Affine fractional Orlicz--Sobolev inequalities and optimal star bodies

Abstract

In this paper, we establish an affine fractional Orlicz--Sobolev inequality by replacing the Euclidean fractional Orlicz seminorm with an energy obtained by minimizing over origin-symmetric star bodies of prescribed volume. Under finite lower and upper power growth assumptions, a subcritical upper growth bound, and a monotonicity condition on $t^{n/s+2}Φ''(t)$, the resulting inequality has the same best constant as its Euclidean counterpart. The power case recovers the affine fractional Sobolev inequality of Haddad and Ludwig with the corresponding Luxemburg normalization. To establish this inequality, we investigate the underlying geometric extremal problem. For every nonzero smooth function of compact support, the existence and uniqueness of an optimal star body are proved under the growth assumptions alone, and its radial function is characterized by an equation involving $Φ(t)+(s/n)tΦ'(t)$. We also establish an anisotropic rearrangement inequality, together with its equality conditions under additional strictness assumptions. Moreover, the normalized local limit is investigated. The optimized energies converge to the local energy, and the radial functions of the minimizing bodies converge uniformly on the sphere. The limiting modular is determined by $\overlineΦ(t)=\int_0^tΦ(τ)\,dτ/τ$. Finally, we compare optimization over star bodies with optimization in the affine orbit.