Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$.
Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{S}'(\mathbb{R}^2)$ such that, for any test function $\varphi \in \mathcal{S}(\mathbb{R})$, we have $$ \langle f(t,\cdot), \varphi(t) \rangle \in B_{p_1,q_1}^{s_1}(\mathbb{R}) \quad \text{ and } \quad \langle f(\cdot , x), \varphi(x) \rangle \in B_{p_2,q_2}^{s_2}(\mathbb{R}) ?$$ That is, $f$ has a given Besov regularity with respect to the first coordinate, and another one with respect to the second coordinate?
I am interested by a family of function spaces with nice structure, that has similar properties than the Besov spaces: (quasi-)Banach structure, characterization using wavelet bases, etc.
This question is motivated by analysis of solutions of PDEs where space $x$ and time $t$ should be treated somehow separately.
