Counterexamples to elliptic Harnack inequality for isotropic unimodal L\'{e}vy processes

Until now, it has been an open question whether every subordinated Brownian motion (SBM) satisfies the elliptic Harnack inequality (EHI). In this paper, we show that the answer is ``no." In our first theorem, we show that if $X=(X_t)_{t \geq 0}$ is an isotropic unimodal L\'{e}vy process, and $X$ satisfies certain criteria (involving the jump kernel of $X$ and the distribution of the location upon first exiting balls of various sizes) then $X$ does not satisfy EHI. (Note that the class of isotropic unimodal L\'{e}vy processes is larger than the class of SBMs.) We then check that many specific SBMs do indeed satisfy the criteria, and thus do not satify EHI.