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.     

Poincaré's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set . Additionally, we study the specific case of monomial weights , showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.     

On the Brézis Nirenberg Stampacchia-type theorems and their applications

In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game.     

Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.     

Quasilinear elliptic problems with combined critical Sobolev–Hardy terms

In this paper, by investigating the effect of the subcritical terms and the coefficients of the singular terms, some existence results for quasilinear elliptic problems involving combined critical Sobolev–Hardy terms are obtained via variational methods.     

A remark on the Brézis–Browder theorem in finite dimensional spaces

Brézis and Browder gave an important characterization of maximal monotonicity for linear operators, in reflexive Banach spaces. In this paper, we show that this characterization also preserves the maximal p-monotonicity, in finite dimensional spaces.     

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.     

On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo–Nirenberg and Sobolev in ${\mathbb{R}}^d$, formulated in terms of the Laplacian $\Delta$ and of the fractional powers $D^n?{\sqrt{?\Delta }}^n$ with real $n?0$; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ${?}^2$ case where, for all sufficiently regular $f:{\mathbb{R}}^d\to \mathbb{C}$, the norm $6D^{j}f{6}_{{?}^r}$ is bounded in terms of $6f{6}_{{?}^2}$ and $6D^{n}f{6}_{{?}^2}$, for $1∕r=1∕2?(?n?j)∕d$, and suitable values of $j,n,?$ (with $j,n$ possibly noninteger). In the special cases $?=1$ and $?=j∕n+d∕2n$ (i.e., $r=+\infty$), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ${?}^2$ case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.     

Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents

We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where ${ \mu \ge 0, 0 < s < 2, 0 \in \partial \Omega}$ and Ω is a bounded domain in R ^N . We prove that if ${N \ge 7, a(0) > 0}$ and all the principle curvatures of ?Ω at 0 are negative, then the above problem has infinitely many solutions.     

Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Ricerche di Matematica - Let $$X = {X_1,X_2, ldots ,X_m}$$ be a system of smooth vector fields in $${{mathbb R}^n}$$ satisfying the Hörmander’s finite rank condition. We prove the...     

Hardy–Littlewood–Sobolev inequality on the parabolic biangle

The Ramanujan Journal - We define the heat semigroup associated with a system of bivariate Jacobi polynomials which are orthogonal with respect to a probability measure on the parabolic biangle...