose the econometric analysis of New Keynesian DSGE models under an alternative expectations generating paradigm, which can be regarded as an intermediate position between rational expectations and learning, nameley an adapted version of the "Quasi-Rational" Expectatations (QRE) hypothesis. Given the agents' statistical model, we build a pseudo-structural form from the baseline system of Euler equations, imposing that the length of the reduced form is the same as in the `best' statistical model.