Let $(X,\mu)$, $(Y,\nu)$ be measure spaces and $\tau_0 \in ]0,\pi[$ such that for all subsets $A \subseteq X$, $B \subseteq Y$ with finite measure there is a constant $c(A,B)$ such that
$\log\left\vert \int_B T_z (\chi_A)d\nu\right\vert \leqslant c(A,B)e^{\tau_0\vert \mathrm{Im}z\vert}$
where $T_z$ is some linear operator defined on all finitely simple functions on $X$ with values in the set of all measurable functions on $Y$. Let $f := \sum_{j = 1}^N a_j \chi_{A_j}$, $g := \sum_{k = 1}^M b_k \chi_{B_k}$ be finitely simple complex functions. I want to show, that
$\log\left\vert \int_Y T_z (f)gd\nu\right\vert \leqslant C(f,g)e^{\tau_0\vert \mathrm{Im}z\vert}$
with
$C(f,g) = \log(MN) + \sum_{j = 1}^N\sum_{k = 1}^M (c(A_j,B_k) + \vert \log\vert a_jb_k\vert\vert)$
What I have so far is
$\log\left\vert \int_Y T_z (f)gd\nu\right\vert = \log \left\vert\sum_{j = 1}^N\sum_{k = 1}^M a_jb_k \int_Y T_z(\chi_{A_j})\chi_{B_k}\right\vert \leqslant \log \left(\sum_{j = 1}^N\sum_{k = 1}^M \vert a_jb_k\vert \left\vert\int_Y T_z(\chi_{A_j})\chi_{B_k}\right\vert\right)$
How do I proceed? I just have to get the logarithm somehow inside to use the estimate above.
Here is my answer:
$\log \left| \sum_{j = 1}^n\sum_{k = 1}^m a^{P(z)}_j b_j^{Q(z)} e^{i\alpha_j} e^{i\beta_k} \int_YT_z(\chi_{A_j})(y)\chi_{B_k}(y)d\nu(y)\right|\\ \leqslant \log \left( \sum_{j = 1}^n\sum_{k = 1}^m a_j^{p/p_0 + p/p_1} b_k^{q'/q'_0 + q'/q'_1} \left|\int_{B_k} T_z(\chi_{A_j}) d\nu\right|\right)\\\leqslant \log\left( \sum_{j = 1}^n\sum_{k = 1}^m a_j^{p/p_0 + p/p_1} b_k^{q'/q'_0 + q'/q'_1} e^{c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}} \right)\\ \leqslant \log\left( \sum_{j = 1}^n\sum_{k = 1}^m e^{\left|\log\left(a_j^{p/p_0 + p/p_1} b_k^{q'/q'_0 + q'/q'_1}\right)\right| + c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}} \right)\\ \leqslant \log\left( mn e^{\sum_{j = 1}^n\sum_{k = 1}^m\log\left(a_j^{p/p_0 + p/p_1} b_k^{q'/q'_0 + q'/q'_1}\right) + c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}} \right)\\ = \log\left( mn \right) + \sum_{j = 1}^n\sum_{k = 1}^m\left|\log\left(a_j^{p/p_0 + p/p_1} b_k^{q'/q'_0 + q'/q'_1}\right)\right| + c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}\\ = \log\left( mn \right) + \sum_{j = 1}^n\sum_{k = 1}^m\left( \frac{p}{p_0} + \frac{p}{p_1}\right)\left|\log\left(a_j\right)\right| + \left( \frac{q'}{q'_0} + \frac{q'}{q'_1} \right) \left|\log\left( b_k\right)\right| + c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}\\ \leqslant \log\left( mn \right)e^{\tau_0 \left| \Im z\right|} + \sum_{j = 1}^n\sum_{k = 1}^m\left( \frac{p}{p_0} + \frac{p}{p_1}\right)\left|\log\left(a_j\right)\right|e^{\tau_0 \left| \Im z\right|} + \left( \frac{q'}{q'_0} + \frac{q'}{q'_1} \right) \left|\log\left( b_k\right)\right|e^{\tau_0 \left| \Im z\right|} + c(A_j,B_k)e^{\tau_0 \left| \Im z\right|}$
since $\displaystyle e^{\tau_0 \left| \Im z\right|} \geqslant 1$ by $\tau_0 \in [0,\pi)$.