The smooth small ball inequality conjecture

Fix a continuous, non-constant function $\varphi$ supported on $[-1/2,1/2]$ with mean zero. For a dyadic interval $I$, define

$$\varphi_I(x)=\varphi\left(\frac{x-c(I)}{|I|}\right),$$

and for a dyadic rectangle $R=R_1\times\cdots\times R_d$ define $\varphi_R(x_1,\ldots,x_d)=\prod_{j=1}^d\varphi_{R_j}(x_j)$. For coefficients $\alpha(R)$ and $d\ge3$, the conjectured estimate is

$$2^{-n}\sum_{|R|=2^{-n}}|\alpha(R)|\lesssim n^{(d-2)/2}\left\|\sum_{|R|\ge2^{-n}}\alpha(R)\varphi_R\right\|_{L^\infty}.$$

Smooth small ball inequality conjecture. The displayed inequality should hold, with an implied constant depending only on $d$ and $\varphi$. This is a smooth analogue of the Haar small ball conjecture; the paper proves a weaker estimate with exponent $(d-1)/2-\eta$ under additional assumptions on $\varphi$.