Let X={X1,X2,...,Xm} be a system of smooth vector fields in R^n satisfying the Hörmander's finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space G associated to system X (1?BRK(x)dx?BR|u|tK(x)dx)1/t<=CR??1?BR1K(x)dx?BR|Xu|2K(x)dx??1/2, where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt's class A_2 and Gehring's class G_?, where ? is a suitable exponent related to the homogeneous dimension.

We study an eigenvalue problem involving a fully anisotropic elliptic differential operator in arbitrary Orlicz-Sobolev spaces. The relevant equations are associated with constrained minimization problems for integral functionals depending on the gradient of competing functions through general anisotropic N-functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called \Delta_2-condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.

The optimal Orlicz target space and the optimal rearrangement- invariant target space are exhibited for embeddings of fractional-order Orlicz- Sobolev spaces W^{s,A}(R^n). Related Hardy type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of the Bourgain-Brezis-Mironescu theorem on the limit as s->1^- and of the Maz'ya-Shaposhnikova theorem on the limit as s ->0^+ are established. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained by symmetrization techniques. The anisotropy of the principal part of the operator is governed by a general n-dimensional Young function of the gradient which is not necessarily of polynomial type and need not satisfy the $\Delta_2$-condition.

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N-function, which is not necessarily of power-type and need not satisfy the $\Delta_2$ nor the $\nabla_2$ -condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions--in the approximable sense--is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L^1-data.

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N-function, which is not necessarily of power-type and need not satisfy the ? nor the ? -condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions--in the approximable sense--is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L-data.

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

We are concerned with a priori estimates, in rearrangement form, for weak solutions to fully anisotropic, nonlinear elliptic equations with lower-order terms whose prototype is \begin{equation*} \left\{ \begin{array} [c]{lll} -\hbox{\rm div} \; (a(x, u, \nabla u)) + b(u)=f(x) & \qquad\hbox{\rm in\ } \Omega \\ u=0 & \qquad\text{on}\;\partial\Omega. \end{array} \right. \end{equation*} Here, $\Omega$ is an open bounded set in $\mathbb{R}^{N}$, with $N\geq2$, $a(x, \eta, \xi)$ is a Carath\'{e}odory function fulfilling \begin{equation*} a(x,\eta,\xi)\cdot\xi\geq\Phi\left( \xi\right) \qquad \text{ for } \left( \eta,\xi\right) \in\mathbb{R}\times\mathbb{R}^{N}, \; \text{ for a. e. } x\in\Omega, \end{equation*} where $\Phi :\mathbb{R}^{N}\rightarrow\left[ 0,+\infty\right[ $ is an $N-$dimensional Young function, and $b:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous and strictly increasing function such that $b\left( 0\right)=0$. Finally, $f:\Omega \rightarrow\mathbb{R}$ is a nonnegative measurable function enjoying suitable integrability conditions.

A sharp integrability condition on the right-hand side of the p-Laplace system for all its solutions to be continuous is exhibited. Their uniform continuity is also analyzed and estimates for their modulus of continuity are provided. The relevant estimates are shown to be optimal as the right-hand side ranges in classes of rearrangement-invariant spaces, such as Lebesgue, Lorentz, Lorentz-Zygmund, and Marcinkiewicz spaces, as well as some customary Orlicz spaces

Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semidiscretization method in time after providing estimates for solutions to anisotropic elliptic problems with zero-order terms.

We obtain a comparison result for solutions to nonlinear fully anisotropic elliptic problems by means of anisotropic symmetrization. As consequence we deduce a priori estimates for norms of the relevant solutions.

We prove estimates for weak solutions to a class of Dirichlet problems associated to anisotropic elliptic equations with a zero order term.

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

The problem is addressed of the maximal integrability of the gradient of solutions to quasilinear elliptic equations, with merely measurable coefficients, in two variables. Optimal results are obtained in the framework of Orlicz spaces, and in the more general setting of all rearrangement-invariant spaces. Applications to special instances are exhibited, which provide new gradient bounds, or improve certain results available in the literature. (C) 2016 Elsevier Ltd. All rights reserved.

Estimates for solutions to homogeneous Dirichlet problems for a class of elliptic equations with zero order term in the form L(u) = g(x, u) + f (x),where the operator L fulfills an anisotropic elliptic condition, are established. Such estimates are obtained in terms of solutions to suitable problems with radially symmetric data, when no sign conditions on g are required.

We correct an error in the proof of Theorem 4.1 of the paper "Boundedness of solutions to anisotropic variational problems" [Comm. Part. Diff. Eq. 36 (2011); 470-486].

The problem is addressed of the maximal integrability of the gradient of solutions to quasilinear elliptic equations, with merely measurable coefficients, in two variables. Optimal results are obtained in the framework of Orlicz spaces, and in the more general setting of all rearrangement-invariant spaces. Applications to special instances are exhibited, which provide new gradient bounds, or improve certain results available in the literature.

Solutions to the n-Laplace equation with a right-hand side f are considered. We exhibit the largest rearrangement-invariant space to which f has to belong for every local weak solution to be continuous. Moreover, we find the optimal modulus of continuity of solutions when f ranges in classes of rearrangement-invariant spaces, including Lorentz, Lorentz-Zygmund and various standard Orlicz spaces.