Fractional Hardy–Sobolev–Mazʹya Inequalities

• Fractional Hardy–Sobolev–Mazʹya inequalities are functional inequalities that blend sharp potential estimates with nonlocal Sobolev embeddings to control boundary singularities.
• They extend classical Hardy and Sobolev results to arbitrary, weighted, and non-Euclidean domains using explicit constant formulas and advanced ground-state representations.
• These inequalities have significant applications in spectral theory, geometric analysis, and potential theory, providing tools for studying nonlocal operators and trace conditions.

The fractional Hardy–Sobolev–Mazʹya inequalities are a rigorous class of functional inequalities combining the sharp potential estimate of the (fractional) Hardy inequality with the nonlocal regularity and embedding properties of fractional Sobolev spaces. These inequalities unify and extend classical results by controlling both boundary singularities and critical Sobolev norms in arbitrary domains, model Euclidean settings, weighted geometries, and even non-Euclidean spaces. They find principal applications in spectral theory, geometric analysis, potential theory, and the calculus of variations for nonlocal operators.

1. Foundational Definitions and Classical Fractional Inequalities

Let $\Omega \subset \mathbb{R}^N$ be open, $0$1\le p<\infty$. The fractional Sobolev space $W^{s,p}(\Omega)$ consists of $u \in L^p(\Omega)$ with finite Gagliardo seminorm:

$$[u]_{W^{s,p}(\Omega)}^p = \iint_{\Omega\times\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy.$$

The classical fractional Hardy inequality asserts, for suitable $\Omega$ (convex, John domains), the existence of a sharp constant $D_{N,p,s}$ such that

$$\iint_{\Omega\times\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy \ge D_{N,p,s}\int_\Omega \frac{|u(x)|^p}{d(x)^{sp}}\,dx,$$

with $d(x)=\mathrm{dist}(x,\partial\Omega)$ (Dyda et al., 2011).

Separately, the fractional Sobolev embedding yields

$$[u]_{W^{s,p}(\Omega)}^p + \|u\|_{L^p(\Omega)}^p \ge C_S\left(\int_\Omega |u(x)|^{p^*_s}\,dx\right)^{p/p^*_s},$$

where $p^*_s = Np/(N-sp)$ (Dyda, 2010).

2. Sharp Fractional Hardy–Sobolev–Mazʹya Inequalities in Model Domains

The prototypical fractional Hardy–Sobolev–Mazʹya inequality fuses the two estimates above:

$$\iint_{\Omega\times\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy \ge D_{N,p,s}\int_\Omega \frac{|u(x)|^p}{d(x)^{sp}}\,dx + o_{N,p,s}\left(\int_\Omega |u(x)|^{p^*_s}\,dx\right)^{p/p^*_s}$$

for arbitrary open $\Omega$, $N>sp$, $p\ge2$, $u\in C_c^\infty(\Omega)$, retaining the sharp Hardy constant $D_{N,p,s}$ (Dyda et al., 2011).

On half-spaces $\mathbb{R}^N_+$ and balls $\mathbb{B}^N$, the same paradigm applies, with explicit formulas for the best constants in terms of integrals over spheres and hypergeometric functions (Dyda, 2010, Sloane, 2010, Lu et al., 2023). In particular, for $\mathbb{R}^N_+$ and $p=2$, the constant

$$K_{N,\alpha} = \pi^{N/2} \frac{\Gamma(1+\alpha/2)}{\Gamma((N+\alpha)/2)}$$

arises in the fractional Hardy term.

On hyperbolic spaces $\mathbb{H}^n$, Lu & Yang prove fractional Hardy–Sobolev–Mazʹya inequalities with constants coinciding with the sharp Euclidean fractional Sobolev constants for $n\ge3$, $(n-1)/2\le\gamma<n/2$ (Lu et al., 2023). The conformal covariance of the Helgason–Fourier-transformed GJMS operators enables transfer of sharp inequalities between hyperbolic space, half-spaces, and balls.

3. Weighted and Singular Hardy–Sobolev–Mazʹya Inequalities

Generalizations consider weights and singularities located on submanifolds. For $$x=(x_k, x_{d-k}) \in \mathbb{R}^k\times\mathbb{R}^{d-k}$$ and a flat submanifold $K = \{x_k=0\}$, introduce weights $a, B$:

$$[u]_{W^{s,p}_{a,B;k}}^p = \iint_{\mathbb{R}^d\times\mathbb{R}^d} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}\,|x_k|^a\,|y_k|^B}\,dx\,dy,$$

and the corresponding Hardy potential $|x_k|^{-sp+a+B}$ (Kijaczko et al., 24 Mar 2025).

The weighted fractional Hardy–Sobolev–Mazʹya inequality assumes the form:

$$[u]_{W^{s,p}_{a,B;k}}^p - C\int_{\mathbb{R}^d} \frac{|u(x)|^p}{|x_k|^{sp-a-B}}\,dx \ge C_0\left(\int_{\mathbb{R}^d} |x_k|^{q(\theta-1)} |u(x)|^q dx\right)^{p/q},$$

where $\theta = 1 + \frac{sp-d}{d}$ and $q\in[p, p^*]$ (Kijaczko et al., 24 Mar 2025).

Logarithmic versions (when the singular set is a point, $K=\{0\}$) involve additional factors, e.g. $\ln(R/|x|)^{q(\theta-1)}$, to ensure integrability (Kijaczko et al., 24 Mar 2025).

Weighted inequalities on convex domains and half-spaces with general weights $\alpha, \beta$ are established, with optimal constants characterized via spherical integrals (Dyda et al., 2022).

4. Remainder Terms and Extremality

For $p\ge2$, the difference between the Gagliardo seminorm and the Hardy potential can be expressed as a non-negative remainder controlling further regularity:

$$\iint\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}\,dx\,dy - C\int\frac{|u(x)|^p}{|x|^{sp}}\,dx \ge c_p\iint\frac{|v(x)-v(y)|^p}{|x-y|^{d+sp}\,|x|^\eta\,|y|^\eta} dx dy,$$

with $v(x) = |x|^{-\lambda}u(x)$ and $c_p>0$ (Dyda et al., 2022).

For $1$v^{p/2}$, with optimal constants $C_p=2^{p-2}(p-1)$ (Dyda et al., 2023). This quadratic term controls oscillations across the ground-state profile and underpins the embedding into $L^q$-spaces.

Extremals for these inequalities, i.e. functions attaining the minimum in the associated variational quotient, have been analyzed and proven to exhibit precise symmetry (moving-plane arguments) and asymptotic behavior near singularities and infinity (Mallick, 2018).

5. Trace Inequalities, Fractional Laplacians, and Extensions

Fractional Hardy–Sobolev–Mazʹya inequalities extend to trace inequalities involving the spectral fractional Laplacian $(-\Delta)^s$ on bounded and unbounded domains:

$$\langle(-\Delta)^s f, f\rangle_{L^2(\Omega)} \ge d_s\int_\Omega \frac{f(x)^2}{d(x)^{2s}}\,dx + C\left(\int_\Omega |f(x)|^{2^*_s} dx\right)^{2/2^*_s},$$

with $d_s$ and $C$ sharp for mean-convex or convex $\Omega$ (Filippas et al., 2011, Filippas et al., 2014).

Caffarelli–Silvestre-type extensions and the analysis of ground-state test functions via ODE methods are central in establishing sharp trace inequalities (Nguyen, 2016, Filippas et al., 2011). Logarithmic Sobolev and Hardy inequalities arise in radial cases and involve explicit digamma-function and gamma-function evaluations for the best constants (Nguyen, 2016).

6. General Domains, John Domains, and Weighted Inequality Frameworks

Advanced geometric contexts such as unbounded John domains and arbitrary convex domains have been addressed. In such domains, weighted Hardy–Sobolev–Mazʹya inequalities are proven under geometric constraints (e.g. Assouad dimension), allowing distance-to-boundary weights and two-weight Riesz-potential representations (Dyda et al., 2017, Dyda et al., 2022). Here, the extension from balls/half-spaces to arbitrary shapes is achieved via slicing and averaging, coarea-type reductions, and robust ground-state representations.

7. Open Problems and Future Directions

Several open questions remain, including:

• Extension of full-weighted fractional Hardy–Sobolev–Mazʹya to $1Dyda et al., 2023).
• Optimality and explicit determination of extremal functions in subquadratic and supercritical regimes (Mallick, 2018).
• Generalization to curved singular sets, Riemannian manifolds, and operators of higher order or nonlocal character (Kijaczko et al., 24 Mar 2025).
• Interplay between boundary geometry (mean convexity, curvature) and sharp constants in trace settings (Filippas et al., 2011, Filippas et al., 2014).

Resolution of the fractional Hardy–Sobolev–Mazʹya conjecture for the entire $0Filippas et al., 2011), unifying the Euclidean, non-Euclidean, and weighted cases.

Table: Key Inequalities and Constants

Setting	Inequality Formulation	Best Constant Expression
$\mathbb{R}^n_+$, $p\ge2$	$$\int\int \dots - D \int \dots \ge C (\int \dots)^{p/q}$$	$D_{N,p,s}$ via Loss–Sloane/Frank–Seiringer
Ball $\mathbb{B}^n$	$$\int\int \dots - K \int \dots \ge c (\int \dots)^{2/2^*}$$	$K_{n,\alpha}$ by integral over sphere
Weighted, submanifold $K$	$$[u]_{W^{s,p}_{a,B;k}}^p - C \int \dots \ge C_0 (\dots)$$	$C(d,s,p,k,a,B)$ in terms of Beta/Gamma integrals
Trace, $(-\Delta)^s$	$$( -\Delta )^s f \ge d_s \dots + C (\dots)^{2/2^*_s}$$	$d_s$, $k_{s}$ via Gamma functions

Details and explicit all-formula versions are found in (Dyda et al., 2011, Dyda et al., 2022, Lu et al., 2023, Filippas et al., 2011, Kijaczko et al., 24 Mar 2025, Dyda et al., 2023).