| Abstract: |
| The study of Hardy inequalities started in 1925 with a seminal paper by G.H. Hardy. For $s\in (0,1), p \in (1,\frac{N}{s})$ and $w \in L^1_{\text{loc}}(\mathbb{R}^{N})$, we study the following fractional Hardy inequality
\begin{align} \label{pr1}
\displaystyle \int_{\mathbb{R}^N} |w(x)||u(x)|^p\, \mathrm{d}x \leq C \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}} \mathrm{d}x\mathrm{d}y:= \|u\|_{s,p}^p\,, \ \forall u \in \mathcal{D}^{s,p}(\mathbb{R}^N), \tag{FH}
\end{align}
for some positive constant $C=C(N,s,p)$ depending on $N,s$ and $p$ only. The space $\mathcal{D}^{s,p}(\mathbb{R}^N)$ is the completion of $C_c^\infty(\mathbb{R}^N)$ with respect to the norm $\|\cdot\|_{s,p}$. The set of all $w$ satisfying \eqref{pr1} is denoted by $\mathcal{H}_{s,p}(\mathbb{R}^N)$. The space $\mathcal{H}_{s,p}(\mathbb{R}^N)$ admits a Banach function space structure using Maz`ya-type characterization of capacity functions. Our aim is also to look for the least possible constant such that equality holds in \eqref{pr1} for some $u \in \mathcal{D}^{s,p}(\mathbb{R}^N)$. The attainment of the least possible constant in \eqref{pr1} depends on the compactness of the map ${W}(u)= \int_{{\mathbb{R}^{N}}} |w| |u|^p\, \mathrm{d}x$ on $\mathcal{D}^{s,p}(\mathbb{R}^N)$. The Banach function space structure of $\mathcal{H}_{s,p}(\mathbb{R}^N)$ and the concentration-compactness type arguments help us in characterizing the compactness of the map ${W}(u)= \int_{{\mathbb{R}^{N}}} |w| |u|^p\, \mathrm{d}x$ on $\mathcal{D}^{s,p}(\mathbb{R}^N)$. As an application, we study the qualitative and quantitative behavior of the eigenvalues of the weighted eigenvalue problem $(-\Delta)_{p}^{s}u = \lambda w |u|^{p-2}u ~~\text{in}~\mathbb{R}^{N}$. |
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