Bañuelos's sharp bound conjecture for stable Lévy multipliers

Let $n\geq 2$, $0<r<\infty$, and let $\varphi\in L^\infty(\mathbb{S}^{n-1})$ satisfy $\|\varphi\|_\infty\leq 1$. Define the multiplier

$$m_r(\xi)=\frac{\int_{\mathbb{S}^{n-1}}|\xi\cdot\theta|^r\varphi(\theta)\,d\sigma(\theta)}{\int_{\mathbb{S}^{n-1}}|\xi\cdot\theta|^r\,d\sigma(\theta)}.$$

Let $T_{m_r}$ be the corresponding Fourier multiplier operator, and set $p^*=\max\{p,p/(p-1)\}$.

Bañuelos's conjecture. For every $1<p<\infty$, $T_{m_r}$ is bounded on $L^p(\mathbb{R}^n)$ and

$$\|T_{m_r}f\|_p\leq(p^*-1)\|f\|_p\quad\text{for all }f\in L^p(\mathbb{R}^n).$$

The conjecture appeared in work cited as Bañuelos [Ban1] and would extend the sharp martingale-transform bound to the larger class of multipliers with arbitrary positive exponent $r$. The source notes that boundedness for $r>2$ and $p\ne2$ is itself an interesting open problem, and that this conjecture would imply Iwaniec's conjecture for the Beurling–Ahlfors transform.