arXiv subjects

Vivek Sahu

Publications and source records attributed to Vivek Sahu.

14 recordsLinked to original sources

arXiv2026

Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality

We create counterexamples to show that Pólya-Szegő type of inequality fails for the weighted nonlocal seminorm \begin{equation*} [u]_{W^{s,p, α}(\mathbb{R}^d)} := \left( \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|x|^{α_{1} p} |y|^{α_{2} p} |u(x)-u(y)|^{p}}{|x-y|^{d+sp}} \, dx \, dy \right)^{\frac{1}{p}} \ , \end{equation*} in all three regimes $α_1 +α_2 < 0, \, α_1 +α_2 = 0$, and $α_1 +α_2 >0$. These counterexamples are such that it completely settles the first two cases for all parameters. We also provide necessary and sufficient conditions on the parameters for $ [u]_{W^{s,p, α}(\mathbb{R}^d)} < \infty$, thus extending an earlier known result of [Serena Dipierro and Enrico Valdinoci, A density property for fractional weighted Sobolev spaces]. We also provide another proof of the fractional C-K-N type inequality, for most of the parameters (Hardy and Sobolev are included), using a familiar technique which is involved in the proof of fractional Sobolev inequality of [Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences mathematiques] together with a symmetrization technique.

math.AP↗

arXiv2026

Logarithmic fractional Hardy inequalities

We establish sharp fractional Hardy inequalities in the critical regimes $sp=d$ and $sp=1$, with explicit logarithmic weight functions. In the whole-space setting, for $p>d\ge 1$, $sp=d$, we prove that \begin{equation*} \int_{\mathbb R^d} \int_{\mathbb R^d} \frac{|u(x)-u(y)|^p}{|x-y|^{2d}}\,dx\,dy \geq \left(\frac{p-1}{p}\right)^p \int_{B_R}|u(x)|^p W_{p,d,R}(x)\,dx, \end{equation*} for $u\in C_c(B_R)$, where $W_{p,d,R}$ is a logarithmic-type weight and the constant is sharp. We also show that the corresponding inequality with the simpler weight $|x|^{-d}(\log(R/|x|))^{-p}$ cannot hold with any positive constant. Furthermore, we derive two-sided estimates and asymptotic formulae for $W_{p,d,R}$, including its limiting connection with the classical Hardy inequality. In the half-space $\mathbb{R}^{d}_{+}$, we establish an analogous sharp fractional Hardy inequality in the critical case $sp=1$, with an explicit logarithmic weight depending on the distance to the boundary. We also investigate the asymptotic behaviour of this weight, obtaining estimates that describe its behaviour near the boundary. These results elucidate the role of logarithmic weights in critical fractional Hardy inequalities in both whole-space and half-space settings.

math.AP↗

arXiv2026

Hardy-Sobolev-Maz'ya inequalities for $\boldsymbol{1<p<2}$

We establish Hardy--Sobolev--Maz'ya inequalities in the subquadratic range $1<p<2$ for singularities supported on affine subspaces and for arbitrary domains. We first prove a Hardy--Sobolev--Maz'ya inequality for singularities supported on affine subspaces of arbitrary codimension $1\leq k<N$, showing that the deficit in the sharp Hardy inequality controls a suitable weighted Sobolev norm. Combining the same convexity estimate with the one-dimensional approach of Frank and Loss in [9], we also establish Hardy--Sobolev--Maz'ya inequalities on arbitrary domains with the mean distance to the boundary and thereby solve a problem posed in [9]. These results complete the previously unresolved subquadratic range $1<p<2$ and, together with the known results for $p\geq 2$, provide a unified Hardy--Sobolev--Maz'ya theory for all $1<p<N$ associated with plane singularities and general domains.

math.AP↗

arXiv2026

Quantitative stability for weighted Hardy inequalities: Local and nonlocal cases

We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for $p>1$, $α\in[0,1)$, $0<p-αp<N$, and $u\in C_c^1(\mathbb{R}^N)$ normalized by $\int_{\mathbb{R}^N}|u|^p|x|^{αp-p}\,dx=1$, we prove \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^p|x|^{αp}\,dx-\Big(\frac{N-p+αp}{p}\Big)^p\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{p-αp}}\,dx \ge C\,\operatorname{dist}_α(u,\mathcal{M}_α)^{\max\{4,2p\}}, \end{equation*} where $\mathcal{M}_α$ is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for $s\in(0,1)$, $α=α_1+α_2\ge0$, $α_1p,α_2p\in(-N,sp)$, and $0<sp-αp<N$, the fractional Hardy deficit satisfies \begin{equation*} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}|x|^{α_1p}|y|^{α_2p}\,dx\,dy-\mathcal{C}\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{sp-αp}}\,dx \ge C\,\operatorname{dist}_{s,α}(u,\mathcal{M}_{s,α})^{\max\{4,2p\}}. \end{equation*} For $p\ge2$ the distance is measured on $u$, while for $1<p<2$ it is measured on a power-type transformation of $u$. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré--Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between $p\ge2$ and $1<p<2$ stems from the convexity estimate used to obtain a remainder term from the deficit.

math.AP↗

arXiv2026

Trudinger-Moser type inequality in fractional Sobolev space with singularity on smooth submanifold

We prove a Trudinger-Moser type inequality in fractional Sobolev spaces with singularities on smooth compact sets of codimension $k$, where $1 < k < d$ and $sp = d$. The singular term is given by the inverse $d$-th power of the distance to the submanifold. The proof is based on a fractional Hardy inequality adapted to smooth submanifolds, and we show the sharpness of the constant. We also establish the equivalence of two natural fractional Sobolev spaces vanishing on the singular set.

math.AP↗

arXiv2026

Improved quantitative stability for the critical Hardy inequality

We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N^{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg norm of the critical exponential Orlicz space $\operatorname{Exp}L^{\frac{N}{N-1}}(Ω)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N^{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.

math.AP↗

arXiv2026

Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis

A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$ δ_{s,p}(u):=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy-\mathcal{C}_{N,s,p}\int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx\ge0, $$ for all $u\in W^{s,p}(\mathbb{R}^N)$. The optimal constant is explicitly known. We investigate quantitative refinements of this inequality. Our first result shows that, under the normalization $ \int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx=1,$ the inequality \[ δ_{s,p}(u)\gtrsim\bigl(\mathrm{dist}_{s,p}(u,\mathcal{Z})\bigr)^α, \] holds, where $α=\max\{4,2p\}$, $\mathcal{Z}$ denotes the family of ``virtual'' extremals, and the distance is measured in Marcinkiewicz (weak-$L^{p_s^*}$) space. The stability exponent remains constant for $p\le2$, while it depends on $p$ for $p>2$. Our approach is based on a localized Poincaré-Sobolev inequality combined with suitable rescaling and Lorentz embeddings. We exploit a decomposition of the nonlocal energy together with Lorentz estimates, which enables us to control the deficit $δ_{s,p}(u)$ in terms of the distance to $\mathcal{Z}$. The method also applies to the local case $s=1$, the argument is rearrangement-free and the exponent in the stability estimate improves the existing literature. For $p=2$, via an Emden-Fowler correspondence and pseudo-differential operators, we show that the nonlocal Hardy deficit coincides with the local one and obtain quantitative stability on $\mathbb{R}\times\mathbb{S}^{N-1}$ using the diagonalization of the fractional Hardy quadratic form due to Frank, Lieb, and Seiringer. As an application, we establish a Hardy-Heisenberg-type uncertainty principle in the nonlocal setting, which appears to be new in the literature.

math.AP↗

arXiv2026

Fractional Sobolev logarithmic inequalities

We establish new Euclidean Sobolev logarithmic inequalities in the framework of fractional Sobolev spaces and their weighted version. Our approach relies on a interpolation inequality, which can be viewed as a fractional Caffarelli-Kohn-Nirenberg type inequality. We further relate the optimal constant in this interpolation inequality to a corresponding variational problem. These results extend classical Sobolev logarithmic inequalities to the nonlocal Euclidean framework and provide new tools for analysis in fractional Sobolev spaces.

math.AP↗

arXiv2026

Fractional boundary Hardy inequality for the critical cases

We establish generalised fractional boundary Hardy-type inequality, in the spirit of Caffarelli-Kohn-Nirenberg inequality for different values of $s$ and $p$ on various domains in $\mathbb{R}^d, ~ d \geq 1$. In particular, for Lipschitz bounded domains any values of $s$ and $p$ are admissible, settling all the cases in subcritical, supercritical and critical regime. In this paper we have solved the open problems posed by Dyda for the critical case $sp =1$. Moreover we have proved the embeddings of $W^{s,p}_{0}(Ω)$ in subcritical, critical and supercritical uniformly without using Dyda's decomposition. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case.

math.AP↗

arXiv2025

Fractional Hardy inequality with singularity on submanifold

We establish fractional Hardy inequality on bounded domains in $\mathbb{R}^{d}$ with inverse of distance function from smooth boundary of codimension $k$, where $k=2, \dots,d$, as weight function. The case $sp=k$ is the critical case, where optimal logarithmic corrections are required. All the other cases of $sp k$ are also addressed.

math.AP↗

arXiv2025

Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold

We investigate the sharp constant for weighted fractional Hardy inequalities with the singularity on a flat submanifold of codimension $k$, where $1\leq k<d$. We also prove a weighted fractional Hardy inequality with a remainder. Using this result, we extend and derive a weighted version of the fractional Hardy-Sobolev-Maz'ya inequality with singularities on a flat submanifold. Furthermore, we obtain a weighted logarithmic fractional Hardy-Sobolev-Maz'ya inequality in the case of a singularity at the origin and we show that in this case, the fractional Hardy-Sobolev-Maz'ya inequality does not hold.

math.AP↗

arXiv2024

The Trudinger type inequality in fractional boundary Hardy inequality

We establish Trudinger-type inequality in the context of fractional boundary Hardy-type inequality for the case $sp=d$, where $p>1, ~ s \in (0,1)$ on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$. In particular, we establish fractional version of Trudinger-type inequality with an extra singular function, namely $d$-th power of the distance function from $\partial Ω$ in the denominator of the integrand. The case $d=1$, as it falls in the category $sp=1$, becomes more delicate where an extra logarithmic correction is required together with subtraction of an average term.

math.AP↗

arXiv2024

Weighted fractional Hardy inequalities with singularity on any flat submanifold

We extend the work of Dyda and Kijaczko by establishing the corresponding weighted fractional Hardy inequalities with singularities on any flat submanifolds. While they derived weighted fractional Hardy inequalities with singularities at a point and on a half-space, we generalize these results to handle singularities on any flat submanifold of codimension $k$, where $1<k<d$. Furthermore, we also address the critical case $sp=k+α+ β$ and establish weighted fractional Hardy inequality with appropriate logarithmic weight function.

math.AP↗

arXiv2024

Boundary Hardy inequality on functions of bounded variation

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~Ω$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(Ω)$, $$\int_Ω \frac{|u(x)|^{p}}{δ^{p}_Ω(x)} dx \leq C\int_Ω |\nabla u(x) |^{p}dx,$$ where $δ_Ω(x)$ is the distance function from $Ω^c$. In this article, we address the long standing open question on the case $p=1$ by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces $W^{s,1}(Ω)$ and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as $s\rightarrow 1^{-}$ plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case $p=1$.

math.AP↗