Optimal embeddings for fractional Orlicz-Sobolev spaces into (generalized) Campanato spaces on the Euclidean space are exhibited. Embeddings into vanishing Campanato spaces are also characterized. Sharp embeddings into BMO(R-n) and VMO(R-n) are derived as special instances. Dissimilarities to corresponding embeddings for classical fractional Sobolev spaces are pointed out.

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on Rn to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

Some recent results on the theory of fractional Orlicz–Sobolev spaces are surveyed. They concern Sobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, and criteria for compact embeddings. The limits of these spaces when the smoothness parameter s ∈ (0, 1) tends to either of the endpoints of its range are also discussed. This note is based on recent papers of ours, where additional material and proofs can be found.

A necessary and sufficient condition for fractional Orlicz–Sobolev spaces to be continuously embedded into L∞(Rn) is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into L∞(Rn) fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.

A necessary and sucient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into L1(Rn) is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also oered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into L1(Rn) fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.

An optimal embedding theorem for fractional Orlicz-Sobolev spaces into Orlicz spaces will be surveyed. A new embedding for the same fractional spaces into generalized Campanato spaces will be also presented. This is a joint work, in progress, with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

The optimal Orlicz target space and the optimal rearrangement- invariant target space are exhibited for embeddings of fractional-order Orlicz-Sobolev spaces. Both the subcritical and the supercritical regimes are considered. In particular, in the latter case the relevant Orlicz-Sobolev spaces are shown to be embedded into the space of bounded continuous functions in R^n. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

The optimal Orlicz target space and the optimal rearrangement- invariant target space are exhibited for embeddings of fractional-order Orlicz-Sobolev spaces. Both the subcritical and the supercritical regimes are considered. In particular, in the latter case the relevant Orlicz-Sobolev spaces are shown to be embedded into the space of bounded continuous functions in Rn. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

Let X = {X-1, X-2,..., X-m} be a system of smooth vector fields in R-n satisfying the Hormander's finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Caratheodory space G associated to system X(1/integral(BR) K(x) dx integral(BR) vertical bar u vertical bar(t) K (x) dx)(1/t) <= C R (1/integral(BR) 1/K(x) dx integral(BR) vertical bar Xu vertical bar(2)/K(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 tau, where tau is a suitable exponent related to the homogeneous dimension.

The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz-Sobolev spaces in $R^n$. An improved embedding with an Orlicz-Lorentz target space, which is optimal in the broader class of all rearrangement-invariant spaces, is also established. Both spaces of order s in (0, 1), and higher-order spaces are considered. Related Hardy type inequalities are proposed as well.

Some recent results on the theory of fractional Orlicz-Sobolev spaces are surveyed. They concernSobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, andcriteria for compact embeddings. The limits of these spaces when the smoothness parameter s in (0, 1) tendsto either of the endpoints of its range are also discussed. This note is based on the papers [1, 2, 3, 4], whereadditional material and proofs can be found.

Extended versions of the Bourgain-Brezis-Mironescu theorems on the limit as s->1^- of the Gagliardo-Slobodeckij fractional seminorm are established in the Orlicz space setting. The results hold for fractional Orlicz-Sobolev spaces built upon general Young functions, as well. The case of Young functions with an asymptotic linear growth is also considered in connection with the space of functions of bounded variation. An extended version of the Maz'ya-Shaposhnikova theorem on the limit as s->0^+ of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. The result holds in fractional Orlicz-Sobolev spaces associated with Young functions fulfilling the \Delta_2-condition, and, as shown by counterexamples, it may fail if this condition is dropped. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

We establish versions for fractional Orlicz-Sobolev seminorms, built upon Young functions, 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^-, dealing with classical fractional Sobolev spaces. As regards the limit as s ->1^-, Young functions with an asymptotic linear growth are also considered in connection with the space of functions of bounded variation. Concerning the limit as s->0^+, Young functions fulfilling the \Delta_2-condition are admissible. Indeed, counterexamples show that our result may fail if this condition is dropped. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

The optimal Orlicz target space and the optimal rearrangement-invariant tar- get 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 frac- tional 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.

The existence of eigenfunctions for a class of fully anisotropic elliptic equations is established. The relevant equations are associated with constrained minimization problems for integral func- tionals depending on the gradient of competing functions through general anisotropic Young 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. In particular, our analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces. This is a joint work with G. di Blasio and F. Feo.

Optimal embeddings for fractional-order Orlicz{Sobolev spaces into Campanato type s- paces are offered. This is a joint work in progress with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

The existence of eigenfunctions for a class of fully anisotropic elliptic equations is estab- lished. The relevant equations are associated with constrained minimization problems for inte- gral functionals depending on the gradient of competing functions through general anisotropic Young functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called 2-condition. In particular, our analysis re- quires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces. This is a joint work with G. di Blasio and F. Feo [1].

Compact embeddings on domain for fractional Orlicz-Sobolev spaces are exhibited.

An extended version of the Maz'ya-Shaposhnikova theorem on the limit as s -> 0+ of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz-Sobolev spaces associated with Young functions satisfying the \Delta2-condition, and, as shown by counterexamples, it may fail if this condition is dropped.

Extended versions of the Bourgain-Brezis-Mironescu theorems on the limit as s->1^- of the Gagliardo-Slobodeckij fractional seminorm are established in the Orlicz space setting. Our results hold for fractional Orlicz-Sobolev spaces built upon general Young functions, and complement those of [13], where Young functions satisfying the $\Delta_2$ and the $\nabla_2$ conditions are dealt with. The case of Young functions with an asymptotic linear growth is also considered in connection with the space of functions of bounded variation.