Fractional Triebel–Lizorkin regularity conjecture for principal Beltrami solutions

Let $0<s<2$ and $1<p<\infty$ satisfy $s<\frac{2}{p}$. Let $\mu,\nu\in F^s_{p,q}\cap L^\infty_c$ satisfy the ellipticity condition, and let $f$ be the principal solution to the Beltrami equation. Here $p_\kappa$ and $p_\kappa'$ denote the exponents associated with the ellipticity parameter $\kappa$. Regularity conjecture. If

$$\frac{1}{p}<\frac{1}{p_\kappa'},$$

then

$$\bar{\partial}f\in F^s_{p,q}\qquad\text{for every }\frac{1}{q}>\frac{1}{p}+\frac{1}{p_\kappa}.$$

The conjecture seeks the optimal Triebel–Lizorkin regularity of the derivative of a principal Beltrami solution in the subcritical range; the stated result would improve the previously known critical-space estimates under the indicated exponent restriction.