[Paper Review] Necessary Conditions for Fractional Hardy-Sobolev's Inequalities

E. Ostrovsky, L. Sirota|arXiv (Cornell University)|Aug 5, 2011

Advanced Harmonic Analysis Research26 references3 citations

TL;DR

This paper establishes necessary conditions for fractional Hardy-Sobolev inequalities in multidimensional Euclidean spaces using the dilation method and Grand Lebesgue space theory. It derives scaling identities that characterize the homogeneity of the inequalities and provides sharp lower bounds for the optimal constants in various settings, including mixed-norm and surface variants, with precise asymptotic behavior near critical exponents.

ABSTRACT

In this short article we obtain some necessary conditions for a so-called fractional Hardy-Sobolev's inequalities in multidimensional case. We also give some examples to show the sharpness of these inequalities.

Motivation & Objective

• To derive necessary conditions for the validity of fractional Hardy-Sobolev inequalities in multidimensional domains, particularly in $\mathbb{R}^d$.
• To establish sharp lower bounds for the optimal constants $K_{HS}(p,q)$, $K_{M;HS}(p,q,r)$, $K_{S;HS}(p,q)$, and $K_{DD;HS}(p,s)$ in various inequality types.
• To generalize classical Lebesgue-Riesz spaces to Grand Lebesgue spaces and analyze the behavior of the inequalities in these broader function spaces.
• To investigate the sharpness of the inequalities through explicit counterexamples and asymptotic analysis near critical exponents.
• To extend the results to mixed-norm and anisotropic Grand Lebesgue spaces, providing a unified framework for the inequalities.

Proposed method

• Applies the dilation method of G. Talenti to derive scaling identities that must hold for any non-constant $u \in C_0^\infty(\mathbb{R}^d)$ satisfying the inequality.
• Uses the norm equivalence $\|f\|_{G\psi} \asymp \sup_{p \in (a,b)} \frac{|f|_p}{\psi(p)}$ in Grand Lebesgue spaces to characterize the boundedness of operators.
• Introduces the operators $\delta_\lambda[u](x,y) = \frac{u(x)-u(y)}{|x-y|^{\lambda d}}$ and $S_\lambda[u](x) = \frac{u(x)}{|x|^{\lambda d}}$ to relate difference and pointwise norms.
• Defines the domain $R_r(\alpha(1),\alpha(2),\beta,\mu;d)$ as the set of exponents $(p,q)$ satisfying a homogeneity condition derived from scaling.
• Constructs the function $\psi_5(r) = \inf_{(p,q)\in R_r} \left[ \nu(p,q) K_{M;HS}(p,q) \right]$ to bound the Grand Lebesgue norm of $S_\lambda u$.
• Employs factorization and anisotropic norm structures in $L_{\vec{p}}$ spaces to extend results to mixed-norm settings.

Experimental results

Research questions

• RQ1 What scaling conditions must hold for a fractional Hardy-Sobolev inequality to be valid in $\mathbb{R}^d$?
• RQ2 What are the necessary conditions on the exponents $p,q,\alpha(1),\alpha(2),\beta,\mu$ for the inequality to hold uniformly for all $u \in C_0^\infty(\mathbb{R}^d)$?
• RQ3 How do the optimal constants behave asymptotically as $p \to 1/\lambda$ in Grand Lebesgue space settings?
• RQ4 Can the mixed-norm Hardy-Sobolev inequality be characterized via anisotropic Grand Lebesgue norms?
• RQ5 What is the sharpness of the constant 1 in the Grand Lebesgue space inequality $\|S_\lambda u\|_{G\psi_2} \leq \|\delta_\lambda u \cdot |x-y|^{-\lambda d}\|_{G\psi_1}$?

Key findings

• The necessary condition $\frac{d - \mu}{q} = \frac{2d + \alpha(1) + \alpha(2) - \beta}{p}$ must hold for the ordinary fractional Hardy-Sobolev inequality (1.1a) in $\mathbb{R}^d$.
• The constant 1 in the Grand Lebesgue space inequality $\|S_\lambda u\|_{G\psi_2} \leq \|\delta_\lambda u \cdot |x-y|^{-\lambda d}\|_{G\psi_1}$ is optimal, as shown by counterexample using $u_0(x) = |\log|x|| \cdot I(|x| \leq 1)$.
• For $\psi_4(p)$ satisfying $\lim_{p \to 1/\lambda} \frac{p \psi_4(p)}{|p - 1/\lambda|^\lambda} = 0$, the ratio $\frac{\|S_\lambda u_0\|_{G\psi_4}}{\|\delta_\lambda u_0\|_{G\psi_1}} \to \infty$, proving the sharpness of the bound.
• The inequality $\|S_\lambda u\|_{G\psi_5} \leq \|\delta_\lambda u \cdot |x-y|^{-\lambda d}\|_{G\nu}$ holds with $\psi_5(r) = \inf_{(p,q) \in R_r} \left[ \nu(p,q) K_{M;HS}(p,q) \right]$, establishing a sharp embedding in mixed-norm Grand Lebesgue spaces.
• The function $\psi_3(p) = \frac{p \psi_2(p)}{|1/\lambda - p|}$ yields a weakly exact estimate: $\|S_\lambda u\|_{G\psi_3} \leq C \|\delta_\lambda u\|_{G\psi_1}$, but the constant cannot be improved uniformly.
• The results extend to anisotropic Grand Lebesgue spaces via the norm $|f|_{\vec{p}}$ and the space $G_Q(\nu)$, where $\nu(p)$ is a positive continuous function on a domain $Q \subset \mathbb{R}^d$.