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Let $Tf\left(x\right)=p.v.{ʃ}_{ℝ¹}{e}^{iP(x-y)}f\left(y\right)/(x-y)dy$, where P is a real polynomial on ℝ. It is proved that T is bounded on the weighted H¹(wdx) space with w ∈ A₁.

Let $(B_t,t\in [0,1])$ be a linear Brownian motion starting from 0 and denote by $(L_t\left(x\right),t\ge 0,x\in ℝ)$ its local time. We prove that the spatial trajectories of the Brownian local time have the same Besov-Orlicz regularity as the Brownian motion itself (i.e. for all t>0, a.s. the function $x\to L_t\left(x\right)$ belongs to the Besov-Orlicz space $B_{M_2,\infty }^{1/2}$ with $M_2\left(x\right)=e^{{\left|x\right|}^2}-1$). Our result is optimal.