A New Approach to Grand and Small Norms in Discrete Lebesgue Spaces

We present a new approach to the definition of Grand and Small discrete Lebesgue spaces. Previously, we developed such an approach in the continuous case. The fundamental difference of our studies is that we base the definition of norms on extrapolation theory, and our approach permits including the extreme cases $p=1$ and $p=\infty$ into consideration, which is the main result of the study presented here. The Small space for $p=\infty$ is realized as the sum
\begin{equation*} l^{s}(\xi)\equiv \sum _{p \in [2, \infty)} \xi(p) l^{p} \end{equation*}
of spaces with a certain fundamental function $\xi$, and the Grand space $l^{g}(\zeta)$ for $p=1$ is given by the product
\begin{equation*} l^{g}(\zeta)\equiv \bigcap_{p \in (1, 2]} \zeta(p) l^{p} \end{equation*}
of spaces with a certain fundamental function $\zeta$. As one of the main results, we show that if the function $\xi$ satisfies the $\Delta_2$ (doubling) condition, then the space $l^{s}(\xi)$ coincides, up to norm equivalence, with the discrete Lorentz space $\lambda(\psi)$, where $\psi(k) \simeq \xi(1/\ln k)$. We also show that if the function $\zeta$ satisfies the $\Delta_2$ condition, then the space $l^{g}(\zeta)$ coincides, up to norm equivalence, with the discrete Marcinkiewicz space $m(\psi)$, where $\psi(k) \simeq k/(\zeta(1/\ln k))$. We expect that our general new construction of the norms in Grand and Small discrete Lebesgue spaces will imply further studies of the spaces and operators in these spaces in such a general setting.