A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications

A generalization of the classical Caffarelli–Kohn–Nirenberg inequality is obtained in the setting of Orlicz–Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a nonhomogeneous and degenerate/singular elliptic PDE.     

A Caffarelli–Kohn–Nirenberg type inequality on Riemannian manifolds

We establish a generalization to Riemannian manifolds of the Caffarelli–Kohn–Nirenberg inequality. The applied method is based on the use of conformal Killing vector fields and E. Mitidieri’s approach to Hardy inequalities.     

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

Fractional Caffarelli–Kohn–Nirenberg inequalities

We establish a full range of Caffarelli–Kohn–Nirenberg inequalities and their variants for fractional Sobolev spaces.     

Smooth metric measure spaces with weighted Poincaré inequality

In this paper, we study smooth metric measure space (M, g, e ^?f dv) satisfying a weighted Poincaré inequality and establish a rigidity theorem for such a space under a suitable Bakry–Émery curvature lower bound. We also consider the space of f-harmonic functions with finite energy and prove a structure theorem.     

Modulus and the Poincaré inequality on metric measure spaces

The purpose of this paper is to develop the understanding of modulus and the Poincaré inequality, as defined on metric measure spaces. Various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces. Consequently, modulus is shown to be upper semi-continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces. Moreover, several competing definitions for the Poincaré inequality are shown to coincide, if the underlying measure is doubling. One such characterization considers only continuous functions and their continuous upper gradients, and extends work of Heinonen and Koskela. Applications include showing that the p-Poincaré inequality (with a doubling measure), for p1, persists through to the limit of a sequence of converging pointed metric measure spaces — this extends results of Cheeger. A further application is the construction of new doubling measures in Euclidean space which admit a 1-Poincaré inequality. Mathematics Subject Classification (2000):31C15, 46E35.     

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.     

New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.     

Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities

We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings.     

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.     

A Weil–Petersson type metric on spaces of metric graphs

In this note, we discuss an analogue of the Weil–Petersson metric for spaces of metric graphs and some of its properties.     

Gagliardo–Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces

The aim of this paper is to give a Gagliardo–Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces over spaces of homogeneous type.