Aims-To evaluate whether endothelin-1 is involved in the pathology of idiopathic pulmonary fibrosis (IPF). Methods-Plasma endothelin-1 concentrations were evaluated in 37 patients with IPF and 27 normal controls by radioimmunoassay. In addition, expression of endothelin-1 in lung tissue was evaluated in biopsy specimens obtained from four patients with IPF. Three biopsy specimens of normal lung were used as controls. Endothelin-1 immunoreactivity was detected using immunohistochemistry. Results-Elevated endothelin-l plasma concentrations were found in patients with IPF compared with controls and a positive correlation was found with duration ofdisease. No significant difference was observed between treated and untreated patients with IPF. Increased endothelin-1 immunoreactivity was found in lungs of three of four patients with IPF. Endothelin-l positive cells consisted mainly of small vessel endothelial cells. Some scattered macrophages were also positive. Conclusions-Elevated plasma concentrations and expression of endothelin-1 in lung tissue are suggestive of increased production of endothelin-1 in at least a proportion of patients with IPF. Consequently, endothelin-l activity could play a role in the fibrogenic process of the disease. (J7 Clin Pathol 1995;48:330-334)

We consider a class of ultraparabolic differential equations that satisfy the Hörmander's hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser's iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.

We consider a class of second order ultraparabolic differential equations in the formwhere A = (a ij ) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, under the assumption that the operator m i=1 X 2 i + X 0 − ∂ t is hypoelliptic and the vector fields X 1 , . . . , X m and X 0 − ∂ t are invariant with respect to a suitable homogeneous Lie group. We adapt the Moser's iterative methods to the non-Euclidean geometry of the Lie groups and we prove an L ∞ loc bound of the solution u in terms of its L p loc norm. We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator m i,j =1 X i (a ij X j ) + X 0 − ∂ t . The bound is given in terms of the value function of an optimal control problem related to the vector fields X 1 , . . . , X m and X 0 − ∂ t . Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients a ij .

In this paper we are concerned with Harnack inequalities for non-negative solutions u : → R to a class of second order hypoelliptic ultraparabolic partial differential equations in the formwhere is any open subset of R N+1 , and the vector fields X 1 , . . . , X m and X 0 − ∂ t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x 0 , t 0 ) ∈ we give a geometric sufficient condition on the compact sets K ⊆ for which the Harnack inequalityholds for all non-negative solutions u to the equation L u = 0. We also compare our result with an abstract Harnack inequality from potential theory.

We obtain two-sided bounds for the density of stochastic processes satisfying a weak Hörmander condition. In particular we consider the cases when the support of the density is not the whole space and when the density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hörmander's theorem). The proofs of our lower bounds are based on Harnack inequalities for positive solutions of PDEs whereas the upper bounds are derived from the probabilistic representation of the density given by the Malliavin calculus.We will particularly focus on processes satisfying a weak Hörmander condition, that is Rank(Lie{Y 1 , · · · , Y n , −∂ t }(x)) < N + 1, ∀ x ∈ R N . This means that the first order vector field Y 0 (or equivalently the drift term of the SDE) is needed to span all the directions.As leading examples we have in mind processes of the form