# Sylvester-Cayley vector partitions algorithm and the Gaussian polynomials

Boris Y. Rubinstein
Stowers Institute for Medical Research
1000 50th St., Kansas City, MO 64110, U.S.A

###### Abstract

We extend an algorithm suggested in 1858 by Sylvester [12] and implemented in
1860 by Cayley [4] for a problem of double partitions and apply it to
derivation of explicit expressions for coefficients of the Gaussian
polynomials through convolution of restricted partition functions.

Keywords: double partitions, Gaussian polynomials, Bernoulli polynomials of
higher order, Eulerian numbers.

2010 Mathematics Subject Classification: 11P82.

##  1 Partition functions

###  1.1 Scalar partition function

Consider a problem of finding a number of nonnegative integer solutions of a
Diophantine equation

$\sum_{i=1}^{m}x_{i}d_{i}={\bf x}\cdot{\bf d}=s,\quad{\bf
d}=\\{d_{1},d_{2},\ldots,d_{m}\\}.$ (1)

A restricted partition function $W(s,{\bf d})$ solving the above problem is a
number of partitions of an integer $s$ into positive integers
$\\{d_{1},d_{2},\ldots,d_{m}\\}$, each not greater than $s$. The generating
function for $W(s,{\bf d})$ has a form

$G(t,{\bf d})=\prod_{i=1}^{m}\frac{1}{1-t^{d_{i}}}=\sum_{s=0}^{\infty}W(s,{\bf
d})\;t^{s}\;.$ (2)

In other words the partition function equals a constant term in Taylor
expansion of the following expression

$W(s,{\bf
d})=\mbox{const}_{t}\left[t^{-s}\prod_{i=1}^{m}(1-t^{d_{i}})^{-1}\right].$ (3)

Sylvester proved [13] a statement about splitting of the partition function
into periodic and non-periodic parts and showed that the restricted partition
function may be presented as a sum of the Sylvester waves [10]

$W(s,{\bf d})=\sum_{j=1}W_{j}(s,{\bf d})\;,$ (4)

where summation runs over all distinct factors $j$ of the elements $d_{i}$.
The wave $W_{j}(s,{\bf d})$ is a quasipolynomial in $s$ closely related to
prime roots $\rho_{j}$ of unity; it is a coefficient of ${t}^{-1}$ in the
series expansion in ascending powers of $t$ of the generator

$F_{j}(s,t)=\sum_{\rho_{j}}\frac{\rho_{j}^{-s}e^{st}}{\prod_{k=1}^{m}\left(1-\rho_{j}^{d_{k}}e^{-d_{k}t}\right)}\;.$
(5)

The summation is made over all prime roots of unity $\rho_{j}=\exp(2\pi in/j)$
for $n$ relatively prime to $j$ (including unity) and smaller than $j$. It was
shown [10] that it is possible to express the Sylvester wave as a finite sum
of the Bernoulli polynomials of higher order.

###  1.2 Vector partition function

Consider a function $W({\bf s},{\bf D})$ counting the number of integer
nonnegative solutions ${\bf x}\geq 0$ to the linear system ${\bf D}\cdot{\bf
x}={\bf s}$, where ${\bf D}$ is a nonnegative integer $l\times m$ matrix. The
vector partition function (VPF) $W({\bf s},{\bf D})$ is a natural
generalization of the restricted partition function to the vector argument.

The generating function for the VPF reads

$G({\bf t},{\bf D})=\prod_{i=1}^{m}\frac{1}{1-{\bf t}^{{\bf
c}_{i}}}=\sum_{{\bf s}}W({\bf s},{\bf D}){\bf t^{s}},\quad{\bf
t^{s}}=\prod_{k=1}^{l}t_{k}^{s_{k}},\quad{\bf t}^{{\bf
c}_{i}}=\prod_{k=1}^{l}t_{k}^{d_{ki}},$ (6)

where ${\bf c}_{i},\ 1\leq i\leq m,$ denotes the $i$-th column of the matrix
${\bf D}$. Note that some elements $d_{ki}$ might equal zero.

The partition problems that include inequalities can be reduced to the VPF
problem as follows. Assume that we have a system of $k$ linear equations and
$l-k$ linear inequalities for $m>l$ integer nonnegative variables $x_{i}$:

$\sum_{i=1}^{m}d_{ji}x_{i}=s_{j},\ (1\leq j\leq
k),\quad\sum_{i=1}^{m}d_{ji}x_{i}\leq n_{j},\ (k+1\leq j\leq l).$ (7)

Introduce $l-k$ new integer variables $x_{i},\ (k+1\leq i\leq l)$ that enter
additively all above equations and inequalities with corresponding factors
$d_{ji}=0$ for $1\leq j\leq k$, and $d_{ji}=1$ for $k+1\leq j\leq l$, and
transform all inequalities into equations

$\sum_{i=1}^{m+l-k}d_{ji}x_{i}=s_{j},\ (1\leq j\leq
k),\quad\sum_{i=1}^{m+l-k}d_{ji}x_{i}=n_{j},\ (k+1\leq j\leq l).$ (8)

It is easy to see that the number of integer nonnegative solutions of (8)
equals that of (7), and thus any system of linear equations with constraints
can be reduced to the standard vector partition problem.

###  1.3 Sylvester-Cayley method of compound partitions

The problem of vector partitions has a long history and J.J. Sylvester made a
significant contribution to its solution. He put forward [12] an idea to
reduce VPF into a sum of scalar partitions (2) (see also [8]). The reduction
is an iterative process based on the variable elimination in the generating
function (6). Sylvester considered a specific double partition problem as an
illustration of his method and determined regions (chambers) of a plane
$\\{s_{1},s_{2}\\}$ each having a unique expression for VPF valid in this
region only. He showed that the expressions in the adjacent chambers coincide
at their common boundary (see also [11]).

This approach was successfully applied by A.Cayley [4] to double partitions
subject to some restrictions on the positive elements of matrix ${\bf D}$ (the
columns ${\bf c}_{i}$ are linearly independent and the inequality
$c_{i2}<s_{2}+2$ should hold for all $1\leq i\leq m$).

In this manuscript we present the result for double partitions obtained by
Cayley and extend it to apply to the Gaussian polynomial problem.

##  2 Double partitions

Introduce an augmented matrix ${\bf E}$ obtained by prepending the column
vector ${\bf s}$ to the matrix ${\bf D}$, so that $e_{10}=s_{1}=r$ and
$e_{20}=s_{2}=\rho$. Denote the rest elements of first row of ${\bf E}$ as
$e_{1i}=d_{1i}=b_{i},\ (1\leq i\leq m)$, and of the second row by
$e_{2i}=d_{2i}=\beta_{i},\ (1\leq i\leq m)$. For simplicity assume that all
$\beta_{i}>0$, and perform a partial fraction expansion (PFE) step to present
$G({\bf t},{\bf D})$ as sum of $m$ fractions

$G({\bf t},{\bf D}){\bf t}^{-{\bf s}}={\bf t}^{-{\bf
s}}\prod_{i=1}^{m}\frac{1}{1-t_{1}^{b_{i}}t_{2}^{\beta_{i}}}=\sum_{i=1}^{m}T_{i}({\bf
t}),\quad T_{i}=\frac{A_{i}({\bf t}){\bf t}^{-{\bf
s}}}{1-t_{1}^{b_{i}}t_{2}^{\beta_{i}}}.$ (9)

Then the solution reads as a sum of the terms

$U_{i}=\mbox{const}_{\bf t}\left[T_{i}({\bf t})\right].$ (10)

Cayley showed that the first term $U_{1}$ corresponding to elimination of the
second column of matrix ${\bf E}$ can be written as

$U_{1}=\mbox{const}_{t_{1}}\left[t_{1}^{-(r\beta_{1}-b_{1}\rho)}\prod_{j\neq
1}^{m}(1-t_{1}^{b_{j}\beta_{1}-b_{1}\beta_{j}})^{-1}\right].$ (11)

Comparing this to (3) we see that $U_{1}$ corresponds to
$W(r\beta_{1}-b_{1}\rho,{\bf d}_{1})$, where the elements $d_{1j}$ of the
vector ${\bf d}_{1}$ are given by $d_{1j}=b_{j}\beta_{1}-b_{1}\beta_{j},\
j\neq 1$. Effectively, this result corresponds to elimination [6] of the first
unknown $x_{1}$ in the system ${\bf D}\cdot{\bf x}={\bf s}$ that leads to a
single Diophantine equation ${\bf d}_{1}\cdot{\bf
x}^{\prime}=r\beta_{1}-b_{1}\rho$, where ${\bf x}^{\prime}$ is the vector of
unknowns obtained from ${\bf x}$ by dropping $x_{1}$.

Similar transformations allow to find all $m$ contributions to obtain

$W({\bf s},{\bf D})=\sum_{i=1}^{m}U_{i}=\sum_{i=1}^{m}W(L_{i},{\bf
d}_{i}),\quad L_{i}=r\beta_{i}-b_{i}\rho,\quad
d_{ij}=b_{j}\beta_{i}-b_{i}\beta_{j},\ j\neq i,$ (12)

where both $L_{i}$ and $d_{ij}$ as the determinants of $2\times 2$ matrices
made of the columns $\\{{\bf c}_{0},\ {\bf c}_{i}\\}$ and $\\{{\bf c}_{j},\
{\bf c}_{i}\\}$, respectively. Cayley mentioned [4] that the contribution of
each term in (12) is nonzero only when $L_{i}$ is nonnegative, thus
reintroducing the notion of chambers of the vector partition. He also pointed
out that when some elements of the vector ${\bf d}_{i}$ are negative (say,
$d_{ij_{k}}<0$ for $1\leq k\leq K$) one has

$W(L_{i},{\bf d}_{i})=(-1)^{K}W(L_{i}-\sum_{k=1}^{K}|d_{ij_{k}}|,|{\bf
d}_{i}|),\quad|{\bf d}_{i}|=\\{|d_{ij}|\\}.$ (13)

Consider now a case when one or more elements in one (say, the second) row of
the matrix ${\bf D}$ are zeroes. Denote the number of nonzero elements as
$m_{0}$, and without loss of generality we assume that $\beta_{i}\neq 0$ for
$1\leq i\leq m_{0}$, and $\beta_{i}=0$ for $m_{0}+1\leq i\leq m$. Using the
above result for the columns with $\beta_{i}=0$ we observe that each of them
leads to a contribution $W(L_{i},{\bf d}_{i})=0$ that vanishes as its argument
$L_{i}=-b_{i}\rho<0$ is negative. Thus we obtain

$W({\bf s},{\bf D})=\sum_{i=1}^{m_{0}}W(L_{i},{\bf d}_{i}),\quad
L_{i}=d_{i0}=r\beta_{i}-b_{i}\rho,\quad d_{ij}=b_{j}\beta_{i}-b_{i}\beta_{j},\
j\neq i.$ (14)

##  3 Gaussian polynomials and their coefficients

The Gaussian polynomial coefficients $P_{m}^{n}(s)$ are defined as a number of
nonnegative integer solutions of a linear Diophantine equation with a
constrain

$\sum_{i=1}^{m}ix_{i}=s,\quad\sum_{i=1}^{m}x_{i}\leq n.$ (15)

The coefficients $P_{m}^{n}(s)$ of the Gaussian polynomial $G_{m}^{n}(t)$
satisfy [14]

$G_{m}^{n}(t)=\frac{G_{m}(t)G_{n}(t)}{G_{m+n}(t)}=\sum_{s=0}^{mn}P_{m}^{n}(s)t^{s},\quad
G_{m}(t)=\prod_{i=1}^{m}\frac{1}{1-t^{i}},$ (16)

where $G_{m}(t)$ denotes the generating function for the restricted partition
function $W_{m}(s)=W(s,\hat{\bf d}^{m}=\\{1,2,\ldots,m\\})$. The
quasipolynomials $P_{m}^{n}(s)$ have a finite order $mn$ and the following
properties [14]

$\displaystyle P_{m}^{n}(0)=P_{m}^{n}(mn)=1,\quad P_{m}^{n}(s)=0,\ \
\mbox{for}\ \ s>mn,$ $\displaystyle
P_{m}^{n}(s)=P_{n}^{m}(s)=P_{m}^{n}(mn-s),\quad
P_{m}^{n}(mn/2-s)=P_{m}^{n}(mn/2+s),$ (17) $\displaystyle
P_{m}^{n}(s)-P_{n}^{m}(s-1)\geq 0,\ \ \mbox{for}\ \ 0\leq s\leq mn/2.$

Cayley [3] considered a problem of computation of the Gaussian polynomial
coefficients and found the generating functions (16) and (19, see below). Some
interesting results about explicit expression of $P_{m}^{n}(s)$ through the
regular partition functions were reported recently in [7].

###  3.1 Sylvester-Cayley algorithm for Gaussian polynomial coefficients

The problem (15) is a particular case of (7) for $k=1,\ l=2$, and $d_{1i}=i,\
d_{2i}=1$, thus it can be transformed into a VPF problem

$\sum_{i=0}^{m}ix_{i}=s,\quad\sum_{i=0}^{m}x_{i}=n,$ (18)

and dealt with using the Sylvester-Cayley algorithm.

Write (14) for

${\bf D}=\left(\begin{array}[]{ccccc}1&1&1&\ldots&1\\\
0&1&2&\ldots&m\end{array}\right),\quad{\bf s}=\left(\begin{array}[]{c}n\\\
s\end{array}\right)\ ,$

that determines a generating function for $P_{m}^{n}(s)$ with fixed integers
$m$ and $n$

$\prod_{i=0}^{m}\frac{1}{1-at^{i}}=\sum_{n=0}^{\infty}G_{m}^{n}(t)a^{n}=\sum_{n=0}^{\infty}\sum_{s=0}^{mn}P_{m}^{n}(s)a^{n}t^{s}.$
(19)

In (14) we have $m_{0}=m,\ r=n,\ \rho=s,\ b_{i}=1,\beta_{i}=i$ and obtain
$L_{i}=ni-s$ and $d_{ij}=\beta_{i}-\beta_{j}=i-j$. Thus the set ${\bf d}_{i}$
can be written as a union of $\hat{\bf d}^{i}$ and $-\hat{\bf d}^{m-i}$. We
then observe from (14)

$P_{m}^{n}(s)=\sum_{i=1}^{m}w_{i}(m,n;s),\quad w_{i}(m,n;s)=W(ni-s,-\hat{\bf
d}^{m-i}\cup\hat{\bf d}^{i}),$ (20)

where the $i$-th term in the above sum contributes for $0\leq s\leq ni$. This
immediately leads to a conclusion that $P_{m}^{n}(s)=0$ for $s>nm$ as all $m$
terms in (20) vanish. By the representation (20) the Gaussian polynomial
$P_{m}^{n}(s)$ has $m$ chambers bounded by lines $s=ni,\ 0\leq i\leq m$.
Specifically, in the $k$-th chamber bounded by the lines $s=n(m-k)$ and
$s=n(m-k+1)$ one has to retain only $k$ terms in the sum with $m-k+1\leq i\leq
m$. Using (13) we arrive at

$w_{i}(m,n;s)=(-1)^{m-i}W(ni-s-s_{m-i},\hat{\bf d}^{i}\cup\hat{\bf
d}^{m-i}),\quad s_{m}=m(m+1)/2,$ (21)

where the partition function $W$ can be computed using the algorithm discussed
in [10]. It is convenient to introduce $P_{m}^{n}(r,s)$ that explicitly
accounts for the summands entering $P_{m}^{n}(s)$ in the $r$-th chamber

$P_{m}^{n}(r,s)=\sum_{i=r}^{m}w_{i}(m,n;s),\quad(r-1)n\leq s\leq rn.$ (22)

###  3.2 Convolution of restricted partitions

Consider a general term $W(s,\hat{\bf d}^{k_{1}}\cup\hat{\bf d}^{k_{2}})$ in
(21). Its generating function reads $G(t)=G_{k_{1}}(t)G_{k_{1}}(t)$ and we
obtain

$\sum_{s=0}W(s,\hat{\bf d}^{k_{1}}\cup\hat{\bf
d}^{k_{2}})t^{s}=\sum_{s_{1}=0}\sum_{s_{2}=0}W_{k_{1}}(s_{1})W_{k_{2}}(s_{1})t^{s_{1}+s_{2}},$

leading to a discrete convolution (the Cauchy product)

$W(s,\hat{\bf d}^{k_{1}}\cup\hat{\bf
d}^{k_{2}})=\sum_{k=0}^{s}W_{k_{1}}(k)W_{k_{2}}(s-k),$

and setting here $k_{1}=i,\ k_{2}=m-i$ we find

$W(s,\hat{\bf d}^{i}\cup\hat{\bf
d}^{m-i})=\sum_{k=0}^{s}W_{i}(k)W_{m-i}(s-k).$

Substituting it in (21) and (20) we obtain

$P_{m}^{n}(s)=\sum_{i=1}^{m}\sum_{k=0}^{ni-s-
s_{m-i}}\\!\\!\\!\\!\\!(-1)^{m-i}W_{i}(k)W_{m-i}(ni-s-s_{m-i}-k),$

where the chamber boundaries remain unchanged. Noticing that
$W_{0}(s)=\delta_{s,0}$ we rewrite the above expression as

$P_{m}^{n}(s)=W_{m}(mn-s)+\sum_{i=1}^{m-1}\sum_{k=0}^{ni-s-
s_{m-i}}\\!\\!\\!\\!\\!(-1)^{m-i}W_{i}(k)W_{m-i}(ni-s-s_{m-i}-k),$ (23)

where the first term in r.h.s. contributes to all chambers. The partial
expression $P_{m}^{n}(r,s)$ for the $r$-th chamber reads

$P_{m}^{n}(r,s)=W_{m}(mn-s)+\sum_{i=r}^{m-1}\\!\\!\\!\sum_{k=0}^{\ \ ni-s-
s_{m-i}}\\!\\!\\!\\!\\!(-1)^{m-i}W_{i}(k)W_{m-i}(ni-s-s_{m-i}-k),\ (r-1)n\leq
s\leq rn.$ (24)

The relations (17) imply a symmetry $s\leftrightarrow mn-s$ leading to an
equivalent representation

$P_{m}^{n}(r,s)=W_{m}(s)+\sum_{i=1}^{r-1}\\!\\!\\!\sum_{k=0}^{\ \ s-in-
s_{i}}\\!\\!\\!\\!\\!(-1)^{i}W_{m-i}(k)W_{i}(s-ni-s_{i}-k),\ (r-1)n\leq s\leq
rn,$ (25)

and we obtain a counterpart of (23)

$P_{m}^{n}(s)=W_{m}(s)+\sum_{i=1}^{m-1}\\!\\!\\!\sum_{k=0}^{\ \ s-in-
s_{i}}\\!\\!\\!\\!\\!(-1)^{i}W_{m-i}(k)W_{i}(s-ni-s_{i}-k).$ (26)

Comparison of (24) to (25) leads to a symmetry relation

$P_{m}^{n}(r,s)=P_{m}^{n}(m+1-r,mn-s),$ (27)

that reduces the computation of $P_{m}^{n}(s)$ to the first
$\lfloor(m+1)/2\rfloor$ chambers.

The formulas (25, 27) present a closed form of the Gaussian polynomial
coefficients as a superposition of the restricted partitions discrete
convolutions. Below we illustrate the above approach by deriving explicit
expressions of $P_{m}^{n}(s)$ for small $m=3,4$ (see also [5]). More
cumbersome expressions required for derivation of $P_{5}^{n}(s)$ and
$P_{6}^{n}(s)$ are presented in Appendix.

###  3.3 Explicit formula for $P_{3}^{n}(s)$

Use (23) or (26) to find

$P_{3}^{n}(s)=W_{3}(s)+\sum_{i=1}^{2}\\!\\!\\!\sum_{k=0}^{\ \ s-in-
s_{i}}\\!\\!\\!\\!\\!(-1)^{i}W_{3-i}(k)W_{i}(s-ni-s_{i}-k),$

or

$P_{3}^{n}(s)=W_{3}(3n-s)+\sum_{i=1}^{2}\sum_{k=0}^{ni-s-
s_{3-i}}\\!\\!\\!\\!\\!(-1)^{i+1}W_{i}(k)W_{3-i}(ni-s-s_{3-i}-k),$

and we have $W_{1}(s)=1$, $s_{2}=3,\ s_{1}=1$ to obtain

$\displaystyle P_{3}^{n}(1,s)$ $\displaystyle=$ $\displaystyle
W_{3}(s)=W_{3}(3n-s)+\sum_{k=0}^{n-s-3}W_{2}(k)-\\!\\!\\!\\!\\!\sum_{k=0}^{2n-s-1}\\!\\!\\!\\!\\!W_{2}(k),\quad
0\leq s\leq n,$ $\displaystyle P_{3}^{n}(2,s)$ $\displaystyle=$ $\displaystyle
W_{3}(3n-s)-\\!\\!\\!\\!\sum_{k=0}^{2n-s-1}\\!\\!\\!W_{2}(k),\quad n\leq s\leq
2n,$ $\displaystyle P_{3}^{n}(3,s)$ $\displaystyle=$ $\displaystyle
W_{3}(3n-s),\quad 2n\leq s\leq 3n.$ (28)

Recall that $W_{2}(s)=s/2+3/4+(-1)^{s}/4$ and find

$\Sigma_{2}(s)=\sum_{k=0}^{s}W_{2}(k)=\frac{(s+1)(s+3)}{4}+\frac{1+\cos\pi
s}{8},$

to arrive at the final expression

$P_{3}^{n}(s)=\left\\{\begin{array}[]{lcl}W_{3}(s),&&0\leq s\leq n,\\\
W_{3}(3n-s)-\Sigma_{2}(2n-s-1),&&n\leq s\leq 2n,\\\ W_{3}(3n-s),&&2n\leq s\leq
3n,\end{array}\right.$ (29)

where

$W_{3}(s)=\frac{47}{72}+\frac{s}{2}+\frac{s^{2}}{12}+\frac{1}{8}\cos\pi
s+\frac{2}{9}\cos\frac{2\pi s}{3}.$ (30)

Note that in [5] the authors completely describe $P_{3}^{n}(s)$ using 36
explicit expressions while our representation requires only three formulas in
(29). It indicates that the approach based on Sylvester-Cayley algorithm
employing the idea of chambers provides more compact and clear result.

###  3.4 Explicit formula for $P_{4}^{n}(s)$

Start with

$P_{4}^{n}(s)=W_{4}(s)+\sum_{i=1}^{3}\\!\\!\\!\sum_{k=0}^{\ \ s-in-
s_{i}}\\!\\!\\!\\!\\!(-1)^{i}W_{4-i}(k)W_{i}(s-ni-s_{i}-k),$

and use $s_{3}=6$ to write

$P_{4}^{n}(s)=W_{4}(s)-\\!\\!\\!\sum_{k=0}^{\ \
s-n-1}\\!\\!\\!\\!\\!W_{3}(k)+\\!\\!\\!\sum_{k=0}^{\ \
s-2n-3}\\!\\!\\!\\!\\!W_{2}(k)W_{2}(s-2n-3-k)-\\!\\!\\!\sum_{k=0}^{\ \
s-3n-6}\\!\\!\\!\\!\\!W_{3}(s-3n-6-k),$ (31)

where

$W_{4}(s)=\frac{2s^{3}+30s^{2}+135s+175}{288}+\frac{(s+5)}{32}\cos\pi
s+\frac{1}{8}\cos\frac{\pi s}{2}+\frac{2}{27}\left(\cos\frac{2\pi
s}{3}-\cos\frac{2\pi(s+1)}{3}\right).$ (32)

###  3.5 Maximal coefficient of Gaussian polynomial

The problem of computation of the maximal coefficient of Gaussian polynomial
was addressed recently in [5] (also L. Fel, private communication).

First note that by (17) the maximal coefficient $p_{m}^{n}$ is given by
$p_{m}^{n}=P_{m}^{n}(mn/2)$ for even $mn$ and $p_{m}^{n}=P_{m}^{n}((mn+1)/2)$
for odd $mn$. For even $m=2k$ we have to compute $P_{m}^{n}(s)$ at $s=kn$ that
belongs to the $(k+1)$-th chamber where $kn\leq s\leq(k+1)n$. For odd $m=2k-1$
the argument of $P_{m}^{n}$ belongs to the middle $k$-th chamber with
$(k-1)n\leq s\leq kn$. Thus we have three cases

$p_{m}^{n}=\left\\{\begin{array}[]{lcl}P_{2k}^{n}(k+1,kn),&&m=2k,\\\
P_{2k-1}^{2r}(k,2kr-r),&&m=2k-1,\ n=2r,\\\ P_{2k-1}^{2r-1}(k,2kr-
k-r+1),&&m=2k-1,\ n=2r-1.\end{array}\right.$ (33)

Use here $k=2$ and for $m=3$ we find $p_{3}^{2r}=P_{3}^{2r}(2,3r)$, and
$p_{3}^{2r-1}=P_{3}^{2r-1}(2,3r-1)$; with $m=4$ we obtain
$p_{4}^{n}=P_{4}^{n}(3,2n)$.

Consider first the case $m=3$. For even $n=2r$ we obtain $s=3r$, so that
$3n-s=3r$ and $2n-s-1=r-1$ leading to
$P_{3}^{2r}(2,3r)=W_{3}(3r)-\Sigma_{2}(r-1)$. For odd $n=2r-1$ we use
$s=3r-1$, that gives $3n-s=3r-2$ and $2n-s-1=r-1$ producing
$P_{3}^{2r}(2,3r-1)=W_{3}(3r-2)-\Sigma_{2}(r-1)$. After algebraic
transformations we arrive at

$p_{3}^{2r}=\frac{1}{2}\left[(r+1)^{2}+\cos\frac{\pi r}{2}\right],\quad
p_{3}^{2r-1}=\frac{r(r+1)}{2}.$ (34)

For $m=4$ we find

$P_{4}^{n}(3,2n)=W_{4}(2n)+\\!\\!\sum_{k=0}^{n-1}W_{3}(k),$

that reduces to

$p_{4}^{n}=\frac{(2n+5)^{3}}{288}+\frac{(2n+5)}{32}+\frac{3}{16}\cos\pi
n+\frac{4}{27}\cos\frac{2\pi n}{3}+\frac{4}{27}\sin\frac{\pi(8n+1)}{6}.$ (35)

Turning to case $m=5$ we use $k=3$ and consider two separate cases – for even
$n=2r$ we obtain $s=5r$, so that $p_{5}^{2r}=P_{5}^{2r}(3,5r)$, while with odd
$n=2r-1$ we have $p_{5}^{2r-1}=P_{5}^{2r-1}(3,5r-2)$. The corresponding
expressions read

$\displaystyle p_{5}^{2r}$ $\displaystyle=$
$\displaystyle\frac{(23r^{2}+69r+49)(r+1)(r+2)}{288}+\frac{425}{1728}+\frac{3(2r+3)}{64}\cos\pi
r+\frac{4}{27}\cos\frac{2\pi r}{3}$ (36) $\displaystyle+$
$\displaystyle\frac{1}{8}\left(\cos\frac{\pi r}{2}+\sin\frac{\pi
r}{2}\right),$ $\displaystyle p_{5}^{2r-1}$ $\displaystyle=$
$\displaystyle\frac{(23r^{2}+46r+36)r(r+2)}{288}-\frac{37}{1728}-\frac{1}{64}\cos\pi
r-\frac{1}{8}\sin\frac{\pi r}{2}$ (37) $\displaystyle+$
$\displaystyle\frac{2}{27}\cos\frac{2\pi
r}{3}-\frac{2}{27}\sin\frac{\pi(8r+1)}{6}.$

Finally, for $m=6$ we again take $k=3$ to obtain $p_{6}^{n}=P_{6}^{n}(4,3n)$
leading to

$\displaystyle p_{6}^{n}$ $\displaystyle=$
$\displaystyle\frac{(2n+7)(66n^{4}+924n^{3}+4606n^{2}+9604n+12061)}{172800}+\frac{2n^{2}+14n+55}{256}\cos\pi
n$ (38) $\displaystyle+$ $\displaystyle\frac{1}{16}\left(\cos\frac{\pi
r}{2}-\sin\frac{\pi r}{2}\right)+\frac{4}{81}\cos\frac{2\pi
n}{3}+\frac{4}{81}\sin\frac{\pi(4n+1)}{6}+\frac{8}{125}\left(\cos\frac{2\pi
n}{5}+\cos\frac{4\pi n}{5}\right)$ $\displaystyle+$
$\displaystyle\frac{4}{125}\left(-\cos\frac{\pi(2n+1)}{5}+\cos\frac{\pi(4n+1)}{5}+2\cos\frac{\pi(8n+1)}{5}\right)$
$\displaystyle+$
$\displaystyle\frac{4}{125}\left(\sin\frac{\pi(4n+1)}{10}+2\sin\frac{\pi(8n+1)}{10}-\sin\frac{\pi(12n+1)}{10}\right).$

It is easy to see from (34-38) that the behavior of the maximal coefficient is
mainly determined by the polynomial part of the corresponding expressions.
This observation leads us to consideration of the polynomial part of Gaussian
polynomial coefficients.

##  4 Polynomial part of Gaussian polynomial coefficients

###  4.1 Polynomial part of partition function

It is known that a leading contribution to the partition function $W(s,{\bf
d})$ for the set of generators ${\bf d}=\\{d_{1},d_{2},\ldots,d_{m}\\}$ is
provided by its polynomial part $w(s,{\bf d})$ that can be expressed through
the Bernoulli polynomials of higher order $B_{k}(s,{\bf d})$ [10]

$w(s,{\bf d})=\frac{B_{m-1}(s+\sigma_{m},{\bf
d})}{(m-1)!\pi_{m}},\quad\sigma_{m}=\sum_{i=1}^{m}d_{i},\quad\pi_{m}=\prod_{i=1}^{m}d_{i},$
(39)

where the Bernoulli polynomials of higher order can be defined in umbral
calculus notation as

$B_{k}(s,{\bf d})=(s+\sum_{i=1}^{m}{}^{i}Bd_{i})^{k},$

after the expansion the replacement $({}^{i}Bd_{i})^{k}\to B_{k}d_{i}^{k}$ is
applied, and $B_{k}$ denotes the $k$-th Bernoulli number. This definition
implies

$B_{k}(s,{\bf d})=\sum_{l=0}^{k}{k\choose l}s^{l}B_{k-l}({\bf d}),$ (40)

and $B_{k}({\bf d})\equiv B_{k}(0,{\bf d})$ denotes the Bernoulli number of
higher order.

###  4.2 Polynomial part of Gaussian polynomial coefficients

It is instructive to consider a polynomial part ${\cal P}_{m}^{n}(s)$ of
quasipolynomial $P_{m}^{n}(s)$

${\cal P}_{m}^{n}(s)=\sum_{i=1}^{m}{\cal W}_{i}(m,n;s),$ (41)

where ${\cal W}_{i}(m,n;s)$ denotes the polynomial part of $w_{i}(m,n;s)$
defined in (21). For $\hat{\bf d}^{m}$ we have $\sigma_{m}=s_{m},\pi_{m}=m!$
and we find that

$w(s,\hat{\bf d}^{i}\cup\hat{\bf
d}^{m-i})=\frac{B_{m-1}(s+s_{i}+s_{m-i},\hat{\bf d}^{i}\cup\hat{\bf
d}^{m-i})}{(m-1)!i!(m-i)!}.$

Use here the replacement $s\to ni-s-s_{m-i}$ to obtain

${\cal P}_{m}^{n}(s)=\sum_{i=1}^{m}(-1)^{m-i}\frac{B_{m-1}(ni-s+s_{i},\hat{\bf
d}^{i}\cup\hat{\bf d}^{m-i})}{(m-1)!i!(m-i)!}.$ (42)

The polynomial part ${\cal P}_{m}^{n}(r,s)$ of the solution in $r$-th chamber
reads

${\cal
P}_{m}^{n}(r,s)=\sum_{i=r}^{m}(-1)^{m-i}\frac{B_{m-1}(ni-s+s_{i},\hat{\bf
d}^{i}\cup\hat{\bf d}^{m-i})}{(m-1)!i!(m-i)!},\quad(r-1)n\leq s\leq rn.$ (43)

##  5 Polynomial part of maximal coefficient

The last result leads to expressions for the polynomial part of the maximal
coefficient of the Gaussian polynomial. For small $m\leq 6$ we have

$\displaystyle{\cal P}_{2}^{n}(2,n)=\frac{2n+3}{4},$ $\displaystyle{\cal
P}_{3}^{2r}(2,3r)=\frac{(r+1)^{2}}{2}+\frac{1}{36},\quad{\cal
P}_{3}^{2r-1}(2,3r-1)=\frac{(3r+1)(3r+2)}{18},$ $\displaystyle{\cal
P}_{4}^{n}(3,2n)=\frac{(2n+5)^{3}}{288},$ (44) $\displaystyle{\cal
P}_{5}^{2r}(3,5r)=\frac{(r+1)(r+2)(23r^{2}+69r+49)}{288}+\frac{571}{14400},$
$\displaystyle{\cal
P}_{5}^{2r-1}(3,5r-2)=\frac{(r+2)r(23r^{2}+46r+36)}{288}+\frac{89}{1600},$
$\displaystyle{\cal
P}_{6}^{n}(4,3n)=\frac{(2n+7)(66n^{4}+924n^{3}+4606n^{2}+9604n+7511)}{172800}.$

###  5.1 Polynomial part for even $m$

For even $m=2k$ the maximal coefficient equals

$\displaystyle{\cal P}_{2k}^{n}(k+1,kn)$ $\displaystyle=$
$\displaystyle\sum_{i=k+1}^{2k}(-1)^{i}\frac{B_{2k-1}(ni-nk+s_{i},\hat{\bf
d}^{i}\cup\hat{\bf d}^{2k-i})}{(2k-1)!i!(2k-i)!}$ (45) $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}(-1)^{j}\frac{B_{2k-1}(n(k-j)+s_{2k-j},\hat{\bf
d}^{j}\cup\hat{\bf d}^{2k-j})}{(2k-1)!j!(2k-j)!}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}(-1)^{j}{2k\choose
j}\frac{B_{2k-1}(n(k-j),-\hat{\bf d}^{2k-j}\cup\hat{\bf
d}^{j})}{(2k-1)!(2k)!},$

where we employ the relation $B_{n}(s,-{\bf d})=B_{n}(s+\sigma_{m},{\bf d})$.
Use here (40) to write

$\displaystyle{\cal
P}_{2k}^{n}(k+1,kn)=\frac{1}{(2k-1)!(2k)!}\sum_{p=0}^{2k-1}{2k-1\choose
p}n^{p}\sum_{j=0}^{k}(-1)^{j}{2k\choose j}B_{2k-p-1}(-\hat{\bf
d}^{2k-j}\cup\hat{\bf d}^{j})(k-j)^{p}.$ (46)

###  5.2 Polynomial part for odd $m=2k-1$ and even $n=2r$

In this case the maximal coefficient reads

$\displaystyle{\cal P}_{2k-1}^{2r}(k,(2k-1)r)$ $\displaystyle=$
$\displaystyle\sum_{i=k}^{2k-1}(-1)^{i+1}{2k-1\choose
i}\frac{B_{2k-2}(2r(i-k)+r,-\hat{\bf d}^{i}\cup\hat{\bf
d}^{2k-1-i})}{(2k-2)!(2k-1)!}$ (47) $\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}\frac{B_{2k-2}(2r(k-1-j)+r,-\hat{\bf d}^{2k-1-j}\cup\hat{\bf
d}^{j})}{(2k-2)!(2k-1)!},$

and we find with (40)

$\displaystyle{\cal
P}_{2k-1}^{2r}(k,(2k-1)r)=\frac{1}{(2k-2)!(2k-1)!}\sum_{p=0}^{2k-2}{2k-2\choose
p}r^{p}$ $\displaystyle\times\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}B_{2k-p-2}(-\hat{\bf d}^{2k-1-j}\cup\hat{\bf d}^{j})(2(k-1-j)+1)^{p}.$ (48)

###  5.3 Polynomial part for odd $m=2k-1$ and $n=2r-1$

We observe

$\displaystyle{\cal P}_{2k-1}^{2r-1}(k,(2k-1)r-k+1)$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{i=k}^{2k-1}(-1)^{i+1}{2k-1\choose
i}\frac{B_{2k-2}((2r-1)(i-k)+r-1,-\hat{\bf d}^{i}\cup\hat{\bf
d}^{2k-1-i})}{(2k-2)!(2k-1)!}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}\frac{B_{2k-2}((2r-1)(k-1-j)+r-1,-\hat{\bf d}^{2k-1-j}\cup\hat{\bf
d}^{j})}{(2k-2)!(2k-1)!},$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}\frac{B_{2k-2}(2r(k-1-j)+r+(j-k),-\hat{\bf d}^{2k-1-j}\cup\hat{\bf
d}^{j})}{(2k-2)!(2k-1)!},$

leading to

$\displaystyle{\cal
P}_{2k-1}^{2r-1}(k,(2k-1)r-k+1)=\frac{1}{(2k-2)!(2k-1)!}\sum_{p=0}^{2k-2}{2k-2\choose
p}r^{p}$ $\displaystyle\times\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}B_{2k-p-2}(j-k,-\hat{\bf d}^{2k-1-j}\cup\hat{\bf d}^{j})(2(k-1-j)+1)^{p}.$
(49)

##  6 Leading term of maximal coefficient

Consider derivation of a general expression for the leading term of the
maximal coefficient. The leading in $n$ term in the summand in (45) evaluates
to

$(-1)^{j}n^{2k-1}{2k\choose j}\frac{(k-j)^{2k-1}}{(2k-1)!(2k)!},$

so that the leading term $L_{n,k}$ of ${\cal P}_{2k}^{n}(k+1,kn)$ reads

$L_{kn}=\frac{n^{2k-1}}{(2k-1)!(2k)!}\sum_{j=0}^{k}(-1)^{j}{2k\choose
j}(k-j)^{2k-1},$ (50)

where the sum evaluates to $A_{2k-1,k}$ being a particular case of the
Eulerian numbers of type A [2]

$A_{n,k}=\sum_{j=0}^{k}(-1)^{j}{n+1\choose j}(k-j)^{n}.$ (51)

These numbers used in a definition of the polylogarithm function

$Li_{-n}(\rho)=\sum_{k=1}^{\infty}k^{n}\rho^{k}=\frac{1}{(1-\rho)^{n+1}}\sum_{i=0}^{n}A_{n,i}\rho^{n-i},\quad\rho\neq
1,$

which in its turn relates to the Eulerian polynomials $H_{n}(\rho)$ defined
through the generating function

$\frac{1-\rho}{e^{t}-\rho}=\sum_{n=0}^{\infty}H_{n}(\rho)\frac{t^{n}}{n!},\quad
H_{n}(\rho)=(-1)^{n}\frac{1-\rho}{\rho}Li_{-n}(\rho).$

It is worth to note that the function $H_{n}(\rho)$ can be extended into the
Eulerian polynomials of higher order used in the derivation [10] of the
partition function $W(s,{\bf d}^{m})$. From (50, 51) it follows that for even
$m$ the leading term $L_{mn}$ of $P_{m}^{n}(m/2+1,mn/2)$ reads

$L_{mn}=\frac{n^{m-1}A_{m-1,m/2}}{(m-1)!m!},$

producing $L_{2n}=n/2,\ L_{4n}=n^{3}/36,\ L_{6n}=11n^{5}/14400$ that coincide
with the leading term of the corresponding expressions in (44).

In case of odd $m=2k-1$ we observe from (44) that $L_{2k-1,2r-1}=L_{2k-1,2r}$
(it is also can be seen by comparing (48) to (49)), so that it is sufficient
to consider the case of even $n=2r$ only. Using (48) we obtain

$\displaystyle
L_{2k-1,2r}=\frac{r^{2k-2}}{(2k-2)!(2k-1)!}\sum_{j=0}^{k-1}(-1)^{j}{2k-1\choose
j}(2(k-1-j)+1)^{2k-2}.$ (52)

The sum in (52) evaluates to the central MacMahon numbers $B_{2k-1,k}$ being a
particular case of the Eulerian numbers $B_{n,k}$ of type B that satisfy [1]
(also see Ch.11 in [9])

$B_{n,k}=(2(n-k)+1)B_{n-1,k-1}+(2k+1)B_{n-1,k},\quad
B_{n,k}=\sum_{j=0}^{k}(-1)^{k-j}{n+1\choose k-j}(2j+1)^{n},$

and we find

$L_{2k-1,2r}=\frac{r^{2k-2}B_{2k-1,k}}{(2k-2)!(2k-1)!},\quad\rightarrow\quad
L_{mn}=\frac{(n/2)^{m-1}B_{m,(m+1)/2}}{(m-1)!m!},$

producing $L_{3,2r}=r^{2}/2,\ L_{5,2r}=23r^{4}/288$ coinciding with the
leading term of the corresponding expressions in (44).

Collecting the above results we obtain a general expression for the leading
term of the maximal coefficient of the Gaussian polynomial through the
Eulerian numbers

$L_{mn}=\frac{1}{(m-1)!m!}\left\\{\begin{array}[]{lcl}n^{m-1}A_{m-1,m/2},&&\mbox{even}\
m,\\\ (n/2)^{m-1}B_{m,(m+1)/2},&&\mbox{odd}\ m,\ \mbox{even}\ n,\\\
((n+1)/2)^{m-1}B_{m,(m+1)/2},&&\mbox{odd}\ m,n.\end{array}\right.$ (53)

## Acknowledgments

The author thanks L.Fel for drawing attention to the Gaussian polynomials
problem and numerous fruitful discussions.

##  Appendix A Computation of $P_{5}^{n}(s)$ and $P_{6}^{n}(s)$

As it follows from (20) and (21) the solution for $m=5$ can be constructed
using only three basic expressions. Namely, we have

$\displaystyle W(s,\hat{\bf d}^{2}\cup\hat{\bf d}^{3})$ $\displaystyle=$
$\displaystyle\frac{s^{4}+18s^{3}+112s^{2}+279s+220}{288}+\frac{37}{1728}+\frac{2s+9}{64}(-1)^{s}+\frac{2}{27}\cos\frac{2\pi
s}{3},$ $\displaystyle W(s,\hat{\bf d}^{1}\cup\hat{\bf d}^{4})$
$\displaystyle=$
$\displaystyle\frac{s^{4}+22s^{3}+166s^{2}+495s+465}{576}+\frac{25}{3456}+\frac{2s+11}{128}(-1)^{s}$
$\displaystyle+$
$\displaystyle\frac{2}{27}\sin\frac{(4s+1)\pi}{6}+\frac{1}{16}\left(\cos\frac{\pi
s}{2}+\sin\frac{\pi s}{2}\right),$ $\displaystyle W(s,\hat{\bf d}^{5})$
$\displaystyle=$
$\displaystyle\frac{s^{4}+30s^{3}+310s^{2}+1275s+1687}{2880}+\frac{41}{86400}+\frac{2s+15}{128}(-1)^{s}+\frac{2}{27}\cos\frac{2\pi
s}{3}$ (A1) $\displaystyle+$ $\displaystyle+\frac{1}{16}\left(\cos\frac{\pi
s}{2}+\sin\frac{\pi s}{2}\right)+\frac{2}{25}\left(\cos\frac{2\pi
s}{5}+\cos\frac{4\pi s}{5}\right).$

Then we use (21) to find $w_{i}(5,n;s)$ to insert into (20) or (22) for
explicit expression of $P_{5}^{n}(s)$ or $P_{5}^{n}(r,s)$ respectively.

The solution for $m=6$ is produced based on the four following expressions

$\displaystyle W(s,\hat{\bf d}^{3}\cup\hat{\bf d}^{3})$ $\displaystyle=$
$\displaystyle\frac{6s^{5}+180s^{4}+2020s^{3}+10440s^{2}+24299s+19650}{25920}+\frac{s+6}{64}(-1)^{s}$
$\displaystyle+$ $\displaystyle\frac{6s+34}{243}\cos\frac{2\pi
s}{3}+\frac{4}{243}\sin\frac{\pi(4s+1)}{6},$ $\displaystyle W(s,\hat{\bf
d}^{2}\cup\hat{\bf d}^{4})$ $\displaystyle=$
$\displaystyle\frac{12s^{5}+390s^{4}+4720s^{3}+26130s^{2}+64458s+54275}{69120}+\frac{2s^{2}+26s+75}{512}(-1)^{s}$
$\displaystyle+$ $\displaystyle\frac{2}{81}\cos\frac{2\pi
s}{3}+\frac{2}{81}\sin\frac{\pi(4s+1)}{6}+\frac{1}{32}\left(\cos\frac{\pi
s}{2}+\sin\frac{\pi s}{2}\right),$ $\displaystyle W(s,\hat{\bf
d}^{1}\cup\hat{\bf d}^{5})$ $\displaystyle=$
$\displaystyle\frac{6s^{5}+240s^{4}+3560s^{3}+24000s^{2}+71325s+70888}{86400}+\frac{s+8}{128}(-1)^{s}$
$\displaystyle+$ $\displaystyle\frac{2}{81}\cos\frac{2\pi
s}{3}+\frac{2}{81}\sin\frac{\pi(4s+1)}{6}+\frac{1}{16}\sin\frac{\pi
s}{2}+\frac{2}{125}\left(\cos\frac{2\pi s}{5}+\cos\frac{4\pi s}{5}\right)$
$\displaystyle+$
$\displaystyle\frac{2}{125}\left(\cos\frac{\pi(2s+1)}{5}+3\cos\frac{\pi(6s+1)}{5}+2\cos\frac{\pi(8s+1)}{5}\right)$
$\displaystyle-$
$\displaystyle\frac{2}{125}\left(2\sin\frac{\pi(8s+1)}{10}+\sin\frac{\pi(12s+1)}{10}+3\sin\frac{\pi(16s+1)}{10}\right),$
$\displaystyle W(s,\hat{\bf d}^{6})$ $\displaystyle=$
$\displaystyle\frac{12s^{5}+630s^{4}+12320s^{3}+110250s^{2}+439810s+598731}{1036800}+\frac{6s^{2}+126s+581}{4608}(-1)^{s}$
(A2) $\displaystyle+$ $\displaystyle\frac{6s+61}{486}\cos\frac{2\pi
s}{3}+\frac{2}{243}\sin\frac{\pi(4s+1)}{6}+\frac{1}{18}\cos\frac{\pi s}{3}$
$\displaystyle+$ $\displaystyle\frac{1}{32}\left(\cos\frac{\pi
s}{2}+\sin\frac{\pi s}{2}\right)+\frac{2}{125}\left(\cos\frac{2\pi
s}{5}+\cos\frac{4\pi s}{5}\right)$ $\displaystyle+$
$\displaystyle\frac{2}{125}\left(\cos\frac{\pi(2s+1)}{5}+3\cos\frac{\pi(6s+1)}{5}+2\cos\frac{\pi(8s+1)}{5}\right)$
$\displaystyle-$
$\displaystyle\frac{2}{125}\left(2\sin\frac{\pi(8s+1)}{10}+\sin\frac{\pi(12s+1)}{10}+3\sin\frac{\pi(16s+1)}{10}\right).$

Now employ (21) to obtain $w_{i}(6,n;s)$ and then use it in (20) or (22) for
explicit expression of $P_{6}^{n}(s)$ or $P_{6}^{n}(r,s)$ respectively.

## References

  * [1] A. Borowiec and W. Mlotkowski, New Eulerian Numbers of Type D, arXiv:1509.03758v1 [math.CO].
  * [2] L. Carlitz, Eulerian Numbers and Polynomials, Mathematics Magazine, 32 (1959), 247-260.
  * [3] A. Cayley, Researches on the Partition of Numbers, Phil. Trans. Roy.l Soc. London, CXLV (1855), 127-140 [Coll. Math. Papers, II (1891), 235-249].
  * [4] A. Cayley, On a Problem of Double Partitions, Philosophical Magazine, XX (1860), 337-341 [Coll. Math. Papers, IV (1891), 166-170].
  * [5] A. Castillo, S. Flores, A. Hernandez, B. Kronholm, A. Larsen and A. Martinez, Quasipolynomials and Maximal Coefficients of Gaussian Polynomials, Ann. Comb., 15 (2019), 589-611.
  * [6] L.E. Dickson, History of the Theory of Numbers, II, (1920), Ch. III, 123.
  * [7] L.G. Fel, Gaussian Polynomials and Restricted Partition Function with Constraints, arXiv:1611.09931v1 [math.CO].
  * [8] G.B. Mathews, On the Partition of Numbers, Proc. London Math. Soc., 28 (1896-97), 486-490.
  * [9] K.T. Petersen, Eulerian Numbers, Birkhauser (2015).
  * [10] B.Y. Rubinstein, Expression for Restricted Partition Function through Bernoulli Polynomials, Ramanujan Journal, 15 (2008), 177-185.
  * [11] B. Sturmfels, Note on Vector Partition Functions, J. Combinat. Theory, A72 (1995), 302-309.
  * [12] J. J. Sylvester, On the Problem of the Virgins, and the General Theory of Compound Partitions, Philosophical Magazine, XVI (1858), 371-376 [Coll. Math. Papers, II (1908), 113-117].
  * [13] J. J. Sylvester, On Subinvariants, i.e. Semi-invariants to Binary Quantics of an Unlimited Order. With an Excursus on Rational Fractions and Partitions, American Journal of Mathematics, V (1882), 79-136 [Coll. Math. Papers, III (1909), 568-622].
  * [14] J. J. Sylvester, A Constructive Theory of Partitions, Arraged in Three Acts, an Interact and an Exodion, American Journal of Mathematics, V (1882), 251-330; VI (1884), 334-336 [Coll. Math. Papers, IV (1912), 1-83].


# Pointwise convergence of sequential Schrödinger means

Chu-Hee Cho, Hyerim Ko, Youngwoo Koh and Sanghyuk Lee  Department of
Mathematical Sciences and RIM, Seoul National University, Seoul 08826,
Republic of Korea<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Department
of Mathematics Education, Kongju National University, Kongju 32588, Republic
of Korea<EMAIL_ADDRESS>
###### Abstract.

We study pointwise convergence of the fractional Schrödinger means along
sequences $t_{n}$ which converge to zero. Our main result is that bounds on
the maximal function $\sup_{n}|e^{it_{n}(-\Delta)^{\alpha/2}}f|$ can be
deduced from those on $\sup_{0<t\leq 1}|e^{it(-\Delta)^{\alpha/2}}f|$ when
$\\{t_{n}\\}$ is contained in the Lorentz space $\ell^{r,\infty}.$
Consequently, our results provide seemingly optimal results in higher
dimensions, which extend the recent work of Dimou–Seeger, and Li–Wang–Yan to
higher dimensions. Our approach based on a localization argument also works
for other dispersive equations and provides alternative proofs of previous
results on sequential convergence.

###### Key words and phrases:

pointwise convergence, Schrödinger operator.

###### 2010 Mathematics Subject Classification:

Primary 42B25, 35Q41, 35S10

##  1\. Introduction

Let $\alpha>0$. We consider the fractional Schrödinger operator

(1.1)
$e^{it(-\Delta)^{\alpha/2}}f(x)=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{i(x\cdot\xi+t|\xi|^{\alpha})}\widehat{f}(\xi)d\xi.$

A classical problem posed by Carleson [5] is to determine the optimal
regularity $s$ for which

(1.2) $\displaystyle\lim_{t\rightarrow
0}e^{it(-\Delta)^{\alpha/2}}f=f\quad\text{a.e. }\quad\forall f\in H^{s},$

where $H^{s}$ denotes the inhomogeneous Sobolev spaces of order $s$ with its
norm
$\|f\|_{H^{s}(\mathbb{R}^{d})}=\|(1+|\cdot|^{2})^{s/2}\widehat{f}\,\|_{L^{2}(\mathbb{R}^{d})}$.
The case $\alpha=2$ has been extensively studied until recently. When $d=1$,
it was shown by the work of Carleson [5] and Kenig–Dahlberg [10] that (1.2)
holds true if and only if $s\geq 1/4$. In higher dimensions, the problem
turned out to be more difficult. Progress was made by contributions of
numerous authors. Sjölin [33] and Vega [40] independently obtained (1.2) for
$s>1/2$. In particular, further improvement on required regularity was made by
Moyua–Vargas–Vega [26], Tao–Vargas [39] when $d=2$, and convergence for
$s>{(2d-1)}/{4d}$ was shown by Lee [19] for $d=2$ and Bourgain [3] in higher
dimensions. Bourgain [4] showed that (1.2) holds only if $s\geq{d}/{2(d+1)}$.
The lower bound was shown to be sufficient for (1.2) by Du–Guth–Li [12] for
$d=2$, and Du–Zhang [14] for $d\geq 3$ except for the endpoint case
$s={d}/{2(d+1)}$ (also, see [23] for earlier results and references therein).

In general, (1.2) continues to be true for $\alpha>1$ if $s>{d}/{2(d+1)}$ (see
e.g., [19, 7], also see [25]). If $\alpha=1$, it is easy to show (1.2) holds
if and only if $s>1/2$ in all dimensions ([9, 42]). When $0<\alpha<1$, it is
known that (1.2) holds if $s>\alpha/4$ and the convergence fails if
$s<\alpha/4$ in $\mathbb{R}^{1}$. However, only partial results are known in
higher dimensions, i.e., (1.2) holds true for $s>\alpha/2$ and fails for
$s<\alpha/4$ when $d\geq 2$ (see [41, 42]).

### Convergence along sequences

Recently, pointwise convergence along sequences was considered by several
authors [11, 35, 36, 37, 38]. More precisely, studied was the problem to
determine the regularity exponent $s$ such that, for a given sequence
$\\{t_{n}\\}$ satisfying $\lim_{n\rightarrow\infty}t_{n}=0$,

(1.3)
$\lim_{n\rightarrow\infty}e^{it_{n}(-\Delta)^{\alpha/2}}f(x)=f(x)\quad\text{a.e.
}x,\quad\forall f\in H^{s}.$

Naturally, one may expect that the more rapidly the sequence $\\{t_{n}\\}$
converges to zero, the less regularity is required to guarantee almost
everywhere convergence. To quantify how fast the sequence converges to zero,
the sequences in $\ell^{r}(\mathbb{N})$ and $\ell^{r,\infty}(\mathbb{N})$ were
considered, where $\ell^{r,\infty}(\mathbb{N})$ denotes the Lorentz space

$\ell^{r,\infty}(\mathbb{N}):=\big{\\{}t_{n}\,:\,\sup_{\delta>0}\delta^{r}\\#\\{n\in\mathbb{N}:|t_{n}|\geq\delta\\}<\infty\big{\\}}$

for $r<\infty$. Note that $\\{n^{-b}\\}\in\ell^{r,\infty}(\mathbb{N})$ if and
only if $b\geq 1/r$. In particular, Dimou and Seeger [11] studied the almost
everywhere convergence problem in $\mathbb{R}^{1}$ using
$\ell^{r,\infty}(\mathbb{N})$. They proved that (1.3) holds for all $f\in
H^{s}$ if and only if $s\geq\min\\{\frac{r\alpha}{4r+2},\frac{1}{4}\\}$ (when
$\alpha>1$), $s\geq\frac{r\alpha}{4r+2}$ (when $0<\alpha<1$), and
$s\geq\frac{r}{2(r+1)}$ (when $\alpha=1$) for a strictly decreasing convex
sequence $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$. There are also results
in higher dimensions by Sjölin [35], Sjölin and Strömberg [36, 37, 38].
Recently, Li, Wang, and Yan [21], relying on the bilinear approach in [19],
obtained some partial results for the case $d=\alpha=2$ without assuming that
$\\{t_{n}\\}$ decreases.

### Maximal estimates

In the study of pointwise convergence the associated maximal functions play
important roles. By a standard argument (1.2) follows if we have

(1.4) $\displaystyle\big{\|}\sup_{0<t\leq
1}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1))}\leq C\|f\|_{H^{s}},$

where $B(x,r)=\\{y\in\mathbb{R}^{d}:|x-y|<r\\}$. Likewise, (1.3) follows from
the estimate

(1.5)
$\displaystyle\big{\|}\sup_{t_{n}}\big{|}e^{it_{n}(-\Delta)^{\alpha/2}}f\big{|}\big{\|}_{L^{2}(B(0,1))}\leq
C\|f\|_{H^{s}},$

which is, in fact, essentially equivalent to (1.3) by Stein’s maximal
principle. Our first result shows that the maximal estimate (1.5) can be
deduced from (1.4) when $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$.

######  Theorem 1.1.

Let $d\geq 1$, $\alpha>0$, $s_{*}>0$ and $0<r<\infty$. Suppose (1.4) holds for
$s\geq s_{*}$. Then, if $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$, (1.5)
holds provided

(1.6) $\displaystyle
s\geq\min\Big{\\{}\frac{r\alpha}{r\min\\{\alpha,1\\}+2s_{*}}s_{*},\
s_{*}\Big{\\}}.$

Thanks to Theorem 1.1 we can improve the previous results and obtain seemingly
optimal results for the convergence of sequential Schrödinger means in higher
dimensions. For $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$ and $d=1$, the
known estimates (1.4) ([5, 41]) and Theorem 1.1 give (1.5) for
$s\geq\min(r\alpha/(4r+2),1/4)$ when $\alpha>1$, and for $s>r\alpha/(4r+2)$
when $0<\alpha<1$. This recovers the result (sufficiency part except the
endpoint case when $0<\alpha<1$) in [11] without the assumption that
$\\{t_{n}\\}$ decreases.

In higher dimensions $d\geq 2$, by combining Theorem 1.1 and recent progress
on the maximal bounds, i.e., (1.4) for $\alpha>1$ and $s>\frac{d}{2(d+1)}$
[12, 14, 7], we have the estimate (1.5) for

(1.7) $s>\min\big{\\{}\frac{\alpha dr}{2(d+1)r+2d},\frac{d}{2(d+1)}\big{\\}}$

whenever $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$. As a consequence, we
have the following result on pointwise convergence.

######  Corollary 1.2.

Let $d\geq 2$, $\alpha>1$ and $0<r<\infty$. For any sequence
$\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$, (1.3) holds for all $f\in
H^{s}(\mathbb{R}^{d})$ if (1.7) holds.

This improves the previous results in [21, 37]. We expect that the regularity
exponents given in (1.7) is sharp up to the endpoint case. However, we are not
able to verify it for the moment.

######  Remark 1.3.

As mentioned before, when $0<\alpha<1$ and $d\geq 2$, it is known that (1.2)
holds if $s>\alpha/2$ ([42]). Thus, Theorem 1.1 yields (1.3) for
$s>\frac{r\alpha}{2(r+1)}$. The implication in Theorem 1.1 also works for more
general operators (see Remark 2.3). In particular, one can also recover the
result of Li, Wang and Yan [22] for the nonelliptic Schrödinger operator by
combining Theorem 1.1 with the results in [31].

######  Remark 1.4.

For the wave operator, i.e., $\alpha=1$, (1.3) holds true if and only if
$s\geq\frac{r}{2(r+1)}$ for $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$. When
$d=1$, this was shown in [11]. In higher dimensions, one can show it using
Theorem 1.1 (also Corollary 3.2). The sharpness can be obtained by following
the argument in [11]. We remark that (1.3) is closely related to $L^{p}$
boundedness of the spherical maximal operator given by taking the supremum
over more general sets (see [32, 1, 30]).

### Localization argument

The proof of Theorem 1.1 relies on a localization argument. We briefly explain
our approach. Via Littlewood-Paley decomposition, the proof of (1.3) can be
reduced to showing

(1.8)
$\displaystyle\big{\|}\sup_{t_{n}}\big{|}e^{it_{n}(-\Delta)^{\alpha/2}}f\big{|}\big{\|}_{L^{2}(B(0,1))}\leq
CR^{s}\|f\|_{L^{2}(\mathbb{R}^{d})}$

where $\widehat{f}$ is supported in $\mathbb{A}_{R}:=\\{\xi:R\leq|\xi|\leq
2R\\}$ (see Section 3). In the previous work [11, 35, 36, 37] the estimate
(1.8) was obtained by relying on the kernel estimates. In contrast, we deduce
(1.8) directly from (1.4). Clearly, (1.4) gives

(1.9) $\displaystyle\big{\|}\sup_{0<t\leq
1}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1))}\leq
CR^{s}\|f\|_{L^{2}(\mathbb{R}^{d})}$

for $R\geq 1$ whenever $\widehat{f}$ is supported in $\mathbb{A}_{R}$. Using
the estimate and a localization argument, we first obtain from (1.9) a
temporally localized maximal estimate

(1.10) $\displaystyle\big{\|}\sup_{t\in
I}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1))}\leq
C^{\prime}(1+R^{\alpha}|I|)^{\max\\{s,\frac{s}{\alpha}\\}}\|f\|_{L^{2}(\mathbb{R}^{d})}$

for $R\geq 1$ and any subinterval $I\subset[0,1]$ with $|I|\leq R^{1-\alpha}$
when $\widehat{f}$ is supported in $\mathbb{A}_{R}$. Moreover, the converse
implication from (1.10) to (1.9) is also true as long as $R^{-\alpha}<|I|\leq
R^{1-\alpha}$ (see Lemma 2.2 for detail). Once we have (1.10), we can obtain
(1.5) by following the argument in [11].

If the exponent $s$ in the estimate (1.9) is sharp, then the same is true for
the estimate (1.10). For instance, when $\alpha=2$, (1.9) holds for
$s>{d}/2(d+1)$, which is optimal up to the endpoint case, and hence so does
(1.10) for the same $s$. When $\alpha>1$ and $|I|\geq R^{1-\alpha}$, one can
see the exponent s in (1.10) can not be smaller than that in (1.9) using the
localization lemma in [19] (cf. [8, 20, 29]).

To show the implication from (1.9) to (1.10), we adapt the idea of temporal
localization lemma in [19, 8]. We establish a spatial localization lemma
(Lemma 2.4), which plays a crucial role in proving Theorem 1.1. More
precisely, we show that the local-in-spatial estimate (1.9) can be extended to
the global-in-spatial estimate with the same regularity exponent. After a
suitable scaling, we obtain the temporal localized estimate (1.10) from the
global-in-spatial estimate.

### Extension to fractal measure

Maximal estimates relative to general measures (instead of the Lebesgue
measure) have been used to get more precise description on the pointwise
behavior of the Schrödinger mean $e^{it(-\Delta)^{\alpha/2}}f$. For a given
sequence $\\{t_{n}\\}$ converging to zero, we consider

$D^{\alpha,d}(f,\\{t_{n}\\})=\big{\\{}x\in\mathbb{R}^{d}:e^{it_{n}(-\Delta)^{\alpha/2}}f(x)\not\to
f(x)\quad~{}\text{as}\quad~{}t_{n}\rightarrow 0\big{\\}}$

and set

$\mathfrak{D}^{\alpha,d}(s,r)=\sup_{f\in
H^{s},~{}\\{t_{n}\\}\in\ell^{r,\infty}}\dim_{H}D^{\alpha,d}(f,\\{t_{n}\\}),$

where $\dim_{H}$ denotes the Hausdorff dimension. One can compare
$\mathfrak{D}^{\alpha,d}(s,r)$ with the dimension of the divergence set

$\mathfrak{D}^{\alpha,d}(s)=\sup_{f\in
H^{s}}\dim_{H}\big{\\{}x\in\mathbb{R}^{d}:e^{it_{n}(-\Delta)^{\alpha/2}}f(x)\not\to
f(x)\quad~{}\text{as}\quad~{}t\rightarrow 0\big{\\}}.$

The bounds on $\mathfrak{D}^{\alpha,d}(s)$ can be obtained by the maximal
estimate relative to general measures (see, for example, [2, 14, 17]), to
which the fractal Strichartz estimates studied in [6, 18, 13] are closely
related (also see [24, 43, 15]).

The implication in Theorem 1.1 also extends to the maximal estimates relative
to general fractal measures, so we can make use of the known estimates for the
$L^{2}$-fractal maximal estimates and the fractal Strichartz estimates to
obtain upper bounds on $\mathfrak{D}^{\alpha,d}(s,r)$, $0<r<\infty$. We
discuss it in detail in Section 3.2.

#### Organization of the paper

In Section 2, we deduce from (1.4) temporally localized maximal estimates in
Lemma 2.2 (relative to general measure) which are to be used to prove Theorem
1.1. We prove Theorem 1.1 and discuss upper bounds on the dimension of
divergence sets in Section 3.

#### Notations

Throughout this paper, a generic constant $C>0$ depends only on dimension $d$,
which may change from line to line. If a constant depends on some other values
(e.g. $\epsilon$), we denote it by $C_{\epsilon}$. The notation $A\lesssim B$
denotes $A\leq CB$ for a constant $C>0$, and we denote by $A\sim B$ if
$A\lesssim B$ and $B\lesssim A$. We often denote $L^{2}(\mathbb{R}^{d})$ by
$L^{2}$, and similarly ${H}^{s}(\mathbb{R}^{d})$ by ${H}^{s}$.

##  2\. Temporally localized maximal estimates

In this section, we prove that the estimates (1.9) and (1.10) are equivalent.
For later use, we consider the equivalence in a more general setting, that is
to say, in the form of estimates relative to fractal measures (Lemma 2.2). To
do this, we recall the following.

######  Definition 2.1.

Let $0<\gamma\leq d$ and let $\mu$ be a nonnegative Borel measure. We say
$\mu$ is $\gamma$-dimensional if there is a constant $C_{\mu}$ such that

(2.1) $\displaystyle\mu(B(x,r))\leq
C_{\mu}r^{\gamma},\quad\forall(x,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}.$

By $\langle\mu\rangle_{\gamma}$ we denote the infimum of such a constant
$C_{\mu}$.

We first deduce a temporally localized maximal estimate from the estimate

(2.2) $\displaystyle\big{\|}\sup_{0<t\leq
1}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),d\mu)}\leq
CR^{\frac{d-\gamma}{2}+s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}$

which holds whenever $\widehat{f}$ is supported on $\mathbb{A}_{R}$.

######  Lemma 2.2.

Let $p\geq 2$, $R\geq 1$, $\alpha>0$, $0<\gamma\leq d$. Let $I\subset[0,1]$ be
an interval such that $|I|\leq\min\\{R^{1-\alpha},1\\}$. Suppose that (2.2)
holds for some $s$ whenever $\widehat{f}$ is supported in $\mathbb{A}_{R}$ and
$\mu$ is a $\gamma$-dimensional measure in $\mathbb{R}^{d}$. Then, for any
$\gamma$-dimensional measure $\widetilde{\mu}$ in $\mathbb{R}^{d}$, there is a
constant $C^{\prime}>0$ such that

(2.3) $\displaystyle\big{\|}\sup_{t\in
I}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),d\widetilde{\mu})}\leq
C^{\prime}\langle\widetilde{\mu}\rangle_{\gamma}^{\frac{1}{2}}R^{\frac{d-\gamma}{2}}(1+R^{\alpha}|I|)^{\max\\{s,\frac{s}{\alpha}\\}}\|f\|_{L^{2}}$

holds whenever $\widehat{f}$ is supported on $\mathbb{A}_{R}$. Conversely, if
(2.3) holds for $\widehat{f}$ supported on $\mathbb{A}_{R}$, $\widetilde{\mu}$
is $\gamma$-dimensional, and interval $I\subset[0,1]$ satisfies
$R^{-\alpha}<|I|\leq\min\\{R^{1-\alpha},1\\}$, then there exists $C>0$ such
that (2.2) holds whenever $\widehat{f}$ is supported on $\mathbb{A}_{R}$ and
$\mu$ is $\gamma$-dimensional.

######  Remark 2.3.

By a simple modification of our argument, Lemma 2.2 can be extended to a class
of evolution operators $e^{itP(D)}$ as long as

$|\partial_{\xi}^{\beta}P(\xi)|\lesssim|\xi|^{\alpha-|\beta|},\quad\forall\beta$

and $|\nabla P(\xi)|\gtrsim|\xi|^{\alpha-1}$ hold (see [8]). Hence, an
analogue of Theorem 1.1 holds true for $e^{itP(D)}$. A typical example of such
an operator is the non elliptic Schrödinger operator
$e^{it(\partial_{x_{1}}^{2}-\partial_{x_{2}}^{2}\pm\partial_{x_{3}}^{2}\pm\dots\pm\partial_{x_{d}}^{2})}$.

The rest of the section is devoted to the proof of Lemma 2.2, for which we
first consider spatial localization.

###  2.1. Spatial localization

By adapting the argument in [19, 8], we prove a spatial localization lemma
exploiting rapid decay of the kernel.

######  Lemma 2.4.

Let $\alpha>0$ and $r\geq 1$. Let $\mu$ be a $\gamma$-dimensional measure in
$\mathbb{R}^{d}$. For $R\geq 1$, we set $I_{R}=[0,R]$. Suppose that

(2.4)
$\big{\|}e^{it(-\Delta)^{\alpha/2}}f\big{\|}_{L_{x}^{2}(B(0,R),d\mu;L_{t}^{r}(I_{R}))}\leq
CR^{s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}$

holds for some $s\in\mathbb{R}$ whenever $\widehat{f}$ is supported in
$\mathbb{A}_{1}$. Then, there exists a constant $C_{1}>0$ such that

(2.5)
$\big{\|}e^{it(-\Delta)^{\alpha/2}}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d},d\mu;L_{t}^{r}(I_{R}))}\leq
C_{1}R^{s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}$

holds whenever $\widehat{f}$ is supported in $\mathbb{A}_{1}$.

###### Proof.

Let $P$ be a projection operator defined by
$\mathcal{F}(Pg)(\xi)=\beta(|\xi|)\widehat{g}(\xi)$ where $\beta\in
C_{c}(2^{-1},2^{2})$ and $\beta=1$ on $[1,2]$. Let $\\{B\\}$ be a collection
of finitely overlapping balls of radius $R$ which cover $\mathbb{R}^{d}$.
Denote $\widetilde{B}=B(a,10\alpha 2^{2|\alpha-1|}R)$ if $B=B(a,R)$. Then, we
note that
$\|F\|_{L^{2}(\mathbb{R}^{d})}^{2}\lesssim\sum_{B}\|F\|_{L^{2}(B)}^{2}$.

Since $Pf=f$, by Minkowski’s inequality we have

$\big{\|}e^{it(-\Delta)^{\alpha/2}}f\big{\|}_{L_{x}^{2}(\mathbb{R}^{d},d\mu;L_{t}^{r}(I_{R}))}^{2}\lesssim(\mathcal{L}_{1}+\mathcal{L}_{2}),$

where

$\displaystyle\mathcal{L}_{1}$
$\displaystyle=\sum_{B}\big{\|}e^{it(-\Delta)^{\alpha/2}}P(\chi_{\widetilde{B}}f)\big{\|}_{L_{x}^{2}(B,d\mu;L_{t}^{r}(I_{R}))}^{2},$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=\sum_{B}\big{\|}e^{it(-\Delta)^{\alpha/2}}P(\chi_{\widetilde{B}^{\mathrm{c}}}f)\big{\|}_{L_{x}^{2}(B,d\mu;L_{t}^{r}(I_{R}))}^{2}.$

Note that $e^{it(-\Delta)^{\alpha/2}}Pf=K(\cdot,t)\ast f$ where $K$ is given
by

$K(x,t)=\int e^{i(x\cdot\xi+t|\xi|^{\alpha})}\beta(|\xi|)\,d\xi.$

By integration by parts, it is easy to see $|K(x,t)|\leq
C_{N}R^{-N}(1+|x|)^{-N}$ for any $N\geq 1$ if $|x|>10\alpha 2^{2|\alpha-1|}R$
and $|t|\leq R$. Thus, we have

$\|e^{it(-\Delta)^{\alpha/2}}(\chi_{\widetilde{B}^{\mathrm{c}}}f)(x)\|_{L_{t}^{r}(I_{R})}\leq
C_{N}R^{-N}(1+|\cdot|)^{-N}\ast|f|(x)$

for any $N\geq 0$ whenever $x\in B$. Taking $N$ large enough, we get

$\mathcal{L}_{2}\leq
CR^{2s}\|(1+|\cdot|)^{-(d+\gamma)}\ast|f|\|_{L_{x}^{2}(\mathbb{R}^{d},d\mu)}^{2}.$

By Schur’s test it follows that
$\|(1+|\cdot|)^{-(d+\gamma)}\ast|f|\|_{L_{x}^{2}(B,d\mu)}\leq
C\langle\mu\rangle^{\frac{1}{2}}\|f\|_{2}$. Therefore, we only need to
consider $\mathcal{L}_{1}$.

Applying (2.4) on each $B$, we obtain

$\textstyle\mathcal{L}_{1}\lesssim
R^{2s}\langle\mu\rangle\sum_{B}\|\chi_{\widetilde{B}}f\|_{L^{2}}^{2}\leq
CR^{2s}\langle\mu\rangle\|f\|_{L^{2}}^{2}.$

The last inequality follows since the balls $\widetilde{B}$ overlap finitely.
This completes the proof. ∎

###  2.2. Proof of Lemma 2.2

To prove Lemma 2.2, we invoke an elementary lemma.

######  Lemma 2.5 ([17]).

Let $\mu$ be a $\gamma$-dimensional measure in $\mathbb{R}^{d}$. If
$\widehat{F}$ is supported on $B(0,R)$, then $\|F\|_{L^{2}(d\mu)}\leq
CR^{\frac{d-\gamma}{2}}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|F\|_{L^{2}(\mathbb{R}^{d})}.$

By translation and Plancherel’s theorem, we may assume that $I=[0,\delta]$
with $\delta\leq\min\\{R^{1-\alpha},1\\}$. We may further assume
$R^{-\alpha}<\delta$ since (2.3) follows by the Sobolev embedding and Lemma
2.5 if $\delta\leq R^{-\alpha}$.

For a givne $\gamma$-dimensional measure $\mu$, we denote by $\mu_{R}$ the
measure defined by the relation 111$\mu_{R}$ is a positive Borel measure by
the Riesz representation theorem.

(2.6) $\displaystyle\int F(x)\,d\mu_{R}(x)=R^{\gamma}\int F(Rx)\,d\mu(x),\quad
F\in C_{0}(\mathbb{R}^{d}).$

It is easy to see that $\mu_{R}$ is a $\gamma$-dimensional measure in
$\mathbb{R}^{d}$ such that

$\mu_{R}(B(x,r))\leq C\langle\mu\rangle_{\gamma}r^{\alpha}$

for some $C>0$. Changing variables $(x,t)\rightarrow(R^{-1}x,R^{-\alpha}t)$
and $\xi\rightarrow R\xi$, we see that (2.2) is equivalent to

(2.7)
$\displaystyle\textstyle\big{\|}\sup_{t\in[0,R^{\alpha}]}|e^{it(-\Delta)^{\alpha/2}}f_{R}|\big{\|}_{L^{2}(B(0,R),d\mu_{R})}\leq
CR^{s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f_{R}\|_{L^{2}}$

where $\widehat{f_{R}}(\xi)=R^{\frac{d}{2}}\widehat{f}(R\xi)$. Note that
$\|f_{R}\|_{2}=\|f\|_{2}$ and $\widehat{f}_{R}$ is supported on
$\mathbb{A}_{1}$. Let us denote $R^{\prime}=\min\\{R,R^{\alpha}\\}$. We claim
that the estimate (2.7) is equivalent to the seemingly weaker estimate

(2.8)
$\displaystyle\textstyle\big{\|}\sup_{t\in[0,R^{\prime}]}|e^{it(-\Delta)^{\alpha/2}}g|\big{\|}_{L^{2}(B(0,R^{\prime}),d\mu_{R})}\leq
CR^{s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|g\|_{L^{2}}$

for $R\geq 1$ whenever $\widehat{g}$ is supported on $\mathbb{A}_{1}$. To show
this, we only need to prove that (2.8) implies (2.7) since the converse is
trivially true. When $\alpha>1$, the implication from (2.8) to (2.7) was shown
in [8] (also, see [19, 25]) when $\mu_{R}$ is the Lebesgue measure and
$\alpha$ is an integer. It is easy to see that the argument in [8] works for
general $\gamma$-dimensional measure $\mu_{R}$. When $0<\alpha\leq 1$, using
Lemma 2.4 with $R$ replaced by $R^{\alpha}$, we get (2.7) from (2.8). This
proves the claim.

We now show that (2.2) and (2.3) are equivalent. Recall that we are assuming
that $R^{-\alpha}<\delta$. Changing variables
$(x,t)\rightarrow(R^{-1}x,R^{-\alpha}t)$, $\xi\rightarrow R\xi$ in (2.3) as
above, we see that (2.3) is equivalent to

(2.9)
$\displaystyle\big{\|}\sup_{t\in[0,R^{\alpha}\delta]}|e^{it(-\Delta)^{\alpha/2}}f_{R}|\big{\|}_{L^{2}(B(0,R),d\widetilde{\mu}_{R})}\leq
C^{\prime}(R^{\alpha}\delta)^{\max\\{s,\frac{s}{\alpha}\\}}\langle\widetilde{\mu}\rangle_{\gamma}^{\frac{1}{2}}\|f_{R}\|_{L^{2}}.$

By Lemma 2.4 with $R$ replaced by $R^{\alpha}\delta$, (2.9) follows from

(2.10)
$\displaystyle\big{\|}\sup_{t\in[0,R^{\alpha}\delta]}|e^{it(-\Delta)^{\alpha/2}}f_{R}|\big{\|}_{L^{2}(B(0,R^{\alpha}\delta),d\widetilde{\mu}_{R})}\leq
C^{\prime}(R^{\alpha}\delta)^{\max\\{s,\frac{s}{\alpha}\\}}\langle\widetilde{\mu}\rangle_{\gamma}^{\frac{1}{2}}\|f_{R}\|_{L^{2}}.$

Thus, (2.10) and (2.9) are trivially equivalent. Therefore, to show the
equivalence of (2.2) and (2.3), it is enough to prove that of (2.8) and
(2.10). Indeed, it is clear that (2.10) follows from (2.8) by replacing
$R^{\prime}$ in (2.8) with $R^{\alpha}\delta$. Conversely, if we replace
$R^{\alpha}\delta$ in (2.10) with $R^{\prime}$, we get (2.8) as long as
$\delta>R^{-\alpha}$. ∎

##  3\. Maximal estimate for sequential Schrödinger means

In this section, we prove Theorem 1.1 and obtain results regarding upper
bounds on the dimension of the divergence set of
$e^{it_{n}(-\Delta)^{\alpha/2}}f$. The results are consequence of extension of
the maximal estimates to general measure. See Section 3.2.

###  3.1. $L^{2}$-maximal estimates

Making use of Lemma 2.2, we deduce the maximal estimates for the sequential
Schrödinger mean from the estimate (2.2).

######  Proposition 3.1.

Let $R\geq 1$, $\alpha>0$, and $0<r<\infty$. Suppose that (2.2) holds for some
$s=s_{*}>0$ whenever $\mu$ is a $\gamma$-dimensional measure in
$\mathbb{R}^{d}$ and $\text{supp\\! }\widehat{f}\subset\mathbb{A}_{R}$. Let

(3.1)
$\displaystyle\tilde{s}_{*}=\frac{r\alpha}{r\min\\{\alpha,1\\}+2s_{*}}s_{*}.$

Then, if $\\{t_{n}\\}\in\ell^{r,\infty}$, there is a constant $C>0$ such that

(3.2)
$\displaystyle\|\sup_{n}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\|_{L^{2}(B(0,1),d\mu)}\leq
CR^{\frac{d-\gamma}{2}+s}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}$

holds for $s\geq\min\\{s_{*},\tilde{s}_{*}\\}$ whenever $\mu$ is
$\gamma$-dimensional and $\text{supp\\! }\widehat{f}\subset\mathbb{A}_{R}$.

When $\alpha>1$, Proposition 3.1 is meaningful only for
$r<2s_{\ast}/(\alpha-1)$.

###### Proof of Proposition 3.1.

We may assume $\tilde{s}_{*}\leq s_{*}$ since (3.2) trivially holds for $s\geq
s_{*}$ by the maximal estimate (2.2). For $0<\delta<1$, let us set

$\mathrm{N}(\delta)=\\{n\in\mathbb{N}:~{}t_{n}<\delta\\}.$

Since $\\{t_{n}\\}\in\ell^{r,\infty}(\mathbb{N})$, there is a uniform constant
$C_{0}>0$ such that

(3.3) $|\mathrm{N}(\delta)^{\mathrm{c}}|\leq C_{0}\delta^{-r}.$

Then, it follows that

$\big{\|}\sup_{n}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),\mu)}\leq\mathcal{I}+\mathcal{I\\!\\!I},$

where

$\displaystyle\mathcal{I}$
$\displaystyle=\big{\|}\sup_{n\in\mathrm{N}(\delta)}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),d\mu)},$
$\displaystyle\mathcal{I\\!\\!I}$
$\displaystyle=\big{\|}\sup_{n\in\mathrm{N}(\delta)^{\mathrm{c}}}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),d\mu)}.$

We consider $\mathcal{I}$ first. Since
$\sup_{n\in\mathrm{N}(\delta)}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\leq\sup_{0<t\leq\delta}|e^{it(-\Delta)^{\alpha/2}}f|$,
by Lemma 2.2 we have

(3.4) $\displaystyle\mathcal{I}\leq
CR^{\frac{d-\gamma}{2}}(R^{\alpha}\delta)^{\max\\{s_{*},\frac{s_{*}}{\alpha}\\}}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}$

provided that $\text{supp\\! }\widehat{f}\subset\mathbb{A}_{R}$ and
$R^{-\alpha}\leq\delta\leq\min\\{R^{-\alpha+1},1\\}$. To handle
$\mathcal{I\\!\\!I}$, we first note that

$\|e^{it_{n}(-\Delta)^{\alpha/2}}f\|_{L^{2}(B(0,1),d\mu)}\lesssim
R^{\frac{d-\gamma}{2}}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{2},$

which follows by Lemma 2.5 and Plancherel’s theorem. So, by the embedding
$\ell^{2}\subset\ell^{\infty}$, combining the above estimate and (3.3), we
obtain

$\textstyle\mathcal{I\\!\\!I}\lesssim\big{(}\sum_{n\in\mathrm{N}(\delta)^{\mathrm{c}}}\big{\|}e^{it_{n}(-\Delta)^{\alpha/2}}f\big{\|}_{L^{2}(B(0,1),d\mu)}\big{)}^{1/2}\lesssim
C_{0}^{\frac{1}{2}}\delta^{-\frac{r}{2}}R^{\frac{d-\gamma}{2}}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{L^{2}}.$

Now we prove (3.2) by optimizing the estimates with a suitable choice of
$\delta$. When $\alpha\geq 1$, we take $\delta=R^{-2s_{*}\alpha/(r+2s_{*})}$,
which gives (3.2) for $s\geq rs_{*}\alpha/(r+2s_{*}).$ When $0<\alpha<1$, we
choose $\delta=R^{-2s_{*}\alpha/(r\alpha+2s_{*})}$ and obtain (3.2) for $s\geq
rs_{*}\alpha/(r\alpha+2s_{*}).$ In both cases, one can easily check
$R^{-\alpha}\leq\delta\leq\min\\{R^{-\alpha+1},1\\}$ for $r$ and $s_{*}$
satisfying $\tilde{s}_{*}\leq s_{*}$. Indeed, if $\alpha>1$, then $\delta\leq
R^{-\alpha+1}$ since $2s_{*}+r\geq r\alpha$. When $0<\alpha\leq 1$, we have
$\delta\leq 1$ since $s_{*}>0$. ∎

Theorem 1.1 is an immediate consequence of the following.

######  Corollary 3.2.

Let $0<\gamma\leq d$ and $0<r<\infty$. Suppose

(3.5)
$\displaystyle\big{\|}\sup_{0<t<1}|e^{it(-\Delta)^{\alpha/2}}f|\big{\|}_{L^{2}(B(0,1),d\mu)}\leq
C\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+s_{*}}}$

holds for some $0<s_{*}$ whenever $\mu$ is a $\gamma$-dimensional measure in
$\mathbb{R}^{d}$. Then, if $\\{t_{n}\\}\in\ell^{r,\infty}$, there is a
constant $C^{\prime}>0$ such that

(3.6)
$\displaystyle\|\sup_{n\in\mathbb{N}}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\|_{L^{2}(B(0,1),d\mu)}\leq
C^{\prime}\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+s}}$

holds for $s\geq\min\\{s_{*},\tilde{s}_{*}\\}$ where $\tilde{s}_{*}$ is given
by (3.1).

The estimate (3.5) clearly implies (2.2). However, to prove Corollary 3.2, we
need to remove the frequency localization in the estimate (3.2) so that the
right hand side of (3.2) is replaced by
$C\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+s}}$.
This can be achieved by adapting the argument in [11].

###### Proof of Corollary 3.2.

As before, we may assume $\tilde{s}_{*}\leq s_{*}$. It suffices to show (3.6)
for $s\geq\tilde{s}_{*}$. Let us choose a smooth function $\beta\in
C_{0}^{\infty}((1/2,2))$ such that $\sum_{k}\beta(2^{-k}\cdot)=1$ and set
$\beta_{0}=\sum_{k\leq 0}\beta(2^{-k}\cdot)$. Let $P_{k}$, $k\geq 0$, be the
projection operator defined by
$\widehat{P_{k}f}(\xi)=\beta(2^{-k}|\xi|)\widehat{f}(\xi)$, $k\geq 1$, and
$\widehat{P_{0}f}(\xi)=\beta_{0}(|\xi|)\widehat{f}(\xi)$.

For $\ell\geq 0$, we set

$\mathrm{N}_{\ell}=\big{\\{}n\in\mathbb{N}:2^{-2(\ell+1)\tilde{s}_{*}/r}<t_{n}\leq
2^{-2\ell\tilde{s}_{*}/r}\big{\\}}.$

For each $\ell\geq 0$, we write $f=\sum_{0\leq k<\ell}P_{k}f+\sum_{k\geq
0}P_{\ell+k}f$. So, we have

$\Big{\|}\sup_{n\in\mathbb{N}}|e^{it_{n}(-\Delta)^{\alpha/2}}f|\Big{\|}_{L^{2}(B(0,1),d\mu)}\leq\mathrm{I}+\mathrm{I\\!I},$

where

$\displaystyle\mathrm{I}$ $\displaystyle=\sup_{\ell\geq
0}\Big{\|}\sup_{n\in\mathrm{N}_{\ell}}\big{|}e^{it_{n}(-\Delta)^{\alpha/2}}(\sum_{0\leq
k<\ell}P_{k}f)\big{|}\Big{\|}_{L^{2}(B(0,1),d\mu)},$
$\displaystyle\mathrm{I\\!I}$ $\displaystyle=\sup_{\ell\geq
0}\Big{\|}\sup_{n\in\mathrm{N}_{\ell}}\big{|}\sum_{k\geq
0}e^{it_{n}(-\Delta)^{\alpha/2}}P_{\ell+k}f\big{|}\Big{\|}_{L^{2}(B(0,1),d\mu)}.$

We consider $\mathrm{I\\!I}$ first. Since $\\{t_{n}\\}\in\ell^{r,\infty}$, it
follows that $|\mathrm{N}_{\ell}|\lesssim 2^{2\tilde{s}_{*}\ell}$. As before,
by the embedding $\ell^{2}\subset\ell^{\infty}$ and then applying Lemma 2.5
and Plancherel’s theorem, we get

$\Big{\|}\sup_{n\in\mathrm{N}_{\ell}}\big{|}\sum_{k\geq
0}e^{it_{n}(-\Delta)^{\alpha/2}}P_{\ell+k}f\big{|}\Big{\|}_{L^{2}(B(0,1),d\mu)}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sum_{k\geq
0}\Big{(}\sum_{\ell\geq
0}2^{(d-\gamma)(\ell+k)+2\tilde{s}_{*}\ell}\|P_{\ell+k}f\|_{L^{2}}^{2}\Big{)}^{\frac{1}{2}}.$

Thus, we obtain

(3.7)
$\displaystyle\mathrm{I\\!I}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sum_{k\geq
0}2^{-\tilde{s}_{*}k}\|f\|_{H^{\frac{d-\gamma}{2}+\tilde{s}_{*}}}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+\tilde{s}_{*}}}.$

We now turn to $\mathrm{I}$. Note that ${2\tilde{s}_{*}}<{r\alpha}$ and
decompose

$\sum_{0\leq k<\ell}P_{k}f=\sum_{0\leq
k<\frac{2\tilde{s}_{*}}{r\alpha}\ell}P_{k}f+\sum_{0<k\leq\frac{r\alpha-2\tilde{s}_{*}}{r\alpha}\ell}P_{\ell-k}f.$

By the Minkowski inequality, we have
$\mathrm{I}\leq\mathrm{I}_{a}+\mathrm{I}_{b}$ where

$\displaystyle\mathrm{I}_{a}$ $\displaystyle=\sum_{k\geq
0}\big{\|}\sup_{\ell>\frac{r\alpha}{2\tilde{s}_{*}}k}\,\sup_{n\in\mathrm{N}_{\ell}}|e^{it_{n}(-\Delta)^{\alpha/2}}P_{k}f|\big{\|}_{L^{2}(B(0,1),d\mu)},$
$\displaystyle\mathrm{I}_{b}$ $\displaystyle=\sup_{\ell\geq
0}\Big{\|}\sup_{n\in\mathrm{N}_{\ell}}\big{|}e^{it_{n}(-\Delta)^{\alpha/2}}(\sum_{0<k\leq\frac{r\alpha-2\tilde{s}_{*}}{r\alpha}\ell}P_{\ell-k}f)\big{|}\Big{\|}_{L^{2}(B(0,1),d\mu)}.$

Regarding $\mathrm{I}_{a}$, note that
$\bigcup_{\ell>\frac{r\alpha}{2\tilde{s}_{*}}k}\mathrm{N}_{\ell}\subset[0,2^{-\alpha
k}]$. By (3.5) and Lemma 2.2 with $R=2^{k}$ and $I=[0,2^{-\alpha k}]$, we
obtain

(3.8)
$\displaystyle\mathrm{I}_{a}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sum_{k\geq
0}2^{\frac{d-\gamma}{2}k}\|P_{k}f\|_{2}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+s}}$

for $s>0.$

To handle $\mathrm{I}_{b}$, we note that $t_{n}\in
J_{\ell}:=[0,2^{-2\ell\tilde{s}_{*}/r}]$ if $n\in\mathrm{N}_{\ell}$, and
$2^{(\ell-k)\alpha}|J_{\ell}|\geq 1$ since
$k\leq\frac{r\alpha-2\tilde{s}_{*}}{r\alpha}\ell$. By Lemma 2.2 with
$R=2^{\ell-k}$ and $I=J_{\ell}$, it follows that

$\mathrm{I}_{b}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sup_{\ell\geq
0}\sum_{0\leq k\leq\frac{r\alpha-2\tilde{s}_{*}}{r\alpha}\ell}\
(2^{\ell-k})^{\frac{d-\gamma}{2}}\big{(}2^{(\ell-k)\alpha-2\ell\tilde{s}_{*}/r}\big{)}^{\max\\{s_{*},\frac{s_{*}}{\alpha}\\}}\|P_{\ell-k}f\|_{L^{2}}.$

Using (3.1) and the fact that
$\min\\{\alpha,1\\}\times\max\\{s_{*},\frac{s_{*}}{\alpha}\\}=s_{*}$, one can
easily see that
$\big{(}2^{(\ell-k)\alpha-2\ell\tilde{s}_{*}/r}\big{)}^{\max\\{s_{*},\frac{s_{*}}{\alpha}\\}}=2^{-k\alpha\max\\{s_{*},\frac{s_{*}}{\alpha}\\}}2^{\ell\tilde{s}_{*}}.$
So, by the embedding $\ell^{2}\subset\ell^{\infty}$ and Minkowski’s
inequality, we get

$\displaystyle\mathrm{I}_{b}$
$\displaystyle\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sum_{k\geq
0}\Big{(}\sum_{\ell>\frac{r\alpha}{r\alpha-2\tilde{s}_{*}}k}(2^{\ell-k})^{d-\gamma}2^{2\ell\tilde{s}_{*}}2^{-2k\max\\{\alpha
s_{*},s_{*}\\}}\|P_{\ell-k}f\|_{L^{2}}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\sum_{k\geq
0}2^{(\tilde{s}_{*}-\max\\{\alpha
s_{*},s_{*}\\})k}\|f\|_{H^{\frac{d-\gamma}{2}+\tilde{s}_{*}}}.$

Thus, we have
$\mathrm{I}_{b}\lesssim\langle\mu\rangle_{\gamma}^{\frac{1}{2}}\|f\|_{H^{\frac{d-\gamma}{2}+\tilde{s}_{*}}}.$
Combining this and the estimates (3.7) and (3.8), we obtain (3.6). ∎

###  3.2. Dimension of divergence set

Via the implication in Corollary 3.2, we can obtain upper bounds on the
divergence set $\mathfrak{D}^{\alpha,d}(s,r)$ making use of the known
estimates for the maximal Schrödinger operator $f\to\sup_{0<t\leq
1}|e^{it(-\Delta)^{\alpha/2}}f|$. We start with recalling the following lemma
([27, 17]).

######  Lemma 3.3.

Suppose

(3.9) $\displaystyle\|e^{it(-\Delta)^{\alpha/2}}f\|_{L^{2}(B(0,1),d\nu)}\leq
C^{\prime}\langle\nu\rangle_{\gamma}^{\frac{1}{2}}R^{\frac{d-\gamma}{2}+s}\|f\|_{L^{2}(\mathbb{R}^{d})},$

holds for some $s\in\mathbb{R}$ whenever $\text{supp\\!
}\widehat{f}\subset\mathbb{A}_{R}$ and $\nu$ is a $\gamma$-dimensional measure
in $\mathbb{R}^{d+1}$. Then there exists $C>0$ such that (2.2) holds whenever
$\mu$ is a $\gamma$-dimensional measure in $\mathbb{R}^{d}$.

In what follows, we summarize the currently available maximal estimates (2.2)
which can be obtained by the best known fractal Strichartz estimates (3.9)
combined with Lemma 3.3.

######  Proposition 3.4.

Let $d\geq 1$, $\alpha\in(0,1)\cup(1,\infty)$, and $\mu$ be a
$\gamma$-dimensional measure in $\mathbb{R}^{d}$. Then (2.2) holds for
$s>s_{\alpha}(\gamma,d)$ where

$\displaystyle
s_{\alpha}(\gamma,d)=\begin{cases}\min\big{\\{}\max\big{\\{}0,~{}\frac{\gamma}{2}-\frac{d}{4}\big{\\}},~{}\frac{\gamma}{2(d+1)}\big{\\}},&\quad\text{when
}\alpha>1,\\\
\min\big{\\{}\max\\{0,\alpha(\frac{\gamma}{2}-\frac{d}{4})\big{\\}},\frac{\alpha}{2}\big{\\}},&\quad\text{when
}0<\alpha<1.\end{cases}$

When $\alpha>1$, the estimate (2.2) and (3.9) for $s>s_{\alpha}(\gamma,d)$
were already obtained in [2, 24, 14]). The estimate for $0<\alpha<1$ also can
be shown by the standard argument in [34, 15] and Lemma 3.3. However, since
the latter case is less well known, we provide the proof for the convenience
of the reader.

###### Proof of Proposition 3.4 for $0<\alpha<1$.

By Lemma 3.3 we only need to prove (3.9) for $s>s_{\alpha}(\gamma,d)$.

Let $\sigma$ be the surface measure on
$\\{(\xi,|\xi|^{\alpha}):1/2\leq|\xi|\leq 2\\}$. In [34, 15], it was shown
that

(3.10) $\displaystyle\int|\widehat{\nu}(R\eta)|^{2}\,d\sigma(\eta)\lesssim
I_{\gamma}(\nu)R^{-\beta}$

holds for $\beta=\max\\{\min\\{\gamma,\frac{d}{2}\\},\gamma-1\\}$ whenever
$\nu$ is a $\gamma$-dimensional measure in $\mathbb{R}^{d+1}$. Here
$I_{\gamma}(\nu)$ denote the $\gamma$-dimensional energy of $\nu$. Let
$\nu_{\lambda}$ be the rescaled measure defined by the relation (2.6) with $d$
replaced by $d+1$. Then, it is easy to see (see e.g., [17, 18]) that (3.10)
implies the estimate

(3.11)
$\displaystyle\|e^{it(-\Delta)^{\alpha/2}}g\|_{L^{2}(B(0,\lambda),d\nu_{\lambda})}\leq
C\lambda^{s}\langle\nu\rangle_{\gamma}^{\frac{1}{2}}\|g\|_{2}$

for $s>({\gamma-\beta})/2$ whenever $\nu$ is $\gamma$-dimensional and
$\widehat{g}$ is supported on $\mathbb{A}_{1}$ ([43, 16]). Therefore, we have
(3.11) for $s>s_{\alpha}(\gamma,d)/\alpha$.

Now we take $\lambda=R^{\alpha}$ in (3.11). Then, applying Lemma 2.4 with $R$
replaced by $R^{\alpha}$, we have

$\displaystyle\|e^{it(-\Delta)^{\alpha/2}}g\|_{L^{2}(B(0,R)\times[0,R^{\alpha}],d\nu_{R^{\alpha}})}\leq
CR^{s}\langle\nu\rangle_{\gamma}^{\frac{1}{2}}\|g\|_{2}$

for $s>s_{\alpha}(\gamma,d)$. By rescaling $\xi\rightarrow R^{-1}\xi$ and
$(x,t)\rightarrow(Rx,R^{\alpha}t)$, we see that (3.9) holds for
$s>s_{\alpha}(\gamma,d)$. ∎

We recall the maximal estimate (2.2) for the wave operator shown in [2, 17].
(See also [8, 18] for the fractal Strichartz estimates (3.9)).

######  Proposition 3.5.

Let $I$ be a subinterval in $[0,1]$. Then (2.2) holds with $\alpha=1$ for
$s>s_{1}(\gamma,d)$ where

$\displaystyle s_{1}(\gamma,d)$ $\displaystyle=\begin{cases}\ \ 0,\ \ \ \ \ &\
\quad 0<\gamma\leq\frac{1}{2},\\\\[2.15277pt] \ \frac{\gamma}{2}-\frac{1}{4},\
\ \ \ \ &\,\quad\frac{1}{2}<\gamma\leq 1,\\\\[2.15277pt] \ \
\frac{\gamma}{4},\ \ \ \ \ &\ \quad 1<\gamma\leq 2,\\\ \end{cases}\ \
\quad\text{ for }d=2;$ $\displaystyle s_{1}(\gamma,d)$
$\displaystyle=\begin{cases}\ \ 0,&\ \ \ \quad
0<\gamma\leq\frac{d-1}{2},\\\\[2.15277pt]
\frac{-d+1+2\gamma}{8},&\quad\frac{d-1}{2}<\gamma\leq\frac{d+1}{2},\\\\[2.15277pt]
\ \frac{\gamma-1}{2(d-1)},&\quad\frac{d+1}{2}<\gamma\leq
d,\end{cases}\quad\text{ for }d\geq 3.$

By Proposition 3.1, the estimates in Proposition 3.4 and 3.5 give the
corresponding estimates for $\sup_{n}|e^{it_{n}(-\Delta)^{\alpha/2}}f|$ with
$\\{t_{n}\\}\in\ell^{r,\infty}$ relative to $\gamma$-dimensional measures.
Then, by a standard argument (see [2]), one can obtain upper bounds on the
Hausdorff dimension of the divergence sets. We summarize the results as
follows:

######  Corollary 3.6.

Let $\alpha>0$, $d\geq 1$, $r\in(0,\infty)$, and $0<\gamma\leq d$. Let
$s_{\ast}=s_{\alpha}(\gamma,d)$ which is given in Proposition 3.4 and 3.5.
Then, $\mathfrak{D}^{\alpha,d}(s,r)\leq\gamma$ if
$s>2^{-1}(d-\gamma)+\min\\{s_{\ast},\tilde{s}_{\ast}\\}$.

### Acknowledgement

This work was supported by the NRF (Republic of Korea) grants No.
2020R1I1A1A01072942 (Cho), No. 2022R1I1A1A01055527 (Ko), No.
2022R1F1A1061968(Koh), and No. 2022R1A4A1018904 (Lee).

## References

  * [1] T. Anderson, K. Hughes, J. Roos, A. Seeger, $L^{p}\rightarrow L^{q}$ bounds for spherical maximal operators, Math. Z. 297 (2021), 1057–1074.
  * [2] J. A. Barceló, J. Bennett, A. Carbery, K. M. Rogers, On the dimension of divergence sets of dispersive equations, Math. Ann. 349 (2011), no. 3, 599–622.
  * [3] J. Bourgain, On the Schrödinger maximal function in higher dimension, Proceedings of the Steklov Institute of Math. 2013. 280 (2012), 46–60.
  * [4] by same author, A note on the Schrödinger maximal function, J. Anal. Math. 130 (2016), 393–396.
  * [5] L. Carleson, Some analytic problems related to statistical mechanics, Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md, 1979), Lecture Notes in Math., 779, Springer, Berlin, (1980), 5–45.
  * [6] C.-H. Cho, S. Ham, S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal. 150 (2017), 61–75.
  * [7] C.-H. Cho, H. Ko, Pointwise convergence of the fractional Schrödinger equation in $\mathbb{R}^{2}$, Taiwanese J. Math. 26 (2022), 177-200, DOI: 10.11650/tjm/210904.
  * [8] C.-H. Cho, S. Lee, A. Vargas, Problems on pointwise convergence of solutions to the Schrödinger equation, J. Fourier Anal. Appl. 18 (2012), 972–994.
  * [9] M. Cowling, Pointwise behavior of solutions to Schrödinger equations, Harmonic analysis (Cortona, 1982), 1983, 83–90.
  * [10] B. E. J. Dahlberg, C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis (Minneapolis, Minn, 1981), Lecture Notes in Math. 908, 205–209.
  * [11] E. Dimou, A. Seeger, On pointwise convergence of Schrödinger means, Mathematika, 66 (2020), 356–372.
  * [12] X. Du, L. Guth, X. Li, A sharp maximal Schrödinger estimate in $\mathbb{R}^{2}$, Ann. of Math. 186 (2) (2017), 607–640.
  * [13] X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, R. Zhang, Weighted restriction estimates and application to Falconer distance set problem, Amer. J. Math. 143 (2021), 175–211.
  * [14] X. Du, R. Zhang, Sharp $L^{2}$ estimate of Schrödinger maximal function in higher dimensions, Ann. of Math. 189 (2019), 837–861.
  * [15] M. B. Erdoğan, A note on the Fourier transform of fractal measures, Math. Res. Lett. 11 (2004), 299–313.
  * [16] by same author, On Falconer’s distance set conjecture, Rev. Mat. Iberoam. 22 (2006), no. 2, 649–662.
  * [17] S. Ham, H. Ko, S. Lee, Dimension of divergence set of the wave equation, Nonlinear Anal. 215 (2022), 112631, https://doi.org/10.1016/j.na.2021.112631.
  * [18] T. L. J. Harris, Improved decay of conical averages of the Fourier transform, Proc. Amer. Math. Soc. 147 (2019), 4781–4796.
  * [19] S. Lee, On pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}^{2}$, International Math. Research Notices. (2006), 21 (Art. ID 32597).
  * [20] S. Lee, K. Rogers, The Schrödinger equation along curves and the quantum harmonic oscillator, Adv. Math. 229 (2012), 1359–1379.
  * [21] W. Li, H. Wang, D. Yan, Pointwise Convergence for sequences of Schrödinger means in $\mathbb{R}^{2}$, arXiv:2010.08701.
  * [22] , Sharp convergence for sequences of nonelliptic Schrödinger means, arXiv:2011.10160.
  * [23] R. Lucá, K. Rogers, Average decay for the Fourier transform of measures with applications, J. Eur. Math. Soc. 21 (2019), 465–506.
  * [24] P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika, 34 (1987), 207–228.
  * [25] C. Miao, J. Yang, J. Zheng, An improved maximal inequality for $2D$ fractional order Schrödinger operators, Studia Mathematica. 230 (2015), 121–165
  * [26] A. Moyua, A. Vargas, L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, International Mathematics Research Notices 1996 (1996), 793–815.
  * [27] D. Oberlin, R. Oberlin, Spherical means and pinned distance sets, Commun. Korean Math. Soc. 30 (2015), 23–34.
  * [28] L. B. Pierce, On Bourgain’s counterexample for the Schrödinger maximal function, Quart. J. Math., 71:1309–1344, (2020).
  * [29] K. M. Rogers, A local smoothing estimate for the Schrödinger equation, Adv. Math. 219 (2008), 2105–2122.
  * [30] J. Roos, A. Seeger, Spherical maximal functions and fractal dimensions of dilation sets, arXiv:2004.00984.
  * [31] K. M. Rogers, A. Vargas, L. Vega, Pointwise convergence of solutions to the nonelliptic Schrödinger equation, Indiana University Mathematics Journal, (2006), 1893–1906.
  * [32] A. Seeger, S. Wainger, J. Wright, Pointwise convergence of spherical means, Math. Proc. Camb. Phil. Soc. 118 (1) (1995), 115–124.
  * [33] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Mathematical Journal 55 (1987), 699–715.
  * [34] by same author, Estimates of averages of Fourier transforms of measures with finite energy, Ann. Acad. Sci. Fenn. Math. 22 (1997), 227–236.
  * [35] by same author, Two Theorems on Convergence of Schrödinger Means, J. Fourier Anal. Appl. 25 (2019), no. 4, 1708–1716.
  * [36] P. Sjölin, J. O. Strömberg, Convergence of sequences of Schrödinger means, J. Math. Anal. Appl. 483 (2020), 123580, 23 pp.
  * [37] by same author, Schrödinger means in higher dimensions, J. Math. Anal. Appl. 504 (2021), 125353, 32 pp.
  * [38] by same author, Analysis of Schrödinger means, Ann. Fenn. Math. 46 (2021), 389–394.
  * [39] T. Tao, A. Vargas, A bilinear approach to cone multipliers. II. Applications, Geometric and Functional Analysis 10 (2000), 216–258.
  * [40] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proceedings of the American Mathematical Society, 102 (1988), 874–878.
  * [41] B. Walther, Maximal estimates for oscillatory integrals with concave phase, Contemp. Math. 189 (1995), 485–495.
  * [42] by same author, Some $L^{p}(L^{\infty})$\- and $L^{2}(L^{2})$-estimates for oscillatory Fourier transforms, Appl. Numer. Harmon. Anal. (1999), 213–231.
  * [43] T. Wolff, Decay of circular means of Fourier transforms of measures, Int. Math. Res. Not. 10 (1999), 547–567.