Weak Lq space ($L^{q,\infty }$)

For $1<q<\infty$, the weak Lq space consists of measurable functions for which $A_q(f)={\sup }_{t>0}t|\{|f|>t\}{|}^{1/q}$ is finite. This distribution-function expression is a quasi-norm. The equivalent norm $N_q(f)={\sup }_{0<|E|<\infty }|E{|}^{-1/q^{\prime }}{\int }_E|f|$ satisfies $A_q\le N_q\le q^{\prime }{A}_q$, with $q^{\prime }=q/(q-1)$. The layer cake representation bounds the integral by ${\int }_0^{\infty }\min (|E|,(A_q/t{)}^q)dt=q^{\prime }{A}_q|E{|}^{1/q^{\prime }}$. The lower bound uses finite-measure subsets of a superlevel set.

The integral expression for the weak Lq space is homogeneous, positive definite up to equality almost everywhere, and obeys the triangle inequality by integrating $|f+g|\le |f|+|g|$ over each set. Its exponent is $-1/q^{\prime }$, not $-1/q$. The sharp comparison constant with the distribution-function quasi-norm is $q^{\prime }$; the power function $x^{-1/q}$ on $(0,\infty )$ attains that ratio on initial intervals.