holds. By substituting (3.16) into Cauchy’s integral formula, we obtain

$\displaystyle H^{-s/2}=$
$\displaystyle\mathop{\mathrm{diag}}\\{(a^{jj}_{-s/2})^{W}(x,D_{x})\\}+\mathop{\mathrm{diag}}\\{(c^{jj}_{n})^{W}(x,D_{x})\\}$
$\displaystyle\quad\quad-\frac{1}{2\pi
i}\int_{\Gamma}z^{-s/2}(z-H)^{-1}(R^{*}_{n}(\bar{z})+S^{*}_{n}(\bar{z}))dz$

for some $c^{jj}_{n}\in S(m^{-s/2}h^{n+1},g)$. For the first and second terms,
by the Calderòn-Vaillancourt theorem, we obtain

$\|\langle
x\rangle\\{(\mathcal{D}+\beta)^{2}\\}^{s/2}V(\mathop{\mathrm{diag}}\\{(a^{jj}_{-s/2})^{W}(x,D_{x})\\}+\mathop{\mathrm{diag}}\\{(c^{jj}_{n})^{W}(x,D_{x})\\})\langle
x\rangle\|_{\mathcal{B}(\mathcal{H})}\lesssim 1.$

For the third term, we compute as

$\displaystyle\int_{\Gamma}\langle x\rangle\langle
D\rangle^{s}I_{4}Vz^{-s/2}(z-H)^{-1}(R^{*}_{n}(\bar{z})+S^{*}_{n}(\bar{z}))\langle
x\rangle dz$ $\displaystyle=\int_{\Gamma}z^{-s/2}(\langle x\rangle\langle
D\rangle^{s}I_{4}V\langle D\rangle^{-s})\cdot(\langle D\rangle^{s}I_{4}\langle
H\rangle^{-s/2})\cdot(\langle H\rangle^{s/2}(z-H)^{-1}\langle H\rangle^{1/2})$
$\displaystyle\quad\quad\quad\quad\quad\cdot(\langle H\rangle^{-1/2}\langle
D\rangle I_{4})\cdot(\langle
D\rangle^{-1}(R^{*}_{n}(\bar{z})+S^{*}_{n}(\bar{z}))\langle x\rangle)dz.$

Here, in the integrand, the first factor is bounded on $\mathcal{H}$ by the
Calderòn-Vaillancourt theorem. The second and fourth factors are bounded on
$\mathcal{H}$ since we have
$(\mathcal{D}+\beta+V)^{2}\approx(\mathcal{D}+\beta)^{2}$. The third factor is
bounded on $\mathcal{H}$ uniformly in $z\in\Gamma$ since $H\gtrsim\gamma$
holds. Concerning the fifth factor, by the Calderòn-Vaillancourt theorem and

$S_{n}(z)\in OPS(\langle\xi\rangle(m+\lambda)^{-1}\langle
x\rangle^{-2},g),\quad R_{n}(z)\in OPS(m(m+\lambda)^{-1}h^{n+1},g),$

we have

$\|\langle D\rangle^{-1}(R^{*}_{n}(\bar{z})+S^{*}_{n}(\bar{z}))\langle
x\rangle\|_{\mathcal{B}(\mathcal{H})}\lesssim(1+\lambda)^{-1}$

for sufficiently large $n\in\mathbb{N}$. Thus we obtain

$\left\|\int_{\Gamma}\langle x\rangle\langle
D\rangle^{s}I_{4}Vz^{-s/2}(z-H)^{-1}(R^{*}_{n}(\bar{z})+S^{*}_{n}(\bar{z}))\langle
x\rangle dz\right\|_{\mathcal{B}(\mathcal{H})}\lesssim 1$

and hence (3.15) is proved. ∎

##  Appendix A Strichartz estimate for wave equation

In this section, we give an abstract perturbation theorem concerning the
Strichartz estimates for wave equations and apply it to some equations. The
Strichartz estimates for wave equations with potentials are proved in many
papers. For example, in [BPST], inverse square type potentials are treated.
Furthermore, in [DF], [D1] and [D2], wave equations with magnetic potentials
are treated. See references therein for more informations. Our method is
essentially the same as that in [D1] but it can be also applied to higher
order or fractional wave equations. Furthermore we also treat the endpoint
case, which is not included in the above references. In the following theorem,
settings are as in [BM].

######  Theorem A.1 (non endpoint case).

Suppose $\mathcal{H}$ is a Hilbert space and self-adjoint operators $H,H_{0}$
and densely defined closed operators $v_{1},v_{2}$ satisfy

$D(H)\cup D(H_{0})\subset D(v_{1})\cap D(v_{2})\quad\text{and}\quad\langle
Hu,v\rangle=\langle H_{0}u,v\rangle+\langle v_{2}u,v_{1}v\rangle$

for all $u,v\in D(H)\cap D(H_{0})$. Furthermore we assume $H$ and $H_{0}$ are
injective, $v_{1}$ is $H_{0}$-smooth, $v_{2}P_{\mathrm{ac}}(H)$ is $H$-smooth,
$H\geq 0$, $H_{0}\geq 0$ and
$\|H^{1/4}u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$. Let
$\mathcal{A}$ be a Banach space and assume $(\mathcal{H},\mathcal{A})$ is a
Banach couple (see [BM] for its definition). If

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H_{0}}}u\|_{L^{p}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$
(A.1)

holds for some $\alpha\in\mathbb{R}$ and $p>2$, we obtain

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{p}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}+\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}$
(A.2)

for the same $p,\alpha$.

###### Proof.

By the Duhamel formula we have

$\displaystyle e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u=$ $\displaystyle\cos
t\sqrt{H_{0}}P_{\mathrm{ac}}(H)u-i\frac{\sin
t\sqrt{H_{0}}}{\sqrt{H_{0}}}\sqrt{H}P_{\mathrm{ac}}(H)u$
$\displaystyle\quad\quad-\int_{0}^{t}\frac{\sin(t-s)\sqrt{H_{0}}}{\sqrt{H_{0}}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds$

and hence by using (A.1),

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{p}_{t}\mathcal{A}}$
$\displaystyle\lesssim\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}+\|H^{1/4}_{0}({\sqrt{H_{0}}}^{-1}\sqrt{H})P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}$
$\displaystyle\quad\quad+\left\|H^{\alpha-1/2}_{0}\sin
t\sqrt{H_{0}}\int_{0}^{t}\cos
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{p}_{t}\mathcal{A}}$
$\displaystyle\quad\quad+\left\|H^{\alpha-1/2}_{0}\cos
t\sqrt{H_{0}}\int_{0}^{t}\sin
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{p}_{t}\mathcal{A}}$

holds. For the second term, we have

$\displaystyle\|H^{1/4}_{0}({\sqrt{H_{0}}}^{-1}\sqrt{H})P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$

since we assume
$\|H^{1/4}u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$. In order
to estimate the third term we compute as

$\displaystyle\left|\left\langle\int_{0}^{\infty}H^{-1/4}_{0}\cos
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds,g\right\rangle\right|$
$\displaystyle=\left|\int_{0}^{\infty}\left\langle
v_{2}e^{-is\sqrt{H}}P_{\mathrm{ac}}(H)u,v_{1}H^{-1/4}_{0}\cos
s\sqrt{H_{0}}\right\rangle\right|$
$\displaystyle\lesssim\|v_{2}e^{-is\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{H}}\|g\|_{\mathcal{H}}.$

Here, in the last line, we have used the $H_{0}$-smoothness of $v_{1}$ and
Lemma 3.2. Then by the duality and (A.1), we obtain

$\displaystyle\left\|H^{\alpha-1/2}_{0}\sin t\sqrt{H_{0}}\int_{0}^{\infty}\cos
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{p}_{t}\mathcal{A}}$
$\displaystyle\lesssim\left\|\int_{0}^{\infty}H^{-1/4}_{0}\cos
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{\mathcal{H}}$
$\displaystyle\lesssim\|v_{2}e^{-is\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{H}}.$

Since $p>2$, by the Christ-Kiselev lemma ([CK]), we have

$\displaystyle\left\|H^{\alpha-1/2}_{0}\sin t\sqrt{H_{0}}\int_{0}^{t}\cos
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{p}_{t}\mathcal{A}}$
$\displaystyle\lesssim\|v_{2}e^{-is\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{H}}\lesssim\|H^{1/4}u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}.$

In the last line, we have used the $H$-smoothness of
$v_{2}P_{\mathrm{ac}}(H)$. Similarly, we obtain

$\left\|H^{\alpha-1/2}_{0}\cos t\sqrt{H_{0}}\int_{0}^{t}\sin
s\sqrt{H_{0}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{p}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$

and hence we have (A.2). ∎

In addition to the conditions in Theorem A.1, if we have
$\|H^{1/4}_{0}u\|_{\mathcal{H}}\lesssim\|H^{1/4}u\|_{\mathcal{H}}$,

$\displaystyle\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}\lesssim\|H^{1/4}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$

holds. Hence we obtain the following corollary.

######  Corollary A.2.

Under the conditions in Theorem A.1, we assume
$\sigma(H)=\sigma_{\mathrm{ac}}(H)=[0,\infty)$ or
$\|H^{1/4}_{0}u\|_{\mathcal{H}}\lesssim\|H^{1/4}u\|_{\mathcal{H}}$. Then

$\|H^{\alpha}_{0}e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{p}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$

holds.

As our first application of the above corollary, we consider the magnetic
Schrödinger operator. The following example was proved by D’Ancona [D1] but
our proof seems a little simpler.

######  Example A.3.

Assume $d\geq 3$,
$A:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d},V:\mathbb{R}^{d}\rightarrow\mathbb{R}$
satisfy $|A(x)|+|\langle x\rangle V(x)|\lesssim\langle
x\rangle^{-1-\epsilon}$, $\langle
x\rangle^{1+\epsilon^{\prime}}A(x)\in\dot{W}^{1/2,2d}$, $A\in
C^{0}(\mathbb{R}^{d})$ for some $0<\epsilon^{\prime}<\epsilon$. Let
$H=-\Delta+A(x)\cdot D+D\cdot A(x)+V(x)$ or $H=(D+A(x))^{2}+V(x)$ and $H\geq
0$. Here $D=-i\nabla$. If zero is neither an eigenvalue nor a resonance of
$H$, we have

$\displaystyle\||D|^{\frac{1}{q}-\frac{1}{p}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q}_{x}}\lesssim\||D|^{1/2}u\|_{L^{2}_{x}}$
(A.3)

for all non endpoint $\frac{d-1}{2}$ admissible pair $(p,q)$.

###### Proof.

By the assumptions, $P_{\mathrm{ac}}(H)=I$ holds. We have
$\|H^{1/4}u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$ by the
Hardy inequality as before and $H_{0}$-smoothness of $v_{1}$ and
$H$-smoothness of $v_{2}$ follow from the proof of Corollary 3.4. ∎

The next example concerns the inverse square type potentials. The proof is
similar to that of Corollary 1.7 and we omit the details here.

######  Example A.4.

Let $H$ be as in Corollary 1.7 and $(p,q)$ be as in the above example. Then
(A.3) holds.

The next example concerning the critical inverse square potential is similarly
proved as Corollary 3.5 by using Theorem A.1.

######  Example A.5.

Assume $H=-\Delta-\frac{(d-2)^{2}}{4}|x|^{-2}$ and $(p,q)$ is as in the above
examples. Then

$\||D|^{\frac{1}{q}-\frac{1}{p}}P^{\perp}_{\mathrm{rad}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q}_{x}}\lesssim\||D|^{1/2}u\|_{L^{2}_{x}}$

holds.

Next we prove the Strichartz estimates for higher order or fractional wave
equations. The following two examples are proved by using Theorem A.1 and the
estimates for the free operators as in Theorem 1.1 or 2.1.

######  Example A.6.

Assume $H$ is as in Theorem 1.1 with $m\geq 2$ and $H\geq 0$. Then we have

$\||D|^{\frac{m-2}{p}+\frac{m}{2}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\||D|^{m/2}u\|_{L^{2}_{x}}$

for all non endpoint $\frac{d}{2}$ admissible pairs $(p,q)$.

######  Example A.7.

Assume $H$ is as in Theorem 2.1 with $\sigma>1$ and $\langle
Hu,u\rangle\lesssim\langle H_{0}u,u\rangle$ holds. Then

$\||D|^{\frac{\sigma-2}{p}+\frac{\sigma}{2}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\||D|^{\sigma/2}u\|_{L^{2}_{x}}$

holds for all non endpoint $\frac{d}{2}$ admissible pairs $(p,q)$.

The next example concerns the spherically averaged Strichartz estimates, which
were proved for the fractional Schrödinger equations with potentials in [MY1].
Let $L^{2}_{\omega}=L^{2}(S^{d-1},d\omega)$,
$\mathcal{L}^{p}_{r}=L^{p}(\mathbb{R}_{>0};r^{d-1}dr)$,
$\|f\|_{\mathcal{L}^{p}_{r}L^{2}_{\omega}}:=\|\|f(r\omega)\|_{L^{2}_{\omega}}\|_{\mathcal{L}^{p}_{r}}$
and we define
$\|f\|_{B(\mathcal{L}^{p}_{r}L^{2}_{\omega})}:=\|\\{\|\phi_{j}(D)f\|_{\mathcal{L}^{p}_{r}L^{2}_{\omega}}\\}\|_{\ell^{2}}$
for some homogeneous Littlewood-Paley decomposition $\\{\phi_{j}\\}$.

######  Example A.8.

Let $H=(-\Delta)^{\sigma}+V$ be as in Theorem 2.1 satisfying $\langle
Hu,u\rangle\lesssim\langle H_{0}u,u\rangle$, $\sigma\in(1,2)$ and $d\geq 2$.
Suppose $(p,q)$ satisfies $2<p,q\leq\infty$,
$\frac{1}{p}\leq(d-\frac{1}{2})(\frac{1}{2}-\frac{1}{q})$,
$\frac{1}{p}\neq(d-\frac{1}{2})(\frac{1}{2}-\frac{1}{q})$ if $d=2$. Set
$s=-d(\frac{1}{2}-\frac{1}{q})+\frac{\sigma}{p}$. Then we have

$\||D|^{s+\frac{\sigma}{2}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}B(\mathcal{L}^{q}_{r}L^{2}_{\omega})}\lesssim\||D|^{\sigma/2}u\|_{L^{2}_{x}}.$

###### Proof.

The corresponding estimates for the free operator are proved in Appendix A in
[MY1]. The $H_{0}$-smoothness of $v_{1}$ and the $H$-smoothness of $v_{2}$ are
proved in Theorem 1.3 in [MY1]. Hence by using Theorem A.1, we have the
desired estimates. ∎

######  Remark A.9.

If $V$ and $u$ are radially symmetric functions, we have
$\|\phi_{j}(D)u\|_{\mathcal{L}^{q}_{r}L^{2}_{\omega}}\approx\|\phi_{j}(D)u\|_{q}$
and hence $\|u\|_{B(\mathcal{L}^{q}_{r}L^{2}_{\omega})}\gtrsim\|u\|_{q}$ holds
by the Minkowski inequality and the Littlewood-Paley theorem. Therefore, by
Example A.8, we obtain

$\||D|^{s+\frac{\sigma}{2}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q}_{x}}\lesssim\||D|^{\sigma/2}u\|_{L^{2}_{x}}.$

Then, by the Sobolev inequality
$\||D|^{s+\frac{\sigma}{2}}u\|_{q}\gtrsim\||D|^{\frac{\sigma-2}{p}+\frac{\sigma}{2}}u\|_{\tilde{q}}$
for a $\frac{d}{2}$ admissible pair $(p,\tilde{q})$, we obtain

$\||D|^{\frac{\sigma-2}{p}+\frac{\sigma}{2}}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{\tilde{q}}_{x}}\lesssim\||D|^{\sigma/2}u\|_{L^{2}_{x}}.$

Thus Example A.8 is also a refinement of the $L^{p}_{t}L^{q}_{x}$ type
Strichartz estimates.

The next example is the inhomogeneous elliptic operators. Concerning the
Strichartz estimates for the Schrödinger type equations, they are treated in
[MY1].

######  Example A.10.

Assume $H=H_{0}+V$,
$H_{0}=\left(\displaystyle\sum_{j=1}^{\sigma}a_{j}(-\Delta)^{j}\right)^{2}$,
$\sigma\in\mathbb{N}$, $a_{j}\geq 0$, $a_{\sigma}>0$, $\sigma<d/4$ and $H$
satisfies the assumptions in Theorem 2.1 in which $(-\Delta)^{\sigma}$ is
replaced by $H_{0}$ and $|x|^{2\sigma}V\in L^{\infty}$ is replaced by
$|x|^{4\sigma}V\in L^{\infty}$. Furthermore we assume $\langle
Hu,u\rangle\lesssim\langle H_{0}u,u\rangle$. Then we have

$\||D|^{\frac{2\sigma-2}{p}+\sigma}e^{-it\sqrt{H}}u\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\|H^{1/4}_{0}u\|_{L^{2}_{x}}$

for all non endpoint $\frac{d}{2}$ admissible pairs $(p,q)$.

###### Proof.

The Strichartz estimates for the free operator are proved in Appendix A in
[MY1]. We decompose $v_{1}=|x|^{2\sigma}V$ and $v_{2}=|x|^{-2\sigma}$. Then
$H_{0}$-smoothness of $v_{1}$ and $H$-smoothness of $v_{2}$ are proved in
Theorem 5.2 in [MY1]. Hence by repeating the proof of Theorem A.1 with easy
modifications, we obtain the desired estimates. ∎

Next we focus on the endpoint Strichartz estimates.

######  Theorem A.11.

Suppose $\mathcal{H}$ is a Hilbert space and self-adjoint operators $H,H_{0}$
and a densely defined closed operator $V$ satisfy

$\displaystyle D(H)\cup D(H_{0})\subset D(V),\quad H=H_{0}+V,\quad H,H_{0}\geq
0,\quad\|H^{1/4}u\|_{\mathcal{H}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}.$

Let $\mathcal{A}$ be a Banach space and assume $(\mathcal{H},\mathcal{A})$ is
a Banach couple (see [BM] for its definition). We also assume the following
homogeneous and inhomogeneous endpoint Strichartz estimates for $H_{0}$ and
the following smoothing estimate for some $\alpha\in\mathbb{R}$:

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H_{0}}}u\|_{L^{2}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}$
(A.4)
$\displaystyle\left\|H^{\alpha}_{0}\int_{0}^{t}\frac{\sin(t-s)\sqrt{H_{0}}}{\sqrt{H_{0}}}F(s)ds\right\|_{L^{2}_{t}\mathcal{A}}\lesssim\|H^{-\alpha}_{0}F\|_{L^{2}_{t}\mathcal{A}^{\prime}}$
(A.5)
$\displaystyle\|H^{-\alpha}_{0}Ve^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}^{\prime}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}.$
(A.6)

Then we obtain

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}}\lesssim\|H^{1/4}_{0}u\|_{\mathcal{H}}+\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}.$
(A.7)

We note that the same statement as Corollary A.2 holds for the above theorem.

###### Proof.

By the Duhamel formula in Theorem A.1, we have

$\displaystyle\|H^{\alpha}_{0}e^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}}$
$\displaystyle\lesssim\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}+\|H^{1/4}_{0}u\|_{\mathcal{H}}$
$\displaystyle\quad\quad+\left\|H^{\alpha}_{0}\int_{0}^{t}\frac{\sin(t-s)\sqrt{H_{0}}}{\sqrt{H_{0}}}Ve^{-is\sqrt{H}}P_{\mathrm{ac}}(H)uds\right\|_{L^{2}_{t}\mathcal{A}}$
$\displaystyle\lesssim\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}+\|H^{1/4}_{0}u\|_{\mathcal{H}}+\|H^{-\alpha}_{0}Ve^{-it\sqrt{H}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}^{\prime}}$
$\displaystyle\lesssim\|H^{1/4}_{0}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}+\|H^{1/4}_{0}u\|_{\mathcal{H}}.$

Here, in the first inequality, (A.4) is used, in the second inequality, (A.5)
is used and in the last inequality, (A.6) is used. ∎

The first application is concerned with very short range potentials.

######  Example A.12.

Assume $H=-\Delta+V$ satisfies $H\geq 0$, $|V(x)|\lesssim\langle
x\rangle^{-2-\epsilon}$, $\langle
x\rangle^{1/2+\epsilon^{\prime}}V\in\dot{W}^{\frac{1}{d-1},\frac{2d(d-1)}{3d-1}}\cap
L^{\frac{2d(d-1)}{3d-2}}$ for some $\epsilon^{\prime}>\epsilon$ and zero is a
regular point of $H$. Then we have

$\||D|^{-\frac{1}{d-1}}e^{-it\sqrt{H}}u\|_{L^{2}_{t}L^{\frac{2(d-1)}{d-3}}}\lesssim\|u\|_{\dot{H}^{1/2}}$

if $d\geq 4$.

###### Proof.

Since $\langle x\rangle^{-1/2-\tilde{\epsilon}}|D|^{1/2}$ is $H$-smooth and
$\langle Hu,u\rangle\lesssim\langle H_{0}u,u\rangle$ holds, it suffices to
show the boundedness of $|D|^{\frac{1}{d-1}}V|D|^{-1/2}\langle
x\rangle^{1/2+\tilde{\epsilon}}:L^{2}\rightarrow
L^{(\frac{2(d-1)}{d-3})^{\prime}}$ for sufficiently small $\tilde{\epsilon}$.
This follows from the following inequality.

$\displaystyle\||D|^{\frac{1}{d-1}}V|D|^{-1/2}\langle
x\rangle^{1/2+\tilde{\epsilon}}\|_{2\rightarrow(\frac{2(d-1)}{d-3})^{\prime}}$
$\displaystyle=\||D|^{\frac{1}{d-1}}V\langle
x\rangle^{1/2+\epsilon^{\prime}}|D|^{-1/2}(|D|^{1/2}\langle
x\rangle^{-1/2-\epsilon^{\prime}}|D|^{-1/2}\langle
x\rangle^{1/2+\tilde{\epsilon}})\|_{2\rightarrow(\frac{2(d-1)}{d-3})^{\prime}}$
$\displaystyle\lesssim\||D|^{\frac{1}{d-1}}V\langle
x\rangle^{1/2+\epsilon^{\prime}}|D|^{-1/2}\|_{2\rightarrow(\frac{2(d-1)}{d-3})^{\prime}}$
$\displaystyle\lesssim\||D|^{\frac{1}{d-1}}(\langle
x\rangle^{1/2+\epsilon^{\prime}}V)\|_{\frac{2d(d-1)}{3d-1}}\||D|^{-1/2}\|_{2\rightarrow\frac{2d}{d-1}}+\|\langle
x\rangle^{1/2+\epsilon^{\prime}}V\|_{\frac{2d(d-1)}{3d-2}}\||D|^{\frac{1}{d-1}-1/2}\|_{2\rightarrow\frac{2d(d-1)}{d^{2}-2d+2}}$
$\displaystyle\lesssim 1.$

Here, in the third line, we have used the $L^{2}$-boundedness of
$|D|^{1/2}\langle x\rangle^{-1/2-\epsilon^{\prime}}|D|^{-1/2}\langle
x\rangle^{1/2+\tilde{\epsilon}}$ (see [EGS]), in the fourth line, we have used
the fractional Leibniz rule and in the last line, we have used the Sobolev
embedding and our assumptions. ∎

The next example is concerned with higher order operators.

######  Example A.13.

Assume $H$ is as in Theorem 1.1 with $m\geq 2$ and $H\geq 0$. Then we have

$\||D|^{m-1}e^{-it\sqrt{H}}u\|_{L^{2}_{t}L^{\frac{2d}{d-2},2}}\lesssim\||D|^{m/2}u\|_{2}.$

###### Proof.

Since we can decompose $V=v_{1}v_{2}$, $|v_{j}(x)|\lesssim\langle
x\rangle^{-m-\epsilon}$, we obtain

$\displaystyle\||D|^{-(m-1)}Ve^{-it\sqrt{H}}u\|_{L^{2}_{t}L^{\frac{2d}{d+2},2}}$
$\displaystyle\lesssim\||D|^{-(m-1)}v_{1}\|_{2\rightarrow\frac{2d}{d+2},2}\cdot\|v_{2}e^{-it\sqrt{H}}u\|_{L^{2}_{t}L^{2}_{x}}$
$\displaystyle\lesssim\|H^{1/4}u\|_{2}\lesssim\||D|^{m/2}u\|_{2},$

where we have used the Sobolev embedding
$|D|^{-(m-1)}:L^{\frac{2d}{2m+d},2}\rightarrow L^{\frac{2d}{d+2},2}$ and the
Hölder inequality $v_{1}:L^{2}\rightarrow L^{\frac{2d}{2m+d},2}$. Thus by
Theorem A.11, we have the desired estimates. ∎

For the last example, we consider the fractional operators with the Hardy type
potentials. The proof is similar to Example A.13 and we omit the details.

######  Example A.14.

Assume $H$ is as in Theorem 2.1 with $\sigma>1$ and $\langle
Hu,u\rangle\lesssim\langle H_{0}u,u\rangle$. Then we have

$\||D|^{\sigma-1}e^{-it\sqrt{H}}u\|_{L^{2}_{t}L^{\frac{2d}{d-2},2}}\lesssim\||D|^{\sigma/2}u\|_{2}.$

##  Appendix B Strichartz estimate for Klein-Gordon equation

In this section, as Appendix A, we give an abstract perturbation theorem
concerning the Strichartz estimates for the Klein-Gordon equation. See [BC],
[DF], [D2], [LSS] for references closely related to this section. We also give
an example of the Strichartz estimates for the higher order Klein-Gordon
equation. The following theorem is proved by the same way as in the case of
wave equations.

######  Theorem B.1 (non endpoint case).

Suppose $\mathcal{H}$ is a Hilbert space and self-adjoint operators $H,H_{0}$
and densely defined closed operators $v_{1},v_{2}$ satisfy

$D(H)\cup D(H_{0})\subset D(v_{1})\cap D(v_{2})\quad\text{and}\quad\langle
Hu,v\rangle=\langle H_{0}u,v\rangle+\langle v_{2}u,v_{1}v\rangle$

for all $u,v\in D(H)\cap D(H_{0})$. Furthermore we assume $H+1$ and $H_{0}+1$
are injective, $v_{1}$ is $H_{0}$-smooth, $v_{2}P_{\mathrm{ac}}(H)$ is
$H$-smooth, $H+1\geq 0$, $H_{0}+1\geq 0$ and
$\|(H+1)^{1/4}u\|_{\mathcal{H}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}$.
Let $\mathcal{A}$ be a Banach space and assume $(\mathcal{H},\mathcal{A})$ is
a Banach couple (see [BM] for its definition). If

$\displaystyle\|(H_{0}+1)^{\alpha}e^{-it\sqrt{H_{0}+1}}u\|_{L^{p}_{t}\mathcal{A}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}$
(B.1)

holds for some $\alpha\in\mathbb{R}$ and $p>2$, we obtain

$\displaystyle\|(H_{0}+1)^{\alpha}e^{-it\sqrt{H+1}}P_{\mathrm{ac}}(H)u\|_{L^{p}_{t}\mathcal{A}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}+\|(H_{0}+1)^{1/4}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}$
(B.2)

for the same $p,\alpha$.

We give some applications to the Schrödinger operator. The proofs are similar
to those of corresponding estimates for wave equations in Appendix A so we
omit the details.

######  Example B.2.

Let $H$ be as in Corollary A.3 but satisfy $H+1\geq 0$ instead of $H\geq 0$.
Then we have

$\displaystyle\|\langle
D\rangle^{\frac{1}{q}-\frac{1}{p}}e^{-it\sqrt{H+1}}u\|_{L^{p}_{t}L^{q}_{x}}\lesssim\|u\|_{H^{1/2}}$
(B.3)

for all non endpoint $\frac{d-1}{2}$ or $\frac{d}{2}$ admissible pairs
$(p,q)$.

######  Example B.3.

Let $H$ be as in Corollary 1.7 but satisfy $\langle
Hu,u\rangle\lesssim\|u\|^{2}_{H^{1}}$ instead of $\langle
Hu,u\rangle\lesssim\|\nabla u\|^{2}_{2}$. Then (B.3) holds for the same $p,q$.

######  Example B.4.

Let $H=-\Delta-\frac{(d-2)^{2}}{4}|x|^{-2}$. Then

$\|\langle
D\rangle^{\frac{1}{q}-\frac{1}{p}}e^{-it\sqrt{H+1}}P^{\perp}_{\mathrm{rad}}u\|_{L^{p}_{t}L^{q}_{x}}\lesssim\|u\|_{H^{1/2}}$

holds for all non endpoint $\frac{d-1}{2}$ or $\frac{d}{2}$ admissible pairs
$(p,q)$.

Next we consider the higher order Klein-Gordon equations. Set
$P_{m}(\xi):=(|\xi|^{2m}+1)^{1/2}$. Then
$P_{m}(\xi)=1+\frac{1}{2}|\xi|^{2m}+O(|\xi|^{4m})$ as $\xi\rightarrow 0$ and
$P_{m}(\xi)=|\xi|^{m}+O(|\xi|^{-m})$ as $\xi\rightarrow\infty$. Hence the
Strichartz estimates for $P_{m}(D)$ differs depending on high or low energy.

######  Lemma B.5 (Low energy estimate).

For all $m\in\mathbb{N}$, $m\geq 2$ and $\frac{d}{2}$ admissible pairs
$(p,q)$, we have

$\left\||D|^{\frac{2(m-1)}{p}}e^{-itP_{m}(D)}\chi_{\\{|D|\lesssim
1\\}}u\right\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\|u\|_{2}$

###### Proof.

We follow the argument in [MY1]. By the Littlewood-Paley theorem, the
Minkowski inequality and the real interpolation, we have

$\displaystyle\|\chi_{\\{|D|\lesssim
1\\}}u\|_{L^{q,2}}\lesssim\left(\sum_{j\leq
1}\|\phi_{j}(D)\chi_{\\{|D|\lesssim 1\\}}u\|^{2}_{L^{q,2}}\right)^{1/2},$
(B.4)

where $\phi_{j}$ denotes a homogeneous Littlewood-Paley decomposition. Since
we have

$|\nabla
P_{m}(\xi)|\approx|\xi|^{2m-1},\quad|\mathop{\mathrm{Hess}}P_{m}(\xi)|\approx|\xi|^{d(2m-2)}$

on $\\{|\xi|\lesssim 1\\}$, by Theorem 3.1 in [HHZ], we obtain

$\left|\int_{\mathbb{R}^{d}}e^{ix\xi-
itP_{m}(\xi)}|\xi|^{d(m-1)}\chi_{\\{|\xi|\lesssim
1\\}}d\xi\right|\lesssim|t|^{-\frac{d}{2}}.$

Hence we have the dispersive estimate
$\||D|^{d(m-1)}e^{-itP_{m}(D)}\chi_{\\{|D|\lesssim
1\\}}\|_{1\rightarrow\infty}\lesssim|t|^{-\frac{d}{2}}$. Since
$\||2^{-j}D|^{d(1-m)}\phi_{j}(D)\|_{\infty\rightarrow\infty}\lesssim 1$ holds,
we obtain $\|e^{-itP_{m}(D)}\phi_{j}(D)\|_{1\rightarrow\infty}\lesssim
2^{-jd(m-1)}|t|^{-\frac{d}{2}}$ for $j\leq 1$. Thus

$\|\phi_{j}(D)e^{-i2^{-2j(m-1)}(t-s)P_{m}(D)}\phi_{j}(D)\|_{1\rightarrow\infty}\lesssim|t-s|^{-\frac{d}{2}}$

holds and by the Keel-Tao theorem ([KT]), we obtain

$\|e^{-i2^{-2j(m-1)}tP_{m}(D)}\phi_{j}(D)u\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\|u\|_{2}.$

Then, by changing variables, (B.4) and the Minkowski inequality, we obtain the
desired estimates. ∎

######  Lemma B.6 (High energy estimate).

For all $m\in\mathbb{N}$, $m\geq 2$ and $\frac{d}{2}$ admissible pairs
$(p,q)$, we have

$\left\||D|^{\frac{m-2}{p}}e^{-itP_{m}(D)}\chi_{\\{|D|\gtrsim
1\\}}u\right\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\|u\|_{2}$

###### Proof.

Since we have

$|\nabla
P_{m}(\xi)|\approx|\xi|^{m-1},\quad|\mathop{\mathrm{Hess}}P_{m}(\xi)|\approx|\xi|^{d(m-2)}$

on $\\{|\xi|\gtrsim 1\\}$, by the similar argument as above, we obtain
$\|e^{-itP_{m}(D)}\phi_{j}(D)\|_{1\rightarrow\infty}\lesssim
2^{-jd(\frac{m}{2}-1)}|t|^{-\frac{d}{2}}$ for $j\geq 1$. Then the rest of the
argument is similar so we omit the details here. ∎

Now we give an application of Theorem B.1 to Lemma B.6. The proof is similar
to the case of higher order wave equations so we omit it here.

######  Example B.7.

Assume $H=(-\Delta)^{m}+V$, $m\geq 2$ is as in Theorem 1.1 or 2.1 and
satisfies $H+1\geq 0$ and $\sigma(H)=\sigma_{\mathrm{ac}}(H)$. Then

$\|\langle D\rangle^{\frac{m-2}{p}+\frac{m}{2}}\chi_{\\{|D|\gtrsim
1\\}}e^{-it\sqrt{H+1}}u\|_{L^{p}_{t}L^{q,2}_{x}}\lesssim\|\langle
D\rangle^{m/2}u\|_{2}$

holds for all non endpoint $\frac{d}{2}$ admissible pairs $(p,q)$.

Finally we consider the endpoint Strichartz estimates. The following theorem
is proved similarly as the corresponding one for wave equations (see Appendix
A).

######  Theorem B.8.

Suppose $\mathcal{H}$ is a Hilbert space and self-adjoint operators $H,H_{0}$
and a densely defined closed operator $V$ satisfy

$\displaystyle D(H)\cup D(H_{0})\subset D(V),\quad H=H_{0}+V,$ $\displaystyle
H+1,H_{0}+1\geq
0,\quad\|(H+1)^{1/4}u\|_{\mathcal{H}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}.$

Let $\mathcal{A}$ be a Banach space and assume $(\mathcal{H},\mathcal{A})$ is
a Banach couple (see [BM] for its definition). We also assume the following
homogeneous and inhomogeneous endpoint Strichartz estimates for $H_{0}$ and
the following smoothing estimate for some $\alpha\in\mathbb{R}$:

$\displaystyle\|(H_{0}+1)^{\alpha}e^{-it\sqrt{H_{0}+1}}u\|_{L^{2}_{t}\mathcal{A}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}$
$\displaystyle\left\|(H_{0}+1)^{\alpha}\int_{0}^{t}\frac{\sin(t-s)\sqrt{H_{0}+1}}{\sqrt{H_{0}+1}}F(s)ds\right\|_{L^{2}_{t}\mathcal{A}}\lesssim\|(H_{0}+1)^{-\alpha}F\|_{L^{2}_{t}\mathcal{A}^{\prime}}$
$\displaystyle\|(H_{0}+1)^{-\alpha}Ve^{-it\sqrt{H+1}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}^{\prime}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}.$

Then we obtain

$\displaystyle\|(H_{0}+1)^{\alpha}e^{-it\sqrt{H+1}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}\mathcal{A}}\lesssim\|(H_{0}+1)^{1/4}u\|_{\mathcal{H}}+\|(H_{0}+1)^{1/4}P_{\mathrm{ac}}(H)u\|_{\mathcal{H}}.$

We give some examples concerning very short range potentials.

######  Example B.9.

Assume $H=-\Delta+V$ satisfy $|\partial^{\alpha}_{x}V(x)|\lesssim\langle
x\rangle^{-2-\epsilon}$ for all $\alpha\in\mathbb{N}^{d}_{0}$, $H+1\geq 0$ and
zero is neither an eigenvalue nor a resonance of $H$. Then we have

$\|\langle
D\rangle^{-\frac{1}{d}}e^{-it\sqrt{H+1}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}L^{\frac{2d}{d-2}}_{x}}\lesssim\|u\|_{H^{1/2}}+\|P_{\mathrm{ac}}(H)u\|_{H^{1/2}}.$

###### Proof.

By the Hölder inequality, $\langle
x\rangle^{-1-\epsilon^{\prime}}:L^{2}\rightarrow L^{\frac{2d}{d+2}}$ is a
bounded operator and $A:=\langle D\rangle^{\frac{1}{d}}V\langle
D\rangle^{-1/2}\langle x\rangle^{2+\epsilon}:L^{\frac{2d}{d+2}}\rightarrow
L^{\frac{2d}{d+2}}$ is also bounded since the symbol of $A$ is in
$S^{0}_{1,0}$. Then, by Theorem B.8 and the Kato smoothing estimate, we obtain
the desired estimates. ∎

######  Example B.10.

Assume $H=-\Delta+V$ satisfy $|\partial^{\alpha}_{x}V(x)|\lesssim\langle
x\rangle^{-(1+\frac{d}{d-1})-\epsilon}$ for all $\alpha\in\mathbb{N}^{d}_{0}$,
$H+1\geq 0$ and zero is neither an eigenvalue nor a resonance of $H$. Then we
have

$\|\langle
D\rangle^{-\frac{1}{d-1}}e^{-it\sqrt{H+1}}P_{\mathrm{ac}}(H)u\|_{L^{2}_{t}L^{\frac{2(d-1)}{d-3}}_{x}}\lesssim\|u\|_{H^{1/2}}+\|P_{\mathrm{ac}}(H)u\|_{H^{1/2}}.$

###### Proof.

The boundedness of $\langle D\rangle^{\frac{1}{d-1}}V\langle
D\rangle^{-1/2}\langle x\rangle^{1+\epsilon^{\prime}}:L^{2}\rightarrow
L^{(\frac{2(d-1)}{d-3})^{\prime}}$ follows from the boundedness of $\langle
x\rangle^{-(\frac{d}{d-1})-\epsilon^{\prime}}:L^{2}\rightarrow
L^{(\frac{2(d-1)}{d-3})^{\prime}}$ and $\langle
D\rangle^{\frac{1}{d-1}}V\langle D\rangle^{-1/2}\langle
x\rangle^{1+\frac{d}{d-1}+2\epsilon^{\prime}}:L^{(\frac{2(d-1)}{d-3})^{\prime}}\rightarrow
L^{(\frac{2(d-1)}{d-3})^{\prime}}$ by the same reason as above. Then we obtain
the desired estimates by repeating the above proof. ∎

## Acknowledgement

The author thanks his supervisor Kenichi Ito for valuable comments and
discussions. He is partially supported by FoPM, WINGS Program, the University
of Tokyo.

## References

  * [1]
  * [BC] Beceanu, M., Chen, G. Strichartz estimates for the Klein-Gordon equation in $\mathbb{R}^{3+1}$. Pure Appl. Anal. 4 (2022), no. 4, 767–809.
  * [BHLNS] Bez, N., Hong, Y., Lee, S., Nakamura, S., Sawano, Y. On the Strichartz estimates for orthonormal systems of initial data with regularity. Adv. Math. 354 (2019), 106736, 37 pp.
  * [BKS1] Bez, N., Kinoshita, S., Shiraki, S. A note on Strichartz estimates for the wave equation with orthonormal initial data. arXiv preprint arXiv: 2306.14547
  * [BKS2] Bez, N., Kinoshita, S., Shiraki, S. Boundary Strichartz estimates and pointwise convergence for orthonormal systems. arXiv preprint arXiv: 2306.14536
  * [BLN1] Bez, N., Lee, S., Nakamura, S. Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates. Forum Math. Sigma 9 (2021), Paper No. e1, 52 pp.
  * [BLN2] Bez, N., Lee, S., Nakamura, S. Maximal estimates for the Schrödinger equation with orthonormal initial data. Selecta Math. (N.S.) 26 (2020), no. 4, Paper No. 52, 24 pp.
  * [BPST] Burq, N., Planchon, F., Stalker, J.G., Tahvilder-Zadeh, A.S. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53 (2004), no. 6, 1665 –1680.
  * [BM] Bouclet, J. M., Mizutani, H. Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities. Trans. Amer. Math. Soc. 370 (2018), no. 10, 7293–7333.
  * [BS] Birman, M. S., Solomyak, M. Double operator integrals in a Hilbert space. Integral Equations Operator Theory 47 (2003), no. 2, 131–168.
  * [CHP1] Chen, T., Hong, Y., Pavlović. Global well-posedness of the NLS system for infinitely many fermions. Arch. Ration. Mech. Anal. 224 (2017), no. 1, 91–123.
  * [CHP2] Chen, T., Hong, Y., Pavlović. On the scattering problem for infinitely many fermions in dimensions $d\geq 3$ at positive temperature. Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 2, 393–416.
  * [CK] Christ, M., Kiselev, A. Maximal functions associated to filtrations. J. Funct. Anal. 179 (2001), no. 2, 409–425.
  * [D1] D’Ancona, P. Kato smoothing and Strichartz estimates for wave equations with magnetic potentials. Comm. Math. Phys. 335 (2015), no. 1, 1–16.
  * [D2] D’Ancona, P. On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations. Commun. Pure Appl. Anal. 19 (2020), no. 1, 609–640.
  * [DF] D’Ancona, P., Fanelli, L. Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Comm. Partial Differential Equations 33 (2008), no. 4-6, 1082–1112.
  * [EG] Erdoğan, M., Green, W, R. The Lp-continuity of wave operators for higher order Schrödinger operators. Adv. Math. 404 (2022), Paper No. 108450, 41 pp.
  * [EGS] Erdoğan, M. B., Goldberg, M., Schlag, W. Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21 (2009), no. 4, 687–722.
  * [FLLS] Frank, R. L., Lewin, M., Lieb, E. H., Seiringer, R. Strichartz inequality for orthonormal functions. J. Eur. Math. Soc. (JEMS) 16 (2014), no. 7, 1507–1526.
  * [FS] Frank, R. L., Sabin, J. Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Amer. J. Math. 139 (2017), no. 6, 1649–1691.
  * [FL1] Fröhlich, J., Lenzmann, E. Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Comm. Math. Phys. 274 (2007), no. 3, 737–750.
  * [FL2] Fröhlich, J., Lenzmann, E. Blowup for nonlinear wave equations describing boson stars. Comm. Pure Appl. Math. 60 (2007), no. 11, 1691–1705.
  * [FSWY] Feng, H., Soffer, A., Wu, Z., Yao, X. Decay estimates for higher-order elliptic operators. Trans. Amer. Math. Soc. 373 (2020), no. 4, 2805–2859.
  * [GY] Galtbayar, A., Yajima, K. The $L^{p}$-boundedness of wave operators for fourth order Schrödinger operators on $\mathbb{R}^{4}$. arXiv preprint arXiv: 2303.13985.
  * [GK] Gohberg, I., Krein, M. G. Theory and applications of Volterra operators in Hilbert space. Transl. Math. Monogr., Vol. 24 American Mathematical Society, Providence, RI, 1970. x+430 pp.
  * [Ha] Hadama, S. Asymptotic stability of a wide class of stationary solutions for the Hartree and Schrödinger equations for infinitely many particles. arXiv preprint arXiv: 2308.15929.
  * [Hö] Hörmander, L. The analysis of linear partial differential operators. III. Pseudodifferential operators Grundlehren Math. Wiss., 274 Fundamental Principles of Mathematical Sciences Springer-Verlag, Berlin, 1985. viii+525 pp.
  * [Ho] Hoshiya, A. Orthonormal Strichartz estimates for Schrödinger operator and their applications to infinitely many particle systems. arXiv preprint arXiv: 2312. 08314.
  * [HHZ] Huang, T., Huang, S., Zheng, Q. Inhomogeneous oscillatory integrals and global smoothing effects for dispersive equations. J. Differential Equations 263 (2017), no. 12, 8606–8629.
  * [K] Kato, T. Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162 (1965/1966), 258–279.
  * [KY] Kato, T., Yajima, K. Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1 (1989), no. 4, 481–496.
  * [KT] Keel, M., Tao, T. Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955–980.
  * [KS] Kühn, F., Schlling, R L. Convolution inequalities for Besov and Triebel-Lizorkin spaces, and applications to convolution semigroups. Studia Math. 262 (2022), no. 1, 93–119.
  * [L] Lenzmann, E. Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10 (2007), no. 1, 43–64.
  * [LS1] Lewin, M., Sabin, J. The Hartree equation for infinitely many particles I. Well-posedness theory. Comm. Math. Phys. 334 (2015), no. 1, 117–170.
  * [LS2] Lewin, M., Sabin, J. The Hartree equation for infinitely many particles, II: Dispersion and scattering in 2D. Anal. PDE 7 (2014), no. 6, 1339–1363.
  * [LSS] Lee, H., Seo, I., Seok, J. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete Contin. Dyn. Syst. 40 (2020), no. 1, 597–608.
  * [MNO] Machihara, S., Nakanishi, K., Ozawa, T. Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev. Mat. Iberoamericana 19 (2003), no. 1, 179–194.
  * [Ma] Martinez, A. An introduction to semiclassical and microlocal analysis. Universitext Springer-Verlag, New York, 2002. viii+190 pp.
  * [M1] Mizutani, H. Strichartz estimates for Schrödinger equations with slowly decaying potentials. J. Funct. Anal. 279 (2020), no. 12, 108789, 57 pp.
  * [M2] Mizutani, H. Remarks on endpoint Strichartz estimates for Schrödinger equations with the critical inverse-square potential. J. Differential Equations 263 (2017), no. 7, 3832–3853.
  * [MWY] Mizutani, H., Wan, Z., Yao, X. $L^{p}$-boundedness of wave operators for bi-Schrödinger operators on the line. arXiv preprint arXiv: 2201.04758.
  * [MY1] Mizutani, H., Yao, X. Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials. Comm. Math. Phys. 388 (2021), no. 1, 581–623.
  * [MY2] Mizutani, H., Yao, X. Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials. arXiv preprint arXiv: 2004.10115.
  * [MS1] Mondal, S. S., Swain, J. Strichartz inequality for orthonormal functions associated with special Hermite operator. arXiv preprint arXiv: 2103.12586.
  * [MS2] Mondal, S. S., Song, M. Orthonormal Strichartz inequalities for the $(k,a)$-generalized Laguerre operator and Dunkl operator. arXiv preprint arXiv: 2208.12015.
  * [O] Ozawa, T. Remarks on proofs of conservation laws for nonlinear Schrödinger equations. Calc. Var. Partial Differential Equations 25 (2006), no. 3, 403–408.
  * [RS] Rodnianski, I., Schlag, W. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155 (2004), no. 3, 451–513.
  * [Si] Simon, B. Trace ideals and their applications. Second edition Math. Surveys Monogr., 120 American Mathematical Society, Providence, RI, (2005). viii+150 pp
  * [T] Triebel, H. Theory of function spaces. II. Monogr. Math., 84 Birkhäuser Verlag, Basel, 1992. viii+370 pp.
  * [Y1] Yajima, K. Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys.110 (1987), no. 3, 415–426.
  * [Y2] Yajima, K. Schrödinger equations II (in Japanese), Asakura Publishing Co., Ltd., Japan (2014)


# Waiter-Client Clique-Factor Game

Vojtěch Dvořák  Department of Pure Maths and Mathematical Statistics,
University of Cambridge, UK<EMAIL_ADDRESS>
###### Abstract.

Fix two integers $n,k$, with $n$ divisible by $k$, and consider the following
game played by two players, Waiter and Client, on the edges of $K_{n}$.
Starting with all the edges marked as unclaimed, in each round, Waiter picks
two yet unclaimed edges. Client then chooses one of these edges to be added to
Client’s graph, while the other edge is added to Waiter’s graph. Waiter wins
if she eventually forces Client to create a $K_{k}$-factor in Client’s graph.
If she does not manage to do that, Client wins.

For fixed $k$ and large enough $n$, it can be easily shown that Waiter wins if
she plays optimally (in particular, this is an immediate consequence of our
result that for such $n$, Waiter can win quite fast). The question posed by
Clemens et al. is how long the game will last if Waiter aims to win as fast as
she can, Client tries to delay her as much as he can, and they both play
optimally. We denote this optimal number of rounds by
$\tau_{WC}(\mathcal{F}_{n,K_{k}-\text{fac}},1)$. In the present paper, we
obtain the first non-trivial lower bound on this quantity for large $k$.
Together with a simple upper bound following the strategy of Clemens et al.,
this gives

$2^{k/3-o(k)}n\leq\tau_{WC}(\mathcal{F}_{n,K_{k}-\text{fac}},1)\leq
2^{k}\frac{n}{k}+C(k),$

where $C(k)$ is a constant dependent only on $k$ and the $o(k)$ term is
independent of $n$ as well.

##  1\. Introduction

Combinatorial games form a broad and widely studied field. In the most usual
setting, we have two players and a set of rules and we try to determine which
player has a winning strategy if both players play optimally. Commonly, even
for simple looking games, this may be a very difficult question. To the reader
interested in the topic in general, we recommend the book of Beck [2]. The
book of Hefetz, Krivelevich, Stojaković and Szabó [6] about the positional
games (which form a large part of the well studied combinatorial games) is
also a good reference.

One of the well known classes of the combinatorial games are the so-called
Waiter-Client games, originally introduced by Beck [1] under the name Picker-
Chooser games. These are played by two players, Waiter and Client. Initially,
we are given an integer $b\geq 1$, a finite set $X$ and a family of winning
sets $\mathcal{F}$ of the subsets of $X$. All elements of $X$ are initially
marked as unclaimed. In each round, Waiter chooses $b+1$ yet unclaimed
elements of $X$ and offers them to Client. Client picks one of them and adds
it to his graph, while the remaining $b$ elements are added to Waiter’s graph.
Waiter wins if she forces Client to create a winning set $F\in\mathcal{F}$ in
Client’s graph, and otherwise if Client can prevent Waiter from doing so,
Client wins.

Various papers studied, for given $X,b,\mathcal{F}$, which player has a
winning strategy in the corresponding Waiter-Client games. Or, in the cases
when we know that Waiter can win, how fast can she guarantee her victory to
be. For instance, Bednarska-Bzdega, Hefetz, Krivelevich and Łuczak [3] studied
such games with the winning sets being large components or long cycles.
Hefetz, Krivelevich and Tan [7] looked on the Waiter-Client games involving
planarity, colourability and minors, and later [8] the same authors studied a
Hamiltonicity game with a board being a random graph. Krivelevich and Trumer
[9] considered a maximum degree game. Yet more results were obtained by Tan
[10] about colourability and $k$-SAT games.

When $b=1$, we call the corresponding Waiter-Client game unbiased. Clemens et
al. [4] studied several unbiased Waiter-Client games played on the edges of
the complete graph, i.e. with $X=E(K_{n})$.

Assume that for our triple $X,b,\mathcal{F}$, Waiter wins the corresponding
Waiter-Client game. Then we will denote by $\tau_{WC}(\mathcal{F},b)$ the
number of rounds that the game will last when Waiter tries to win as fast as
possible, Client tries to slow her down as much as possible, and they both
play optimally. Note that $X$ does not appear in this notation - but that will
not be an issue, as from now on, we will only consider $X=E(K_{n})$.

One of the games considered by Clemens et al. is the so-called triangle-factor
game, where the winning sets are triangle-factors, i.e. decompositions of the
vertices of $K_{n}$ into disjoint triples, each of which forms a triangle
$K_{3}$ (hence we must in particular insist that $n$ is divisible by $3$). It
is known that for large enough such $n$, Waiter has a winning strategy here.
Clemens et al. [4] moreover proved that for such large $n$ divisible by $3$,
we have
$\frac{13}{12}n\leq\tau_{WC}(\mathcal{F}_{n,K_{3}-\text{fac}},1)\leq\frac{7}{6}n+o(n)$.
The lower bound was later improved by the present author [5], so that the
precise value of $\tau_{WC}(\mathcal{F}_{n,K_{3}-\text{fac}},1)$ is now known
up to the $o(n)$ additive term.

######  Theorem 1.1.

For $n$ divisible by $3$ and large enough that Waiter wins the triangle-factor
game, we have

$(\mathcal{F}_{n,K_{3}-\text{fac}},1)=\frac{7}{6}n+o(n).$

Clemens et al. pose a question what happens in the more general case of the
$k$-clique-factor game for large values of $k$. Here, we insist that $n$, the
number of vertices of our graph, is divisible by $k$, and the winning sets are
$K_{k}$-factors, i.e. decompositions of the vertices of $K_{n}$ into disjoint
sets of size $k$, each of which forms a clique $K_{k}$. Once again, it is easy
to see that for $n$ large enough in terms of $k$, Waiter has a winning
strategy.

Addressing this question of Clemens et al., our main aim in this paper is to
give the first non-trivial lower bound on
$\tau_{WC}(\mathcal{F}_{n,K_{k}-\text{fac}},1)$. Combined with a simple upper
bound (which we expect to be of the correct magnitude) following the strategy
of Clemens et al. for the triangle-factor game, this gives the following
result.

######  Theorem 1.2.

There exist functions $n_{0}(k),C(k)$ such that one has

$2^{k/3-o(k)}n\leq\tau_{WC}(\mathcal{F}_{n,K_{k}-\text{fac}},1)\leq
2^{k}\frac{n}{k}+C(k)$

for $n\geq n_{0}(k)$ and divisible by $k$.

Most of the paper deals with proving the more difficult lower bound. The
strategy of Client that we use is a trivial random one - Client always picks
either element with equal probability and independently of the other rounds.
Once we show that with a positive probability Client can survive a certain
number of rounds (no matter what strategy Waiter uses), that guarantees a
deterministic strategy of Client that ensures he can always survive that many
rounds.

The difficult part of our approach is finding a quantity which we can bound in
the expected value when Client plays randomly, no matter what strategy Waiter
uses, and suitably relate to a $K_{k}$-factor. Such a quantity cannot be
simple like the number of vertices that are part of some $k$-clique in
Client’s graph. Indeed, Waiter can initially create a clique $A$ with $2k-2$
vertices in Client’s graph, and then one by one make each vertex $v$ in the
graph part of a $k$-clique consisting of $v$ and $k-1$ elements of $A$, while
using only about $kn$ rounds in total. So one needs a lot more delicate
definition of events whose probabilities we bound.

The structure of the paper is as follows. To motivate our main proof, in
section 2 we present a simpler and easier to grasp lower bound using similar
ideas. Building on this, we prove the lower bound in Theorem 1.2 in section 3.
In section 4, we use a strategy of Clemens et al. to obtain the upper bound in
Theorem 1.2. And in section 5, we give some final remarks.

##  2\. A simple exponential bound

In this section, we will motivate the proof of our main result by providing a
simple proof along similar lines of a somewhat weaker lower bound (which still
has a term exponential in $k$ in front of $n$).

######  Theorem 2.1.

Fix $k\geq 100$ and consider $n$ divisible by $k$ and large enough that Waiter
has a winning strategy for the corresponding Waiter-Client $k$-clique-factor
game. Then $\tau_{WC}(\mathcal{F}_{n,K_{k}-\text{fac}},1)\geq
2^{k/6-\frac{k}{\log k}}n$.

The strategy of Client will be random and extremely simple - Client always
takes the two edges offered and chooses the one to add to his graph randomly,
with equal probabilities for each. If we can show that with probability at
least $1/2$, no matter what strategy Waiter uses, she does not win against
randomly playing Client after $2^{k/6-\frac{k}{\log k}}\,n$ rounds, then that
also implies Client has a deterministic strategy guaranteeing him to make the
game last at least that long. Note that the choice of the probability $1/2$
for our purpose is arbitrary - we could replace it with any constant $p$ such
that $0<p<1$.

We shall colour the edges of Client’s graph red and the edges of Waiter’s
graph blue. For a given vertex $v$, we shall also denote by $N_{R}(v)$ its red
neighbourhood and by $N_{B}(v)$ its blue neighbourhood.

The proof is based on an idea that after $2^{k/6-\frac{k}{\log k}}n$ rounds,
if there was a red $k$-clique factor already created, many of the cliques $C$
in this $k$-clique factor will contain some vertex $z_{i}$ such that, if we
denote the vertices of $C$ by $z_{1},...,z_{k}$:

  * •

$z_{i}$ has red degree less than $2^{k/6-\frac{k}{2\log k}}$;

  * •

for many pairs $z_{j},z_{l}$, both the red edges $z_{i}z_{j}$ and $z_{i}z_{l}$
were added before the red edge $z_{j}z_{l}$ (here many means
$(\frac{1}{6}-o(1))k^{2}$).

We then bound the probability of this event happening at any vertex $z_{i}$
and conclude by taking expectation that on average, there will not be many
such vertices, which gives the desired contradiction. In the next few pages we
shall formalize this idea and make it work.

Assume that $k\geq 100$, that $n$ is divisible by $k$, that $n$ is large
enough that Waiter has a winning strategy for the corresponding Waiter-Client
$K_{k}$-factor game and that $2^{k/6-\frac{k}{\log k}}\,n$ rounds have passed.
If Waiter already won at this point, there are disjoint red $k$-cliques
$C_{1},...,C_{n/k}$ partitioning the vertices of the graph.

Call a vertex a high degree vertex if it has red degree at least
$2^{k/6-\frac{k}{2\log k}}$, and call it a low degree vertex otherwise. Call a
clique a high degree clique if it contains at least one high degree vertex,
and call it a low degree clique otherwise.

######  Claim 2.2.

Among $C_{1},...,C_{n/k}$, there are at most $\frac{n}{2k}$ high degree
cliques.

###### Proof.

Assume for a contradiction that this is false, so we have at least
$\frac{n}{2k}$ high degree vertices. As every red edge touches at most two
high degree vertices (i.e. its two endpoints), that implies we have at least
$2^{k/6-\frac{k}{2\log k}}\,\frac{n}{4k}$ red edges in our graph. Since only
$2^{k/6-\frac{k}{\log k}}n$ rounds have passed, we have

$2^{k/6-\frac{k}{\log k}}n<2^{k/6-\frac{k}{2\log k}}\,\frac{n}{4k},$

for $k\geq 100$, which is a contradiction. ∎

Next, we need to define several events. For a given vertex $v$, denote by
$X(v)$ the event that $v$ is not a high degree vertex after
$2^{k/6-\frac{k}{\log k}}\,n$ rounds have passed. Denote by $Y(v)$ the event
that:

  * •

there exists a subset $w_{1},...,w_{k-1}$ in $N_{R}(v)$ after
$2^{k/6-\frac{k}{\log k}}\,n$ rounds have passed that forms a red clique;

  * •

moreover, at least $\frac{(k-1)(k-2)}{6}$ edges $w_{i}w_{j}$ within this
subset were added after both edges $vw_{i}$ and $vw_{j}$ have been added.

Finally let $S(v)=X(v)\cap Y(v)$.

The following simple lemma explains why we care about the event $Y(v)$ \- it
is because having a red $K_{k}$-factor, we can guarantee this event to happen
for many vertices.

######  Lemma 2.3.

Start with an empty graph on $k$ vertices and keep adding edges in some order
until our graph is complete. Then there exists a vertex $v$ such that at least
$\frac{(k-1)(k-2)}{6}$ edges $w_{i}w_{j}$ were added after both the edges
$vw_{i}$ and $vw_{j}$ were added.

It is not hard to see that up to the smaller order terms, this result is best
possible - indeed, just adding the edges in random order, it will typically
happen that for each vertex $v$, about $\frac{k^{2}}{6}$ edges $w_{i}w_{j}$
will be added after both the edges $vw_{i}$ and $vw_{j}$ were added.

###### Proof of Lemma 2.3.

Call a pair $(v,w_{i}w_{j})$ consisting of a vertex $v$ and an edge
$w_{i}w_{j}$ good if $w_{i}w_{j}$ is the last edge added to triangle
$vw_{i}w_{j}$. Clearly, there are ${k\choose 3}$ good pairs, so some vertex
$v$ appears in at least $\frac{1}{k}{k\choose 3}=\frac{(k-1)(k-2)}{6}$ good
pairs.

∎

Combining Claim 2.2 and Lemma 2.3, we can conclude that if Waiter won fast,
there in fact must be many vertices $v$ for which the event $S(v)$ occurred.
This is crucial for us, as the probability of the event $S(v)$ happening is
something we can bound efficiently.

######  Claim 2.4.

If Waiter already won after $2^{k/6-\frac{k}{\log k}}\,n$ rounds have passed,
there must exist at least $\frac{n}{2k}$ vertices $v$ such that the event
$S(v)$ has occurred.

###### Proof.

Since by Claim 2.2, there can be at most $\frac{n}{2k}$ high degree cliques
among $C_{1},...,C_{n/k}$, there must be at least $\frac{n}{2k}$ low degree
cliques among them. But by Lemma 2.3 applied to an arbitrary low degree clique
$C_{j}$, there exists a vertex $v_{j}$ of $C_{j}$ for which the event
$Y(v_{j})$ has occurred. As $C_{j}$ is a low degree clique, that means the
event $S(v_{j})$ has occurred as well. Finally, as the cliques
$C_{1},...,C_{n/k}$ are all disjoint, we conclude there must be at least
$\frac{n}{2k}$ such vertices. ∎

So now, we know we can finish the proof of Theorem 2.1 if we can show the
following.

######  Lemma 2.5.

No matter what strategy Waiter uses, after $2^{k/6-\frac{k}{\log k}}\,n$
rounds have passed, we have

$\mathbb{E}\Big{(}\sum_{v}1_{S(v)}\Big{)}\leq\frac{n}{4k}.$

In particular, note that Lemma 2.5 establishes that Waiter can win against
randomly playing Client within $2^{k/6-\frac{k}{\log k}}\,n$ rounds with
probability at most $1/2$.

To start the proof of Lemma 2.5, we define

$Y=\\{\mathbf{y}=(y_{1},...,y_{k-1}):y_{i}\in\mathbb{N},1\leq
y_{1}<...<y_{k-1}\leq 2^{k/6-\frac{k}{2\log k}}\\}.$

Next, for $\mathbf{y}\in Y$ and a vertex $v$, we define the event
$T(v,\mathbf{y})$ as follows. Label the vertices in $N_{R}(v)$ at the time
when $2^{k/6-\frac{k}{\log k}}n$ rounds have passed as $w_{1},...,w_{l}$,
where the vertices $w_{i}$ are ordered by the time when the edge $vw_{i}$
appeared. Then $T(v,\mathbf{y})$ is the event that:

  * •

$v,w_{y_{1}},...,w_{y_{k-1}}$ is a red clique;

  * •

moreover, at least $\frac{(k-1)(k-2)}{6}$ edges $w_{y_{i}}w_{y_{j}}$ were
added after both the edges $vw_{y_{i}}$ and $vw_{y_{j}}$ were added.

We divide the rest of the proof into three claims.

######  Claim 2.6.

For any $\textbf{y}\in Y$ and any $v$, we have
$\mathbb{P}(T(v,\mathbf{y}))\leq 2^{-\frac{k^{2}}{6}+k}$, regardless of the
strategy of Waiter.

###### Proof.

To complete a red clique $v,w_{y_{1}},...,w_{y_{k-1}}$ in such a way that the
event $T(v,\mathbf{y})$ would occur, Waiter has to offer Client at least
$\frac{(k-1)(k-2)}{6}$ edges $w_{y_{i}}w_{y_{j}}$ at the time when both the
edges $vw_{y_{i}}$ and $vw_{y_{j}}$ were already added. If at any time Waiter
offers Client two such edges at the same time, one of the edges receives a
blue colour and then the probability of the clique
$v,w_{y_{1}},...,w_{y_{k-1}}$ to end up all red is zero. Hence, Waiter can
only offer Client one such edge at time, always succeeding with probability
$1/2$ and independently of the other rounds. This gives the bound

$\mathbb{P}(T(v,\mathbf{y}))\leq
2^{-\frac{(k-1)(k-2)}{6}}<2^{-\frac{k^{2}}{6}+k},$

as required.

Note that it is not an issue to us that throughout, we may not yet know what
vertices will the latter ones, like $w_{y_{k-1}}$, be, or how many edges
$w_{y_{i}}w_{y_{j}}$ precisely will be added after both the edges $vw_{y_{i}}$
and $vw_{y_{j}}$ were added. The definition of our event simply guarantees
there must be at least $\frac{(k-1)(k-2)}{6}$ such edges, and at any time that
any of these is offered, we know that if it is coloured blue, then the event
$T(v,\mathbf{y})$ cannot occur anymore. ∎

######  Claim 2.7.

For any $v$, we have

$S(v)\subset\bigcup_{\mathbf{y}\in Y}T(v,\mathbf{y}).$

###### Proof.

For $S(v)$ to occur, $v$ needs to be a low degree vertex. Hence, whatever
subset of $k-1$ vertices in $N_{R}(v)$ sees the event $Y(v)$ happen must
consist of the first $2^{k/6-\frac{k}{2\log k}}$ vertices connected in red to
$v$, and hence also sees the corresponding event $T(v,\mathbf{y})$ happen for
some $\mathbf{y}\in Y$. ∎

Combining Claim 2.6, Claim 2.7 and the union bound, we obtain the final result
that we need.

######  Claim 2.8.

We have $\mathbb{P}(S(v))<\frac{1}{4k}$ for any $v$.

###### Proof.

Using the results above, we have

$\displaystyle\mathbb{P}(S(v))\leq\mathbb{P}\big{(}\bigcup_{\mathbf{y}\in
Y}T(v,\mathbf{y})\big{)}\leq\sum_{\mathbf{y}\in
Y}\mathbb{P}(T(v,\mathbf{y}))\leq|Y|\,2^{-\frac{k^{2}}{6}+k}$
$\displaystyle={2^{k/6-\frac{k}{2\log k}}\choose
k-1}2^{-\frac{k^{2}}{6}+k}<2^{(k/6-\frac{k}{2\log
k})k}2^{-\frac{k^{2}}{6}+k}=2^{k-\frac{k^{2}}{2\log k}}<\frac{1}{4k},$

provided $k\geq 100$. ∎

But now, Lemma 2.5 follows immediately from Claim 2.8, and hence Theorem 2.1
follows as well.

##  3\. An improved lower bound

In this section, we prove the lower bound in Theorem 1.2, with the $o(k)$ term
being $\frac{k}{\log k}$ provided $k\geq 10^{8}$.

The strategy we will use will be like in the previous section, but this time
with a more general definition of the events whose probabilities we are
bounding. Roughly, we will use that if after $2^{k/3-\frac{k}{\log k}}n$
rounds, there was a red $k$-clique factor already created, many of the cliques
$C$ in this $k$-clique factor will contain a vertex $z_{i}$ such that, if we
denote the vertices of $C$ by $z_{1},...,z_{k}$:

  * •

$z_{i}$ has red degree less than $2^{k/3-\frac{k}{2\log k}}$;

  * •

for many pairs $z_{j},z_{l}$, both the vertices $z_{j}$ and $z_{l}$ were in
the red connected component of $z_{i}$ in the graph spanned by
$z_{1},...,z_{k}$ before the red edge $z_{j}z_{l}$ was added (here many means
$(\frac{1}{3}-o(1))k^{2}$).

This improves the bound significantly, though the details get more technical.

As we only change several ingredients in the proof from the previous section
while keeping the rest very similar, we omit the proofs of some claims and
instead reference the proofs of the analogous claims in section 2. We also use
the same notation for the events playing the same role, in the hope that this
will make the connections to the motivating simpler proof in section 2
clearer.

Once again, the strategy of Client will be random - Client always takes the
two edges offered and chooses the one to be added to his graph randomly, with
equal probabilities for each and independently of the other rounds. The aim
will also be as previously - to show that no matter what strategy Waiter uses,
after $2^{k/3-\frac{k}{\log k}}\,n$ rounds have passed, she could have won
with probability at most $1/2$ against randomly playing Client. This
guarantees a desired deterministic strategy of Client to survive
$2^{k/3-\frac{k}{\log k}}\,n$ rounds.

As before, we shall colour the edges of Client’s graph red and the edges of
Waiter’s graph blue, and we shall denote by $N_{R}(v)$ and $N_{B}(v)$ the red
and the blue neighbourhoods of a vertex $v$.

Assume that $k\geq 10^{8}$, that $n$ is divisible by $k$ and large enough that
Waiter has a winning strategy for the corresponding Waiter-Client
$K_{k}$-factor game, and that $2^{k/3-\frac{k}{\log k}}\,n$ rounds have
passed. If Waiter already won at this point, there are disjoint red cliques
$C_{1},...,C_{n/k}$ partitioning the vertices of the graph.

Once again, call a vertex a high degree vertex if it has red degree at least
$2^{k/3-\frac{k}{2\log k}}$, and a low degree vertex otherwise. And call a
clique a high degree clique if it contains at least one high degree vertex,
and a low degree clique otherwise.

######  Claim 3.1.

Among $C_{1},...,C_{n/k}$, there are at most $\frac{n}{2k}$ high degree
cliques.

###### Proof.

The proof is analogous to the proof of Claim 2.2. ∎

We once again define several events. For a given vertex $v$, denote by $X(v)$
the event that $v$ is a low degree vertex after $2^{k/3-\frac{k}{\log k}}\,n$
rounds have passed. Denote by $Y(v)$ the event that:

  * •

there exists a subset $w_{1},...,w_{k-1}$ in $N_{R}(v)$ after
$2^{k/3-\frac{k}{\log k}}\,n$ rounds have passed that forms a red clique;

  * •

moreover, none of the vertices $w_{1},...,w_{k-1}$ is a high degree vertex;

  * •

finally, at least $\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ edges
$w_{i}w_{j}$ were added after both $w_{i}$ and $w_{j}$ were already in the red
connected component of $v$ in the graph spanned by $v,w_{1},...,w_{k-1}$.

Let $S(v)=X(v)\cap Y(v)$.

The lemma that follows corresponds to Lemma 2.3 in the previous section - but
as the result is stronger, we need more care proving it.

######  Lemma 3.2.

Start with an empty graph on $k$ vertices and keep adding edges in some order
until our graph is complete. Then there exists a vertex $v$ such that at least
$\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ edges $w_{i}w_{j}$ were added
after both $w_{i}$ and $w_{j}$ were already in the connected component of $v$.

Let us note that up to the smaller order terms, Lemma 3.2 is tight. Indeed,
consider an initially empty graph on $k=2^{t}$ vertices for some large $t$.
First, create a matching in this graph. After this step we have $2^{t-1}$
connected components $A_{1},...,A_{2^{t-1}}$, each consisting of two vertices.
Next, fill in all the edges between $A_{2i-1}$ and $A_{2i}$ for
$i=1,...,2^{t-2}$, leading to $2^{t-2}$ connected components of four vertices
each, each forming a clique. Continue in this manner until the end, always
halving the number of the connected components and doubling the number of
vertices in each component in each step, and making sure that each connected
component is a clique at the end of each step. It is not hard to see that for
each vertex $v$, $(1+o(1))\frac{k^{2}}{3}$ edges $w_{i}w_{j}$ were added after
both $w_{i}$ and $w_{j}$ were already in the connected component of $v$.

###### Proof of Lemma 3.2.

We will count all the pairs $(v,w_{i}w_{j})$ consisting of the vertex $v$ and
the edge $w_{i}w_{j}$ with the property that the edge $w_{i}w_{j}$ was added
after both $w_{i}$ and $w_{j}$ were already in the connected component of $v$.
If we show there are at least $\frac{k^{3}}{3}-\frac{k^{3}}{(\log k)^{2}}$
such pairs, we are done, as that means some vertex $v_{0}$ is counted in at
least $\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ such pairs.

Call an edge rare if it is one of the first $4(\log k)^{2}$ edges added at
some vertex $z$. Clearly, there are at most $4k(\log k)^{2}$ rare edges.

Further, call an edge connective at $z$ if it has one endpoint at a vertex $z$
and connects a vertex $z$ to a connected component of at least $4(\log k)^{2}$
vertices that $z$ was previously not connected to. Call an edge connective if
there exists some vertex $z_{0}$ such that this edge is connective at $z_{0}$.
Clearly, for any vertex $z$, there can be at most $\frac{k}{4(\log k)^{2}}$
edges connective at $z$ used throughout, since no other vertex $z^{\prime}$
can be in two different connected components that $z$ is connected to by an
edge that is connective at $z$. Hence overall we have at most
$\frac{k^{2}}{4(\log k)^{2}}$ connective edges over all the vertices.

Consider all the unordered triples of distinct vertices $(v_{1},v_{2},v_{3})$
such that none of the edges $v_{1}v_{2}$, $v_{1}v_{3}$ and $v_{2}v_{3}$ is
rare or connective. Then we claim that if the edge $v_{a}v_{b}$,
$a,b\in\\{1,2,3\\}$, was added first out of the edges $v_{1}v_{2}$,
$v_{1}v_{3}$ and $v_{2}v_{3}$ and $c\in\\{1,2,3\\}$ is such that $c\neq a,b$,
then the pairs $(v_{a},v_{b}v_{c})$ and $(v_{b},v_{a}v_{c})$ were counted.
Once we show that, we are done, as summing over all the triples and using that
we have less than

$4k(\log k)^{2}+\frac{k^{2}}{4(\log k)^{2}}<\frac{k^{2}}{3.5(\log k)^{2}}$

rare or connective edges, this gives at least

$2\Big{(}{k\choose 3}-k\frac{k^{2}}{3.5(\log
k)^{2}}\Big{)}>\frac{k^{3}}{3}-\frac{k^{3}}{(\log k)^{2}}$

such pairs, provided $k\geq 10^{8}$.

So we are left to show that if the edge $v_{1}v_{2}$ was added before the
edges $v_{1}v_{3}$ and $v_{2}v_{3}$ were added, and none of the edges
$v_{1}v_{2}$, $v_{1}v_{3}$ and $v_{2}v_{3}$ is rare or connective, then the
pair $(v_{1},v_{2}v_{3})$ was counted (we can without loss of generality
reduce just to this case, as for the rest we can just relabel as needed). If
the pair $(v_{1},v_{2}v_{3})$ was not counted, that would mean either $v_{1}$
was not in the same component as $v_{2}$ or $v_{1}$ was not in the same
component as $v_{3}$ at the time when the edge $v_{2}v_{3}$ was added. As the
edge $v_{1}v_{2}$ was added even before, it is clearly impossible that $v_{1}$
was not in the same component as $v_{2}$. But if $v_{2}$ and $v_{3}$ were not
in the same component at the time when the edge $v_{2}v_{3}$ was added, then
either this edge would have been one of the first $4(\log k)^{2}$ edges added
at $v_{3}$, and hence rare, or it would have connected $v_{2}$ to a new
connected component of at least $4(\log k)^{2}$ vertices (as in particular the
connected component of $v_{3}$ contains the entire neighbourhood of $v_{3}$),
and hence it would have been connective at $v_{2}$. As neither is true by
assumption, we know this also could not have happened, $v_{2}$ and $v_{3}$
(and hence also $v_{1}$ and $v_{3}$) were in the same connected component at
the time when the edge $v_{2}v_{3}$ was added, and we are done. ∎

Having now proven this result, we continue as in the previous section.

######  Claim 3.3.

If Waiter already won after $2^{k/3-\frac{k}{\log k}}n$ rounds have passed,
there must exist at least $\frac{n}{2k}$ vertices $v$ such that the event
$S(v)$ has occurred.

###### Proof.

The proof is analogous to the proof of Claim 2.4. ∎

So now, once again, we know we can finish the proof if we can show the
following lemma.

######  Lemma 3.4.

No matter what strategy Waiter uses, after $2^{k/3-\frac{k}{\log k}}\,n$
rounds have passed, we have

$\mathbb{E}\Big{(}\sum_{v}1_{S(v)}\Big{)}\leq\frac{n}{4k}.$

This time, we have to define a more involved set of labels which will let us
take $S(v)$ as a subset of the union of events whose probabilities we can
easily bound. Let

$\displaystyle
Z=\\{\mathbf{y}=(y_{1},z_{1},y_{2},z_{2},...,z_{k-2},y_{k-1}):y_{i},z_{i}\in\mathbb{N},$
$\displaystyle 1\leq y_{1},...,y_{k-1}\leq 2^{k/3-\frac{k}{2\log k}},1\leq
z_{i}\leq i+1\\}.$

Enumerate the vertices of our graph as $v_{1},...,v_{n}$. Moreover, for each
$v_{i}$, enumerate its red neighbours after $2^{k/3-\frac{k}{\log k}}\,n$
rounds have passed as $w_{i,1},w_{i,2},...,w_{i,t_{i}}$, where the labels
correspond to the order in which these red edges were added.

Next, for $\mathbf{y}\in Z$, let $T(v_{s},\mathbf{y})$ be the following event:

  * •

we have a red clique $x_{1},...,x_{k}$, consisting of low degree vertices
only, where $x_{1}=v_{s}$, $x_{2}=w_{s,y_{1}}$ and we always obtain the next
vertex as follows: if we have $x_{1},...,x_{i+1}$ already chosen, $z_{i}=m$
(for some $1\leq m\leq i+1$) and $x_{m}=v_{d}$, then $x_{i+2}=w_{d,y_{i+1}}$;

  * •

moreover, at least $\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ edges
$x_{i}x_{j}$ were added after both $x_{i}$ and $x_{j}$ were already in the red
connected component of $v_{s}$ in the graph spanned by $x_{1},...,x_{k}$;

  * •

further, if $z_{i}=m$, then $x_{m}$ is the first vertex out of
$x_{1},...,x_{i+1}$ that got connected to $x_{i+2}$;

  * •

finally, there is no pair $x_{i_{1}},x_{i_{2}}$ with $i_{1}<i_{2}$ such that
$x_{i_{2}}$ appeared in the red connected component of $v_{s}$ in the graph
spanned by $x_{1},...,x_{k}$ strictly sooner than $x_{i_{1}}$ (of course, both
could have appeared at the same time, say when a new connected component
consisting of several vertices gets added to the connected component of
$v_{s}$ in this graph).

As in the previous section, we divide the rest of the proof into three claims.

######  Claim 3.5.

For any $\textbf{y}\in Z$, we have $\mathbb{P}(T(v,\mathbf{y}))\leq
2^{-\frac{k^{2}}{3}+\frac{k^{2}}{(\log k)^{2}}}$, regardless of the strategy
of Waiter.

###### Proof.

The proof is very similar to the proof of Claim 2.6 and hence is omitted. The
important thing that makes our argument still work is the fact that once
again, we can identify the vertex $x_{i}$ as soon as it appears in the
connected component of $v$ in the graph spanned by the final clique
$x_{1},...,x_{k}$ that $v$ is part of. So any edges inside the connected
component of $v$ within this clique that are offered and are counted among the
at least $\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ edges crucial for our
argument are known to have this role before they are offered. ∎

Let us note that Claim 3.5 only establishes an upper bound. For many
$\mathbf{y}\in Z$, the event $T(v,\mathbf{y})$ cannot happen at all. Say if
for $i_{1}<i_{2}$, we have $z_{i_{1}}=z_{i_{2}}$ but $y_{i_{1}}>y_{i_{2}}$,
then the corresponding event cannot happen, as it is easy to check that it is
then impossible to fulfill all the encoding rules.

The following claim is once again very similar to Claim 2.7, though this time
bit less immediate.

######  Claim 3.6.

For any $v$, we have

$S(v)\subset\bigcup_{\mathbf{y}\in Z}T(v,\mathbf{y}).$

###### Proof.

It is easy to see that if a red clique $v,w_{1},...,w_{k-1}$ (consisting of
low degree vertices only) was created, which moreover has the property that at
least $\frac{k^{2}}{3}-\frac{k^{2}}{(\log k)^{2}}$ edges $w_{i}w_{j}$ were
added after both $w_{i}$ and $w_{j}$ were already in the connected component
of $v$ in the graph spanned by $v,w_{1},...,w_{k-1}$, then we can encode this
into a suitable event $T(v,\mathbf{y})$. Indeed, just start with $v$ and keep
adding the vertices in order in which they appear in its red connected
component in the graph spanned by $v,w_{1},...,w_{k-1}$, following the rule
during our encoding that if $x_{m}$ is the first vertex out of
$x_{1},...,x_{i+1}$ that got connected to $x_{i+2}$, then we set $z_{i}=m$ and
choose $y_{i+1}$ accordingly. When the entire new connected component is added
in a single round to the connected component of $v$, add its vertices in such
an order that at any point, the graph spanned by $x_{1},...,x_{i+1}$ at that
point in time is connected. As all the vertices $v,w_{1},...,w_{k-1}$ are low
degree ones (and hence the labels $y_{i}$ will be in the required range), it
is easy to check that such an encoding indeed works. ∎

Combining Claim 3.5, Claim 3.6 and the union bound, we once again obtain the
final result.

######  Claim 3.7.

We have $\mathbb{P}(S(v))<\frac{1}{4k}$ for any $v$.

###### Proof.

Using the results above, we have

$\displaystyle\mathbb{P}(S(v))\leq\mathbb{P}\big{(}\bigcup_{\mathbf{y}\in
Z}T(v,\mathbf{y})\big{)}\leq\sum_{\mathbf{y}\in Z}\mathbb{P}(T(v,\mathbf{y}))$
$\displaystyle\leq|Z|\,2^{-\frac{k^{2}}{3}+\frac{k^{2}}{(\log k)^{2}}}\leq
k^{k}2^{(k/3-\frac{k}{2\log k})k}2^{-\frac{k^{2}}{3}+\frac{k^{2}}{(\log
k)^{2}}}<\frac{1}{4k},$

provided $k\geq 10^{8}$. ∎

But now, Lemma 3.4 follows immediately from Claim 3.7, and hence we are also
done proving the lower bound in Theorem 1.2.

##  4\. Upper bound

In this section, we prove the upper bound in Theorem 1.2. Roughly, we follow
the approach of Clemens et al. [4], who used it for the case $k=3$, i.e. the
triangle-factor game, but we spell out the details for the convenience of the
reader.

Assume $k\geq 4$ is fixed, $n$ is divisible by $k$ and is large enough (for
instance $n\geq 4^{8^{k}}$ will do). Once again, colour the edges of Client’s
graph red and the edges of Waiter’s graph blue.

The crucial ingredient of our proof is the following algorithm that Waiter
has.

######  Lemma 4.1.

Given $l\geq 2$ and $2^{l}-1$ vertices $v_{1},v_{2},...,v_{2^{l}-1}$ of an
initially empty board, Waiter can in at most $2^{l}$ rounds of playing on this
board create a red clique $w_{1},...,w_{l}$ with $w_{1}=v_{1}$, and moreover
keep the property that every edge placed so far (whether red or blue) has at
least one endpoint of the form $w_{i}$ for some $1\leq i\leq l$.

###### Proof.

To do that, Waiter uses the following approach. Waiter sets

$w_{1}=v_{1},S_{1}=\\{v_{1},v_{2},v_{3},...,v_{2^{l}-1}\\}.$

After, Waiter offers Client one by one pairs of edges
$(v_{1}v_{2i},v_{1}v_{2i+1})$ for $i=1,...,2^{l-1}-1$. This now gives a subset
of $2^{l-1}-1$ vertices connected to $v_{1}$ in red, and we label this subset
as $S_{2}$ and set $w_{2}$ to be a vertex $v_{j}$ of the smallest index $j$ in
$S_{2}$ (so in this case, this smallest index $j$ will be $2$ or $3$).

We continue iteratively in this manner. Given $S_{j},w_{j}$ (where $j<l$ and
$|S_{j}|=2^{l+1-j}-1$), Waiter offers Client the edges from $w_{j}$ to
$S_{j}\setminus\\{w_{j}\\}$ in pairs, sets $S_{j+1}$ to be the set of the
vertices of $S_{j}\setminus\\{w_{j}\\}$ connected to $w_{j}$ in red and sets
$w_{j+1}$ to be the vertex $v_{m}$ of the smallest index $m$ in $S_{j+1}$.
Note that in particular this ensures that $|S_{j+1}|=2^{l+1-(j+1)}-1$.

Waiter can clearly continue like this until we obtain $S_{l},w_{l}$ and then
$w_{1},...,w_{l}$ is our desired clique.

The number of rounds that had passed before our clique has been created is

$(2^{l-1}-1)+(2^{l-2}-1)+...+(2-1)<2^{l}.$

The property that every edge placed so far has at least one endpoint of the
form $w_{i}$ for some $1\leq i\leq l$ trivially holds as well. ∎

Now we are ready to describe the strategy that Waiter will use to guarantee
winning within the desired number of rounds.

Waiter will proceed in three stages. Throughout, Waiter will update a set $F$
of vertices which is initially empty and has the property at any point in time
that the vertices of $F$ contain a red $K_{k}$-factor.

Stage I. In the first stage, Waiter creates a red clique $R$ on $8^{k}$
vertices in the first $2^{8^{k}}$ rounds and ensures that every edge placed so
far has at least one endpoint in $R$ and the set of vertices $S_{0}$ defined
as

$S_{0}=\\{v:\,v\text{ is an endpoint of at least one red or blue edge}\\}$

satisfies $|S_{0}|<2^{8^{k}}$.

To do that, Waiter simply picks an arbitrary set $S_{0}$ of $2^{8^{k}}-1$
vertices and uses an algorithm from Lemma 4.1.

Denote by $A$ the vertices of our graph not in $S_{0}$ and set
$B=S_{0}\setminus R$ (note that unlike $F$, the sets $A,B,R$ will not be
updated further). After this stage, Waiter still keeps $F=\emptyset$.

Stage II. In this stage, Waiter keeps picking $2^{k}-1$ vertices at time (with
the first vertex $v_{1}$ always being from $B$ and all the other ones from $A$
until all the vertices of $B$ are used, and after using just the vertices from
$A$) and creating a new $K_{k}$ using the algorithm from Lemma 4.1, insisting
as mentioned that the one vertex we have from $B$ is always in the resulting
clique (until we run out of the vertices in $B$, which will happen before this
stage ends, as we insisted that $n\geq 4^{8^{k}}$). Whenever Waiter creates
such a clique, she puts its $k$ vertices into $F$. Waiter does this until
there are less than $2^{k}$ vertices in $A\setminus F$ left.

Stage III. Now we have some vertices $z_{1},...,z_{t}$ for $0\leq t<2^{k}$ in
$A\setminus F$ left. One by one, Waiter offers for each $z_{i}$ pairs of the
edges between $z_{i}$ and the vertices in what is left in $R\setminus F$ to
Client until she creates a clique with one vertex $z_{i}$ and $k-1$ vertices
in $R\setminus F$. Then Waiter takes the $k$ vertices of the resulting clique
and puts them into $F$. Due to our constraints, we can see Waiter has enough
time to do this for all the vertices $z_{1},...,z_{t}$, and what is left of
the graph (i.e. not in $F$) after this process is a subset of $R$, which hence
also decomposes into $k$-cliques and can be just put into $F$ immediately.
Thus, we have created a red $K_{k}$-factor.

In total, we see that Waiter has needed at most
$2^{8^{k}}+2^{k}\,\frac{n}{k}+2^{k}k$ rounds, proving the upper bound in
Theorem 1.2 with $C(k)=2^{8^{k}}+2^{k}k$.

##  5\. Conclusion

The upper bound and the lower bound in Theorem 1.2 are still very far apart.
We suspect that the upper bound is roughly of the correct magnitude, as it is
hard to imagine Waiter could come up with a better strategy than some variant
of the natural one from section 4.

As a first step towards proving the tight result for the lower bound, we
believe one could perhaps try to improve the lower bound in Theorem 1.2 to
$2^{k/2-o(k)}n$. Indeed, the logic behind our belief that this should be
easier than going past this barrier is as follows. In section 2, we obtained
the bound of the form $2^{k/6-o(k)}n=2^{\frac{1}{3}(k/2)-o(k)}n$ as we managed
to include approximately one third of the edges in the low degree cliques into
our argument. We could then improve this to
$2^{k/3-o(k)}n=2^{\frac{2}{3}(k/2)-o(k)}n$ in section 3 since using a more
involved argument, we could make use of about two thirds of the edges in the
low degree cliques (which, as mentioned in the relevant section, is tight for
the particular argument that we use). So if one managed to come up with a
different argument using almost all the edges in the low degree cliques (which
does not seem inconceivable), then there would be a hope for the bound of the
form $2^{k/2-o(k)}n$.

We expect getting past this barrier to be even more difficult and to require a
different approach. Nonetheless, we presume it still should be sufficient to
use a random play as the strategy of Client in this higher range.

## Acknowledgements

The author would like to thank his PhD supervisor Professor Béla Bollobás for
his helpful comments.

## References

  * [1] J. Beck.  Positional games and the second moment method.  Combinatorica, 22(2):169–216, 2002.
  * [2] J. Beck.  Combinatorial Games: Tic-Tac-Toe Theory.  Cambridge University Press, 2008.
  * [3] M. Bednarska-Bzdega, D. Hefetz, M. Krivelevich, and T. Łuczak.  Manipulative Waiters with probabilistic intuition.  Combinatorics, Probability and Computing, 25(6):823–849, 2016.
  * [4] D. Clemens, P. Gupta, F. Hamann, A. Haupt, M. Mikalački, and Y. Mogge.  Fast strategies in Waiter-Client games on ${K}_{n}$.  The Electronic Journal of Combinatorics, 27(3):1–35, 2020.
  * [5] V. Dvořák.  Waiter–Client triangle-factor game on the edges of the complete graph.  European Journal of Combinatorics, 96:103356, 2021.
  * [6] D. Hefetz, M. Krivelevich, M. Stojaković, and T. Szabó.  Positional games.  Springer, 2014.
  * [7] D. Hefetz, M. Krivelevich, and W. E. Tan.  Waiter–Client and Client–Waiter planarity, colorability and minor games.  Discrete Mathematics, 339(5):1525–1536, 2016.
  * [8] D. Hefetz, M. Krivelevich, and W. E. Tan.  Waiter–Client and Client–Waiter Hamiltonicity games on random graphs.  European Journal of Combinatorics, 63:26–43, 2017.
  * [9] M. Krivelevich and N. Trumer.  Waiter-Client maximum degree game.  arXiv preprint arXiv:1807.11109, 2018.
  * [10] W. E. Tan.  Waiter–Client and Client–Waiter colourability and $k$–SAT games.  The Electronic Journal of Combinatorics, pages P2–46, 2017.