A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$

In the paper we prove the weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int _{\mathbb {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$ ∫ R N V φ 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + K ∫ R N φ 2 μ ( x ) d x , for functions $$\varphi $$ φ in a weighted Sobolev space $$H^1_\mu $$ H μ 1 , for a wider class of potentials V than inverse square potentials and for weight functions $$\mu $$ μ of a quite general type. The case $$\mu =1$$ μ = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators $$\begin{aligned} Lu=\varDelta u+\frac{\nabla \mu }{\mu }\cdot \nabla u \end{aligned}$$ L u = Δ u + ∇ μ μ · ∇ u perturbed by singular potentials.

Self-improving properties of discrete Muckenhoupt weights

In this paper, we will provide a complete study of the self-improving properties of the discrete Muckenhoupt class 𝒜 p ⁢ ( 𝒞 ) {\mathcal{A}^{p}(\mathcal{C})} of weights defined on ℤ + {\mathbb{Z}_{+}} . In addition, we will determine the range of the new constants which are related to the original constants via an algebraic equation. For illustration, we will give an example to prove that the results are sharp. The results will be obtained by employing a discrete version of an inequality due to Hardy–Littlewood and a new discrete Hardy-type inequality with negative powers.

Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives

In this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ T = R and $\mathbb{T=N}$ T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

Picone’s identity for -Laplace operator and its applications

UDC 517.9We prove a nonlinear analogue of Picone's identity for -Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.

On a new generalization of some Hilbert-type inequalities

In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.