Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\mathbb{Z}]^2$.
Now, consider the quantity \begin{equation} (-\Delta)^{-3}\Bigl[ fg- \int_{\mathbb{T}^2} fg \Bigr] \in W^{6,1}[\mathbb{T}^2] \end{equation}
Then I wonder if it is possible to find positive constants $C_i$'s independent of $f,g$ such that \begin{equation} \Bigl \lVert (-\Delta)^{-3}\Bigl[ fg- \int_{\mathbb{T}^2} fg \Bigr] \Bigr \rVert_1 \leq \sum_{i=0}^3 C_i\lVert \Delta^{-3+i} f \rVert_2 \lVert \Delta^{-i} g \rVert_2 \end{equation}
I tried to use the Fourier expansion, but the product turns into convolution and things seem to get tricky...All references I looked for deal only with the Leibniz rule for the fractional Laplacian of positive exponent.
Could anyone help me with the negative exponent case?