Fractional integral associated with the Schrödinger operators on variable exponent space

Let $ \mathcal{L} = -\Delta+V $ be the Schrödinger operators on $ \mathbb{R}^n $ with nonnegative potential $ V $ belonging to the reverse Hölder class $ RH_q $ for some $ q \geq \frac{n}{2} $. We prove the boundedness of fractional integral operator $ \mathcal{I}_\alpha $ related to the Schrödinger operators $ \mathcal{L} $ from strong and weak variable exponent Lebesgue spaces into suitable variable exponent Lipschitz type spaces.