Logarithmic inequalities for Fourier multipliers

In the paper we study LlogL estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. We exhibit a class of functions $m:\mathbb R ^d \rightarrow \mathbb C $ , for which the corresponding multipliers $T_m$ satisfy the following estimate: for $K>1$ , any locally integrable function $f$ on $\mathbb R ^d$ and any Borel subset $A$ of $\mathbb R ^d$ , $$\begin{aligned} \int _{A}|T_m f(x)|\,\text{ d}x\le K\int _{\mathbb{R }d}\Psi (|f(x)|)\,\text{ d}x+\frac{|A|}{2(K-1)}, \end{aligned}$$ where $\Psi (t)=(t+1)\log (t+1)-t$ . We also present related lower bounds which arise from considering appropriate examples for the Beurling-Ahlfors operator.