On the interpolation constants for variable Lebesgue spaces

For θ∈(0,1)$\theta \in (0,1)$ and variable exponents p0(·),q0(·)$p_0(\cdot ),q_0(\cdot )$ and p1(·),q1(·)$p_1(\cdot ),q_1(\cdot )$ with values in [1, ∞], let the variable exponents pθ(·),qθ(·)$p_\theta (\cdot ),q_\theta (\cdot )$ be defined by 1/pθ(·):=(1−θ)/p0(·)+θ/p1(·),1/qθ(·):=(1−θ)/q0(·)+θ/q1(·).$$\begin{equation*} 1/p_\theta (\cdot ):=(1-\theta )/p_0(\cdot )+\theta /p_1(\cdot ), \quad 1/q_\theta (\cdot ):=(1-\theta )/q_0(\cdot )+\theta /q_1(\cdot ). \end{equation*}$$The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space Lpj(·)$L^{p_j(\cdot )}$ to the variable Lebesgue space Lqj(·)$L^{q_j(\cdot )}$ for j=0,1$j=0,1$, then ∥T∥Lpθ(·)→Lqθ(·)≤C∥T∥Lp0(·)→Lq0(·)1−θ∥T∥Lp1(·)→Lq1(·)θ,$$\begin{equation*} \Vert T\Vert _{L^{p_\theta (\cdot )}\rightarrow L^{q_\theta (\cdot )}} \le C \Vert T\Vert _{L^{p_0(\cdot )}\rightarrow L^{q_0(\cdot )}}^{1-\theta } \Vert T\Vert _{L^{p_1(\cdot )}\rightarrow L^{q_1(\cdot )}}^{\theta }, \end{equation*}$$where C is an interpolation constant independent of T. We consider two different modulars ϱmax(·)$\varrho ^{\max }(\cdot )$ and ϱsum(·)$\varrho ^{\rm sum}(\cdot )$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that Cmax≤2$C_{\rm max}\le 2$ and Csum≤4$C_{\rm sum}\le 4$, as well as, lead to sufficient conditions for Cmax=1$C_{\rm max}=1$ and Csum=1$C_{\rm sum}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that pj(·)=qj(·)$p_j(\cdot )=q_j(\cdot )$, j=0,1$j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ϱmax(·)=ϱsum(·)$\varrho ^{\rm max}(\cdot )=\varrho ^{\rm sum}(\cdot )$).