Today I came across the notation $L_\text{weak}$ and I don't know what exactly it means. I can only find the weak Lp spaces , but here they don't use the same notation. Is this the same thing?
I'm also a little bit confused by the norm given in the article: $$\|f\|_{p,w}:=\sup_{t>0}t\left(\mu\left\{x \in S: \left|f(x)\right|>t\right\}\right)^\frac{1}{p},$$
it doesn't seem to fit in the context of the computations in which I need it. Are the any common estimates (except for the bound by the $L^p$ norm) I don't see yet?
Thank you for any answer.
Let $(X,\mu)$ and $(Y,V)$ be measure spaces,and let $T$ be an operator from $L^{p}(X,\mu)$ into the space of measurable functions from $Y$ to $C$.We say that $T$ is weak $(p,q),q<\infty,$ if $$ v(\{y\in Y:|Tf(y)|>\lambda\}) \le (\dfrac{C||f||_{p}}{\lambda})^{q} $$
an operator $K$ is of weak type $(2,2)$ means $$ v(\{y\in Y:|Kf(y)|>\lambda\}) \le (\dfrac{C||f||_{2}}{\lambda})^{2}$$ means $$\lambda v(\{y\in Y:|Kf(y)|>\lambda\})^{1/2} \le C||f||_{2}$$ so $$||Kf||_{L^{2},\infty} \le C||f||_{2}$$