Triebel–Lizorkin space

Remarks

A different choice of $\psi_0$ gives the same space with an equivalent quasi-norm, and for $p, q \ge 1$ it is a Banach space. Compare the Besov space $B^s_{p,q}$, where the $L^p$ norm is taken before the sum over $j$. For $p = q$ the two definitions coincide, so $F^s_{p,p} = B^s_{p,p}$.

Example: for $1 < p < \infty$, $F^0_{p,2}(\R^n) = L^p(\R^n)$ with equivalent norms, where $L^p$ is the Lp space. This is the Littlewood–Paley theorem: $\norm{f}_{L^p}$ is comparable with $\norm{(\sum_j \abs{\Delta_j f}^2)^{1/2}}_{L^p}$. More generally, $F^k_{p,2}(\R^n)$ is the Sobolev space $W^{k,p}(\R^n)$ for integers $k \ge 0$ and $1 < p < \infty$.

Non-example: a nonzero constant function $c$ is not in any $F^s_{p,q}(\R^n)$ with $p < \infty$. Here $\Delta_0 c = c$, because $\mathcal{F} c$ is a multiple of the Dirac delta and $\psi_0(0) = 1$, and the constant $\abs{c}$ is not in $L^p(\R^n)$.