Density extension

By | December 25, 2015

Proposition. Let p,q\in[1,\infty) and let T:L^{p}\left(X\right)\rightarrow L^{q}\left(Y\right) be a linear (respectively, nonnegative sublinear) operator defined
on a dense linear subspace D of L^{p}\left(X\right) and taking values in L^{q}\left(Y\right). If

    \[ \left\Vert Tf\right\Vert _{q}\le M\left\Vert f\right\Vert _{p} \]

for all f\in D, then T has a unique extension to a linear operator from L^{p}\left(X\right) to L^{q}\left(Y\right) for which the inequality holds for all f\in L^{p}\left(X\right).

Proof. First, we assume that T is linear. Fix f\in L^{p}\left(X\right). Since D is dense in L^{p}\left(X\right), there is a sequence \left\{ f_{n}\right\} in D such that \left\Vert f_{n}-f\right\Vert _{p}\rightarrow 0 as n\rightarrow\infty. Then note that by linearlity,

    \[ \left|Tf_{n}-Tf_{m}\right|=\left|T\left(f_{n}-f_{m}\right)\right| \]

holds. So

    \[ \left\Vert Tf_{n}-Tf_{m}\right\Vert _{q}\le M\left\Vert f_{n}-f_{m}\right\Vert _{p} \]

holds for any n,m by assumption. Since \left\{ f_{n}\right\} is a Cauchy sequence in L^{p}, \left\{ Tf_{n}\right\} is a Cauchy sequence in L^{q}. So by completeness of L^{q}, \left\{ Tf_{n}\right\} converges in L^{q}, say Tf=\lim_{n\rightarrow\infty}Tf_{n}.

Next, the limit is independent of the approximating sequence. Let \left\{ g_{n}\right\} be another sequence in D that converges to f in L^{p}. Then

    \[ \left\Vert T\left(f_{n}\right)-T\left(g_{n}\right)\right\Vert _{q}\le M\left\Vert f_{n}-g_{n}\right\Vert _{p} \]

and since \left\Vert f_{n}-g_{n}\right\Vert _{p}\leq\left\Vert f_{n}-f\right\Vert _{p}+\left\Vert g_{n}-f\right\Vert _{p}, we conclude that \left\{ T\left(g_{n}\right)\right\} converges to a limit in L^{q} that equals to the limit of \left\{ T\left(f_{n}\right)\right\}.

Finally, if f_{n}\rightarrow f and T\left(f_{n}\right)\rightarrow T\left(f\right) in L^{p},L^{q}, respectively, then \left\Vert f_{n}\right\Vert _{p}\rightarrow\left\Vert f\right\Vert _{p} and \left\Vert Tf_{n}\right\Vert _{q}\rightarrow\left\Vert Tf\right\Vert _{q}.

    \[ \left\Vert Tf\right\Vert _{q}\le M\left\Vert f\right\Vert _{p} \]

holds for all f\in L^{p}.

Next, assume T is nonnegative sublinear operator. Note that

    \[ \left|Tf-Tg\right|\le\left|T\left(f-g\right)\right|. \]

Indeed, T\left(f\right)=T\left(f-g+g\right)\le T\left(f-g\right)+Tg shows

    \[ Tf-Tg\le T\left(f-g\right). \]

Similarly, Tg-Tf\le T\left(g-f\right). Hence

    \[ \left|Tf-Tg\right|\le\left|T\left(f-g\right)\right|=T\left(f-g\right). \]

The rest part are similar.

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