$\displaystyle\leq$ $\displaystyle
4\,c\,c_{1}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\kappa}\,\frac{1}{1-2^{-(\kappa-\nu)}}\frac{1}{(2|z-z_{0}|)^{\kappa-\nu}}=\frac{4\,c\,c_{1}}{2^{\kappa-\nu}-1}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}.$

For the term $I_{4}$ define $\Psi\colon\mathds{R}^{d}\to\mathds{R}$ by
$\Psi(x)=\chi_{r}(x-z)$. Then $z\not\in\mathop{\rm
supp}(\mathds{1}_{\Gamma}-\Psi|_{\Gamma})$. Hence by definition of Schwartz
kernel

$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\int_{\Gamma}K_{g}(z,w)\,(1-\chi_{r}(w-z))\,\Big{(}\varphi(z_{0})-\varphi(z)\Big{)}\,dw$
$\displaystyle=$
$\displaystyle\Big{(}\varphi(z_{0})-\varphi(z)\Big{)}\,([{\cal
N},M_{g}](\mathds{1}_{\Gamma}-\Psi|_{\Gamma}))(z).$

Moreover,

$([{\cal N},M_{g}](\Psi|_{\Gamma}))(z)=({\cal N}((g\,\Psi)|_{\Gamma}))(z)$

and using Lemma 9.2, one estimates $|({\cal N}((g\,\Psi)|_{\Gamma}))(z)|\leq
c^{\prime}$. Furthermore,

$|([{\cal N},M_{g}]\mathds{1}_{\Gamma})(z)|=|({\cal N}(g|_{\Gamma}))(z)|\leq
c^{\prime\prime}$

by Proposition 9.3. Therefore

$|I_{4}|=|\varphi(z)-\varphi(z_{0})|\,|([{\cal
N},M_{g}](\mathds{1}_{\Gamma}-\Psi|_{\Gamma}))(z)|\leq(c^{\prime}+c^{\prime\prime})\,|||\varphi|||_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}.$

For the last term $I_{5}$ note that $\varphi\,[{\cal
N},M_{g}]\mathds{1}_{\Gamma}=\varphi\,{\cal
N}(g|_{\Gamma})-\varphi\,g|_{\Gamma}\cdot{\cal N}\mathds{1}_{\Gamma}$. Now
Proposition 9.3 gives $\|{\cal N}(g|_{\Gamma})\|_{C^{\kappa}(\Gamma)}\leq
c^{\prime\prime}$. Since $g$ has a zero on $\Gamma$, it follows from the mean
value theorem that $\|g|_{\Gamma}\|_{C(\Gamma)}\leq\mathop{\rm diam}\Omega$
and $|||\,g|_{\Gamma}\,|||_{C^{\nu}(\Gamma)}\leq 1+\mathop{\rm diam}\Omega$.
Therefore $\|\,g|_{\Gamma}\,\|_{C^{\nu}(\Gamma)}\leq 1+2\mathop{\rm
diam}\Omega$. Then

$\displaystyle\|\varphi\,[{\cal
N},M_{g}]\mathds{1}_{\Gamma}\|_{C^{\nu}(\Gamma)}$ $\displaystyle\leq$
$\displaystyle 2\|\varphi\|_{C^{\nu}(\Gamma)}\,\|{\cal
N}(g|_{\Gamma})\|_{C^{\kappa}(\Gamma)}+3\|\varphi\|_{C^{\nu}(\Gamma)}\,\|\,g|_{\Gamma}\,\|_{C^{\nu}(\Gamma)}\,\|{\cal
N}\mathds{1}_{\Gamma}\|_{C^{\nu}(\Gamma)}$ $\displaystyle\leq$
$\displaystyle\Big{(}2c^{\prime\prime}+3(1+2\mathop{\rm diam}\Omega)\,\|{\cal
N}\mathds{1}_{\Gamma}\|_{C^{\nu}(\Gamma)}\Big{)}\,\|\varphi\|_{C^{\nu}(\Gamma)}.$

Hence

$|I_{5}|\leq\Big{(}2c^{\prime\prime}+3(1+2\mathop{\rm diam}\Omega)\,\|{\cal
N}\mathds{1}_{\Gamma}\|_{C^{\nu}(\Gamma)}\Big{)}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}$

as required.

Combining the contributions one obtains that there exists a $c_{3}>0$,
independent of $\varphi$, $g$, $z$ and $z_{0}$, such that

$|(T_{g}\varphi)(z)-(T_{g}\varphi)(z_{0})|\leq
c_{3}\,\|\varphi\|_{C^{\nu}(\Gamma)}\,|z-z_{0}|^{\nu}.$

This gives the required semi-norm estimate.

It remains to obtain a suitable $L_{\infty}$ estimate for $T_{g}\varphi$. Let
$z\in\Gamma$ and set $\tilde{g}=g-g(z)\,\mathds{1}_{\mathds{R}^{d}}\in W_{2}$.
Then

$(T_{g}\varphi)(z)=(T_{\tilde{g}}\varphi)(z)=\int_{\Gamma}K_{\tilde{g}}(z,w)\,(\varphi(w)-\varphi(z))\,dw+\varphi(z)\,({\cal
N}(\tilde{g}|_{\Gamma}))(z),$

where we used that $\tilde{g}(z)=0$. Now Lemma 7.4(b) gives

$\displaystyle|\int_{\Gamma}K_{\tilde{g}}(z,w)\,(\varphi(w)-\varphi(z))\,dw|$
$\displaystyle\leq$
$\displaystyle\int_{\Gamma}\frac{c}{|w-z|^{d-1}}\,2\|\varphi\|_{C^{\nu}(\Gamma)}\,|w-z|^{\nu}$
$\displaystyle\leq$ $\displaystyle 2c\,\frac{c_{2}}{1-2^{-\nu}}\,(\mathop{\rm
diam}(\Omega))^{\nu}\,\|\varphi\|_{C^{\nu}(\Gamma)}.$

Moreover,

$|\varphi(z)\,({\cal N}(\tilde{g}|_{\Gamma}))(z)|\leq
c^{\prime\prime}\,\|\varphi\|_{C^{\nu}(\Gamma)}$

by Proposition 9.3. This completes the proof of Step 9.

Since $\\{\Phi|_{\Gamma}:\Phi\in C^{\infty}(\mathds{R}^{d})\\}$ is dense in
$C^{\nu}(\Gamma)$, for all $g\in W_{2}$ there exists a unique bounded
extension $\widetilde{T}_{g}\colon C^{\nu}(\Gamma)\to C^{\nu}(\Gamma)$ of
$T_{g}$ and

$\sup_{g\in W_{2}}\|\widetilde{T}_{g}\|_{C^{\nu}(\Gamma)\to
C^{\nu}(\Gamma)}<\infty.$

Step 2 We prove that the commutator $[{\cal N},M_{g}]$ is defined on $D({\cal
N})$ and extends to a bounded operator on $L_{2}(\Gamma)$. This is Statement
(b) of the theorem for $p=2$.
Choose $\nu=\kappa/2$ in Step 9. Let ${\cal D}=\\{\Phi|_{\Gamma}:\Phi\in
C^{\infty}(\mathds{R}^{d})\\}$. Then there is a $c>0$ such that

$\|\widetilde{T}_{g}\varphi\|_{C^{\nu}(\Gamma)}\leq
c\,\|\varphi\|_{C^{\nu}(\Gamma)}$

for all $\varphi\in{\cal D}$ and all $g\in W_{2}$. Similarly, by duality there
exists a $c^{\prime}>0$ and for all $g\in W_{2}$ there exists a bounded
operator $\widecheck{T}_{g}\colon C^{\nu}(\Gamma)\to C^{\nu}(\Gamma)$ such
that $\|\widecheck{T}_{g}\|_{C^{\nu}(\Gamma)\to C^{\nu}(\Gamma)}\leq
c^{\prime}$ and $\widecheck{T}_{g}\varphi=[{\cal N}^{*},M_{g}]\varphi$ for all
$\varphi\in{\cal D}$. Then

$(\widetilde{T}_{g}\varphi_{1},\varphi_{2})_{L_{2}(\Gamma)}=([{\cal
N},M_{g}]\varphi_{1},\varphi_{2})_{L_{2}(\Gamma)}=-(\varphi_{1},[{\cal
N}^{*},M_{g}]\varphi_{2})_{L_{2}(\Gamma)}=-(\varphi_{1},\widecheck{T}_{g}\varphi_{2})_{L_{2}(\Gamma)}$

for all $\varphi_{1},\varphi_{2}\in{\cal D}$. Hence by density

$(\widetilde{T}_{g}\varphi_{1},\varphi_{2})_{L_{2}(\Gamma)}=-(\varphi_{1},\widecheck{T}_{g}\varphi_{2})_{L_{2}(\Gamma)}$

for all $\varphi_{1},\varphi_{2}\in C^{\nu}(\Gamma)$. By Krein’s lemma (see
[Bra] Theorem 10) for all $g\in W_{2}$ there exists a unique bounded operator
$\widehat{T}_{g}\colon L_{2}(\Gamma)\to L_{2}(\Gamma)$ such that
$\widehat{T}_{g}$ is an extension of $\widetilde{T}_{g}$ and
$\|\widehat{T}_{g}\|_{L_{2}(\Gamma)\to
L_{2}(\Gamma)}\leq\sqrt{c\,c^{\prime}}$. Then $\widehat{T}_{g}$ is an
extension of $T_{g}$ for all $g\in W_{2}$.

Let $g\in W_{2}$. If $\varphi\in{\cal D}$, then $\varphi\in D({\cal N})$ and
$g|_{\Gamma}\,\varphi\in D({\cal N})$ by Corollary 9.5. Moreover,
$\widehat{T}_{g}\varphi=[{\cal N},M_{g}]\varphi={\cal
N}(g|_{\Gamma}\,\varphi)-g|_{\Gamma}\,{\cal N}\varphi$. Let $\Psi\in
C^{\infty}(\mathds{R}^{d})$ and set $v=\Psi|_{\Omega}$. Then $v\in
H^{1}(\Omega)$. Now

$({\cal N}(g|_{\Gamma}\,\varphi),{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}-(g|_{\Gamma}\,{\cal N}\varphi,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}=(\widehat{T}_{g}\varphi,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}.$

So

$\gothic{a}(\gamma(g|_{\Gamma}\,\varphi),v)-\gothic{a}(\gamma(\varphi),g|_{\Omega}\,v)=(\widehat{T}_{g}\varphi,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}.$ (21)

By density and continuity, (21) is valid for all $v\in H^{1}(\Omega)$. The
space ${\cal D}$ is dense in $H^{1/2}(\Gamma)$ and the map $\varphi\mapsto
g|_{\Gamma}\,\varphi$ is bounded from $H^{1/2}(\Gamma)$ into
$H^{1/2}(\Gamma)$. Hence by density, (21) is valid for all $\varphi\in
H^{1/2}(\Gamma)$ and $v\in H^{1}(\Omega)$. Now let $\varphi\in D({\cal N})$.
Then

$\displaystyle\gothic{a}(\gamma(g|_{\Gamma}\,\varphi),v)$ $\displaystyle=$
$\displaystyle\gothic{a}(\gamma(\varphi),g|_{\Omega}\,v)+(\widehat{T}_{g}\varphi,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}$ $\displaystyle=$ $\displaystyle({\cal
N}\varphi,g|_{\Gamma}\,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}+(\widehat{T}_{g}\varphi,{\mathop{\rm
Tr\,}}v)_{L_{2}(\Gamma)}=(\psi,{\mathop{\rm Tr\,}}v)_{L_{2}(\Gamma)}$

for all $v\in H^{1}(\Omega)$, where $\psi=g|_{\Gamma}\,{\cal
N}\varphi+\widehat{T}_{g}\varphi\in L_{2}(\Gamma)$. Hence
$g|_{\Gamma}\,\varphi\in D({\cal N})$ and ${\cal
N}(g|_{\Gamma}\,\varphi)=g|_{\Gamma}\,{\cal N}\varphi+\widehat{T}_{g}\varphi$.
In particular, $\varphi\in D([{\cal N},M_{g}])$ and $[{\cal
N},M_{g}]\varphi=\widehat{T}_{g}\varphi$. We proved that $\widehat{T}_{g}$ is
an extension of $[{\cal N},M_{g}]$ and $\|\widehat{T}_{g}\|_{L_{2}(\Gamma)\to
L_{2}(\Gamma)}\leq\sqrt{c\,c^{\prime}}$ for all $g\in W_{2}$. This completes
the proof of Statement (b).

Step 3 We prove Statement (a).
Let $\varphi\in D({\cal N})\cap C^{\nu}(\Gamma)$. Then
$g|_{\Gamma}\,\varphi\in D({\cal N})$ by Statement (b) and obviously
$g|_{\Gamma}\,\varphi\in C^{\nu}(\Gamma)$. So $g|_{\Gamma}\,\varphi\in D({\cal
N})\cap C^{\nu}(\Gamma)$. Moreover, $[{\cal
N},M_{g}]\varphi=\widehat{T}_{g}\varphi=\widetilde{T}_{g}\varphi$. Since
$\widetilde{T}_{g}$ is bounded from $C^{\nu}(\Gamma)$ into $C^{\nu}(\Gamma)$,
with norm bound uniformly in $g\in W_{2}$, Statement (a) follows.

Step 4 Finally we prove Statement (b) for all $p\in(1,\infty)\setminus\\{2\\}$
and Statement (c).
By Lemma 9.6, the operator $\widehat{T}_{g}$ is a Calderón–Zygmund operator
and hence it is of weak type $(1,1)$. Using interpolation and duality it
extends to a bounded operator on $L_{p}(\Gamma)$ for all $p\in(1,\infty)$ and
the bounds of Statement (b) hold on $L_{p}(\Gamma)$. This completes the proof
of Theorem 1.1. ∎

##  10 Poisson bounds

In this section we prove the Poisson bound stated in Theorem 1.3.

We start by proving that the semigroup is bounded on $L_{\infty}(\Gamma)$.
This result does not need much regularity and it holds on a Lipschitz domain
and bounded measurable real coefficients. We use the next general observation
which allows us to restrict attention to connected components of $\Omega$.

######  Lemma 10.1.

Let $\Omega$ be open, bounded and Lipschitz. Let $\widehat{\Omega}$ be a
connected component of $\Omega$. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl},b_{k},c_{k},c_{0}\in L_{\infty}(\Omega)$. Suppose there is a $\mu>0$
such that

$\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$

for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Suppose the corresponding
elliptic operator with Dirichlet boundary conditions is invertible. Let ${\cal
N}$ be the corresponding Dirichlet-to-Neumann operator on
$L_{2}(\partial\Omega)$. Restricting the coefficients to $\Omega_{0}$ one also
obtains a Dirichlet-to-Neumann operator $\widehat{\cal N}$ on
$L_{2}(\partial\widehat{\Omega})$. Let $t>0$. Then $e^{-t{\cal N}}$ leaves
$L_{2}(\partial\widehat{\Omega})$ invariant and $e^{-t{\cal
N}}\varphi=e^{-t\widehat{\cal N}}\varphi$ for all $\varphi\in
L_{2}(\partial\widehat{\Omega})$, where as usual we identify
$L_{2}(\partial\widehat{\Omega})$ with the elements of $L_{2}(\partial\Omega)$
which vanish on $(\partial\Omega)\setminus\partial\widehat{\Omega}$.

###### Proof.

Let $\gamma$ and $\hat{\gamma}$ denote the harmonic liftings on $\Omega$ and
$\widehat{\Omega}$. If $\varphi\in H^{1/2}(\partial\widehat{\Omega})$, then
$\hat{\gamma}(\varphi)$ is harmonic on $\Omega$, hence
$\gamma(\varphi)=\hat{\gamma}(\varphi)$. Therefore if $\varphi\in
D(\widehat{\cal N})$, then $\varphi\in D({\cal N})$ and ${\cal
N}\varphi=\widehat{\cal N}\varphi$.

If $\lambda\in\mathds{R}$ is large enough and $\varphi\in
L_{2}(\partial\widehat{\Omega})$, then $(\lambda\,I+\widehat{\cal
N})^{-1}\varphi\in D(\widehat{\cal N})\subset D({\cal N})$ and

$(\lambda\,I+{\cal N})\,(\lambda\,I+\widehat{\cal
N})^{-1}\varphi=(\lambda\,I+\widehat{\cal N})\,(\lambda\,I+\widehat{\cal
N})^{-1}\varphi=\varphi.$

So $(\lambda\,I+{\cal N})^{-1}\varphi=(\lambda\,I+\widehat{\cal
N})^{-1}\varphi\in L_{2}(\partial\widehat{\Omega})$ and $(\lambda\,I+{\cal
N})^{-1}$ leaves $L_{2}(\partial\widehat{\Omega})$ invariant. If $t>0$ and
$\varphi\in L_{2}(\partial\widehat{\Omega})$, then the Euler formula gives

$e^{-t{\cal N}}\varphi=\lim_{n\to\infty}(I+{\textstyle\frac{t}{n}}\,{\cal
N})^{-n}\varphi=\lim_{n\to\infty}(I+{\textstyle\frac{t}{n}}\,\widehat{\cal
N})^{-n}\varphi=e^{-t\widehat{\cal N}}\varphi\in
L_{2}(\partial\widehat{\Omega})$

and the proof is complete. ∎

######  Proposition 10.2.

Suppose $\Omega$ is open, bounded and Lipschitz. For all
$k,l\in\\{1,\ldots,d\\}$ let $c_{kl},b_{k},c_{k},c_{0}$ be bounded measurable
and real valued. Suppose there is a $\mu>0$ such that

$\mathop{\rm
Re}\sum_{k,l=1}^{d}c_{kl}(x)\,\xi_{k}\,\overline{\xi_{l}}\geq\mu\,|\xi|^{2}$

for all $\xi\in\mathds{C}^{d}$ and $x\in\Omega$. Suppose the corresponding
elliptic operator with Dirichlet boundary conditions is invertible.

Then there is a $\lambda\geq 0$ and a constant $M\geq 1$ such that

$\|e^{-t{\cal N}_{\lambda}}\varphi\|_{L_{\infty}(\Gamma)}\leq
M\,\|\varphi\|_{L_{\infty}(\Gamma)}$

for all $t>0$ and $\varphi\in L_{\infty}(\Gamma)$, where ${\cal N}_{\lambda}$
is the corresponding Dirichlet-to-Neumann operator with $c_{0}$ replaced by
$c_{0}+\lambda\,\mathds{1}_{\Omega}$.

###### Proof.

There exists a $\lambda>0$ such that the form $\gothic{a}_{\lambda}\colon
H^{1}(\Omega)\times H^{1}(\Omega)\to\mathds{C}$ is coercive, where
$\gothic{a}_{\lambda}(u,v)=\gothic{a}(u,v)+\lambda\,(u,v)_{L_{2}(\Omega)}$ for
all $u,v\in H^{1}(\Omega)$ as before. Let $A_{\lambda}$ be the operator in
$L_{2}(\Omega)$ associated with $\gothic{a}_{\lambda}$. Then $A_{\lambda}$ is
the operator with Neumann boundary conditions. Let
$\lambda_{0}=\inf\\{\mathop{\rm Re}z:z\in\sigma(A_{\lambda})\\}$. Then
$\lambda_{0}\geq 0$. First suppose that $\Omega$ is connected. By [AEG]
Theorem 4.5(b) there exists a unique $f_{0}\in D(A_{\lambda})\cap
C(\overline{\Omega})$ such that $A_{\lambda}f_{0}=\lambda_{0}\,f_{0}$,
$\|f_{0}\|_{L_{2}(\Omega)}=1$ and $f_{0}(x)>0$ for all
$x\in\overline{\Omega}$. Let $\delta=\min\\{f_{0}(x):x\in\overline{\Omega}\\}$
and $M=\max\\{f_{0}(x):x\in\overline{\Omega}\\}$. Set
$\varphi_{0}=f_{0}|_{\Gamma}\in C(\Gamma)$.

Define

$\widetilde{\cal C}=\\{u\in L_{2}(\Omega,\mathds{R}):0\leq u\leq
f_{0}\\}\quad\mbox{and}\quad{\cal C}=\\{\varphi\in
L_{2}(\Gamma,\mathds{R}):0\leq\varphi\leq\varphi_{0}\\}.$

Then $\widetilde{\cal C}$ and ${\cal C}$ are closed convex sets in
$L_{2}(\Omega)$ and $L_{2}(\Gamma)$. The projections $\widetilde{P}\colon
L_{2}(\Omega)\to\widetilde{\cal C}$ and $P\colon L_{2}(\Gamma)\to{\cal C}$ are
given by

$\widetilde{P}u=f_{0}\wedge(\mathop{\rm Re}u)^{+}\quad\mbox{and}\quad
P\varphi=\varphi_{0}\wedge(\mathop{\rm Re}\varphi)^{+}.$

We next show that the semigroup $(e^{-tA_{\lambda}})_{t>0}$ leaves invariant
$\widetilde{\cal C}$. Indeed, let $u\in\widetilde{\cal C}$ and $t>0$. Since
$e^{-tA_{\lambda}}$ is a positive operator by [Ouh] Corollary 4.3, one deduces
that

$0\leq e^{-tA_{\lambda}}u\leq
e^{-tA_{\lambda}}f_{0}=e^{-t\lambda_{0}}\,f_{0}\leq f_{0}$

and $e^{-tA_{\lambda}}u\in\widetilde{\cal C}$.

By [Ouh] Theorem 2.2 1)${}\Rightarrow{}$2) it follows that $\widetilde{P}u\in
H^{1}(\Omega)$ for all $u\in H^{1}(\Omega)$. On the other hand, it follows
easily from the identity $f_{0}\wedge(\mathop{\rm
Re}u)^{+}=\frac{1}{2}(f_{0}+(\mathop{\rm Re}u)^{+}-|f_{0}-(\mathop{\rm
Re}u)^{+}|)$ and the continuity on $H^{1}(\Omega)$ of the maps $u\mapsto|u|$
and $u\mapsto(\mathop{\rm Re}u)^{+}$ that the function
$\widetilde{P}|_{H^{1}(\Omega)}\colon H^{1}(\Omega)\to H^{1}(\Omega)$ is
continuous. If $u\in H^{1}(\Omega)\cap C(\overline{\Omega})$, then
${\mathop{\rm Tr\,}}u=u|_{\Gamma}$ and therefore

$({\mathop{\rm Tr\,}}\circ\widetilde{P})(u)={\mathop{\rm
Tr\,}}(f_{0}\wedge(\mathop{\rm Re}u)^{+})=\varphi_{0}\wedge(\mathop{\rm
Re}u)^{+}|_{\Gamma})=(P\circ{\mathop{\rm Tr\,}})(u).$

Since $H^{1}(\Omega)\cap C(\overline{\Omega})$ is dense in $H^{1}(\Omega)$ one
deduces by continuity that ${\mathop{\rm
Tr\,}}\circ\widetilde{P}=P\circ{\mathop{\rm Tr\,}}$ on $H^{1}(\Omega)$. Using
[EO4] Proposition 3.3 we conclude that the semigroup $(e^{-t{\cal
N}_{\lambda}})_{t>0}$ leaves invariant the closed convex set ${\cal C}$. In
particular, this implies that the semigroup $(e^{-t{\cal N}_{\lambda}})_{t>0}$
is positive. This positivity was already proved in [EO4] Corollary 4.1. Now
let $t>0$ and $\varphi\in L_{\infty}(\Gamma)$ with
$\|\varphi\|_{L_{\infty}(\Gamma)}\leq 1$. Then $\varphi_{0}\,|\varphi|\in{\cal
C}$ and by invariance also $e^{-t{\cal
N}_{\lambda}}(\varphi_{0}\,|\varphi|)\in{\cal C}$, hence $e^{-t{\cal
N}_{\lambda}}(\varphi_{0}\,|\varphi|)\leq\varphi_{0}$. Moreover,
$|\delta\,\varphi|\leq\varphi_{0}\,|\varphi|$. Therefore

$|\delta\,e^{-t{\cal N}_{\lambda}}\varphi|=|e^{-t{\cal
N}_{\lambda}}(\delta\,\varphi)|\leq e^{-t{\cal
N}_{\lambda}}|\delta\,\varphi|\leq e^{-t{\cal
N}_{\lambda}}(\varphi_{0}\,|\varphi|)\leq\varphi_{0}\leq M.$

Hence $|e^{-t{\cal N}_{\lambda}}\varphi|\leq\frac{M}{\delta}$. Consequently
$\|e^{-t{\cal
N}_{\lambda}}\varphi\|_{L_{\infty}(\Gamma)}\leq\frac{M}{\delta}\,\|\varphi\|_{L_{\infty}(\Gamma)}$
for all $\varphi\in L_{\infty}(\Gamma)$.

Now we remove the connectedness assumption. Since $\Omega$ is bounded and
Lipschitz, it has a finite number of connected components. So $\Omega$ is a
disjoint union of a finite number of connected open subsets $\Omega_{k}$.
Using Lemma 10.1 we can apply the above arguments to the restriction of the
semigroup $(e^{-t{\cal N}})_{t>0}$ to $L_{2}(\partial\Omega_{k})$ for each
$k$. One obtains $\|e^{-t{\cal
N}_{\lambda}}\varphi\|_{L_{\infty}(\partial\Omega_{k})}\leq\frac{M_{k}}{\delta_{k}}\,\|\varphi\|_{L_{\infty}(\partial\Omega_{k})}$
for all $\varphi\in L_{\infty}(\Gamma)$ and $t>0$. Here $M_{k}$ and
$\delta_{k}$ are the same quantities as $M$ and $\delta$ but now taken on
$\overline{\Omega_{k}}$. This proves the proposition on $L_{\infty}(\Gamma)$.
∎

Let us mention that we can take $c_{0}$ to be complex. Indeed, by the method
of [EO4] Theorem 6.1 the semigroup $(e^{-t{\cal N}_{\lambda}})$ is dominated
by the Dirichlet-to-Neumann semigroup with $c_{0}$ replaced by $\mathop{\rm
Re}c_{0}$.

For smooth domain and smooth coefficients we can remove the constant $\lambda$
in the proposition. As in the previous sections, we denote by $A_{D}$ the
elliptic operator with Dirichlet boundary conditions.

######  Proposition 10.3.

Suppose $\Omega$ is bounded and of class $C^{1+\kappa}$ for some $\kappa>0$.
Suppose $c_{kl},b_{k},c_{k},c_{0}$ are Hölder continuous on $\Omega$ and real
valued. Suppose also that $0\notin\sigma(A_{D})$. Then there is a constant
$M\geq 1$ such that

$\|e^{-t{\cal N}}\varphi\|_{L_{\infty}(\Gamma)}\leq
M\,\|\varphi\|_{L_{\infty}(\Gamma)}$

for all $t\in(0,1]$ and $\varphi\in L_{\infty}(\Gamma)$.

###### Proof.

Take $\lambda>0$ large enough as in the previous proposition. Let
$\gamma_{\lambda}^{\\#}$ and $K_{\gamma_{\lambda}^{\\#}}$ be as in Lemma 8.2
and Theorem 8.3. By Lemma 7.3 (c), there exists a $\tilde{c}>0$ such that

$|K_{\gamma}(x,z)|\leq\frac{\tilde{c}}{|x-z|^{d-\frac{3}{4}}}\quad\mbox{and}\quad|K_{\gamma_{\lambda}^{\\#}}(x,z)|\leq\frac{\tilde{c}}{|x-z|^{d-\frac{3}{4}}}$

for all $x\in\Omega$ and $z\in\Gamma$. Actually, the estimates hold with $1$
at the place of $\frac{3}{4}$ but then the next argument breaks down if $d=2$.
By applying [Fri] (Lemma 2, Section 4, Chapter 1) we obtain an upper bound for
the kernel $K_{Q}$ of $Q=(\gamma_{\lambda}^{\\#})^{*}\,\gamma$. Specifically,
there is a $c>0$ such that

$|K_{Q}(z,w)|\leq\frac{c}{|z-w|^{d-\frac{3}{2}}}$

for all $z,w\in\Gamma$. Therefore, this kernel is integrable on $\Gamma$ with
respect to each variable with a bound independent of the other variable. This
implies that $Q$ extends to a bounded operator on $L_{\infty}(\Gamma)$. The
formula

${\cal N}={\cal N}_{\lambda}-\lambda\,Q$

from Lemma 8.2 together with Proposition 10.2 imply the $L_{\infty}$-estimate
for the semigroup $(e^{-t{\cal N}})$. ∎

######  Lemma 10.4.

Under the assumptions of Proposition 10.3, the semigroup $(e^{-t{\cal
N}})_{t>0}$ is given by a kernel $K$ and there exist constants $c>0$ and
$\omega_{0}\in\mathds{R}$ such that

$|K_{t}(z,w)|\leq c\,t^{-(d-1)}\,e^{\omega_{0}t}$

for all $t>0$ and $z,w\in\Gamma$.

###### Proof.

We argue as in [EO1], Theorem 2.6. Let $d\geq 3$. By the Sobolev embedding it
follows that $e^{-t{\cal N}}$ maps $L_{2}(\Gamma)$ into
$L_{\frac{2(d-1)}{d-2}}(\Gamma)$ with a norm estimate
$c\,t^{-1/2}\,e^{\omega^{\prime}t}$ for all $t>0$. Since the semigroup is
bounded on $L_{\infty}(\Gamma)$ by Proposition 10.3, this estimate
extrapolates to give

$\|e^{-t{\cal
N}}\|_{2\to\infty}\leq\tilde{c}\,t^{-\frac{d-1}{2}}\,e^{\omega^{\prime\prime}t}$

for all $t>0$ and some constants $\tilde{c}$ and $\omega^{\prime\prime}$. For
this extrapolation argument we refer to [Cou] and [Ouh], Lemma 6.1. The same
argument applies to the adjoint semigroup $(e^{-t{\cal N}^{*}})_{t>0}$, since
${\cal N}^{*}$ is the Dirichlet-to-Neumann operator associated with the
adjoint form $\gothic{a}^{*}$. We conclude by the help of the semigroup
property that $e^{-t{\cal N}}$ maps $L_{1}(\Gamma)$ into $L_{\infty}(\Gamma)$
with the desired estimates.

The case $d=2$ is treated in a similar way, see [EO1], Theorem 2.6. ∎

###### Proof of Theorem 1.3..

Note that by interpolation and the fact that the semigroup $(e^{-t{\cal
N}})_{t>0}$ is bounded on $L_{p}(\Gamma)$ for all $p\in[1,\infty]$ the
corresponding $L_{p}(\Gamma)\to L_{q}(\Gamma)$ estimates for $e^{-t{\cal N}}$
hold for all $1\leq p\leq q\leq\infty$.

To simplify notation, set $S_{t}=e^{-t{\cal N}}$ and consider $t\in(0,1]$. Let
$g\in W_{2}$ and denote by $M_{g}$ the multiplication operator by $g$ as
before. Iterating the formula

$[M_{g},S_{t}]=-\int_{0}^{t}S_{t-s}\,[M_{g},{\cal N}]\,S_{s}\,ds$

for the commutator of $S_{t}$ with $M_{g}$ one obtains by induction for the
commutator $\delta_{g}^{d}(S_{t})=[M_{g},[\ldots,[M_{g},S_{t}]\ldots]]$ of
order $d$ the integral

$\displaystyle\delta_{g}^{d}(S_{t})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{d}(-t)^{k}\sum_{\scriptstyle
j_{1},\ldots,j_{k}\in\mathds{N}\atop\scriptstyle
j_{1}+\ldots+j_{k}=d}\int_{H_{k}}S_{t_{k+1}\,t}\,\delta^{j_{k}}({\cal
N})\,S_{t_{k}\,t}\circ\ldots\circ$ $\displaystyle\hskip 142.26378pt{}\circ
S_{t_{2}\,t}\,\delta^{j_{1}}({\cal
N})\,S_{t_{1}\,t}\,d\lambda_{k}(t_{1},\ldots,t_{k+1}),$

where

$H_{k}=\\{(t_{1},\ldots,t_{k+1})\in(0,\infty)^{k+1}:t_{1}+\ldots+t_{k+1}=1\\},$

$d\lambda_{k}$ denotes Lebesgue measure of the $k$-dimensional surface $H_{k}$
and $\delta^{j}({\cal N})$ is the commutator of order $j$ for all
$j\in\mathds{N}$.

The idea is to estimate the $L_{1}(\Gamma)\to L_{\infty}(\Gamma)$ norm of this
commutator of order $d$. By Theorem 8.3, the kernel $K_{j}$ of
$\delta_{g}^{j}({\cal N})$ can be estimated and there is a $c>0$ such that
$|K_{j}(z,w)|\leq\frac{c}{|z-w|^{d-j}}$ for all $z,w\in\Gamma$ with $z\neq w$.
Therefore, using Riesz potentials, it follows that for all $1<p\leq q<\infty$
with $(d-1)(\frac{1}{p}-\frac{1}{q})\in\\{0,1,\ldots,d-1\\}$ one has
$\|\delta_{g}^{j}({\cal N})\|_{p\to q}\leq c_{p,q}$ for some constant
$c_{p,q}>0$, independent of $g\in W_{2}$, where
$j=1+(d-1)(\frac{1}{p}-\frac{1}{q})$. For the case case $p=1$ and $q=\infty$
(that is for the case $j=d$) one uses the fact that $S_{t}$ is bounded on
$L_{1}(\Gamma)$ and $L_{\infty}(\Gamma)$ uniformly for $t\in(0,1]$ and that
$\delta_{g}^{d}({\cal N})$ is bounded from $L_{1}(\Gamma)$ into
$L_{\infty}(\Gamma)$. The latter follows since its kernel
$(z,w)\mapsto(g(z)-g(w))^{d}\,K_{\cal N}(z,w)$ is bounded by a constant by
Theorem 8.3. Inserting all these estimates in the above formula for
$\delta_{g}^{d}(S_{t})$ gives

$\|\delta_{g}^{d}(S_{t})\|_{1\to\infty}\leq c\,t$ (22)

for some $c>0$, uniformly for all $t\in(0,1]$ and $g\in W_{2}$. All this is
explained in detail in Section 8 of [EO2] and no self-adjointness of the
operator is needed in this step. The estimate (22) gives

$|(g(z)-g(w))^{d}K_{t}(z,w)|\leq c\,t$

for all $t\in(0,1]$ and $z,w\in\Gamma$. The constant $c$ is uniform with
respect to $g\in W_{2}$. Optimising over $g\in W_{2}$ and using Lemma 10.4
gives the Poisson bound. ∎

######  Corollary 10.5.

Adopt the assumptions and notation of Theorem 1.3. Let $S$ be the semigroup
generated by $-{\cal N}$. Then $S$ leaves the space $C(\Gamma)$ invariant. For
all $t>0$ let $T_{t}=S_{t}|_{C(\Gamma)}\colon C(\Gamma)\to C(\Gamma)$. Then
$T=(T_{t})_{t>0}$ is a $C_{0}$-semigroup.

###### Proof.

It follows from [EW] Proposition 5.7 that $S$ leaves the space $C(\Gamma)$
invariant. Let $M\geq 1$ be as in Proposition 10.3. Now let $\Phi\in
C^{\infty}(\mathds{R}^{d})$ and set $\varphi=\Phi|_{\Gamma}$. Then $\varphi\in
D({\cal N})$ and ${\cal N}\varphi\in C(\Gamma)$ by Lemma 9.1(a). Let
$t\in(0,1]$. Then

$\displaystyle\|T_{t}\varphi-\varphi\|_{C(\Gamma)}$ $\displaystyle=$
$\displaystyle\|S_{t}\varphi-\varphi\|_{C(\Gamma)}=\Big{\|}\int_{0}^{t}S_{s}\,{\cal
N}\varphi\,ds\Big{\|}_{C(\Gamma)}\leq\int_{0}^{t}\|S_{s}\,{\cal
N}\varphi\|_{C(\Gamma)}\,ds$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\|T_{s}\|\,\|{\cal N}\varphi\|_{C(\Gamma)}\,ds\leq
M\,t\,\|{\cal N}\varphi\|_{C(\Gamma)}.$

Hence $\lim_{t\downarrow 0}\|T_{t}\varphi-\varphi\|_{C(\Gamma)}=0$. Since
$\\{\Phi|_{\Gamma}:\Phi\in C^{\infty}(\mathds{R}^{d})\\}$ is dense in
$C(\Gamma)$, the corollary follows. ∎

The Dirichlet-to-Neumann operator, as a sectorial operator, generates a
holomorphic semigroup on $L_{2}(\Gamma)$. Let us denote by
$\theta_{0}\in(0,\frac{\pi}{2}]$ the maximum angle of holomorphy. Following
[EO3] and using the holomorphic extension of a small sector of [DR], the
Poisson bound of Theorem 1.3 extends to a Poisson bound for complex time. More
precisely, for any $\theta\in(0,\theta_{0})$, there exist $c>0$ and
$\omega\in\mathds{R}$ such that

$|K_{z}(w_{1},w_{2})|\leq c\,(\mathop{\rm Re}z)^{-(d-1)}\,e^{\omega\mathop{\rm
Re}z}\left(1+\frac{|w_{1}-w_{2}|}{|z|}\right)^{-d}$

for all $z\in\Sigma(\theta)$, the open sector of (half-)angle $\theta$. As a
consequence, this implies that $(e^{-t{\cal N}})_{t>0}$ is also holomorphic on
$L_{p}(\Gamma)$ for all $p\in[1,\infty)$ and on $C(\Gamma)$ with angle
$\theta_{0}$ as on $L_{2}(\Gamma)$. The proofs are the same as in the
symmetric case of [EO3], where one replaces $\frac{\pi}{2}$ by $\theta_{0}$.
Let us only mention that for the holomorphy on $C(\Gamma)$ it is used there
that $e^{-t{\cal N}}L_{\infty}(\Gamma)\subset C(\Gamma)$ for all $t>0$. This
later property is contained in [EW] Proposition 5.7.

## References

  * [AE1] Arendt, W. and Elst, A. F. M. ter, Sectorial forms and degenerate differential operators.  J. Operator Theory 67 (2012), 33–72.
  * [AE2] , Operators with continuous kernels.  Integral Equations Operator Theory 91:45 (2019).
  * [AEG] Arendt, W., Elst, A. F. M. ter and Glück, J., Strict positivity for the principal eigenfunction of elliptic operators with various boundary conditions.  Adv. Nonlinear Stud. 20 (2020), 633–650.
  * [AM] Arendt, W. and Mazzeo, R., Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup.  Commun. Pure Appl. Anal. 11 (2012), 2201–2212.
  * [APL] Astala, K., Päivärinta, L. and Lassas, M., Calderón’s inverse problem for anisotropic conductivity in the plane.  Comm. Partial Differential Equations 30 (2005), 207–224.
  * [Aus] Auscher, P., Regularity theorems and heat kernels for elliptic operators.  J. London Math. Soc. 54 (1996), 284–296.
  * [BR] Behrndt, J. and Rohleder, J., An inverse problem of Calderón type with partial data.  Comm. Partial Differential Equations 37 (2012), 1141–1159.
  * [BE] Binz, T. and Elst, A. F. M. ter, Dynamic boundary conditions for divergence form operators with Hölder coefficients.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2024).  In press.
  * [BMN] Behrndt, J., Malamud, M. M. and Neidhardt, H., Scattering matrices and Dirichlet-to-Neumann maps.  J. Funct. Anal. 273 (2017), 1970–2025.
  * [Bra] Bramanti, M., Singular integrals in nonhomogeneous spaces: $L^{2}$ and $L^{p}$ continuity from Hölder estimates.  Rev. Mat. Iberoam. 26 (2010), 347–366.
  * [CM] Coifman, R. R. and Meyer, Y., Commutateurs d’intégrales singulières et opérateurs multilinéaires.  Ann. Inst. Fourier, Grenoble 28 (1978), 177–202.
  * [Cou] Coulhon, T., Dimensions of continuous and discrete semigroups on the $L^{p}$-spaces.  In Semigroup theory and evolution equations, Delft 1989, Lecture Notes in Pure and Appl. Math. 135, 93–99. Dekker, New York, 1991.
  * [DE] Denis, C. and Elst, A. F. M. ter, The Stokes Dirichlet-to-Neumann operator.  J. Evol. Equ. 24 (2024), Paper No. 22.
  * [DR] Duong, X. T. and Robinson, D. W., Semigroup kernels, Poisson bounds, and holomorphic functional calculus.  J. Funct. Anal. 142 (1996), 89–129.
  * [EO1] Elst, A. F. M. ter and Ouhabaz, E. M., Analysis of the heat kernel of the Dirichlet-to-Neumann operator.  J. Funct. Anal. 267 (2014), 4066–4109.
  * [EO2] , Dirichlet-to-Neumann and elliptic operators on $C^{1+\kappa}$-domains: Poisson and Gaussian bounds.  J. Differential Equations 267 (2019), 4224–4273.
  * [EO3] , Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions.  J. Evol. Eq. 19 (2019), 21–31.
  * [EO4] , The diamagnetic inequality for the Dirichlet-to-Neumann operator.  Bull. Lond. Math. Soc. 54 (2022), 1978–1997.
  * [ER] Elst, A. F. M. ter and Rehberg, J., Hölder estimates for second-order operators on domains with rough boundary.  Adv. Diff. Equ. 20 (2015), 299–360.
  * [EW] Elst, A. F. M. ter and Wong, M. F., Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators.  J. Evol. Equ. 20 (2020), 1195–1225.
  * [Fri] Friedlander, L., Some inequalities between Dirichlet and Neumann eigenvalues.  Arch. Rational Mech. Anal. 116 (1991), 153–160.
  * [Fri] Friedman, A., Partial differential equations of parabolic type.  Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
  * [GKLP] Girouard, A., Karpukhin, M., Levitin, M. and Polterovich, I., The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript.  J. Spectr. Theory 12 (2022), 195–225.
  * [HZ] Hofmann, S. and Zhang, G., $L^{2}$ estimates for commutators of the Dirichlet-to-Neumann map associated to elliptic operators with complex-valued bounded measurable coefficients on $\mathds{R}^{n+1}_{+}$.  J. Math. Anal. Appl. 504 (2021), Paper No. 125408.
  * [KLS] Kenig, C. E., Lin, F. and Shen, Z., Periodic homogenization of Green and Neumann functions.  Comm. Pure Appl. Math. 67 (2014), 1219–1269.
  * [McS] McShane, E. J., Extension of range of functions.  Bull. Amer. Math. Soc. 40 (1934), 837–842.
  * [Lan] Lannes, D., The water waves problem.  Mathematical Surveys and Monographs 188. Amer. Math. Soc., Providence, RI, 2013.
  * [LU] Lee, J. M. and Uhlmann, G., Determining anisotropic real-analytic conductivities by boundary measurements.  Comm. Pure Appl. Math. 42 (1989), 1097–1112.
  * [Ouh] Ouhabaz, E. M., Analysis of heat equations on domains.  London Mathematical Society Monographs Series 31. Princeton University Press, Princeton, NJ, 2005.
  * [Ouh2] , A “milder” version of Calderón’s inverse problem for anisotropic conductivities and partial data.  J. Spectr. Theory 8 (2018), 435–457.
  * [She] Shen, Z., Commutator estimates for the Dirichlet-to-Neumann map in Lipschitz domains.  In Li, J., Li, X. and Lu, G., eds., Some topics in harmonic analysis and applications, Advanced Lectures in Mathematics 34, 369–384. International Press; Higher Education Press, Somerville, MA; Beijing, 2016.
  * [Uhl] Uhlmann, G., Electrical impedance tomography and Calderón’s problem.  Inverse Problems 25 (2009), Paper No. 123011.
  * [XZZ] Xu, Q., Zhao, W. and Zhou, S., Commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains.  J. Math. Anal. Appl. 458 (2018), 219–241.


# Please, not another note about Generalized Inverses

Philipp Wacker University of Canterbury, Christchurch, New Zealand,
<EMAIL_ADDRESS>
(May 2023)

###### Abstract

We prove some statements of left- and right-continuous variants of generalized
inverses of non-decreasing real functions.

##  1 Introduction

This short manuscript fills a few theoretical gaps in the recorded knowledge
about generalized inverses (also called quantile functions in the context of
probability theory), and corrects a few inaccuracies in the existing
literature. While there is a certain overlap with parts of [Vic99, KMP99,
Fen+12, EH13, La ̵15, KR15], none of these give the full picture of
generalized inverses, and there are persistent errors that need rectification.
Finally, this note111this is a popular title for communicating results about
generalized inverses, see the references section. presents some (as far as the
author is aware) new insights about the exact form of $T\circ T^{-1}$ and
$T^{-1}\circ T$, where $T^{-1}$ is a generalized inverse.

Notation: In the following, we define $f(x+)=\lim_{\varepsilon\searrow
x}f(x+\varepsilon)$ and $f(x-)=\lim_{\varepsilon\searrow x}f(x-\varepsilon)$
for a function $f:\mathbb{R}\to\mathbb{R}$. Similarly,
$f(-\infty)=\lim_{x\to-\infty}f(x)$ and $f(\infty)=\lim_{x\to+\infty}f(x)$. A
non-decreasing function is said to be a map $f:\mathbb{R}\to\mathbb{R}$ such
that $x<y$ implies $f(x)\leq f(y)$. We denote by
$\overline{\mathbb{R}}=\mathbb{R}\cup\\{-\infty,\infty\\}$ the set of the
extended real numbers.

We start by defining generalized inverses.

######  Definition 1 (generalized inverse).

Let $T:\mathbb{R}\to\mathbb{R}$ be a non-decreasing function where we set
$T(-\infty)=\lim_{x\to-\infty}T(x)$ and $T(\infty)=\lim_{x\to\infty}T(x)$.
Then the generalized inverses $T^{+}:\mathbb{R}\to\bar{\mathbb{R}}$ and
$T^{-}:\mathbb{R}\to\bar{\mathbb{R}}$ of $T$ are defined by

$\displaystyle T^{+}(y)$ $\displaystyle=\inf\\{x\in\mathbb{R}:T(x)>y\\}$ (1)
$\displaystyle T^{-}(y)$ $\displaystyle=\inf\\{x\in\mathbb{R}:T(x)\geq y\\}.$
(2)

with the convention that $\inf\emptyset=\infty$ and $\inf\mathbb{R}=-\infty$.

######  Remark 1.

[Fen+12] proved that we can equivalently write
$T^{+}(y)=\sup\\{x\in\mathbb{R}:T(x)\leq y\\}$ and
$T^{-}(y)=\sup\\{x\in\mathbb{R}:T(x)<y\\}$, as long as we make sure that the
domain of $T$ is the whole of $\mathbb{R}$.

######  Remark 2.

$T^{+}$ and $T^{-}$ are the right- and left-continuous generalized inverses of
$T$, in the sense outlined by lemma 1(c) below.

##  2 Useful statements for working with generalized inverses

We follow up with a list of elementary properties of $T^{+}$. Parts (a)–(e)
are a generalization of [EH13, Proposition 1] to the case of both $T^{+}$ and
$T^{-}$. Part (f) is proven in [KMP99, section 2]. Parts (g) is similar to [La
̵15, Proposition 4.2], and prove and sharpen all results to cover both the
case $T^{+}$ and $T^{-}$. (h) and (j) are new and show what we can say if $T$
is left- or right-continuous. Parts (i) and (k) correct a mistake in [La ̵15,
Proposition 4.3] (see remark below), and generalize the statement to handle
$T^{-}$, as well.

######  Lemma 1.

Let $T:\mathbb{R}\to\mathbb{R}$ be a nondecreasing map.

  1. (a)

    1. (1)

$T^{+}(y)=-\infty$ if and only if $T(x)>y$ for all $x\in\mathbb{R}$.

    2. (2)

$T^{+}(y)=\infty$ if and only if $T(x)\leq y$ for all $x\in\mathbb{R}$.

    3. (3)

$T^{-}(y)=-\infty$ if and only if $T(x)\geq y$ for all $x\in\mathbb{R}$.

    4. (4)

$T^{-}(y)=\infty$ if and only if $T(x)<y$ for all $x\in\mathbb{R}$.

  2. (b)

$T^{+}$ and $T^{-}$ are nondecreasing.

  3. (c)

$T^{+}$ is right-continuous, and $T^{-}$ is left-continuous. For all
$y\in\mathbb{R}$,

    1. (1)

$T^{+}(y-)=T^{-}(y-)=T^{-}(y)$

    2. (2)

$T^{-}(y+)=T^{+}(y+)=T^{+}(y)$

  4. (d)

$T^{+}$ is continuous at $y$ if and only if $T^{-}$ is continuous at $y$.

  5. (e)

$T^{-}(y)\leq T^{+}(y)$

  6. (f)

$T^{-}(y)=T^{+}(y)$ if and only if $\mathrm{Card}(T^{-1}(\\{y\\})\leq 1$.

  7. (g)

The following relations hold:

    1. (1)

If $y\leq T(x)$, then $T^{-}(y)\leq x$. Equivalently,
if $x<T^{-}(y)$, then $T(x)<y$.

    2. (2)

If $y<T(x)$, then $T^{+}(y)\leq x$. Equivalently,
if $x<T^{+}(y)$, then $T(x)\leq y$.

    3. (3)

$T^{-}(T(x))\leq x$

    4. (4)

If $y>T(x)$, then $T^{-}(y)\geq x$. Equivalently,
if $x>T^{-}(y)$, then $T(x)\geq y$.

    5. (5)

If $y\geq T(x)$, then $T^{+}(y)\geq x$. Equivalently,
if $x>T^{+}(y)$, then $T(x)>y$.

    6. (6)

$T^{+}(T(x))\geq x$.

    7. (7)

If $T^{+}(T(x))=T^{-}(T(x))$, then $T^{+}(T(x))=T^{-}(T(x))=x$

    8. (8)

$T(T^{+}(y)-)\leq y$

  8. (h)

Let $T$ be right-continuous at $x$. Then the following relations hold:

    1. (1)

If $y>T(x)$, then $T^{-}(y)>x$. Equivalently,
if $x\geq T^{-}(y)$, then $T(x)\geq y$.

    2. (2)

If $y>T(x)$, then $T^{+}(y)>x$. Equivalently,
if $x\geq T^{+}(y)$, then $T(x)\geq y$.

    3. (3)

$y\leq T(x)$ if and only if $T^{-}(y)\leq x$.

  9. (i)

If $T$ is right-continuous at $x=T^{+}(y)$, then $T(T^{+}(y))\geq y$. If $T$
is right-continuous at $x=T^{-}(y)$, then $T(T^{-}(y))\geq y$.

  10. (j)

Let $T$ be left-continuous at $x$. Then the following relations hold:

    1. (1)

If $y<T(x)$, then $T^{-}(y)<x$. Equivalently,
if $x\leq T^{-}(y)$, then $T(x)\leq y$.

    2. (2)

If $y<T(x)$, then $T^{+}(y)<x$. Equivalently,
if $x\leq T^{+}(y)$, then $T(x)\leq y$.

    3. (3)

$y\geq T(x)$ if and only if $T^{+}(y)\geq x$.

  11. (k)

If $T$ is left-continuous at $x=T^{+}(y)$, then $T(T^{+}(y))\leq y$. If $T$ is
left-continuous at $x=T^{-}(y)$, then $T(T^{-}(y))\leq y$.

  12. (l)

If $T$ is continuous at $T^{+}(y)$, then $T(T^{+}(y))=y$. If $T$ is continuous
at $T^{-}(y)$, then $T(T^{-}(y))=y$.

  13. (m)

If $T$ is constant on an interval $I=(x_{1},x_{2})$, then for all $x\in I$ we
have $T^{+}(T(x))>x>T^{-}(T(x))$.

  14. (n)

We define the left-continuous and right-continuous versions $T_{l}(x):=T(x-)$
and $T_{r}(x):=T(x+)$ of $T$. Then $T_{l}^{+}=T_{r}^{+}$ as well as
$T_{l}^{-}=T_{r}^{-}$.

###### Proof.

(a), (b), (c), (d) and (e) follow immediately from elementary properties of
the infimum as well as the monotonicity of $T$. We just prove part of (c) for
illustration: Let $A_{0}=\\{x:T(x)>y\\}$ and
$A_{\varepsilon}=\\{x:T(x)>y+\varepsilon\\}$. Then
$A_{0}=\bigcup_{\varepsilon>0}A_{\varepsilon}$ and thus

$\displaystyle T^{+}(y)$ $\displaystyle=\inf A_{0}=\inf_{\varepsilon>0}\inf
A_{\varepsilon}=\inf_{\varepsilon>0}T^{+}(y+\varepsilon)$
$\displaystyle=\lim_{\varepsilon\searrow 0}T^{+}(y+\varepsilon).$

Regarding (g):

  1. (g).(1)

Follows directly from definition and the infimum: Let $T(x)\geq y$, then $x\in
A:=\\{\xi\in\mathbb{R}:T(\xi)\geq y\\}$, i.e. $T^{-}(y)=\inf A\leq x$.

  2. (g).(2)

Follows directly from definition and the infimum: Let $T(x)>y$, then $x\in
A:=\\{\xi\in\mathbb{R}:T(\xi)>y\\}$, i.e. $T^{+}(y)=\inf A\leq x$

  3. (g).(3)

This follows from (g).(1), by setting $y=T(x)$.

  4. (g).(4)

We assume that $y>T(x)$. Thus for any $\xi\in\mathbb{R}$ with the property
that $T(\xi)\geq y$, we have $T(\xi)>T(x)$, i.e. $\xi>x$ by monotonicity of
$T$. Since $A\subset B$ implies $\inf A\geq\inf B$, this shows
$T^{-}(y)=\inf\\{\xi\in\mathbb{R}:T(\xi)\geq
y\\}\geq\inf\\{\xi\in\mathbb{R}:\xi>x\\}=x$.

  5. (g).(5)

We assume that $y\geq T(x)$. Thus for any $\xi\in\mathbb{R}$ with the property
that $T(\xi)>y$, we have $T(\xi)>T(x)$, i.e. $\xi>x$ by monotonicity of $T$.
Since $A\subset B$ implies $\inf A\geq\inf B$, this shows
$T^{+}(y)=\inf\\{\xi\in\mathbb{R}:T(\xi)>y\\}\geq\inf\\{\xi\in\mathbb{R}:\xi>x\\}=x$.

  6. (g).(6)

This follows from (g).(5), by setting $y=T(x)$

  7. (g).(7)

Follows from (g).(3) and (g).(6) .

  8. (g).(8)

Clearly, $x:=T^{+}(y)-\varepsilon<T^{+}(y)$, thus by (g).(2),
$T(x)=T(T^{+}(y)-\varepsilon)\leq y$, which shows the statement via
$\varepsilon\to 0$.

Regarding (h):

  1. (h).(1)

We prove the equivalent characterization: Let $x\geq T^{-}(y)$. We choose a
sequence $\xi_{n}\searrow x$, with $\xi_{n}>x$, hence $\xi_{n}>T^{-}(y)$, and
thus $T(\xi_{n})\geq y$ (by (g).(4)). Using right continuity of $T$, we see
that $T(x)=\lim_{n}T(\xi_{n})\geq y$.

  2. (h).(2)

This follows from (h).(1) and (e)

  3. (h).(3)

is a direct implication of (g).(1) and (h).(1)

(i) follows from (h).(1) and (h).(2) by setting $x=T^{-}(y)$ and $T^{+}(y)$,
respectively.

(j) is proven quite similarly to (h).

(k) is proven quite similarly to (i).

Regarding (l): Assuming continuity of $T$ at $T^{+}(y)$ or $T^{-}(y)$,
respectively, the statement follows from an application of (i) and (k).

(m) is proven as follows. Since $T(x)=y$ for all $x\in(x_{1},x_{2})$, (g).(1)
and (g).(5) imply that $T^{-}(y)\leq x$ for all $x\in I$ as well as
$T^{+}(y)\geq x$ for all $x\in I$, i.e. by taking the limit $x\to x_{1/2}$, we
have $T^{-}(y)\leq x_{1}$ and $T^{+}(y)\geq x_{2}$. Thus, for any $x\in I$, we
get $T^{+}(T(x))=T^{+}(y)\geq x_{2}>x>x_{1}\geq T^{-}(y)=T^{-}(T(x))$.

Lastly we prove (n). Let $T$ be an arbitrary non-decreasing function, and
$T_{r}$ as above. We fix an arbitray $y$ and set $M_{l}=\\{x:T(x-)>y\\}$ and
$M_{r}=\\{x:T(x+)>y\\}$, and thus $T_{l}^{+}(y)=\inf M_{l}$ and
$T_{r}^{+}(y)=\inf M_{r}$. By elementary inclusion $M_{l}\subseteq M_{r}$,
i.e. $T_{l}^{+}(y)\geq T_{r}^{+}(y)$. It remains to show the opposite
inequality. If $M_{l}=M_{r}$, then the statement follows directly. Otherwise,
let $x^{\star}\in M_{r}\setminus M_{l}$ which means that $T(x^{\star}-)\leq y$
and $T(x^{\star}+)>y$. We will now show that $x^{\star}$ is a lower bound for
$M_{r}$. Indeed, let $x<x^{\star}$, then $T(x+)\leq T(x^{\star}-)\leq y$, i.e.
$x\not\in M_{r}$. Since $M_{l}\subseteq M_{r}$, this also shows that
$x^{\star}$ is a lower bound for $M_{l}$. Now we show that
$x^{\star}\in\overline{M_{l}}$. Indeed, if we set $\varepsilon>0$, then
$T((x^{\star}+\varepsilon)-)\geq T(x^{\star}+)>y$, i.e. for all
$\varepsilon>0$, $x_{\varepsilon}:=x^{\star}+\varepsilon\in M_{l}$. Since
$x_{\varepsilon}\searrow x^{\star}$, $x^{\star}\in\overline{M_{l}}$. By
inclusion, $x^{\star}\in\overline{M_{r}}$, as well. $x^{\star}$ being a lower
bound and a limit point proves that $x^{\star}=\inf M_{l}$, and
$x^{\star}=\inf M_{r}$, i.e. $x^{\star}=T_{l}^{+}(y)=T_{r}^{+}(y)$. The
version of with $T^{-}$ is proven analogously.

∎

######  Remark 3.

[EH13, Section 3.2] comments on the fact that there are errors in previous
work on generalized inverses and constructs a series of four statements in
published manuscripts with counterexamples for why they are wrong, but without
providing a way of resolving these contradictions. This manuscript does: (i),
(k), and (l) give the exact conditions for what we can say about $T\circ
T^{+}$ as well as $T\circ T^{-}$. In particular, the counterexamples from
[EH13] do not apply here, since $\infty$ is not a point of (right- or
left-)continuity of $T$ (it does not even make sense to think about this).

There is one particular application that is especially interesting in
practice, which is the inverse sampling method for univariate random
variables. This is a well-known fact, but it is an elementary direct result of
the previous lemma.

######  Corollary 1.

Let $(\Omega,\mathcal{B},\mathbb{P})$ be a probability space and
$X:\Omega\to\mathbb{R}$ a random variable with $F_{X}$ being its cumulative
distribution function. Then the push-forward of a uniform random variable
$U([0,1])$ under the generalized inverse $F_{X}^{-}$ is the law of $X$. This
means that we can generate independent samples $x_{i}\sim U([0,1])$, plug them
into $F_{X}^{-}$, and the $\\{F_{X}^{-}(x_{i})\\}$ will be samples from $X$.

###### Proof.

Since $F_{X}$ is right-continuous, we know (from Lemma 1(h).(3)) that $y\leq
F_{X}(x)$ if and only if $F_{X}^{-}(y)\leq x$. We compute the cumulative
distribution function of $F_{X}^{-}(U)$:

$\displaystyle\mathbb{P}(\mathcal{F}_{X}^{-}(U)\leq\lambda)$
$\displaystyle=\mathbb{P}(U\leq F_{X}(\lambda))$
$\displaystyle=F_{X}(\lambda),$

since the cumulative distribution function of $U$ is given by
$\mathbb{P}(U\leq r)=r$ (for $r\in[0,1]$). This means that the law of
$F_{X}^{-}(U)$ is identical to the law of $X$. ∎

##  3 Jumps and Plateaus

The following two lemmata are an adaptation of [Vic99, Lemma 2], and can also
be found in [La ̵15, Proposition 4.3], but because the former is concerned
with $T^{-}$ instead of $T^{+}$, discusses “third order terms” like
$T(T^{+}(T(x)))>T(x)$ instead of “second order terms” like $T^{+}(T(x))>x$,
and does not prove maximality of the half-open intervals involved, and the
latter has a typo (and refers to the proof to the former instead of providing
a direct proof), we give a proof of this statement for completeness’ sake.
Additionally, we fix an error in the literature (see remark 4 below).

The following statements relate plateaus and jumps of $T$ and $T^{\pm}$ to one
another. For a visualization of these connections, see [La ̵15, KR15].

######  Lemma 2.

Let $T$ be nondecreasing.

  1. (a)

If

$T^{+}(y)>T^{-}(y)$ (3)

then for any $x\in(T^{-}(y),T^{+}(y))=I$, both

$\displaystyle T(x)$ $\displaystyle=y\text{ and }$ (4) $\displaystyle
T^{+}(T(x))$ $\displaystyle>x>T^{-}(T(x))$ (5)

and there is no greater interval than $I$ of the same type such that (4)
holds.

  2. (b)

Conversely, if either

    1. (1)

there is a proper interval $I=(x_{1},x_{2})$ such that for all $x_{0}\in I$,
$T(x_{0})=y$ or

    2. (2)

for some $x_{0}$, we have $T^{+}(T(x_{0}))>x_{0}$, then with $y:=T(x_{0})$, or

    3. (3)

for some $x_{0}$, we have $T^{-}(T(x_{0}))<x_{0}$, then with $y:=T(x_{0})$,

then

$T^{+}(y)>T^{-}(y)$ (6)

  3. (c)

For any given $x$, the following two statements are equivalent:

     * •

$T\equiv y$ on a proper interval $(x_{1},x_{2})$.

     * •

$T^{+}(y)>T^{-}(y)$.

###### Proof.

We start by proving (a). Let $x\in(T^{-}(y),T^{+}(y))$, then
$T^{-}(y)<x<T^{+}(y)$, i.e. by an application of Lemma 1(g).(2) and (g).(4),
$y\leq T(x)\leq y$, which shows that $T$ is indeed constant on the interval
$(T^{-}(y),T^{+}(y))$. Now we show maximality. If $x>T^{+}(y)$, then $T(x)>y$
by Lemma 1(g).(5). Similarly, if $x<T^{-}(y)$, then $T(x)<y$ by (g).(1). This
shows that there is no larger half-open interval $[a,b)$ on which $T\equiv y$.
The relation (5) is a direct implication of (4) (which we just proved to be
true) and lemma Lemma 1(m).

Regarding (b): We prove that (b).(1) implies (6). This is a direct consequence
of Lemma 1(m):

Now (b).(2) also implies (6): By Lemma 1(g).(3), $T^{-}(T(x_{0}))\leq
x_{0}<T^{+}(T(x_{0}))$. Similarly for (b).(3) (via Lemma 1(g).(6)).

(c) follows from a combination from the two other statements. ∎

######  Lemma 3.

Let $T$ be nondecreasing.

  1. (a)

If

$T(x+)>T(x-)$ (7)

then for any $y\in(T(x-),T(x+))=I$, both

$\displaystyle T^{+}(y)$ $\displaystyle=x=T^{-}(y)$ (8)

and there is no greater interval than $I$ of the same type such that either
equality in (8) holds.

  2. (b)

Conversely, if there is a proper interval $I=(y_{1},y_{2})$ such that either

    1. (1)

for all $y\in I$, $T^{+}(y)=x$, or

    2. (2)

for all $y\in I$, $T^{-}(y)=x$, then

$T(x+)>T(x-)$ (9)

  3. (c)

For any given $x$, the following two statements are equivalent:

     * •

$T(x+)>T(x-)$.

     * •

$T^{+}\equiv y\equiv T^{-}$ on a proper interval $(y_{1},y_{2})$.

###### Proof.

We start by proving (a). Let $T(x+)>T(x-)$. Then for any $y\in(T(x-),T(x+))$,
we have that for any $\varepsilon>0$, $T(x-\varepsilon)<y<T(x+\varepsilon)$,
i.e. (using again the relevant statements in lemma 1) $x-\varepsilon\leq
T^{-}(y)$ and $T^{+}(y)\leq x+\varepsilon$. By letting $\varepsilon\to 0$, we
obtain the statement. Maximality is proven similarly: Take $y>T(x+)$, i.e.
there exists $\varepsilon>0$ such that $T(x+\varepsilon)<y$, and thus
$T^{-}(y)\geq x+\varepsilon$, which shows that $y$ is not an element of the
set on which $T^{-}\equiv x$. Since $T^{+}\geq T^{-}$, this also proves that
$y$ is not an element of the set on which $T^{+}\equiv x$. Maximality from
below is proven in the same way.

We now prove (b), assuming (b).(1), i.e. $T^{+}(y)\equiv x$ on
$(y_{1},y_{2})$. For any $\varepsilon>0$, $T^{+}(y)<x+\varepsilon$, i.e.
$T(x+\varepsilon)>y$ for all $y\in(y_{1},y_{2})$. Thus, $T(x+\varepsilon)\geq
y_{2}$ and $T(x+)\geq y_{2}$. In the same way we prove $T(x-)\leq y_{1}$: For
any $\varepsilon>0$, $x-\varepsilon<T^{-}(y)$, i.e. $T(x-\varepsilon)<y$ for
all $y\in(y_{1},y_{2})$. Thus, $T(x-\varepsilon)\leq y_{1}$ and $T(x-)\leq
y_{1}$. All in all, this proves the statement since $T(x-)\leq y_{1}<y_{2}\leq
T(x+)$. The statement (b).(2) $\Rightarrow$ (9) is proven in a similar
fashion.

(c) follows from a combination from the two other statements. ∎

######  Remark 4.

Lemma 3(c) and Lemma 2(c) are a correction of [La ̵15, Proposition 4.3]. In
fact, it is not true that $T(T^{+}(y))>T(T^{+}(y)-)$ implies $T(T^{+}(y))>y$
or that $T^{+}(T(x))>T^{+}(T(x)-)$ implies $T^{+}(T(x))>x$, as Figure 1 shows:
We set $x=x_{2}$. Then $T(x)=y$, and $T^{+}(T(x))=T^{+}(y)=x_{2}=x$, i.e. the
first condition in [La ̵15, Proposition 4.3, 1.)] holds. But, $T(x)=y\in
H(T)$, since $y$ is a plateau of $T$. This is in contradiction to the
statement of [La ̵15, Proposition 4.3, 1.)].

Figure 1: Counterexample: $T^{+}(T(x))>T^{+}(T(x)-)$ but $T^{+}(T(x))=x$.

##  4 Inversion Statements

The remaining statements classify exactly what can be said about $T\circ
T^{\pm}$ and $T^{\pm}\circ T$ under suitable continuity assumptions.

######  Lemma 4.

Let $X=\\{x_{i}\\}$ be the (ordered) list of all discontinuities of $T$, where
we denote $y_{i}^{+}=T(x_{i}+)$ and $y_{i}^{-}=T(x_{i}-)$. Then

$T(T^{+}(y))=\begin{cases}T(x_{i}),&\text{ for }y\in(y_{i}^{-},y_{i}^{+})\\\
y,&\text{ for }y\not\in\bigcup_{i}[y_{i}^{-},y_{i}^{+}]\end{cases}$
$T(T^{-}(y))=\begin{cases}T(x_{i}),&\text{ for }y\in(y_{i}^{-},y_{i}^{+})\\\
y,&\text{ for }y\not\in\bigcup_{i}[y_{i}^{-},y_{i}^{+}]\end{cases}$

Let $Y=\\{y_{i}\\}$ be the (ordered) list of all discontinuities of $T^{\pm}$,
where we denote $x_{i}^{+}=T^{+}(y_{i})$ and $x_{i}^{-}=T^{-}(y_{i})$. Then

$T^{+}(T(x))=\begin{cases}x_{i}^{+}=T^{+}(y_{i}),&\text{ for
}x\in(x_{i}^{-},x_{i}^{+})\\\ x,&\text{ for
}x\not\in\bigcup_{i}[x_{i}^{-},x_{i}^{+}]\end{cases}$
$T^{-}(T(x))=\begin{cases}x_{i}^{-}=T^{-}(y_{i}),&\text{ for
}x\in(x_{i}^{-},x_{i}^{+})\\\ x,&\text{ for
}x\not\in\bigcup_{i}[x_{i}^{-},x_{i}^{+}]\end{cases}$

###### Proof.

We show the characterization for $T\circ T^{+}$: Let first
$y\in(y_{i}^{-},y_{i}^{+})$. Then Lemma 3(a) proves $T^{+}(y)=x_{i}=T^{-}(y)$,
i.e. $T(T^{+}(y))=T(x_{i})=T(T^{-}(y))$. If
$y\not\in\bigcup_{i}[y_{i}^{-},y_{i}^{+}]$, then $T^{+}(y)\not\in X$, or
otherwise $T^{+}(y)=x_{j}$ for some $j$, which would be in contradiction to
the maximality of the set $(y_{j}^{-},y_{j}^{+})$ in Lemma 3(a). Similarly,
$T^{-}(y)\not\in X$. Thus $T$ is continuous at $T^{+}(y)$ and at $T^{-}(y)$,
i.e. $T(T^{\pm}(y))=y$ by virtue of Lemma 1(l).

Regarding $T^{+}\circ T$: If $i<j$, $y_{i}<y_{j}$ and thus
$x_{i}^{+}=T^{+}(y_{i})\leq T^{-}(y_{j})=x_{j}^{-}$, so $x_{i}^{+}<x_{j}^{-}$
and thus the intervals $(x_{i}^{-},x_{i}^{+})$ are disjoint from another. Let
$x\in(x_{i}^{-},x_{i}^{+})=(T^{-}(y_{i}),T^{+}(y_{i}))$. Then by Lemma 2(a),
$T^{+}(T(x))=T^{+}(y_{i})=x_{i}^{+}$ and $T^{-}(T(x))=T^{-}(y_{i})=x_{i}^{-}$.
On the other hand, let $x\not\in\bigcup_{i}[x_{i}^{-},x_{i}^{+}]$. Then
$T(x)\not\in Y$, because otherwise $T(x)=y_{j}$ for some $j$, and then
$(x_{j}^{-},x_{j}^{+})=(T^{+}(y_{i}-),T^{+}(y_{i}))$ would not be the greatest
interval possible, in contradiction to Lemma 2(a). This means that $T^{+}$ is
continuous at $T(x)$, and thus (because $T^{-}$ and $T^{+}$ are left- and
right-continuous versions of one another, see Lemma 1(c))
$T^{+}(T(x))=T^{-}(T(x))$. By Lemma 1(g).(7), $T^{+}(T(x))=x=T^{-}(T(x))$. ∎

######  Remark 5.

Note that there is no statement about the edge cases $T(T^{\pm}(y_{i}^{\pm}))$
and $T^{\pm}(T(x_{i}^{\pm}))$. In particular, while $T\circ T^{+}=T\circ
T^{-}$ on the two sets considered in the statement of Lemma 4, it is entirely
possible that, e.g., $T(T^{+}(y_{i}^{+}))\neq T(T^{-}(y_{i}^{+}))$. The
remaining values depend on the type of continuity of $T$ at those edge points.
Assuming global (left- or right-)continuity of $T$ allows us to precisely
characterize the invertibility interaction between $T$ and $T^{\pm}$, and
close the gaps in Lemma 4

######  Lemma 5.

Let $T$ be nondecreasing and continuous from the right. We denote by
$X=\\{x_{i}\\}$ the (ordered) list of all discontinuities of $T$, i.e.
$y_{i}^{+}:=T(x_{i})>T(x_{i}-)=:y_{i}^{-}$ and $T(x)=T(x-)$ for $x\not\in X$.
We denote by $Y=\\{y_{i}\\}$ the (ordered) list of plateaus of $T^{\pm}$, i.e.
for each $y_{i}$ there exists a proper (maximal in the set of half-open
intervals) interval $I_{i}=[x_{i}^{-},x_{i}^{+})$ such that $T(x)\equiv y_{i}$
for all $x\in I_{i}$. Then

$\displaystyle T(T^{+}(y))$ $\displaystyle=\begin{cases}y_{i}^{+},&\text{ for
}y\in[y_{i}^{-},y_{i}^{+})\\\ y,&\text{ else }\end{cases}$ $\displaystyle
T^{+}(T(x))$ $\displaystyle=\begin{cases}x_{i}^{+},&\text{ for
}x\in[x_{i}^{-},x_{i}^{+})\\\ x,&\text{ else }\end{cases}$

Let $T$ be nondecreasing and continuous from the left. We denote by
$X=\\{x_{i}\\}$ the (ordered) list of all discontinuities of $T$, i.e.
$y_{i}^{+}:=T(x_{i}+)>T(x_{i})=:y_{i}^{-}$ and $T(x+)=T(x)$ for $x\not\in X$.
We denote by $Y=\\{y_{i}\\}$ the (ordered) list of plateaus of $T^{\pm}$, i.e.
for each $y_{i}$ there exists a proper (maximal in the set of half-open
intervals) interval $I_{i}=(x_{i}^{-},x_{i}^{+}]$ such that $T(x)\equiv y_{i}$
for all $x\in I_{i}$. Then

$\displaystyle T(T^{-}(y))$ $\displaystyle=\begin{cases}y_{i}^{-},&\text{ for
}y\in(y_{i}^{-},y_{i}^{+}]\\\ y,&\text{ else }\end{cases}$ $\displaystyle
T^{-}(T(x))$ $\displaystyle=\begin{cases}x_{i}^{-},&\text{ for
}x\in(x_{i}^{-},x_{i}^{+}]\\\ x,&\text{ else }\end{cases}$

###### Proof.

This follows directly from the fact that the concatenation of right-
continuous, nondecreasing functions is again right-continuous and non-
decreasing (and similarly for left-continuous functions), so we can fill the
gaps in our knowledge of, say, $T\circ T^{+}$ by taking limits from the right,
etc. ∎

##  5 Conclusion

This manuscript tries to unify, organize, generalize, and correct some
statements about generalized inverses. Since this is just the latest work in a
long succession of notes claiming to do just that, we close with only cautious
optimism of having done so successfully.

## References

  * [EH13] Paul Embrechts and Marius Hofert  “A note on generalized inverses”  In _Mathematical Methods of Operations Research_ 77.3 Springer, 2013, pp. 423–432
  * [Fen+12] Changyong Feng, Hongyue Wang, Xin M Tu and Jeanne Kowalski  “A note on generalized inverses of distribution function and quantile transformation”  Scientific Research Publishing, 2012
  * [KMP99] Erich Peter Klement, Radko Mesiar and Endre Pap  “Quasi-and pseudo-inverses of monotone functions, and the construction of t-norms”  In _Fuzzy Sets and Systems_ 104.1 Elsevier, 1999, pp. 3–13
  * [KR15] Thomas Kämpke and Franz Josef Radermacher  “The Generalized Inverse of Distribution Functions”  In _Income Modeling and Balancing_ Springer, 2015, pp. 9–28
  * [La ̵15] Arnaud La Fortelle  “A study on generalized inverses and increasing functions Part I: generalized inverses”  In _hal-01255512_ , 2015
  * [Vic99] Peter Vicenik  “A note to a construction of t-norms based on pseudo-inverses of monotone functions”  In _Fuzzy Sets and Systems_ 104.1 Elsevier, 1999, pp. 15–18