\left(
\partial _iv\left( x\right) \right) dx,\;\left\| u\right\| _{H_0^1\left(
{\cal O}\right) }=\left( \left\| u\right\| _2^2+\left\| \nabla
u\right\| _2^2\right) ^{\frac 12}.$$As usual we shall denote $H^{-1}({\ensuremath{\mathcal O}})$ its dual space.\
We shall denote by $ H_{loc}^1
({\ensuremath{\mathcal O}})$ the space of functions which are locally square integrable in $ \mathcal{O}$ and which admit first order derivatives that are also locally square integrable.

For each $t>0$ and for all real numbers $p,\,q\geq 1$, we denote by $L^{p,q}([0,t]\times {\cal O})$ the space of (classes of) measurable functions $u:[0,t]\times {\cal O}\longrightarrow \mathbb{{R}}$ such that $$\Vert u\Vert _{p,q;\,t}:=\left( \int_0^t\left( \int_{{\cal O}}|u(s,x)|^p\,dx\right) ^{q/p}\,ds\right) ^{1/q}$$ is finite. The limiting cases with $p$ or $q$ taking the value $\infty $ are also considered with the use of the essential sup norm.\
Now we introduce some other spaces of functions and discuss a certain duality between them. Like in [@DMS05] and [@DMS09], for self-containeness, we recall the following definitions:\
Let $(p_1,q_1)$, $(p_2,q_2)$ $\in[1,\infty]^2$ be fixed and set $$I=I\left( p_1,q_1,p_2,q_2\right) :=\left\{ \left( p,q\right)
\in \left[ 1,\infty \right] ^2/\;\exists \;\rho \in \left[
0,1\right] s.t.\right.$$ $$\left. \frac 1p=\rho \frac
1{p_1}+\left( 1-\rho \right) \frac 1{p_2},\frac 1q=\rho \frac
1{q_1}+\left( 1-\rho \right) \frac 1{q_2}\right\} .$$ This means that the set of inverse pairs $\left( \frac 1p,\frac 