On Doob’s inequality and Burkholder’s inequality in weak spaces
Tóm tắt
Let X be a Banach function space over a nonatomic probability space. We associate with X the weak space
$${\mathrm {w}}\text {-}{X}$$
, which is a quasi-Banach space defined in a natural way. We give necessary and sufficient conditions for Doob’s inequality in
$${\mathrm {w}}\text {-}{X}$$
to be valid, and necessary and sufficient conditions for Burkholder’s inequality in
$${\mathrm {w}}\text {-}{X}$$
to be valid.
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