We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of 
$\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution 
$u(x,t)$. Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to 
$L^p-$logarithmic Sobolev inequalities for 
$\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$. As a byproduct, Sobolev affine trace-type inequalities for 
$\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for 
$Q_{\nu /2}(\mathbb {R}^n)$ are established.
We construct new optimal 
$L^{p}$ Hardy-type inequalities for elliptic Schrödinger-type operators with a potential term.
$\boldsymbol {L^p}$
-HARDY AND
$\boldsymbol {L^p}$
-RELLICH TYPE INEQUALITIES WITH REMAINDER TERMS
In this paper we obtain some improved
$L^p$
-Hardy and
$L^p$
-Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.
We establish existence of weighted Hardy and Rellich inequalities on the spaces 
$L_{p}(\unicode[STIX]{x1D6FA})$, where 
$\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with 
$K$ a closed convex subset of 
$\mathbf{R}^{d}$. Let 
$\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of 
$\unicode[STIX]{x1D6FA}$ and 
$d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to 
$\unicode[STIX]{x1D6E4}$. We consider weighting functions 
$c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with 
$c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and 
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form and the Rellich inequalities are given by 
$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$ with 
$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$
$H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$. The constants 
$b_{p},d_{p}$ depend on the weighting parameters 
$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.
In this paper we study the problem
where Ω = ℝ
N
or Ω = B
1, N ⩾ 3, p > 1 and 
. Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.
Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operator
is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality
holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.
We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrödinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces 
and H10 of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r(2−N)/2 and 
of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.
Let 
$\mathbb{G}$ be a step-two nilpotent group of 
$\text{H}$-type with Lie algebra 
$\mathfrak{G}\,=\,V\,\oplus \,\text{t}$. We define a class of vector fields 
$X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter 
$k\,\ge \,1$, and we consider the corresponding 
$p$-Laplacian operator 
${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$. For 
$k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ are the left invariant vector fields corresponding to an orthonormal basis of 
$V$; for $\mathbb{G}$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator 
${{L}_{p,\,k}}$ and as an application, we get a Hardy type inequality associated with 
$X$.
In this paper, we establish several weighted 
${{L}^{p}}\left( 1\,<\,p\,\infty \right)$ Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates.
