Related papers: Sharp Hardy type inequalities on the Carnot Group
We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.
We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the…
By employing harmonic analysis techniques, we derive weak-type Caffarelli-Kohn-Nirenberg inequalities under natural parameter conditions. A key feature of these weak-type versions is that they remain valid even at critical parameter values…
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega_\alpha(f,t)_q$ and $\omega_\beta(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and…
In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by…
We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy…
In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big\|\widehat{f \sigma}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim…
This article is concerned with a class of elliptic equations on Carnot groups depending of one real positive parameter and involving a critical nonlinearity. As a special case of our results we prove the existence of at least one nontrivial…
In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these…
We characterize the rigidity of Carnot groups in the class of $C^2$ contact maps in terms of complex characteristics. Furthermore, we obtain a Liouville type theorem for Carnot groups which states that 1-quasiconformal maps form finite…
The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Krist\'aly, Huang, and Zhao [Trans. Am. Math.…
In this paper, we proved a sharp Minkowski type inequality in Cartan-Hadamard 3-spaces by harmonic mean curvature flow and improves the known estimates for total mean curvature in hyperbolic 3-space. In particular, we sharpened…
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in…
In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of…
We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.
In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} \Delta(|x|^{-\gamma}\Delta u)=|x|^\gamma u^{\frac{N+4+3\gamma}{N-4-\gamma}},\quad u\geq 0 \quad…
We investigate the weighted fractional order Hardy inequality $$ \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partial\Omega)^{-\alpha}\text{dist}(y,\partial\Omega)^{-\beta}\,dy\,dx\geq…
We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form \[ \int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla…
In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold $M$. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic…
In this paper we prove an isoperimetric inequality for holomorphic functions in the unit polydisc $\mathbf U^n$. As a corollary we derive an inclusion relation between weighted Bergman and Hardy spaces of holomorphic functions in the…