#  $L^{p}$ continuity of eigenprojections for 2-d Dirichlet Laplacians under
perturbations of the domain

Ryan L. Acosta Babb and James C. Robinson

###### Abstract

We generalise results by Lamberti and Lanza de Cristoforis, (2005) concerning
the continuity of projections onto eigenspaces of self-adjoint differential
operators with compact inverses as the (spatial) domain of the functions is
perturbed in $\mathbb{R}^{2}$. Our main case of interest is the Dirichlet
Laplacian. We extend these results from bounds from $H_{0}^{1}$ to $H_{0}^{1}$
to bounds from $L^{p}$ to $L^{p}$.

##  1 Introduction

It is a classical consequence of the Uniform Boundeness Principle that the
norm convergence of Fourier series is equivalent to the uniform boundedness of
the operator norms of the partial sums (see Grafakos, 2014a, , Chap. 4). In
dimensions 2 or higher, Fefferman’s celebrated theorem (Fefferman,, 1971) on
the unboundedness of the ball multiplier in $L^{p}$ has as a consequence that
we cannot choose the partial sums in any way we want, since convergence is
sensitive to the choice of “truncation” for these infinite series.

For concreteness, let us fix our dimension $d=2$ and consider the lattice
$\mathbb{Z}^{2}$, which serves to index the Fourier coefficients of a periodic
function $f:[0,1]^{2}\to\mathbb{R}$. Then there are many ways in which we can
group these indices in order to determine the cutoff point for a partial sum
of the series

$\sum_{(m,n)\in\mathbb{Z}^{2}}\widehat{f}(m,n)\mathrm{e}^{2\pi\mathrm{i}(mx+ny)}.$

A natural choice would be to truncate by the size of the eigenvalues
associated to each exponential (viewed as an eigenfunction of the Laplacian).
Up to constants, this would mean fixing $N>0$ and summing over all $(m,n)$
with $m^{2}+n^{2}<N^{2}$. Geometrically, we are using expanding concentric
circles in the lattice to determine the cutoff. As we mentioned earlier,
Fefferman’s theorem shows that this method never yields $L^{p}$ convergence
outside the “trivial” case $p=2$. It is a straightforward extension of this
result that other “curved” cutoffs, such as ellipses, will also fail to yield
convergence (see Grafakos, 2014b, , Chap. 5 and its exercises for a
discussion).

On the other hand, taking squares or rectangles to mark the cutoff does work
for all $1<p<\infty$, as is well-known (see Grafakos, 2014a, , Chap. 4).
Results by Córdoba, (1977) show that other polygonal regions also generate
“good” cutoffs within the range $4/3<p<4$.

So far we have been interested in the geometry of “frequency space”, i.e. the
lattice $\mathbb{Z}^{2}$. All results above concern functions defined on the
torus, i.e. periodic functions on the square. The scaling properties of
Fourier series imply that all results carry over to functions defined on
rectangles $[0,a]\times[0,b]$. Recently, one of the authors has transferred
$L^{p}$ convergence results to functions defined on a certain class of
triangular domains (Acosta Babb, 2023a, ). Another case of interest is
functions defined on the disc, which, to the best of our knowledge, remains
open (see Acosta Babb, 2023b, , for a discussion of some of the issues).

One way in which we could explore the issue of convergence in other planar
domains is by perturbation. The idea is to take a planar region $\Omega$ and
deform it into another region $\widetilde{\Omega}$. We then study how the
eigenfunctions of the Laplacian and the projection operators onto eigenspaces
change alongside this change to $\Omega$. The hope is that, for sufficiently
small perturbations we can carry over convergence results on a domain where
they are known (e.g. a square, a triangle) to new domains where the problem is
still unsolved. The importance of having such $L^{p}$ convergence results for
problems in PDE is discussed in Fefferman et al., (2022). They note that a
general understanding of $L^{p}$ convergence for eigenfunction expansions is
still open, prompting our current efforts.

We now discuss the setup of the perturbation approach in some more detail,
following Lamberti and Lanza de Cristoforis, (2005).

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, open and connected domain in
the plane (so, in particular, the Poincaré inequality holds on $\Omega$). We
perturb $\Omega$ by a map $\varphi\colon\Omega\to\widetilde{\Omega}$ close to
the identity in a sense to be made precise later. We are interested in
relating the Dirichlet eigenproblem on $\widetilde{\Omega}$ back to that on
$\Omega$. We wish to compare the two operators by applying them to functions
defined on $\Omega$. Thus, “$\Delta$” will denote the Laplacian acting on
$u\in H_{0}^{1}(\Omega)$, while “$\Delta_{\varphi}$” denotes the Laplacian
acting on $u\circ\varphi^{-1}\in H_{0}^{1}(\widetilde{\Omega})$, and then
“pulled back” via $\varphi$ to $H_{0}^{1}(\Omega)$. Thus,
$\Delta-\Delta_{\varphi}$ is well-defined as an operator acting on functions
defined on $\Omega$.

We list the (positive) eigenvalues of $-\Delta$, _including multiplicities_ ,
as

$0<\lambda_{1}\leqslant\lambda_{2}\leqslant\lambda_{3}\leqslant\ldots.$

Similarly, we denote the eigenvalues of $-\Delta_{\varphi}$ by
“$\widetilde{\lambda}_{j}$”.

Note that, since $0$ is not an eigenvalue, the Laplacian and its inverse yield
equivalent eigenvalue problems:

$-\Delta u=\lambda u\quad\text{if and only
if}\quad-\Delta^{-1}u=\lambda^{-1}u.$ (1)

We will therefore concentrate on the latter problem, since the inverse
operators are compact and self-adjoint (see Lamberti and Lanza de
Cristoforis,, 2005, Lemma 3.5).

We will need to impose a further technical condition on the eigenspaces we
consider. Fix a finite subset $F\subset\mathbb{N}$ and denote by “$P_{F}$” the
orthogonal projection in $L^{2}$ onto the eigenfunctions $u_{j}$ associated to
the $\lambda_{j}$ with $j\in F$. Note, in particular, that $u_{j}\neq u_{j+1}$
with $\lambda_{j}=\lambda_{j+1}$ if $\lambda_{j}$ has a multiplicity of at
least 2.

######  Definition 1.

We say that $F$ _splits_ an eigenvalue $\lambda$ if there are indices
$j,k\in\mathbb{N}$ such that $\lambda=\lambda_{j}=\lambda_{k}$ and $j\in F$
but $k\notin F$.

In terms of the projection $P_{F}$, this means that $P_{F}u_{j}=u_{j}$ but
$P_{F}u_{k}=0$ even though $-\Delta u_{j}=\lambda u_{j}$ and $-\Delta
u_{k}=\lambda u_{k}$. We need to rule out eigenvalue splitting in order to
apply Theorem 4 (see the remark following Theorem 2 for further details).

In Section 2 we set up the necessary notation and discuss the perturbation of
the domain in some detail. In Section 3 we prove that under certain
assumptions on $\varphi$, $F$ will not split eigenvalues of $\Delta_{\varphi}$
if it did not split eigenvalues of $\Delta$. Section 4 contains the main
estimates, as well as some auxiliary continuity results for the eigenvalues of
the Laplacian. Finally, in Section 5 we state and prove our main result for
the Dirichlet Laplacian: $L^{p}$ to $L^{p}$ boundedness of the projections
under perturbations of the domain $\Omega$.

Throughout we will use the notation $A(s)\lesssim B(s)$ to mean that there is
a constant $C>0$, which does not depend on $s$, such that

$A(s)\leqslant B(s)\quad\text{for all}\quad s.$

##  2 Preliminaries

We begin with a homeomorphism $\varphi\colon\Omega\to\widetilde{\Omega}$ and
we wish to study operators on both of these domains. Since these operators act
on different function spaces, e.g. $H_{0}^{1}(\Omega)$ and
$H_{0}^{1}(\widetilde{\Omega})$, we cannot compare them directly.

Let $u,v\in H_{0}^{1}(\Omega)$. Their $L^{2}$ inner product is

$\left\langle{u,v}\right\rangle_{L^{2}(\Omega)}=\int_{\Omega}uv\,\mathrm{d}{x}.$
(2)

On the other hand, $u\circ\varphi^{-1},v\circ\varphi^{-1}\in
H_{0}^{1}(\widetilde{\Omega})$, with $L^{2}$ inner product

$\displaystyle\left\langle{u\circ\varphi^{-1},v\circ\varphi^{-1}}\right\rangle_{L^{2}(\widetilde{\Omega})}$
$\displaystyle=\int_{\varphi(\Omega)}u(\varphi^{-1}(y))v(\varphi^{-1}(y))\,\mathrm{d}{y}$
$\displaystyle=\int_{\Omega}u(x)v(x)\left|\det{D\varphi}(x)\right|\,\mathrm{d}{x}=:Q_{\varphi}[u,v].$
(3)

If we denote the standard inner product on $L^{2}(\Omega)$ by “$Q$”, then we
can compare the inner products from Eqs. 2 and 2:

$(Q-Q_{\varphi})[u,v]=\int_{\Omega}uv(1-\left|\det{D\varphi}\right|)\,\mathrm{d}{x}\quad\text{for
all}\quad u,v\in H_{0}^{1}(\Omega).$ (4)

We begin by recording a key result in Lamberti and Lanza de Cristoforis,
(2005) which we are going to generalise in this paper. Let $H$ be a Hilbert
space equipped with two different bilinear forms, $Q$ and $\widetilde{Q}$.
Suppose that $T$ and $\widetilde{T}$ are two compact operators from $H$ to
itself and self-adjoint on $(H,Q)$ and $(H,\widetilde{Q})$, respectively. Let
$F\subset\mathbb{N}$ be a finite set of indices and denote by $P_{F}[Q,T]$ the
projection

$P_{F}[Q,T]u:=\sum_{j\in F}Q[u,u_{j}]u_{j}$

where $u_{j}$ are eigenfunctions of $T$ corresponding to the indices $j\in F$.

######  Theorem 2 (Lamberti and Lanza de Cristoforis, (2005)).

Let $F\subset\mathbb{N}$ be a finite set of indices that does not split
eigenvalues of $T$ or $\widetilde{T}$. Then, there is a positive constant
$C=C(T,\widetilde{T},F)$ such that

$\left\lVert P_{F}[Q,T]-P_{F}[\widetilde{Q},\widetilde{T}]\right\rVert_{H\to
H}\leqslant C\left(\left\lVert Q-\widetilde{Q}\right\rVert_{H\times
H\to\mathbb{R}}+\left\lVert T-\widetilde{T}\right\rVert_{H\to H}\right).$ (5)

######  Remark.

The main tool in the proof is Kato’s integral formula for a projection:

$P_{F}[Q,T]=\mathrm{Re}\left(-\frac{1}{2\pi\mathrm{i}}\int_{\gamma[F,T]}(T-\xi
I)^{-1}\,\mathrm{d}{\xi}\right).$

Here $\gamma[F,T]$ is a contour encircling the portion of the spectrum of $T$
which contains the eigenvalues in $F$.

As we can see, the integral formula is “blind” to multiplicities: it only
differentiates the values of the eigenvalues, and there is no way of keeping
track of exactly which eigenfunctions we want to keep in our projection. This
is the reason behind the no-splitting condition, since if our set $F$ were to
split an eigenvalue, then Kato’s formula would not actually give us the
projection $P_{F}[T,Q]$, but rather a projection onto a larger space which
includes all eigenfunctions for every eigenvalue “in $F$”.

Our first goal is to extend Theorem 2 from Hilbert spaces to $L^{p}$. We must
keep in mind that there are three Laplacians in play here: the Laplacian
$\Delta=\Delta_{\Omega}$ acting on $C_{c}^{\infty}(\Omega)$; the Laplacian
$\Delta_{\widetilde{\Omega}}$ acting on $C_{c}^{\infty}(\widetilde{\Omega})$;
and a transformed Laplacian $\Delta_{\varphi}$ acting also on
$C_{c}^{\infty}(\Omega)$ as follows:

$\Delta_{\varphi}f(x):=(\Delta_{\widetilde{\Omega}}(f\circ\varphi^{-1}))(\varphi(x))\quad\text{for
all}\quad f\in C_{c}^{\infty}(\Omega).$

That is: the function $f\circ\varphi^{-1}$ is defined on $\widetilde{\Omega}$,
which we then differentiate using $\Delta_{\widetilde{\Omega}}$, and finally
pull back to $\Omega$ by composing the result with $\varphi$.

We will denote the eigenfunctions of $\Delta_{\Omega}$ and
$\Delta_{\widetilde{\Omega}}$ respectively by $u_{j}$ and $\widetilde{u}_{j}$.
The eigenfunctions of $\Delta_{\varphi}$ are
$u^{\varphi}_{j}:=\widetilde{u}_{j}\circ\varphi$ (see Lamberti and Lanza de
Cristoforis,, 2005, Theorem 3.10).

We are now in a position to distinguish three “partial sum” operators of
interest:

  1. (i)

Partial sums on $\Omega$, which are expansions in $L^{2}(\Omega)$ in terms of
the eigenfunctions $u_{j}$ of $\Delta_{\Omega}$ relative to $\Omega$:

$P_{F}f=\sum_{j\in
F}\left\langle{u_{j},f}\right\rangle_{\Omega}u_{j}\quad\text{for all}\quad
f\in L^{2}(\Omega).$ (6)

  2. (ii)

Partial sums on $\widetilde{\Omega}$, which are expansions in
$L^{2}(\widetilde{\Omega})$ in terms of the eigenfunctions $\widetilde{u}_{j}$
of $\Delta_{\widetilde{\Omega}}$ relative to $\widetilde{\Omega}$:

$\widetilde{P}_{F}g=\sum_{j\in
F}\left\langle{\widetilde{u}_{j},g}\right\rangle_{\widetilde{\Omega}}\widetilde{u}_{j}\quad\text{for
all}\quad g\in L^{2}(\widetilde{\Omega}).$ (7)

  3. (iii)

Transformed partial sums on $\Omega$ which are “pull-backs” of
$\widetilde{S}_{F}$ to $L^{2}(\Omega)$. These are expansions with respect to
the eigenfunctions $u^{\varphi}_{j}$ of $\Delta_{\varphi}$ in $L^{2}(\Omega)$,
now equipped with the inner product $Q_{\varphi}$:

$P^{\varphi}_{F}f=\sum_{j\in
F}Q_{\varphi}[u^{\varphi}_{j},f]u^{\varphi}_{j}\quad\text{for all}\quad f\in
L^{2}(\Omega).$ (8)

The goal is to use $P^{\varphi}_{F}$ to transfer bounds on $P_{F}\colon
L^{p}(\Omega)\to L^{p}(\Omega)$ to bounds on $\widetilde{P}_{F}\colon
L^{p}(\widetilde{\Omega})\to L^{p}(\widetilde{\Omega})$. This “transference”
result is encapsulated in the following lemma.

######  Lemma 3 (Transference from $\Omega$ to $\widetilde{\Omega}$).

Let $\varphi\colon\Omega\to\widetilde{\Omega}$ be (weakly) differentiable,
invertible and satisfy the bounds

$0<\inf_{\Omega}\left|\det{D\varphi}\right|\leqslant\sup_{\Omega}\left|\det{D\varphi}\right|<\infty.$

Suppose that we have bounds of the form

$\left\lVert P_{F}\right\rVert_{L^{p}(\Omega)\to L^{p}(\Omega)}\leqslant A$

and

$\left\lVert P_{F}-P_{F}^{\varphi}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\leqslant B.$

Then, we have the bound

$\left\lVert\widetilde{P}_{F}\right\rVert_{L^{p}(\widetilde{\Omega})\to
L^{p}(\widetilde{\Omega})}\leqslant\left(\frac{\sup\left|\det{D\varphi}\right|}{\inf\left|\det{D\varphi}\right|}\right)^{1/p}(A+B).$

###### Proof.

Fix $f\in L^{p}(\Omega)$ and let $g:=f\circ\varphi^{-1}\in
L^{p}(\widetilde{\Omega})$. Then, by a change of variables in the integral
defining the inner product on $\widetilde{\Omega}$, we have

$\widetilde{P}_{F}g=\sum_{j\in
F}\left(\int_{\Omega}\widetilde{u}_{j}(\varphi(x))f(x)\left|\det{D\varphi(x)}\right|\,\mathrm{d}{x}\right)\widetilde{u}_{j}=\sum_{j\in
F}Q_{\varphi}[\widetilde{u}_{j}\circ\varphi,f]\widetilde{u}_{j}.$

Taking $L^{p}(\widetilde{\Omega})$ norms and changing variables again:

$\displaystyle\left\lVert\widetilde{P}_{F}f\right\rVert_{L^{p}({\widetilde{\Omega}})}^{p}$
$\displaystyle=\int_{\widetilde{\Omega}}\left|\sum_{j\in
F}Q_{\varphi}[\widetilde{u}_{j}\circ\varphi,f]\widetilde{u}_{j}(y)\right|^{p}\,\mathrm{d}{y}$
$\displaystyle=\int_{\Omega}\left|\sum_{j\in
F}Q_{\varphi}[\widetilde{u}_{j}\circ\varphi,f]\widetilde{u}_{j}(\varphi(x))\right|^{p}\left|\det{D\varphi(x)}\right|\,\mathrm{d}{x}$
$\displaystyle\leqslant\left\lVert\left|\det{D\varphi(x)}\right|\right\rVert_{L^{\infty}({\Omega})}\left\lVert\sum_{j\in
F}Q_{\varphi}[u^{\varphi}_{j},f]u^{\varphi}_{j}\right\rVert_{L^{p}({\Omega})}^{p},$

whence we arrive at

$\left\lVert\widetilde{P}_{F}g\right\rVert_{L^{p}({\widetilde{\Omega}})}\leqslant\left\lVert\left|\det{D\varphi(x)}\right|\right\rVert_{L^{\infty}({\Omega})}^{1/p}\left\lVert
P^{\varphi}_{F}f\right\rVert_{L^{p}({\Omega})}.$

The result now follows by applying the triangle inequality to
$P^{\varphi}_{F}f=P_{F}f+P^{\varphi}_{F}f-P_{F}f$ and noticing that

$\left\lVert
f\right\rVert_{L^{p}({\Omega})}^{p}\leqslant\sup_{\widetilde{\Omega}}\left|\det{D\varphi}^{-1}\right|\left\lVert
g\right\rVert_{L^{p}({\widetilde{\Omega}})}^{p}.\qed$

In the following, abstract theorem, we will think of
$T=(\Delta_{\Omega})^{-1}$ with $P_{F}[Q,T]=P_{F}$, and
$\widetilde{T}=(\Delta_{\varphi})^{-1}$ with $\widetilde{Q}=Q_{\varphi}$, so
$P_{F}[\widetilde{Q},\widetilde{T}]=P^{\varphi}_{F}$.

######  Theorem 4.

Let $T$ and $\widetilde{T}$ be bounded operators from $L^{p}$ to $L^{p}$ with
discrete spectra $(\mu_{j})_{j\in\mathbb{N}}$ and
$(\widetilde{\mu}_{j})_{j\in\mathbb{N}}$, respectively. Let
$F\subset\mathbb{N}$ be a finite set of indices that does not split
eigenvalues of $T$ or $\widetilde{T}$. Then, there is a positive constant
$C=C(T,\widetilde{T},F)$ such that

$\displaystyle\left\lVert
P_{F}[Q,T]-P_{F}[\widetilde{Q},\widetilde{T}]\right\rVert_{L^{p}\to
L^{p}}\leqslant C\left(\left\lVert Q-\widetilde{Q}\right\rVert_{L^{2}\times
L^{2}\to\mathbb{R}}+\left\lVert T-\widetilde{T}\right\rVert_{L^{2}\to
L^{2}}\right)$ $\displaystyle\times\left(1+\left\lVert
T-\widetilde{T}\right\rVert_{L^{p}\to L^{p}}+\left\lVert
T-\widetilde{T}\right\rVert_{L^{2}\to L^{2}}+\left\lVert
Q-\widetilde{Q}\right\rVert_{L^{2}\times L^{2}\to\mathbb{R}}\right).$

######  Remark.

We will discuss the precise dependence of the constant $C$ on $F$ after the
proof.

###### Proof.

We adapt the proof of Theorem 2 to the $L^{p}$ setting. Without loss of
generality, we may assume that the set $F$ of indices is connected in the
following sense: if $j\in F$ and $j+2\in F$, then $j+1\in F$. When this is not
the case, we simply split $F$ into its finitely many connected components and
apply the following argument to each one.

Recall Kato’s integral formula for a projection:

$P_{F}[T,Q]=\mathrm{Re}\left(-\frac{1}{2\pi\mathrm{i}}\int_{\gamma[F,T]}(T-\xi
I)^{-1}\,\mathrm{d}{\xi}\right).$

Here $\gamma[F,T]$ is a contour encircling the portion of the spectrum of $T$
which contains the eigenvalues in $F$. More specifically, we define the
quantity

$d[T,F]=\min_{j\in F,k\notin F}\left|\mu_{j}-\mu_{k}\right|$

and choose the point $c=\tfrac{1}{2}\left(\mu_{\min{F}}-\mu_{\max{F}}\right)$.
As our contour $\gamma$, we take the circle centred at $c$ with radius
$c+\tfrac{1}{2}d[T,F]$.

We next prove the following technical claim:

$\int_{\gamma}(T-\xi)^{-1}f\,\mathrm{d}{\xi}=\int_{\gamma}\sum_{j\in
F}\frac{1}{\mu_{j}-\xi}\left\langle{u_{j},f}\right\rangle
u_{j}\,\mathrm{d}{\xi}.$ (9)

It suffices to check that (Conway,, 2007, VII,§4)

$\int_{\gamma}\left\langle{(T-\xi)^{-1}f,g}\right\rangle\,\mathrm{d}{\xi}=\int_{\gamma}\left\langle{\sum_{j\in
F}\frac{1}{\mu_{j}-\xi}\left\langle{u_{j},f}\right\rangle
u_{j},g}\right\rangle\,\mathrm{d}{\xi}$

for all $f\in C_{c}^{\infty}$ and $g\in L^{q}$ with $1/p+1/q=1$, since
$C_{c}^{\infty}$ in dense in $L^{p}$.

For $f\in C_{c}^{\infty}$, we can write
$f=\sum\left\langle{u_{j},f}\right\rangle u_{j}$ and apply the bounded
operator $(T-\xi)^{-1}$:

$\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\left\langle{(T-\xi)^{-1}f,g}\right\rangle\,\mathrm{d}{\xi}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\sum\frac{\left\langle{u_{j},f}\right\rangle}{\mu_{j}-\xi}\left\langle{u_{j},g}\right\rangle\,\mathrm{d}{\xi}$
$\displaystyle=\sum\left\langle{u_{j},f}\right\rangle\left\langle{u_{j},g}\right\rangle\left(\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\frac{1}{\mu_{j}-\xi}\,\mathrm{d}{\xi}\right).$

By the classical Cauchy integral formula, the integral in brackets is $1$ if
$j\in F$ and $0$ otherwise. Hence, the last expression is equal to

$\frac{1}{2\pi\mathrm{i}}\int_{\gamma}\left\langle{\sum_{j\in
F}\frac{1}{\mu_{j}-\xi}\left\langle{u_{j},f}\right\rangle
u_{j},g}\right\rangle\,\mathrm{d}{\xi},$

and the claim follows.

The next step is to obtain a bound on the operator norm of

$R_{F}(\xi)f:=\sum_{j\in
F}\frac{1}{\mu_{j}-\xi}\left\langle{u_{j},f}\right\rangle u_{j}.$

Once again, taking $f\in C_{c}^{\infty}$ we have

$\left\lVert R_{F}(\xi)f\right\rVert_{L^{p}}\leqslant\sum_{j\in
F}\frac{\left\lVert u_{j}\right\rVert_{L^{p}}\left\lVert
u_{j}\right\rVert_{L^{q}}}{\left|\mu_{j}-\xi\right|}\left\lVert
f\right\rVert_{L^{p}}\leqslant\frac{C[T,F]}{d[T,F]}\left\lVert
f\right\rVert_{L^{p}},$

by construction of $\gamma$, where

$C[T,F]:=C[F,T]=\sum_{j\in F}\left\lVert u_{j}\right\rVert_{L^{p}}\left\lVert
u_{j}\right\rVert_{L^{q}}.$

Combining this estimate with Kato’s formula and Eq. 9, we obtain a bound on
$P_{F}[T,Q]-P_{F}[\widetilde{T},\widetilde{Q}]$ of the following form (see
Lamberti and Lanza de Cristoforis,, 2005, eq. (17)):

$\left\lVert
P_{F}[Q,T]-P_{F}[\widetilde{Q},\widetilde{T}]\right\rVert_{L^{p}\to
L^{p}}\lesssim\frac{C[T,F]}{d[T,F]}\max_{j\in
F}\left|\mu_{j}-\widetilde{\mu}_{j}\right|\\\
+\frac{C[T,F]C[\widetilde{T},F]}{d[T,F]d[\widetilde{T},F]}\max_{j\in
F}\left|\mu_{j}-\widetilde{\mu}_{j}\right|\left(\left\lVert
T-\widetilde{T}\right\rVert_{L^{p}\to L^{p}}+\max_{j\in
F}\left|\mu_{j}-\widetilde{\mu}_{j}\right|\right).$ (10)

Lamberti and Lanza de Cristoforis, (2005) show that

$\left|\mu_{j}-\widetilde{\mu}_{j}\right|\lesssim\left\lVert
Q-\widetilde{Q}\right\rVert_{L^{2}\times L^{2}\to\mathbb{R}}+\left\lVert
T-\widetilde{T}\right\rVert_{L^{2}\to L^{2}}\quad\text{for all}\quad
j\in\mathbb{Z}.$

Now, $\min\\{d[T,F],d[\widetilde{T},F]\\}$ and
$\max{C[T,F],C[\widetilde{T},F]}$ can be treated as a positive constant
(depending on $F$), so the quotients above are bounded. Thus, combining the
last two displays yields the result. ∎

######  Remark.

We will take “$T=-\Delta^{-1}$” on the square, so the eigenvalues decay as
$\mu_{j}=O(1/j^{2})$. We consider the set
$F=\\{j:\lambda_{j}=\mu_{j}^{-1}\leqslant N^{2}\\}$. Hence, the constants
$d[T,F]$ decay at a rate $O(1/j^{3})$, which means that the constants
involving $1/d[T,F]$ blow up at least as fast as $N^{3}$ (see Example 25).
Furthermore, the operators $T$ and $\widetilde{T}$ will be “close” in a sense
to be made precise below, and in a such a way that the eigenvalues
$\widetilde{\mu}_{j}$ and $\mu_{j}$ remain comparable (see Theorem 2 and Lemma
15 in Section 4). Hence, $d[T,F]\approx d[\widetilde{T},F]$, so

$\frac{1}{d[T,F]d[\widetilde{T},F]}\approx N^{6}.$

On the other hand, consider the constant

$C[F,T]=\sum_{j\in F}\left\lVert u_{j}\right\rVert_{L^{p}}\left\lVert
u_{j}\right\rVert_{L^{q}}.$

Without loss of generality, assume that $1<p<2<q<\infty$. Then, normalising to
$\left\lVert u_{j}\right\rVert_{L^{2}}=1$,

$\left\lVert u_{j}\right\rVert_{L^{p}}\left\lVert
u_{j}\right\rVert_{L^{q}}\lesssim\left\lVert
u_{j}\right\rVert_{L^{2}}\left\lVert
u_{j}\right\rVert_{H^{1}}\lesssim\mu_{j}^{-1/2}=O(N),$

so $C[F,T]=O(N^{2})$. Hence, we can expect the constant $C$ in Theorem 4 to
blow up roughly like $N^{8}$ in this case.

######  Remark.

When $p=2$, the constant

$C[F,T]=\sum_{j\in F}\left\lVert u_{j}\right\rVert_{L^{p}}\left\lVert
u_{j}\right\rVert_{L^{q}}$

can be improved to $1$ (see Lamberti and Lanza de Cristoforis,, 2005, eq.
(13)). Indeed,

$\left\lVert(T-\xi)^{-1}f\right\rVert^{2}_{L^{2}}=\sum\frac{\left|\left\langle{u_{j},f}\right\rangle\right|^{2}}{\left|\mu_{j}-\xi\right|^{2}}\leqslant\frac{1}{d[T,F]^{2}}\left\lVert
f\right\rVert^{2}_{L^{2}}.$

In their paper, Lamberti and Lanza de Cristoforis, (2005) claim to obtain this
bound from the following inequality (Taylor and Lay,, 1986, VI.3, Theorem
3.1):

$\left\lVert(T-\xi)^{-1}\right\rVert\leqslant\frac{1}{d(\xi,V(T))},$

$d(\xi,V(T))$ is the distance of $\xi\in\gamma$ to the _numerical range_ of
the operator $T$,

$V(T):=\left\\{(Tx,x):\left\lVert x\right\rVert=1\right\\}.$

However, in a Hilbert Space (Halmos,, 1982, p. 116), $V(T)$ is the closed
convex hull of the spectrum of $T$. Since $T$ is compact, $0$ lies in the
closure of the spectrum, and so the contour $\gamma$ as described will
necessarily intersect the numerical range when it crosses the real axis.

Our alternative calculation has the advantage of generalising to $L^{p}$, at
the cost of increasing the size of the constants.

Theorem 4 gives us bounds on the projections in terms of bounds on the
operator norms of the difference $T-\widetilde{T}$. The rest of the paper will
focus on obtaining bounds on $\left\lVert
T-\widetilde{T}\right\rVert_{L^{p}({\Omega})}$ which depend (i) on the change
of variables map $\varphi\colon\Omega\to\widetilde{\Omega}$, and (ii) on the
set $F$.

###  2.1 The inverse of the Laplacian

Under certain assumptions on the regularity of $\Omega$, the Laplacian

$\Delta\colon W^{2,p}(\Omega)\cap W^{1,p}_{0}(\Omega)\to L^{p}(\Omega)$

is invertible; see, for example, Theorem 9.15 in Gilbarg and Trudinger,
(2001).

######  Theorem 5.

Let $\Omega$ be a $C^{1,1}$ domain in $\mathbb{R}^{2}$. Then, if $f\in
L^{p}(\Omega)$ with $1<p<\infty$, the Dirichlet problem $\Delta u=f$ in
$\Omega$ with $u\in W_{0}^{1,p}(\Omega)$ has a unique solution $u\in
W^{2,p}(\Omega)$.

A similar result also holds for a square, as follows from Theorem 4.4.3.7 in
Grisvard, (1985) (whose statement we slightly adapt).

######  Theorem 6.

Assume that $\Omega$ is a polygon with straight boundary edges meeting at
right angles. Then, for each $f\in L^{p}(\Omega)$ there is a unique solution
$u\in W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)$ to the Poisson problem $\Delta
u=f$ with zero boundary conditions.

These are the more “obvious” examples; for a discussion on more general
conditions on $\partial\Omega$ for the $W^{2,p}$-solvability of the Poisson
equation with $L^{p}$ initial data, see Chapter 7 of Maz’ya and Shaposhnikova,
(1985).

In other words: for squares, rectangles or $C^{1,1}$ domains (among others),
the Dirichlet Laplacian is invertible with inverse $\Delta^{-1}\colon L^{p}\to
W^{2,p}\cap W_{0}^{1,p}$.

For brevity, we will henceforth write $X_{p}$ for the Banach space
$W^{2,p}\cap W_{0}^{1,p}$ equipped with the $W^{2,p}$ norm.

######  Definition 7.

Let $\varphi\colon\Omega\to\widetilde{\Omega}$ be an invertible map with
bounded weak derivatives up to second order. Furthermore, we assume that

$\inf_{\Omega}\left|\det{D\varphi}\right|>0.$

We call the triple $(\Omega,\varphi,\widetilde{\Omega})$ _$p$ -admissable_ if
$\varphi$ is as above and $\Delta\colon X_{p}\to L^{p}$ is invertible on both
$\Omega$ and $\widetilde{\Omega}$. We will often simplify this to
“$\varphi\colon\Omega\to\widetilde{\Omega}$ is $p$-admissable” or even
“$\varphi$ is $p$-admissable”.

######  Remark.

We require one more order of differentiation than Lamberti and Lanza de
Cristoforis, (2005), as well as the invertibility of $\Delta$ from $X_{p}$ to
$L^{p}$, which comes “for free” on bounded domains in $\mathbb{R}^{2}$ when
working with $\Delta\colon H^{1}_{0}\to H^{-1}$.

Henceforth, we will assume that $(\Omega,\varphi,\widetilde{\Omega})$ is
$p$-admissable.

######  Definition 8.

Let $Y(\cdot)$ be a function space on $\Omega$ and $\widetilde{\Omega}$, and
let $\varphi\colon\Omega\to\widetilde{\Omega}$ be a change of coordinates. We
define the following operators:

$\displaystyle C_{\varphi}\colon Y(\widetilde{\Omega})$
$\displaystyle\longrightarrow Y(\Omega)$ $\displaystyle v\quad$
$\displaystyle\longmapsto v\circ\varphi$

and

$\displaystyle C_{\varphi^{-1}}\colon Y(\Omega)$ $\displaystyle\longrightarrow
Y(\widetilde{\Omega})$ $\displaystyle u\quad$ $\displaystyle\longmapsto
u\circ\varphi.$

######  Remark.

Note that $(C_{\varphi})^{-1}=C_{\varphi^{-1}}$ and vice versa.

We will denote by $T$ the inverse Laplacian

$T=\Delta^{-1}\equiv(\Delta_{\Omega})^{-1}\colon L^{p}(\Omega)\to
X_{p}(\Omega)\hookrightarrow L^{p}(\Omega).$

We will also denote by $T_{\varphi}$ the inverse of $\Delta_{\varphi}$, i.e.
the map

$T_{\varphi}=C_{\varphi}\circ\Delta_{\widetilde{\Omega}}^{-1}\circ
C_{\varphi^{-1}}\colon L^{p}(\Omega)\to L^{p}(\Omega).$

Thus, we have the following diagram:

${L^{p}(\widetilde{\Omega})}$${X_{p}(\widetilde{\Omega})}$${L^{p}(\widetilde{\Omega})}$${L^{p}(\Omega)}$${X_{p}(\Omega)}$${L^{p}(\Omega)}$$\scriptstyle{\Delta_{\widetilde{\Omega}}^{-1}}$$\scriptstyle{C_{\varphi}}$$\scriptstyle{\Delta_{\Omega}^{-1}}$$\scriptstyle{C_{\varphi^{-1}}}$

where $T_{\varphi}$ is the “up–over–down” path from $L^{p}(\Omega)$ to
$L^{p}(\Omega)$.

We now introduce a slightly modified version of the operator
$\Delta_{\varphi}$.

######  Definition 9.

Let $\mathcal{L}_{\varphi}\colon L^{p}(\Omega)\to L^{p}(\Omega)$ be the
operator defined by

$\mathcal{L}_{\varphi}:=C_{\varphi^{-1}}^{t}\circ\Delta_{\widetilde{\Omega}}\circ
C_{\varphi^{-1}},$

where $C_{\varphi^{-1}}^{t}\colon L^{p}(\widetilde{\Omega})\to L^{p}(\Omega)$
is the operator defined, for $v\in L^{p}(\widetilde{\Omega})$, by

$\left\langle{C_{\varphi^{-1}}^{t}v,w}\right\rangle:=\left\langle{v,C_{\varphi^{-1}}w}\right\rangle\equiv\int_{\widetilde{\Omega}}v(y)w(\varphi^{-1}(y))\,\mathrm{d}{y}\quad\text{for
all}\quad w\in L^{q}(\Omega).$

######  Lemma 10.

The operator $C_{\varphi^{-1}}^{t}$ is given by

$\left[C_{\varphi^{-1}}^{t}v\right](x)=\left|\det{D\varphi(x)}\right|[C_{\varphi}v](x)\quad\text{for
all}\quad v\in L^{p}(\widetilde{\Omega}),$

with inverse

$\left[(C_{\varphi^{-1}}^{t})^{-1}u\right](y)=C_{\varphi}^{-1}\left(\frac{u(x)}{\left|\det{D\varphi(x)}\right|}\right)(y)\quad\text{for
all}\quad u\in L^{p}(\Omega).$

In particular,

$(C_{\varphi^{-1}}^{t})^{-1}=C_{\varphi}^{t}.$ (11)

######  Definition 11.

Define the operator $\mathcal{J}_{\varphi}\colon L^{p}(\Omega)\to
L^{p}(\Omega)$ by

$\mathcal{J}_{\varphi}u(x):=\left|\det{D\varphi(x)}\right|u(x)\quad\text{for
all}\quad u\in L^{p}(\Omega).$

If $\inf_{\Omega}\left|\det{D\varphi}\right|>0$, then

$\mathcal{J}_{\varphi}^{-1}u(x)=\frac{u(x)}{\left|\det{D\varphi(x)}\right|}.$

We can therefore conclude the following.

######  Corollary 12.

We have $T_{\varphi}=\mathcal{L}_{\varphi}^{-1}\circ\mathcal{J}_{\varphi}$.

###### Proof.

First, note that
$\mathcal{L}_{\varphi}=C_{\varphi^{-1}}^{t}\circ\Delta_{\widetilde{\Omega}}\circ
C_{\varphi^{-1}}$ implies that

$\displaystyle(\mathcal{L}_{\varphi})^{-1}$
$\displaystyle=(C_{\varphi^{-1}})^{-1}\circ(\Delta_{\widetilde{\Omega}})^{-1}\circ(C_{\varphi^{-1}}^{t})^{-1}$
$\displaystyle=C_{\varphi}\circ\Delta_{\widetilde{\Omega}}^{-1}\circ
C_{\varphi}^{-1}\circ\mathcal{J}_{\varphi}^{-1}$
$\displaystyle=(\Delta_{\varphi})^{-1}\circ\mathcal{J}_{\varphi}^{-1},$

and so composing with $\mathcal{J}_{\varphi}$ finishes the proof. ∎

These rather formal calculations are justified by the admissibility conditions
on $\varphi$.

######  Remark.

The crucial detail to observe is that we factor the (transformed) inverse
Laplacian $T_{\varphi}$ into two operators: the inverse “Laplacian”
$\mathcal{L}_{\varphi}^{-1}\colon L^{p}(\Omega)\to L^{p}(\Omega)$ and the
embedding $\mathcal{J}_{\varphi}\colon L^{p}(\Omega)\to L^{p}(\Omega)$. This
circuitous route, which is inspired by Lamberti and Lanza de Cristoforis,
(2005), will vastly simplify our proof of 21 below. Essentially, we will need
to transfer integration by parts from $\Omega$ to $\widetilde{\Omega}$. If we
were simply to use $\Delta_{\varphi}$, we would have

$\displaystyle\left\langle{\Delta_{\varphi}u,w}\right\rangle$
$\displaystyle=\left\langle{C_{\varphi}\Delta_{\widetilde{\Omega}}C_{\varphi^{-1}}u,w}\right\rangle$
$\displaystyle=\int_{\Omega}(\Delta_{\widetilde{\Omega}}(u\circ\varphi^{-1}))(\varphi(x))w(x)\,\mathrm{d}{x}$
$\displaystyle=\int_{\widetilde{\Omega}}\Delta_{\widetilde{\Omega}}(u\circ\varphi^{-1})(y)w(\varphi^{-1}(y))\left|\det
D\varphi^{-1}(y)\right|\,\mathrm{d}{y}$
$\displaystyle=-\int_{\widetilde{\Omega}}D_{y}(u\circ\varphi^{-1})(y)\cdot
D_{y}\left[(w\circ\varphi^{-1})\left|\det{D\varphi^{-1}(y)}\right|\right]\,\mathrm{d}{y}.$

Using the operator $\mathcal{L}_{\varphi}$ instead will allow us to circumvent
differentiating the determinant, which is then reintroduced by
$\mathcal{J}_{\varphi}$ in a more convenient place.

##  3 Continuity and splitting of eigenvalues

In Theorem 4 we assumed that the set $F$ does not split eigenvalues of either
operator. In this section, we show that, under reasonable assumptions on
$\varphi$, it is enough to assume that $F$ does not split eigenvalues of
$\Delta$ on $\Omega$. For in this case, continuity results of the eigenvalues
with respect to changes of the domain will ensure that $F$ will not split the
eigenvalues of $\Delta_{\widetilde{\Omega}}$ on $\widetilde{\Omega}$. The next
lemma guarantees that $\Delta_{\widetilde{\Omega}}$ and its pull-back to
$\Omega$ have the same eigenvalues. Thus, if $F$ does not split eigenvalues of
$T$, then it will not split eigenvalues of $T_{\varphi}$, and we can apply
Theorem 4. (Recall that $\lambda>0$ is an eigenvalue of $-\Delta$ if, and only
if, $\frac{1}{\lambda}>0$ is an eigenvalue of $-\Delta^{-1}$.)

######  Lemma 13.

Let $\lambda\in\mathbb{R}$ and $v\in
L^{2}(\widetilde{\Omega})\setminus\\{0\\}$. Then,

$\Delta_{\widetilde{\Omega}}^{-1}v=\lambda v\quad\text{if, and only if,}\quad
T_{\varphi}(C_{\varphi}v)=\lambda C_{\varphi}v.$

###### Proof.

Immediate from the definitions; see Theorem 3.10 in Lamberti and Lanza de
Cristoforis, (2005). ∎

Lemma 13 tells us that the eigenvalues of $\Delta_{\widetilde{\Omega}}^{-1}$
remain unchanged after we “pull them back to $\Omega$”—that is the purpose of
$T_{\varphi}$.

We next check that the eigenvalues of $\Delta^{-1}$ are close to the
eigenvalues of $\Delta_{\widetilde{\Omega}}^{-1}$. This we do via a theorem
due to Courant and Hilbert, (1989) (see p. 423).

######  Theorem 14.

Let $\varphi:\Omega\to\widetilde{\Omega}$ be a bijection with
$\inf_{x\in\Omega}\left|\det{D\varphi}\right|>0$. Then, for any $\eta>0$ there
is an $\varepsilon>0$ such that if

$\left\lVert\operatorname{id}-\varphi\right\rVert_{W^{1,\infty}({\Omega})}\leqslant\varepsilon,$

then

$\left|\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}-1\right|<\eta\quad\text{for
all}\quad j\in\mathbb{N}.$

######  Remark.

Strictly speaking, this result applies to $-\Delta$, not $-\Delta^{-1}$. But
we have seen that the eigenvalue problems for both operators are equivalent
(see Eq. 1 and the surrounding discussion in the Introduction). The proof of
this result uses the variational formulation of the eigenvalue problem. As
Lamberti and Lanza de Cristoforis, (2005) point out, the “variational
eigenvalues” need not coincide with the true eigenvalues. However, this is the
case in bounded domains. (More generally, when the embedding
$H_{0}^{1}(\Omega)\hookrightarrow L^{2}(\Omega)$ is compact; see their
discussion on p. 289.)

######  Lemma 15.

Let $j,k\in F$ be such that $\lambda_{j}\neq\lambda_{k}$. Assume that

$\gamma:=\inf\\{\lambda_{l}-\lambda_{m}:l>m>0\\}>0.$

Then, $\left|\widetilde{\lambda}_{j}-\widetilde{\lambda}_{k}\right|>\gamma/2$,
provided that
$\left\lVert\operatorname{id}-\varphi\right\rVert_{W^{1,\infty}({\Omega})}$ is
small enough.

###### Proof.

Without loss of generality, we may assume that $\lambda_{j}>\lambda_{k}$, and
set $\delta:=\lambda_{j}-\lambda_{k}\geqslant\gamma$. Note that

$\widetilde{\lambda}_{j}-\widetilde{\lambda}_{k}=\lambda_{j}\cdot\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}-\lambda_{k}\cdot\frac{\widetilde{\lambda}_{k}}{\lambda_{k}}=(\lambda_{j}-\lambda_{k})\cdot\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}+\lambda_{k}\left(\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}-\frac{\widetilde{\lambda}_{k}}{\lambda_{k}}\right).$

By Theorem 14,

$1-\eta<\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}<1+\eta,\quad\left|\frac{\widetilde{\lambda}_{j}}{\lambda_{j}}-\frac{\widetilde{\lambda}_{k}}{\lambda_{k}}\right|<2\eta\quad\text{and}\quad\left|\lambda_{k}\right|\leqslant\max_{l\in
F}\left|\lambda_{l}\right|.$

Hence,

$\widetilde{\lambda}_{j}-\widetilde{\lambda}_{k}\geqslant(\lambda_{j}-\lambda_{k})(1-\eta)-2\max_{l\in
F}\left|\lambda_{l}\right|\eta\geqslant\gamma-\eta(\delta+2\max_{l\in
F}\left|\lambda_{l}\right|),$

which is larger than $\gamma/2$, provided that
$\eta<\frac{\gamma}{\delta+2\max_{l\in F}\left|\lambda_{l}\right|}$. ∎

Since there are only finitely many indices in $F$, it follows that, for a
fixed $F$ and a sufficiently small perturbation $\varphi$, the multiplicities
of the eigenvalues of $\Delta_{\widetilde{\Omega}}$ (in $F$) will not decrease
with respect to the multiplicities of the eigenvalues of $\Delta$ (in $F$).

Combining Lemma 15 with Lemma 13, we deduce that $F$ does not split
eigenvalues for $T$ nor $T_{\varphi}$.

######  Corollary 16.

If $\varphi$ is $p$-admissable and
$\left\lVert\operatorname{id}-\varphi\right\rVert_{W^{1,\infty}({\Omega})}$
sufficiently small, then $F$ will not split eigenvalues of $T$ nor
$T_{\varphi}$.

##  4 Bounds on the norms of the operators

###  4.1 Auxiliary bounds

We need bounds on the operators $C_{\varphi}\colon
L^{p}(\widetilde{\Omega})\to L^{p}(\Omega)$ and $C_{\varphi}^{t}\colon
L^{p}(\Omega)\to L^{p}(\widetilde{\Omega})$.

######  Lemma 17.

If $\varphi$ is $p$-admissable, then

$\left\lVert C_{\varphi}\right\rVert_{L^{p}(\widetilde{\Omega})\to
L^{p}(\Omega)}\leqslant\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|^{1/p}},$

and, with $1/p+1/q=1$,

$\left\lVert C_{\varphi}^{t}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\widetilde{\Omega})}\leqslant\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|^{1/q}}.$
(12)

###### Proof.

For any $v\in L^{p}(\widetilde{\Omega})$ we have

$\displaystyle\left\lVert C_{\varphi}v\right\rVert_{L^{p}({\Omega})}^{p}$
$\displaystyle=\int_{\Omega}\left|v(\varphi(x))\right|^{p}\,\mathrm{d}{x}$
$\displaystyle=\int_{\Omega}\left|v(\varphi(x))\right|^{p}\frac{\left|\det{D\varphi}\right|}{\left|\det{D\varphi}\right|}\,\mathrm{d}{x}$
$\displaystyle\leqslant\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\int_{\Omega}\left|v(\varphi(x))\right|^{p}\left|\det{D\varphi}\right|\,\mathrm{d}{x}$
$\displaystyle=\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\int_{\widetilde{\Omega}}\left|v(y)\right|^{p}\,\mathrm{d}{y},$

by the change of variables formula, and (12) immediately follows by a simple
duality argument. ∎

######  Corollary 18.

If $\varphi$ is $p$-admissable, then

$\left\lVert\mathcal{L}_{\varphi}^{-1}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\lesssim\frac{1}{\inf_{\Omega}{\left|\det{D\varphi}\right|}}.$

###### Proof.

Recall that, by Eq. 11, we have

$\mathcal{L}_{\varphi}^{-1}=C_{\varphi}\circ\Delta_{\widetilde{\Omega}}^{-1}\circ
C_{\varphi}^{t}.$

Hence, the result follows from Lemma 17 and the estimate

$\left\lVert\Delta_{\widetilde{\Omega}}^{-1}\right\rVert_{L^{p}(\widetilde{\Omega})\to
L^{p}(\widetilde{\Omega})}\lesssim 1,$

by the $p$-admissability of $\widetilde{\Omega}$. ∎

Next we need some technical lemmas concerning $\varphi$ and its derivatives.
We will write $\varphi=\mathrm{id}+(f,g)$ to mean

$\varphi(x,y)=(x+f(x,y),y+g(x,y)).$

We want to consider $\varphi$ “close to the identity” in the sense that
$\left\lVert\mathrm{id}-\varphi\right\rVert_{W^{2,\infty}({\Omega;\mathbb{R}^{2}})}$
is small. In other words, $f,g$ and their derivatives up to second order have
small $L^{\infty}$ norms.

######  Lemma 19.

Suppose that
$\left\lVert\mathrm{id}-\varphi_{k}\right\rVert_{W^{1,\infty}({\Omega;\mathbb{R}^{2}})}\to
0$ as $k\to\infty$. Then,

$\left\lVert 1-\det{D\varphi_{k}}\right\rVert_{L^{\infty}({\Omega})}\to 0.$

###### Proof.

Let

$D\varphi_{k}=\left(\begin{matrix}a_{k}&b_{k}\\\
c_{k}&d_{k}\end{matrix}\right).$

Then, by assumption,

$D\varphi_{k}-I=\left(\begin{matrix}a_{k}-1&b_{k}\\\
c_{k}&d_{k}-1\end{matrix}\right)\to 0\quad\text{in}\quad L^{\infty}.$

Hence,

$\det{(D\varphi_{k}-I)},(1-a_{k}),(1-b_{k})\to 0,$

and it follows that

$1-\det{(D\varphi_{k}-I)}\equiv 1-(a_{k}d_{k}-b_{k}c_{k})\to 0.\qed$

######  Remark.

Note that we only required $\mathrm{id}-\varphi\in
W^{1,\infty}(\Omega;\mathbb{R}^{2})$ for this lemma. The next result will
require convergence in $W^{2,\infty}$.

######  Lemma 20.

Suppose that
$\left\lVert\mathrm{id}-\varphi_{k}\right\rVert_{W^{2,\infty}({\Omega;\mathbb{R}^{2}})}\to
0$ as $k\to\infty$ and write

$M_{k}:=D\varphi_{k}^{-1}D\varphi_{k}^{-t}\left|\det{D\varphi}\right|-I\equiv\left(\begin{matrix}a_{k}&b_{k}\\\
c_{k}&d_{k}\end{matrix}\right).$

Then, as $k\to\infty$,

  1. (i)

$\left\lVert M_{k}\right\rVert_{L^{\infty}({\Omega})}\to 0$, and

  2. (ii)

$\left\lVert\frac{\partial a_{k}}{\partial x}+\frac{\partial b_{k}}{\partial
y}\right\rVert_{L^{\infty}({\Omega})}$, $\left\lVert\frac{\partial
c_{k}}{\partial x}+\frac{\partial d_{k}}{\partial
y}\right\rVert_{L^{\infty}({\Omega})}\to 0$.

###### Proof.

Recall also that $\varphi_{k}=\mathrm{id}+(f_{k},g_{k})$. For simplicity, we
will drop the $k$ subscripts on $\varphi$, $f$ and $g$, writing
$f_{x}:=\frac{\partial f_{k}}{\partial x}$, and so on. Thus, e.g., “$f_{x}\to
0$” will mean “$\frac{\partial f_{k}}{\partial x}\to 0$ as $k\to\infty$”. Note
that our assumption means precisely that $f_{x},f_{y},g_{x},g_{y}\to 0$, as
well as all second partial derivatives.

By Lemma 19, eventually $\det{D\varphi}>0$ a.e., so we may drop the absolute
value signs. A simple calculation shows that

$D\varphi^{-1}D\varphi^{-t}=\frac{1}{(\det{D\varphi})^{2}}\left(\begin{matrix}g_{x}^{2}+(1+g_{y})^{2}&-(f_{y}+g_{x}+f_{x}g_{y}+f_{y}g_{x})\\\
-(f_{y}+g_{x}+f_{x}g_{y}+f_{y}g_{x})&(1+f_{x})^{2}+f_{y}^{2}\end{matrix}\right)$

and that

$\det{D\varphi}=1+f_{x}+g_{y}+f_{x}g_{y}+f_{y}g_{x}.$

Hence, for example, the top-left entry of $M_{k}$ is

$a_{k}=\frac{g_{x}^{2}+g_{y}^{2}-g_{y}-f_{x}-f_{x}g_{y}-f_{y}g_{x}}{\det{D\varphi}}.$

By Lemma 19, the denominator remains bounded, while the numerator vanishes as
$k\to\infty$ by assumption. Similar calculations for $b,c$ and $d$ establish
(i).

For (ii), we differentiate the last display with respect to $x$ to find

$\frac{\partial a_{k}}{\partial
x}=\frac{p(Df,Dg,D^{2}f,D^{2}g)}{(\det{D\varphi})^{2}},$

where $p$ is a homogeneous polynomial in the first and second derivatives of
$f$ and $g$. Thus, as above, $\partial a_{k}/\partial x\to 0$. We omit the
remaining cases. ∎

###  4.2 Bounds on $T-T_{\varphi}$

With these estimates, we can now control $T-T_{\varphi}$.

######  Proposition 21.

If $\varphi\colon\Omega\to\widetilde{\Omega}$ is $p$-admissable, then

$\displaystyle\left\lVert T-T_{\varphi}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\lesssim\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\left(\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}\right),$

where $K_{\varphi}$ is a constant depending on $D\varphi$ and its derivatives
which vanishes as
$\left\lVert\mathrm{id}-\varphi\right\rVert_{W^{2,\infty}({\Omega})}\to 0$.

###### Proof.

Fix $u\in L^{p}(\Omega)$. Recall that
$T_{\varphi}=\mathcal{L}_{\varphi}^{-1}\circ\mathcal{J}_{\varphi}$. Hence, it
follows from 18 that

$\displaystyle\left\lVert Tu-T_{\varphi}u\right\rVert_{L^{p}({\Omega})}$
$\displaystyle\leqslant\left\lVert\mathcal{L}_{\varphi}^{-1}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\left\lVert\mathcal{J}_{\varphi}u-\mathcal{L}_{\varphi}\Delta^{-1}u\right\rVert_{L^{p}({\Omega})}$
$\displaystyle\lesssim\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\left\lVert\mathcal{J}_{\varphi}u-\mathcal{L}_{\varphi}\Delta^{-1}u\right\rVert_{L^{p}({\Omega})}.$
(13)

To bound the last $L^{p}$ norm, we use duality. For convenience, let
$Au:=\mathcal{J}_{\varphi}u-\mathcal{L}_{\varphi}\Delta^{-1}u$, and compute,
for $w\in C_{c}^{\infty}(\Omega)$ with $\left\lVert
w\right\rVert_{L^{q}({\Omega})}\leqslant 1$:

$\displaystyle\left\langle{Au,w}\right\rangle$
$\displaystyle=\left\langle{\mathcal{J}_{\varphi}u,w}\right\rangle-\left\langle{\mathcal{L}_{\varphi}\Delta^{-1}u,w}\right\rangle$
$\displaystyle\begin{split}&=\int_{\Omega}uw\left|\det{D\varphi}\right|\,\mathrm{d}{x}-\left\langle{\mathcal{L}_{\varphi}\Delta^{-1}u,w}\right\rangle.\end{split}$
(14)

We now unravel the definition of $\mathcal{L}_{\varphi}$, integrate by parts
and change variables:

$\displaystyle\left\langle{\mathcal{L}_{\varphi}[\Delta^{-1}u],w}\right\rangle$
$\displaystyle=\left\langle{C_{\varphi^{-1}}^{t}\Delta_{\widetilde{\Omega}}C_{\varphi}^{-1}[\Delta^{-1}u],w}\right\rangle$
$\displaystyle=\left\langle{\Delta_{\widetilde{\Omega}}C_{\varphi^{-1}}[\Delta^{-1}u],C_{\varphi^{-1}}w}\right\rangle$
$\displaystyle=-\int_{\widetilde{\Omega}}D_{y}\left[\Delta^{-1}u(\varphi^{-1}(y))\right]\cdot
D_{y}\left[w(\varphi^{-1}(y))\right]\,\mathrm{d}{y}$
$\displaystyle=-\int_{\widetilde{\Omega}}D_{x}(\Delta^{-1}u)(\varphi^{-1}(y))D\varphi^{-1}(y)D\varphi^{-t}(y)(D_{x}w)^{t}(\varphi^{-1}(y))\,\mathrm{d}{y}$
$\displaystyle=-\int_{\Omega}D(\Delta^{-1}u)D\varphi^{-1}(D\varphi^{-1})^{t}(Dw)^{t}\left|\det{D\varphi}\right|\,\mathrm{d}{x},$

where for a matrix $M$, $M^{-t}=(M^{-1})^{t}$, and $Dv\cdot Dw=Dv(Dw)^{t}$. We
will now add and subtract

$\pm\int_{\Omega}D(\Delta^{-1}u)\cdot Dw\,\mathrm{d}{x}$

from (4.2) to obtain:

$\displaystyle\left\langle{Au,w}\right\rangle=$
$\displaystyle\int_{\Omega}uw\left|\det{D\varphi}\right|\,\mathrm{d}{x}+\int_{\Omega}D(\Delta^{-1}u)\cdot
Dw\,\mathrm{d}{x}$ (15)
$\displaystyle\begin{split}&\int_{\Omega}D(\Delta^{-1}u)D\varphi^{-1}D\varphi^{-t}(Dw)^{t}\left|\det{D\varphi}\right|\,\mathrm{d}{x}\\\
&\quad-\int_{\Omega}D(\Delta^{-1}u)\cdot Dw\,\mathrm{d}{x}.\end{split}$ (16)

To estimate (15), we integrate by parts in the second summand to see that

$\displaystyle\eqref{eqn:I}$
$\displaystyle=\int_{\Omega}uw\left|\det{D\varphi}\right|\,\mathrm{d}{x}-\int_{\Omega}\Delta(\Delta^{-1}u)w\,\mathrm{d}{x}$
$\displaystyle=\int_{\Omega}uw(\left|\det{D\varphi}\right|-1)\,\mathrm{d}{x}$
$\displaystyle\leqslant\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}\left\lVert
u\right\rVert_{L^{p}({\Omega})}\left\lVert w\right\rVert_{L^{q}({\Omega})}.$
(17)

For the second term, note that

$\eqref{eqn:II}=\int_{\Omega}D(\Delta^{-1}u)\left\\{D\varphi^{-1}(D\varphi^{-1})^{t}\left|\det{D\varphi}\right|-I\right\\}\cdot
Dw\,\mathrm{d}{x},$

where $I$ denotes the $2\times 2$ identity matrix. Writing the expression in
braces as

$M_{\varphi}=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)$

and integrating by parts yields

$\eqref{eqn:II}=-\int_{\Omega}\left[(a_{x}+b_{y},c_{x}+d_{y})\cdot
D(\Delta^{-1}u)+M_{\varphi}\cdot D^{2}(\Delta^{-1}u)\right]w\,\mathrm{d}{x},$

where and $A\cdot B$ is the dot product of the matrices $A$ and $B$, seen as
vectors in $\mathbb{R}^{4}$. Then,

$\displaystyle\eqref{eqn:II}$ $\displaystyle\lesssim
K_{\varphi}\left(\left\lVert
D^{2}(\Delta^{-1}u)\right\rVert_{L^{p}({\Omega})}+\left\lVert
D(\Delta^{-1}u)\right\rVert_{L^{p}({\Omega})}\right)\left\lVert
w\right\rVert_{L^{q}({\Omega})}$ $\displaystyle\leqslant
K_{\varphi}\left\lVert\Delta^{-1}u\right\rVert_{W^{2,p}({\Omega})}\left\lVert
w\right\rVert_{L^{q}({\Omega})}$ $\displaystyle\lesssim K_{\varphi}\left\lVert
u\right\rVert_{L^{p}({\Omega})}\left\lVert w\right\rVert_{L^{q}({\Omega})}.$
(18)

The last line follows from the boundedness of $\Delta^{-1}\colon
L^{p}(\Omega)\to X_{p}(\Omega)$. And $K_{\varphi}$ is a constant depending on
$D\varphi$ and its derivatives, and which vanishes in the required manner by
Lemma 20.

Hence, combining the estimates (17) and (18) with (4.2) and (13) yields the
result. ∎

######  Remark.

If $\varphi\colon\Omega\to\widetilde{\Omega}$ is conformal, then we can regard
it as a holomorphic map from $\mathbb{C}$ to $\mathbb{C}$ whose complex
derivative $\varphi^{\prime}(x+\mathrm{i}y)$ does not vanish at any
$(x,y)\in\Omega$. In such a case we have

$\det{D\varphi}=\left|\varphi^{\prime}\right|^{2}\quad\text{and}\quad\frac{1}{\left|\det{D\varphi}\right|}D\varphi
D\varphi^{t}=I,$

whence it follows that

$D\varphi^{-1}D\varphi^{-t}\left|\det{D\varphi}\right|=I,$

and so Section 4.2 simplifies to

$\displaystyle\left\langle{Au,w}\right\rangle$
$\displaystyle=\int_{\Omega}uw\left|\det{D\varphi}\right|\,\mathrm{d}{x}+\int_{\Omega}D[\Delta^{-1}u]\cdot
Dw\,\mathrm{d}{x}$
$\displaystyle=\int_{\Omega}uw\left(\left|\det{D\varphi}\right|-1\right)\,\mathrm{d}{x}.$

######  Corollary 22.

Suppose that $\varphi\colon\Omega\to\widetilde{\Omega}$ is $p$-admissable,
conformal and differentiable. Then,

$\left\lVert T-T_{\varphi}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\lesssim\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}.$

##  5 Main Result

We can now combine Eq. 4 (the expression for $Q-Q_{\varphi}$) and 21 (the
estimates for $T-T_{\varphi}$) with Theorem 4 to obtain our main result.

Before we state the theorem, we note that

$P_{F}\equiv P_{F}[Q,\Delta]=P_{F}[Q,T],$

since the eigenvalue problems for both operators yield the same
_eigenfunctions_ , as we discussed above. Similarly,

$P_{F}^{\varphi}\equiv
P_{F}[Q_{\varphi},\Delta_{\varphi}]=P_{F}[Q_{\varphi},T_{\varphi}].$

######  Theorem 23.

If $\varphi\colon\Omega\to\widetilde{\Omega}$ is $p$-admissable, and $F$ does
not split eigenvalues of $T$ or $T_{\varphi}$, then

$\left\lVert P_{F}-P_{F}^{\varphi}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\lesssim\\\ C(F)\left(\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+\frac{\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}}{\inf_{\Omega}\left|\det{D\varphi}\right|}\right)\\\
\times\left(1+\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+\frac{\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}}{\inf_{\Omega}\left|\det{D\varphi}\right|}\right),$

where $C(F)$ is the constant from Theorem 4, $K_{\varphi}$ is the constant
from Lemma 20 and the implicit constants depend only on $p$.

Furthermore, if $\varphi_{k}$ is a sequence of $p$-admissable transformations,
such that
$\left\lVert\mathrm{id}-\varphi_{k}\right\rVert_{W^{2,\infty}({\Omega})}\to
0$, then $\left\lVert P_{F}-P_{F}^{\varphi_{k}}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\to 0$ for each fixed $F$ that does not split eigenvalues.

###### Proof.

Under the assumptions on $\Omega$ and $\widetilde{\Omega}$, $T$ and
$T_{\varphi}$ are well-defined compact operators. Hence, Theorem 4 applies to
give us a bound of the form

$\left\lVert
P_{F}[Q,T]-P_{F}[Q_{\varphi},T_{\varphi}]\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\leqslant C(F)A(1+A+B),$ (19)

where

$A:=\left\lVert Q-Q_{\varphi}\right\rVert_{L^{2}\times
L^{2}\to\mathbb{R}}+\left\lVert T-T_{\varphi}\right\rVert_{L^{2}\to
L^{2}}\quad\text{and}\quad B:=\left\lVert T-T_{\varphi}\right\rVert_{L^{p}\to
L^{p}}.$

Now,

$\left|Q[u,v]-Q_{\varphi}[u,v]\right|\leqslant\int_{\Omega}\left|uv\right|\left(1-\left|\det{D\varphi}\right|\right)\,\mathrm{d}{x},$

so

$\left\lVert Q-Q_{\varphi}\right\rVert_{L^{2}(\Omega)\times
L^{2}(\Omega)\to\mathbb{R}}\leqslant\left\lVert\left|\det{D\varphi}\right|-1\right\rVert_{L^{\infty}({\Omega})}.$

And from 21 (since $p=2$ is admissible) we have

$\displaystyle\left\lVert T-T_{\varphi}\right\rVert_{L^{2}(\Omega)\to
L^{2}(\Omega)}\lesssim\frac{1}{\inf_{\Omega}\left|\det{D\varphi}\right|}\left(\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}\right),$

which implies that

$A\lesssim\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+\frac{\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}}{\inf_{\Omega}\left|\det{D\varphi}\right|}.$

As for $B$, we once again apply 21 to $\left\lVert
T-T_{\varphi}\right\rVert_{L^{p}(\Omega)\to L^{p}(\Omega)}$ to find that

$B\lesssim\frac{\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}+K_{\varphi}}{\inf_{\Omega}\left|\det{D\varphi}\right|},$

and the result follows.

The continuity statement is an immediate consequence of Lemma 19 and Lemma 20.
∎

22 gives us the following simplification when $\varphi$ is holomorphic.

######  Corollary 24.

Suppose, in addition, that $\varphi$ is holomorphic. Then,

$\left\lVert P_{F}-P_{F}^{\varphi}\right\rVert_{L^{p}(\Omega)\to
L^{p}(\Omega)}\lesssim C(N)\left\lVert
1-\left|\det{D\varphi}\right|\right\rVert_{L^{\infty}({\Omega})}\times
E_{\varphi},$

where $E_{\varphi}$ remains bounded as $\mathrm{id}-\varphi\to 0$ in
$W^{2,\infty}$.

##  6 Conclusions and open problems

Careful inspection of the constants involved in Theorem 23 and the
corresponding result in Lamberti and Lanza de Cristoforis, (2005) shows that
our estimates are qualitatively the same as theirs, in the sense that all
constants that do not explicitly depend on $\varphi$ and its derivatives
remain the same.

An important limitation of this approach, however, is that it assumes that the
projections $P_{F}$ map onto eigenspaces which do not split eigenvalues.
Eigenvalue splitting is not allowed because of Kato’s projection formula, as
discussed in the remark immediately after Theorem 2.

The following example motivates our interest in relaxing the notion of
splitting.

######  Example 25.

Let $\Omega=[0,1]^{2}$, a unit square; then the eigenvalues of the Dirichlet
Laplacian are $\pi^{2}(m^{2}+n^{2})$ for $(m,n)$ running over
$\mathbb{N}^{2}$. Fix any $N\in\mathbb{N}$. The following results are
classical (see Grafakos, 2014a, ):

  1. (i)

If we take $F_{N}=\\{(m,n):0<m,n\leqslant N\\}$, then $P_{F_{N}}$ is the
partial sum operator corresponding to “square cutoff” truncations of a double
sine-series on the torus. For any value of $1<p<\infty$, these projections are
uniformly bounded in $N$, which is equivalent to saying that the partial sums
$P_{F_{N}}u$ converge to $u$ for any $u\in L^{p}$.

  2. (ii)

If we take $B_{N}=\\{(m,n):m^{2}+n^{2}\leqslant N^{2}\\}$, then $P_{B_{N}}$ is
the partial sum operator corresponding to “circular cutoff” truncations of a
double sine-series on the torus. These partial sums do not converge in any
$L^{p}$ space other than $p=2$.

It would be interesting to transfer the convergence results for (i) to other
domains obtained by perturbing the square by some suitable $\varphi$, as
described here. However, by definition, the sets $B_{N}$ will not split
eigenvalues, while the sets $F_{N}$ will often split eigenvalues. Take, for
instance, the pairs $(5,5)$ and $(1,7)$, both corresponding to the eigenvalue
$50\pi^{2}$. Then $(5,5)\in F_{6}$ but $(1,7)\notin F_{6}$.

12345671234567$m$$n$$F_{6}$$(5,5)$$(7,1)$$(1,7)$ Figure 1: The “box” $F_{6}$
splits the eigenvalue $\lambda=50\pi^{2}$. The region $\Gamma_{6}$ is denoted
by the solid blue arc.

Denote this “bad region” by $\Gamma_{N}$, i.e.

$\Gamma_{N}=\\{(m,n)\in F_{N}:m^{2}+n^{2}=k^{2}+l^{2}\text{ for some
}(k,l)\notin F_{N}\\}.$

Then $F_{N}$ splits eigenvalues if, and only, if $\Gamma_{N}\neq\varnothing$
(see Fig. 1).

It is rather surprising that this “small” region $\Gamma_{N}$ lying between
the circle and the square is responsible for such dramatic differences in the
convergence of the partial sums. Indeed, Fefferman’s proof shows that
curvature is a key part of the failure of $L^{p}$ convergence. On the other
hand, Córdoba’s results show that varying the number of vertices, we can still
guarantee convergence in $L^{p}$ at least in the range $4/3<p<4$.

Our work in this paper has been motivated by the “failure” of the circle to
yield “well-behaved” (i.e. convergent) cutoffs (see Fefferman et al.,, 2022,
concluding remarks). Indeed, it would seem that circular cutoffs are the
“natural choice” to truncate the partial sums since they correspond to
choosing all eigenfunctions whose eigenvalues are bounded by some value. The
square truncations, on the other hand, exploit the “artificial” labelling of
these eigenvalues by pairs of integers. Fefferman himself remarks on his
surprise at his own theorem, since it was conjectured that circular cutoffs
would yield convergence in the region $\frac{4}{3}<p<4$ (see the introduction
to Fefferman,, 1971).

Thus, two open questions of especial interest remain:

  1. (i)

Can we eliminate the “no-splitting” requirement in order to apply our results
to the square cutoffs?

  2. (ii)

Can we improve the constant $C(F)$ in order to get uniform estimates on the
operator norms?

Regarding (ii), we observe that in some simple examples, there is _no_
constant $C(F)$, and so we can immediately recover $L^{p}$ convergence on
$\widetilde{\Omega}$ from $L^{p}$ convergence on $\Omega$.

######  Example 26.

Let $\Omega=[0,\pi]^{2}$ with eigenfunctions $u_{m,n}=\sin(mx)\sin(ny)$, and
$\varphi(x,y)=(ax,by)$ mapping onto $\widetilde{\Omega}=[0,\pi a]\times[0,\pi
b]$. The eigenfunctions on the new domain are
$\widetilde{u}_{m,n}=\sin(mx/a)\sin(ny/b)$. A simple calculation then shows
that

$\left\lVert
P_{F}u-P_{F}^{\varphi}u\right\rVert_{L^{p}({\Omega})}\leqslant\left|1-ab\right|\left\lVert
P_{F}u\right\rVert_{L^{p}({\Omega})},$

and so we can recover $L^{p}$ boundedness of the _square_ projections.

## Acknowledgment

This work was conducted thanks to the EPSRC studentship 2443915 under the
project EP/V520226/1.

## References

  * (1) Acosta Babb, R. L. (2023a).  The $L^{p}$ convergence of Fourier series on triangular domains.  Proceedings of the Edinburgh Mathematical Society, 66(2):453–474.
  * (2) Acosta Babb, R. L. (2023b).  Remarks on the ${L}^{p}$ convergence of Bessel–Fourier series on the Disc.  Comptes Rendues de l’Académie des Sciences - Mathématique, 361:1075–1080.
  * Conway, (2007) Conway, J. B. (2007).  A Course in Functional Analysis.  Graduate Texts in Mathematics. Springer New York.
  * Córdoba, (1977) Córdoba, A. (1977).  The multiplier problem for the polygon.  Annals of Mathematics, 105(3):581–588.
  * Courant and Hilbert, (1989) Courant, R. and Hilbert, D. (1989).  Methods of Mathematical Physics, volume 1 of Methods of Mathematical Physics.  Wiley.
  * Fefferman, (1971) Fefferman, C. L. (1971).  The multiplier problem for the ball.  Annals of Mathematics, 94(2):330–336.
  * Fefferman et al., (2022) Fefferman, C. L., Hajduk, K. W., and Robinson, J. C. (2022).  Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces.  Proceedings of the London Mathematical Society, 125(4):759–777.
  * Gilbarg and Trudinger, (2001) Gilbarg, D. and Trudinger, N. (2001).  Elliptic Partial Differential Equations of Second Order.  Classics in Mathematics. Springer Berlin Heidelberg.
  * (9) Grafakos, L. (2014a).  Classical Fourier Analysis.  Graduate Texts in Mathematics. Springer New York.
  * (10) Grafakos, L. (2014b).  Modern Fourier Analysis.  Graduate Texts in Mathematics. Springer New York.
  * Grisvard, (1985) Grisvard, P. (1985).  Elliptic Problems in Nonsmooth Domains.  Society for Industrial and Applied Mathematics.
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  * Lamberti and Lanza de Cristoforis, (2005) Lamberti, P. D. and Lanza de Cristoforis, M. (2005).  A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator.  Zeitschrift für Analysis und ihre Anwendungen, 24(2):277–304.
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∎

11institutetext:  W.-J. Liu 22institutetext: Y. Zheng 33institutetext: Jiangsu
Engineering Center of Network Monitoring, Nanjing University of Information
Science & Technology, Nanjing 210044, P.R.China
33email<EMAIL_ADDRESS>44institutetext: W.-J. Liu 55institutetext: C.-T Li
66institutetext: Y. Zheng 77institutetext: Y. Xu 88institutetext: Y.-S. Xu
99institutetext: School of Computer and Software, Nanjing University of
Information Science & Technology, Nanjing 210044, P.R.China

# Quantum Privacy-Preserving Price E-Negotiation

Wen-Jie Liu  Chun-Tang Li  Yu Zheng  Yong Xu  Yin-Song Xu

(Received: / Accepted: date)

###### Abstract

Privacy-preserving price e-negotiation (3PEN) is an important topic of secure
multi-party computation (SMC) in the electronic commerce field, and the key
point of its security is to guarantee the privacy of seller’s and buyer’s
prices. In this study, a novel and efficient quantum solution to the 3PEN
problem is proposed, where the oracle operation and the qubit comparator are
utilized to obtain the comparative results of buyer’s and seller’s prices, and
then quantum counting is executed to summarize the total number of products
which meets the trading conditions. Analysis shows that our solution not only
guarantees the correctness and the privacy of 3PEN, but also has lower
communication complexity than those classical ones.

###### Keywords:

Secure multi-party computation privacy-preserving price e-negotiation oracle
operation qubit comparator quantum counting

##  1 Introduction

Negotiation is a dialogue between two or more people or parties intended to
reach a beneficial outcome over one or more issues where a conflict exists
with respect to at least one of these issues. This beneficial outcome can be
for all of the parties involved, or just for one or some of them. Thus,
negotiation plays an important role in the electronic commerce, artificial
intelligence, and even political fields. As we know, electronic
trading(e-trading) has become increasingly pervasive in the business world.
And electronic negotiation (e-negotiation) is a critical process of e-trading
where the negotiating parties with different preferences, criteria, and
constraints try to maximize their gains and reach a consensus on the terms of
trading such as prices, delivery dates, and payment terms CA15 ; AM12 .
E-negotiation is motivated by its potential to provide business partners with
more efficient processes, enabling them to arrive at more satisfying
agreements in less time AN12 ; CA09 .

In addition to efficiency improvements, the security of the exchanged data in
e-negotiation has also been concerned LA00 ; HA02 ; YM05 . Limthanmaphon _et
al_. LA00  proposed a agent-based negotiation model based on the digital
signature, which provides the concurrency control and the transactions
security in e-negotiation. He and Leung HA02  proposed three approaches for
enhancing trust in e-transaction including: enforcement of third parties,
chunking e-negotiation transactions into smaller chunks, and enforcement of
social laws. Yang YM05  proposed a secure protocol to protect data the agent
collects and agent itinerary based on onion routing scheme. These protocols
ensure integrity of agent itinerary, anonymity of non-neighbor hosts in the
agent itinerary, and fault-tolerance of remote hosts. However, they do not
address the privacy of participants.

Secure multi-party computation(SMC) YP82  is a particular kind of cryptography
with the goal of creating methods for parties to jointly compute a function
over their inputs while keeping those private inputs in secret. As a practical
topic of SMC, privacy-preserving price e-negotiation (3PEN) aims to guarantee
the seller’s and the buyer’s privacy in electronic commerce. Based on
distributed SMC, Chakraborty _et al_. CP05  first proposed two privacy-
preserving negotiation protocols (C05): one is non-discriminatory and the
other is discriminatory, where the price and the demands of the buyers are not
disclosed to anyone before entering the final agreement. Al-Jaljouli _et al_.
AS07  proposed a mobile agent-based e-negotiation protocol (A07) which
protects the security properties of privacy, authentication, non-repudiation,
anonymity, and integrity. It sholud be noted that these protocols have
communication complexity $O(N)$, here $N$ is the numbers of products.

With the advent of quantum computation, classical cryptosystems, including the
symmetric AES cryptosystem and the asymmetric RSA cryptosystem, are facing
enormous threatens and challenges from quantum computer. Fortunately, quantum
cryptography can provide the unconditional security, which is guaranteed by
some physical principles of quantum mechanics, to resist against its impact.
On the other hand, quantum parallelism makes it possible to greatly speed up
solving some specific computational tasks, such as quantum factoring SA94  and
quantum searching GA96 . With quantum mechanics utilized in the information
processing, many important research findings are presented in recent decades,
such as quantum key distribution (QKD) BQ84 ; EQ91 ; LM12 , quantum key
agreement (QKA) CQ10 ; HE17 ; LM17 ; LA18 , quantum secure direct
communication (QSDC) LWJ09 ; LZH17 ; LZH18 ; ZJ18 , quantum private comparison
(QPC) YA09 ; LQ13 ; LS14 ; LS014 ; LC14 , quantum sealed-bid auction (QSBA)
NS09 ; LA14 ; LM16 , quantum state remote preparation (QRSP) LWJ15 ; TX18 ;
MM18 , quantum private query (QPQ) GF12 ; LA19  and even quantum machine
learning algorithms LQ013 ; LQ18 ; LA019 . These above findings have shown the
potential power in either the efficiency improvements or the security
enhancements.

In this study, a quantum privacy-preserving price e-negotiation (Q3PEN)
protocol, i.e., an efficient quantum solution to the 3PEN problem, is
proposed. For the convenience of description, we take the two-party
negotiation as an example, and it can be easily extended to the multi-party
scenario. In this protocol, the oracle operations are performed to prepare the
corresponding state, the qubit comparator is applied to obtain the final state
which represents the comparative results of the two-party’s prices, and
finally quantum counting is executed to summarize the number of products which
meets the trading conditions (i.e., the numbers of those the buyer’s bid is
higher than the seller’s selling price surpass a certain threshold). The
purpose of our work is to reduce the communication complexity and protect the
two-party’s privacy.

The rest of this paper is organized as follows. Some definitions about Q3PEN
are given in the next section, and a novel and efficient Q3PEN protocol is
presented in Sect. 3. The correctness, privacy and efficiency analysis of our
protocol are discussed in Sect. 4, and then Sect. 5 concludes the paper and
makes some further discussions.

##  2 Definitions of Q3PEN

Before introducing the procedure of Q3PEN protocol, we firstly make some
definitions. Without loss of generality, we suppose there are two parties, the
buyer Alice and the seller Bob, and the formal definitions of Q3PEN problem
and Q3PEN protocol are given as below.

Definition 1 (Q3PEN problem) Bob has $N$ products with the minimum selling
prices $B$=$(b_{1},b_{2},\cdots,b_{N})$, and Alice wants to buy these products
under the maximum buying prices $A$=$(a_{1},a_{2},\cdots,a_{N})$. Then Alice
and Bob jointly judge whether the number of the products meets the trading
condition $\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$ without
leaking their private prices, where $f({a_{i}},{b_{i}})\in\\{0,1\\}$ is the
comparison function $f({a_{i}},{b_{i}})=({a_{i}}\geq{b_{i}})$, and
$\varepsilon$ is a predefined threshold (in general, $1\ll\varepsilon\leq N$).
If the condition is satisfied, the two parties trade, otherwise they cancel
the negotiation.

Definition 2 (Q3PEN protocol) Alice and Bob input their buying and selling
price set $A$, $B$, respectively. After executing this protocol, they can
obtain the result of
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$, and decide whether
they trade or not. In addition, the protocol should satisfy three properties
as below,

Correctness: Two honest parties Alice and Bob can get the correct result of
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$.

Alice’s Privacy: Bob cannot learn any secret information about Alice’s prices
$A$ except the possible information deduced from the final result of
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$ and his prices $B$.

Bob’s Privacy: Alice cannot get any secret information about Bob’s prices $B$
except the possible information deduced from
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$ and $A$.

##  3 Protocol of Q3PEN

Referring to Definition 2, we propose a six-step Q3PEN protocol as below
(shown in Fig. 1).

Figure 1: The six-step procedure of Q3PEN protocol.

Step 1. Alice and Bob respectively prepare their initial states
$\left|{{\phi_{A}}}\right\rangle$ and $\left|{{\phi_{B}}}\right\rangle$ which
are both in state
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}$, where $d=\left\lceil{\log\left({\mathop{\max}\limits_{i,{\kern
1.0pt}{\kern 1.0pt}k}({a_{j}},{b_{k}})+1}\right)}\right\rceil$ ($1\leq j,k\leq
N$). Then Alice (Bob) applies the oracle operation ${O_{A}}$ (${O_{B}}$) on
$\left|{{\phi_{A}}}\right\rangle$ ($\left|{{\phi_{B}}}\right\rangle$) (shown
in Fig. 2).

$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}\xrightarrow{{{O_{A}}}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle}$
(1)
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}\xrightarrow{O_{B}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle}$
(2)

Then Alice (Bob) sends the result state
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle}$
($\left|{{{\phi^{\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle}$)
to Bob (Alice).

Figure 2: The oracle operation Alice or Bob applied on the initial state
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}$. Here $n=\left\lceil{\log(N+1)}\right\rceil$.

Step 2. After receiving $\left|{{{\phi^{\prime}}_{A}}}\right\rangle$
($\left|{{{\phi^{\prime}}_{B}}}\right\rangle$) from Alice (Bob), Bob (Alice)
further performs the oracle operation ${O_{B}}$ (${O_{A}}$) between
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|{i}\right\rangle}$ in
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle$
($\left|{{{\phi^{\prime}}_{B}}}\right\rangle$) and the auxiliary state
${\left|0\right\rangle^{\otimes d}}$, which is similar to the oracle operation
in Step 1.

$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}\xrightarrow{{{O_{B}}}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle}$
(3)
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle\otimes{{\left|0\right\rangle}^{\otimes
d}}}\xrightarrow{{{O_{A}}}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle\left|{{a_{i}}}\right\rangle}$
(4)

Then Bob (Alice) obtains a new state
$\left|{{{\phi^{\prime\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle}$
($\left|{{{\phi^{\prime\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle\left|{{a_{i}}}\right\rangle}$).

Step 3. Bob (Alice) applies another oracle operation ${O_{f}}$ on
$\left|{{{\phi^{\prime\prime}}_{A}}}\right\rangle$
($\left|{{{\phi^{\prime\prime}}_{B}}}\right\rangle$) and an auxiliary qubit
$\left|0\right\rangle$ (shown in Fig. 3),

Figure 3: The circuit of the oracle operation applied on state
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle\otimes\left|0\right\rangle}$.

$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle\otimes\left|0\right\rangle}\xrightarrow{{{O_{f}}}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle\left|{{\kern
1.0pt}f({a_{i}},{b_{i}})}\right\rangle}{\kern 1.0pt},$ (5)
$\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle\left|{{a_{i}}}\right\rangle\otimes\left|0\right\rangle}\xrightarrow{{{O_{f}}}}\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{b_{i}}}\right\rangle\left|{{a_{i}}}\right\rangle\left|{{\kern
1.0pt}f({a_{i}},{b_{i}})}\right\rangle}{\kern 1.0pt}.$ (6)

Then Bob (Alice) gets the final state
$\left|{{{\phi^{\prime\prime\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle\left|{f({a_{i}},{b_{i}})}\right\rangle}$
($\left|{{{\phi^{\prime\prime\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left|{{b_{i}}}\right\rangle\left|{f({a_{i}},{b_{i}})}\right\rangle}$).
Here, the function $f$ is a qubit comparator defined as below,

$f({a_{i}},{b_{i}})=\left\\{\begin{gathered}1,{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{a_{i}}\geq{b_{i}}\hfill\\\ 0,{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{a_{i}}<{b_{i}}{\kern 1.0pt}\hfill\\\
\end{gathered}\right..$ (7)

And Fig. 4 and Fig. 5 give the corresponding circuit implementation of the
qubit comparator.

Figure 4: The circuit of one-qubit comparator comparing two qubits. Here,
$\times$ is the NOT gate, and $\bullet$ in the Toffloi gate denotes the
negative-control qubit conditional being set to one, while $\oplus$ represents
the target qubit. Figure 5: The circuit of multi-qubit comparator comparing
${a_{i}}$ and ${b_{i}}$. ${U_{c}}$ is the one-qubit comparator described in
Fig. 3, and $\circ$ denotes the positive-control qubit conditional being set
to zero.

Step 4. Bob (Alice) executes the quantum counting algorithm BQ98  on
$\left|{{{\phi^{\prime\prime\prime}}_{A}}}\right\rangle$
($\left|{{{\phi^{\prime\prime\prime}}_{B}}}\right\rangle$) to obtain the
number of those products which meet $f({a_{i}},{b_{i}})=1$, respectively. Then
he (she) will get the estimator of counting result
${t_{B}}=\sum\limits_{i=1}^{N}{f\left({{a_{i}},{b_{i}}}\right)}$
(${t_{A}}=\sum\limits_{i=1}^{N}{f\left({{a_{i}},{b_{i}}}\right)}$).

Step 5. Alice and Bob execute quantum bit string commitment KQ03  to exchange
$n$-bit string ${t_{A}}$ and ${t_{B}}$, (1) Alice (Bob) commits ${t_{A}}$
(${t_{B}}$) by preparing the $logm$-qubit state
${\left|\tau\right\rangle_{A}}$ (${\left|\tau\right\rangle_{B}}$) according to
an error correcting code BQ01  and sending them to Bob(Alice), here $m=cn$ for
a fixed constant $c>1$; (2) Alice(Bob) unveils ${t_{A}}$ (${t_{B}}$) to the
other. (3) Alice (Bob) verifies the validness of the other’s bit string by
measuring the projection onto ${\left|\tau\right\rangle_{B}}$
(${\left|\tau\right\rangle_{A}}$). If he(she) obtains eigenvalue 1, he (she)
accepts the unveiling; otherwise he or she concludes Bob(Alice) cheated. Then,
Alice and Bob check the other’s honesty by comparing the two counting results,
if both of the two parties are honest, the difference of ${t_{A}}$ and
${t_{B}}$ will be less than $\delta$ (i.e.,
$\left|{{t_{A}}-{t_{B}}}\right|\leq\delta$), where $\delta$ is total error of
estimation in quantum counting algorithm.

Step 6. Alice and Bob further check whether the two counting results satisfy
the trading requirement: ${t_{A}}\geq\varepsilon$ and ${t_{B}}\geq\varepsilon$
($1\ll\varepsilon\leq N$). If they meet the requirement, Alice and Bob
continue to trade, otherwise they will cancel this negotiation.

##  4 Correctness, Privacy and Efficiency Analysis

###  4.1 Correctness analysis

Theorem 1. _Assume there exist a constant $\varepsilon$, $1\ll\varepsilon\leq
N$, such that for two participants Alice and Bob executing our protocol can
satisfy the correctness property of Q3PEN protocol defined in Sect. 2, that
is, two honest parties Alice and Bob can get the correct result of whether
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}\geq\varepsilon$._

_Proof:_ Without loss of generality, we suppose the number of products $N=6$,
the threshold $\varepsilon=5$, and Alice’s maximal buying prices
$A=(3,2,5,4,7,6)$, Bob’s minimal selling prices $B=(2,2,5,5,6,6)$, so
$n=\left\lceil{\log(6+1)}\right\rceil=3$,
$d=\left\lceil{\log\left({\mathop{\max}\limits_{i,{\kern 1.0pt}{\kern
1.0pt}k}({a_{j}},{b_{k}})+1}\right)}\right\rceil=\left\lceil{\log(7+1)}\right\rceil=3$.
In Step 1, Alice and Bob prepare states
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle$ and
$\left|{{{\phi^{\prime}}_{B}}}\right\rangle$ through the oracle operations
${O_{A}}$ and ${O_{B}}$,

$\left|{{{\phi^{\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|3\right\rangle+\left|2\right\rangle\left|2\right\rangle+\left|3\right\rangle\left|5\right\rangle+\left|4\right\rangle\left|4\right\rangle+\left|5\right\rangle\left|7\right\rangle+\left|6\right\rangle\left|6\right\rangle),$
(8)
$\left|{{{\phi^{\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|2\right\rangle+\left|2\right\rangle\left|2\right\rangle+\left|3\right\rangle\left|5\right\rangle+\left|4\right\rangle\left|5\right\rangle+\left|5\right\rangle\left|6\right\rangle+\left|6\right\rangle\left|6\right\rangle),$
(9)

and then send them to each other, respectively. After receiving
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle$ and
$\left|{{{\phi^{\prime}}_{B}}}\right\rangle$ in Step 2, Bob(Alice) applies the
oracle operation ${O_{B}}$(${O_{A}}$) again, and they get the new state as
below,

$\begin{gathered}\left|{{{\phi^{\prime\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|3\right\rangle\left|2\right\rangle+\left|2\right\rangle\left|2\right\rangle\left|2\right\rangle+\left|3\right\rangle\left|5\right\rangle\left|5\right\rangle{\kern
1.0pt}{\kern
1.0pt}+\left|4\right\rangle\left|4\right\rangle\left|5\right\rangle{\kern
1.0pt}\hfill\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\left|5\right\rangle\left|7\right\rangle\left|6\right\rangle{\kern
1.0pt}+\left|6\right\rangle\left|6\right\rangle\left|6\right\rangle).\hfill\\\
\end{gathered}$ (10)
$\begin{gathered}\left|{{{\phi^{\prime\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|2\right\rangle\left|3\right\rangle+\left|2\right\rangle\left|2\right\rangle\left|2\right\rangle+\left|3\right\rangle\left|5\right\rangle\left|5\right\rangle+\left|4\right\rangle\left|5\right\rangle\left|4\right\rangle\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}+\left|5\right\rangle\left|6\right\rangle\left|7\right\rangle{\kern
1.0pt}+\left|6\right\rangle\left|6\right\rangle\left|6\right\rangle).\hfill\\\
\end{gathered}$ (11)

Bob (Alice) further applies another oracle operation ${O_{f}}$ on
$\left|{{{\phi^{\prime\prime}}_{A}}}\right\rangle$
($\left|{{{\phi^{\prime\prime}}_{B}}}\right\rangle$), and embed the
comparative results of buying and selling prices into the qubit in the last
position,

$\begin{gathered}\left|{{{\phi^{\prime\prime\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|3\right\rangle\left|2\right\rangle\left|1\right\rangle+\left|2\right\rangle\left|2\right\rangle\left|2\right\rangle\left|1\right\rangle+\left|3\right\rangle\left|5\right\rangle\left|5\right\rangle\left|1\right\rangle\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\left|4\right\rangle\left|4\right\rangle\left|5\right\rangle\left|0\right\rangle{\kern
1.0pt}+\left|5\right\rangle\left|7\right\rangle\left|6\right\rangle{\kern
1.0pt}\left|1\right\rangle+\left|6\right\rangle\left|6\right\rangle\left|6\right\rangle\left|1\right\rangle).\hfill\\\
\end{gathered}$ (12)
$\begin{gathered}\left|{{{\phi^{\prime\prime\prime}}_{B}}}\right\rangle=\frac{1}{{\sqrt{6}}}(\left|1\right\rangle\left|2\right\rangle\left|3\right\rangle\left|1\right\rangle+\left|2\right\rangle\left|2\right\rangle\left|2\right\rangle\left|1\right\rangle+\left|3\right\rangle\left|5\right\rangle\left|5\right\rangle\left|1\right\rangle\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\left|4\right\rangle\left|5\right\rangle\left|4\right\rangle\left|0\right\rangle{\kern
1.0pt}+\left|5\right\rangle\left|6\right\rangle\left|7\right\rangle{\kern
1.0pt}\left|1\right\rangle+\left|6\right\rangle\left|6\right\rangle\left|6\right\rangle\left|1\right\rangle).\hfill\\\
\end{gathered}$ (13)

In Step 4, Bob and Alice further execute quantum counting to get their
counting result
${t_{A}}=\sum\limits_{i=1}^{6}{f\left({{a_{i}},{b_{i}}}\right)}=5$,
${t_{B}}=\sum\limits_{i=1}^{6}{f\left({{a_{i}},{b_{i}}}\right)}=5$,
respectively. Thus, the result value ${t_{A}}={t_{B}}\geq 5$, so the
correctness of the protocol is guaranteed.

###  4.2 Privacy analysis

The privacy property means that both Alice and Bob cannot get the other’s
price information or any secret information deduced from the final result,
which consists of Bob’s or Alice’s privacy.

####  4.2.1 Alice’s privacy

Theorem 2. _Assume $N$ is the number of products,
$n=\left\lceil{\log(N+1)}\right\rceil$, and $m$ is the number of commited bits
($m=cn$, $c>1$), such that Bob cannot learn any secret information about
Alice’s prices $A$, the possible information deduced from
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}$ and his prices $B$. And Bob’s
cheating behavior can be detected by Alice with probability
($1-{\left({\frac{1}{2}}\right)^{m-\log n}}$) very close to 1._

_Proof:_ Suppose Bob is dishonest and he tries to take some strategies to get
information about Alice’s private prices $A$. In our protocol, all the
classical information about $A$ is embedded into the state
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle=\frac{1}{{\sqrt{N}}}\sum\limits_{i=1}^{N}{\left|i\right\rangle\left|{{a_{i}}}\right\rangle}$,
so the most possible attack is that Bob performs a certain projective
measurement on state $\left|{{{\phi^{\prime}}_{A}}}\right\rangle$ when he
received it from Alice in Step 2. Then, Bob will get
$\left|i\right\rangle\left|{{a_{i}}}\right\rangle$ with the same probability
$\frac{1}{N}$ for any $i$. Thus $\left|{{\phi^{\prime}}_{A}}\right\rangle$ can
be characterized by the quantum ensemble
$\varepsilon=\\{{P_{i}},{\rho}(i)\\}$, where ${P_{i}}=\frac{1}{N}$ is the
probability that Bob gets the component ${a_{i}}$ (Assuming that Bob does not
have any prior information on Alice’s prices initially), and

${\rho}(i)=\left|i\right\rangle\left|{{a_{i}}}\right\rangle\left\langle{{a_{i}}}\right|\left\langle
i\right|.$ (14)

According to Holevo’s theorem HP82 , the information that Bob can access by
performing any measurement on $\left|{{\phi^{\prime}}_{A}}\right\rangle$ is
bounded by the entropy:

$\begin{gathered}I\leq
X(\varepsilon)=S({\rho})-\frac{1}{N}\sum\limits_{i=1}^{N}{S({\rho}(i))}\hfill\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}=S({\rho})\hfill\\\
\end{gathered}$ (15)

where ${\rho}=\sum\limits_{i=1}^{N}{\left({\frac{1}{N}{\rho}(i)}\right)}$, so
$I\leq S({\rho})=\log N$. From the above analysis, we can see that Bob can
obtain at most one element of Alice’s prices $A$, i.e., one-$N$th of Alice’s
private information, by any possible measurement. And he will loses the chance
to get the other elements at the same time. What is more, the information that
Bob can extract can be ignored when number $N$ is large enough.

Besides, if Bob has performed the projective measurement to extract some
information about $A$, he will certainly lose the chance to further get his
counting result ${t_{B}}$. However, in order to complete the honest test of
Alice in Step 5, Bob may try to steal as much information as possible from
Alice’s quantum commitment. Bob can extract at most $\log n$ bits from Alice’s
$m$-length quantum bit string ($m=cn$, $c>1$) KQ03 . Furthermore, the
probability that Bob successfully stealing all $n$ bits is at most
$P={\left({\frac{1}{2}}\right)^{m-\log n}}$. Therefore, if Bob extracts
Alice’s partial private information from the state
$\left|{{{\phi^{\prime}}_{A}}}\right\rangle$, his cheating behavior will be
detected by Alice’s honest test with at least the probability of ($1-P$), and
the probability will be very close to $1$ when $n$ is large enough.

####  4.2.2 Bob’s privacy

Theorem 3. _Assume $N$ is the number of products,
$n=\left\lceil{\log(N+1)}\right\rceil$, and $m$ is the number of commited bits
($m=cn$, $c>1$), such that Alice cannot learn any secret information about
Bob’s prices $B$, the possible information deduced from
$\sum\limits_{i=1}^{N}{f({a_{i}},{b_{i}})}$ and his prices $A$. And Alice’s
cheating behavior can be detected by Bob with probability
($1-{\left({\frac{1}{2}}\right)^{m-\log n}}$) very close to 1._

_Proof:_ Similarly, a dishonest Alice can obtain at most one element of Bob’s
private prices $B$ (i.e., ${b_{i}}$) by her local measurement on state
$\left|{{{\phi^{\prime}}_{B}}}\right\rangle$. However, she will certainly lose
the chance to further get the counting result ${t_{B}}$, and her cheating
behavior will be detected by Bob with the probability of
($1-{\left({\frac{1}{2}}\right)^{m-\log n}}$).

###  4.3 Efficiency analysis

The communication cost is one of the key indicators of the efficiency for
communication protocols. In our protocol, the buyer Alice and the seller Bob
exchange two quantum state, $\left|{{\psi_{A}}}\right\rangle$ and
$\left|{{\psi_{B}}}\right\rangle$, with each other in Step 1, where each
quantum state is the $(n+d)$-qubit state. Besides, the counting results
$t_{A}$ and $t_{B}$ are exchanged too in Step 5, where $t_{A}$ and $t_{B}$ all
are the classic $n$-bit message. Therefore, the total communication cost of
our protocol is $2(n+d)$ qubits and $2n$ cbits, where
$n=\left\lceil{\log(N+1)}\right\rceil$,
$d=\left\lceil{\log\left({\mathop{\max}\limits_{i,{\kern 1.0pt}{\kern
1.0pt}k}({a_{j}},{b_{k}})+1}\right)}\right\rceil$ (i.e., $d$ is just relevant
to the range of selling and buying prices and there is no direct correlation
between $d$ and $N$). So, the communication complexity of our protocol is
$O\left({logN}\right)$. However, in the classic C05 protocols CP05 , $2N$
cbits are exchanged between the buyer and the seller, while $4N$ cbits is
required to be transmitted in the A07 protocol AS07  (it should be noted that
the message of product’s price is just one cbit in these classic protocols),
which means both of C05 and A07 protocols hold the linear communication
complexity $O\left({N}\right)$. Obviously, our protocol achieves a great
reduction in the communication complexity.

In order to more intuitively demonstrate the difference of communication costs
between our Q3PEN protocol and the classic C05, A07 protocols, we uniformly
suppose the message of each product’s price is two cbits, i.e., $d=2$. So the
exchanged message of the C05 and A07 protocols are $2N*d=4N$ and $4N*d=8N$
cbits, respectively, while the exchanged qubits/cbits in our protocol are
$4\left\lceil{\log(N+1)}\right\rceil+2d=4\left\lceil{\log(N+1)}\right\rceil+4$.
And Fig. 6 illustrates the communication costs among the classic C05, A07
protocols and our Q3PEN protocol.

Figure 6: The communication costs of the C05, A07 protocols and our Q3PEN
protocol when $d=2$.

##  5 Conclusion

With ingeniously making use of quantum oracle and quantum counting technology,
we propose an efficient quantum privacy-preserving price e-negotiation
protocol. Compared with those previous classic protocols, the communication
costs is greatly reduced, and the two participants’ privacies are perfectly
guaranteed as well. Although the proposed protocol is an example of two-party
price e-negotiation, it can be extended to the multi-party scenario, such as
one-to-many (i.e., one buyer negotiates with $n$ sellers), many-to-one, and
many-to-many.

###### Acknowledgements.

The authors would like to thank the anonymous reviewers and editor for their
comments that improved the quality of this paper. This work is supported by
National Natural Science Foundation of China (Grant Nos. 61672290 and
61802002), Natural Science Foundation of Jiangsu Province(Grant No.
BK20171458), the Six Talent Peaks Project of Jiangsu Province (Grant No.
2015-XXRJ-013), the Practice Innovation Training Program Projects for Jiangsu
College Students (Grant No. 201810300016Z), and the Priority Academic Program
Development of Jiangsu Higher Education Institutions(PAPD).

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