Finite-type Sobolev improving conjecture

Let $n\geq 2$, let $I\subset\mathbb R$ be a compact interval, let $\gamma\colon I\to\mathbb R^n$ be a smooth curve of maximal type $d$, and let $A_{\gamma}$ be its averaging operator. Write $\alpha_{\mathrm{cr}}(p)$ for the critical Sobolev exponent appearing in the paper and define

$$\alpha_{\mathrm{cr}}(d;p):=\min\left\{\alpha_{\mathrm{cr}}(p),\frac{1}{d}\right\}.$$

Finite-type Sobolev improving conjecture. The operator $A_{\gamma}$ maps $L^p$ to $L^p_{\alpha}$ for all $p\geq 2$ and

$$\alpha\leq\alpha_{\mathrm{cr}}(d;p),$$

with strict inequality if

$$\min\{2n-2,d\}\leq p\leq\max\{2n-2,d\}.$$

This conjectures the full Sobolev improving range for averaging over curves of maximal type $d$, including the constraints imposed by the critical exponent and by finite type. The supplied context gives the conjectured range but does not define $\alpha_{\mathrm{cr}}(p)$ or maximal type in the extracted span.