** Weak- and Lorentz spaces **

*For further information on this topic, see Chapter 1 of Grafakos’s text, “Classical Fourier Analysis”.*

**Definition ** * For and integer , we define the ***weak- space** as the vector space of measurable functions on such that

*
** Equivalently, the space consists of such that . *

*Note:* The “” is used in the notation to emphasize that this expression is not a norm (cf discussion on ).

For comparison’s sake, note that if is a measurable function on , then

strongly resembles

and so suggests the following definition.

**Definition ** * For and integer , we define the ***Lorentz space ** as the vector space of measurable functions on for which

*
** is finite. For the remainder of this discussion, we will write to mean . *

**Remark ** * Indeed, and by the above discussion. *

Again is not a norm in general. Nevertheless, for all ,

and

Thus, is a *quasi-norm*. When , this quasi-norm is equivalent to an actual norm, . When and , there cannot be a norm that is equivalent to . Nevertheless, there is a metric which generates the same topology. In either case, we obtain a complete metric space.

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