c\,{\mathsf{d}}_{{\mathbb{X}}}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}\int_{-1}^{1}\int_{-1}^{1}\left[\frac{n^{2{\lambda}+3}}{\big{(}1+n\sqrt{1-\xi_{1}(u,v)^{2}}\big{)}^{{\kappa}}}+\frac{n^{2{\lambda}+3}}{\big{(}1+n\sqrt{1-\xi_{2}(u,v)^{2}}\big{)}^{{\kappa}}}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\big{(}\Sigma_{1}+\Sigma_{2}(u)+\Sigma_{3}(v)\big{)}(1-u^{2})^{\mu-1}(1-v^{2})^{{\gamma}-1}\mathrm{d}u\mathrm{d}v.$

The integrals that contain $\Sigma_{1}$ and $\Sigma_{3}(v)$ can be estimated
exactly as in the proof of Theorem 3.10. The integral that contains
$\Sigma_{2}(u)$ does not cause additional problem and can be handled just as
the integral containing $\Sigma_{3}(v)$. We omit the details. ∎

The case $p=1$ of the following lemma establishes Assertion 3 for
$W_{{\gamma},\mu}^{\varrho}$.

######  Lemma 4.11.

Let $d\geq 2$, ${\gamma}>-\frac{1}{2}$ and $\mu>-\frac{1}{2}$. For
$0<p<\infty$, assume
${\kappa}>\frac{2d+2}{p}+2({\beta}+{\gamma})|\frac{1}{p}-\frac{1}{2}|$. Then
for $(x,t)\in{\mathbb{X}}^{d+1}$,

$\displaystyle\int_{{\mathbb{X}}^{d+1}}\frac{W_{{\gamma},\mu}^{\varrho}(y,s)}{W_{{\gamma},\mu}^{\varrho}(n;y,s)^{\frac{p}{2}}\big{(}1+n{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}((x,t),(y,s))\big{)}^{{\kappa}p}}\mathrm{d}y\mathrm{d}s\leq
cn^{-d-1}W_{{\gamma},\mu}^{\varrho}(n;x,t)^{1-\frac{p}{2}}.$

###### Proof.

Again, it suffices to consider $\varrho=0$. Let $J_{p}$ denote the left-hand
side the inequality to be proved. As in the case of ${\mathbb{X}}_{0}$, we
only need to estimate $J_{2}$. Furthermore, following the proof of Lemma 3.11,
we only need to estimate the integral in $J_{2}$ over either
${\mathbb{X}}_{+}^{d+1}$ or ${\mathbb{X}}_{-}^{d+1}$, which we choose as
${\mathbb{X}}_{+}^{d+1}={\mathbb{V}}^{d+1}$ and denote it by $J_{2,+}$. Then

$\displaystyle
J_{2,+}=\int_{0}^{1}s^{d}\int_{{\mathbb{B}}^{d}}\frac{W_{{\gamma},\mu}^{0}(sy^{\prime},s)}{W_{{\gamma},\mu}^{0}(n;sy^{\prime},s)(1+n{\mathsf{d}}_{{\mathbb{X}}}((x,t),(sy^{\prime},s)))^{2{\kappa}}}\mathrm{d}y^{\prime}\mathrm{d}s.$

Let $x=tx^{\prime}$ and $y=sy^{\prime}$ with
$x^{\prime},y^{\prime}\in{\mathbb{B}}^{d}$. Using
${\mathsf{w}}_{0,{\gamma}}^{0}(t)=(1-t^{2})^{{\gamma}-\frac{1}{2}}$ and
${\mathsf{w}}^{0}_{{\beta},{\gamma}}(n;s)=(1-t^{2}+n^{-2})^{{\gamma}}$, we can
easily verify that

$\frac{W_{{\gamma},\mu}^{0}(y,s)}{W_{{\gamma},\mu}^{0}(n;y,s)}\leq
c\frac{{\mathsf{w}}_{0,{\gamma}}^{0}(s)}{\sqrt{1-\|y^{\prime}\|^{2}}{\mathsf{w}}^{0}_{0,{\gamma}}(n;s)},$

which leads to

$\displaystyle
J_{2,+}\leq\int_{0}^{1}s^{d}\int_{{\mathbb{B}}^{d}}\frac{{\mathsf{w}}_{0,{\gamma}}^{0}(s)}{{\mathsf{w}}^{0}_{0,{\gamma}}(n;s)(1+n{\mathsf{d}}_{{\mathbb{X}}}((x,t),(sy^{\prime},s)))^{2{\kappa}}}\frac{\mathrm{d}y^{\prime}}{\sqrt{1-\|y^{\prime}\|^{2}}}\mathrm{d}s.$

Setting $x=tx^{\prime}$, $X=(x^{\prime},\sqrt{1-\|x^{\prime}\|^{2}})$ and
$Y=(y^{\prime},\sqrt{1-\|y^{\prime}\|^{2}})$, so that
${\mathsf{d}}_{{\mathbb{B}}^{d}}(x^{\prime},y^{\prime})={\mathsf{d}}_{\SS^{d}}(X,Y)$,
we use the identity

(4.15)
$\int_{{\mathbb{B}}^{d}}g\left(y^{\prime},\sqrt{1-\|y^{\prime}\|^{2}}\right)\frac{dy^{\prime}}{\sqrt{1-\|y^{\prime}\|^{2}}}=\int_{\SS_{+}^{d}}g(y)\mathrm{d}\sigma(y),$

where $\SS_{+}^{d}$ denotes the upper hemisphere of $\SS^{d}$, which allows us
to follow the proof of Lemma 3.11 to obtain

$\displaystyle J_{2,+}\,$ $\displaystyle\leq
c\int_{0}^{1}\int_{-1}^{1}\frac{s^{d}{\mathsf{w}}_{0,{\gamma}}^{0}(s)(1-u^{2})^{\frac{d-2}{2}}}{{\mathsf{w}}^{0}_{0,{\gamma}}(n;s)\left(1+n\sqrt{1-tsu-\sqrt{1-t^{2}}\sqrt{1-s^{2}}}\right)^{2{\kappa}}}\mathrm{d}u\mathrm{d}s.$

The integral in the right-hand side with $d$ replaced by $d+1$ appeared in the
proof of Lemma 3.11, and it is bounded by $cn^{-d-1}$ accordingly. ∎

######  Proposition 4.12.

For ${\gamma}\geq 0$, $\mu\geq 0$ and $(x,t)\in{\mathbb{X}}^{d+1}$,

$\int_{{\mathbb{X}}^{d+1}}\left|{\mathbf{L}}_{n}^{E}\big{(}W_{{\gamma},\mu}^{\varrho};(x,t),(y,s)\big{)}\right|^{p}W_{{\gamma},\mu}^{\varrho}(y,s)\mathrm{d}y\mathrm{d}s\leq\bigg{(}\frac{n^{d}}{W_{{\gamma},\mu}^{\varrho}(n;t)}\bigg{)}^{p-1}.$

This follows by applying Lemma 4.11 on the estimate in Theorem 4.9.

We have established Assertions 1 – 3 for
${\mathbf{L}}_{n}^{E}(W_{{\gamma},\mu};\cdot,\cdot)$. The kernel uses,
however, only polynomials that are even in $t$ and in $s$ variable.

######  Corollary 4.13.

For $d\geq 2$, $\varrho\geq 0$, ${\gamma}\geq 0$ and $\mu\geq 0$, the space
$({\mathbb{X}}^{d+1},W_{{\gamma},\mu}^{\varrho},{\mathsf{d}}_{{\mathbb{X}}}^{\varrho})$
is a localizable homogeneous space, where its localized kernels are defined
for polynomials even in $t$ and in $s$ variables.

###  4.5. Maximal ${\varepsilon}$-separated sets on the hyperbolic surface

We give a construction of maximal ${\varepsilon}$-separated on the hyperboloid
and the double cone, following the definition of Definition 2.3. Our
construction follows the one on ${\mathbb{X}}_{0}^{d+1}$ in Section 3.5. We
first need ${\varepsilon}$-separated set on the unit ball ${\mathbb{B}}^{d}$.
We adopt the following notation. For ${\varepsilon}>0$, we denote by
$\Xi_{{\mathbb{B}}}({\varepsilon})$ a maximal ${\varepsilon}$-separated set on
the unit ball ${\mathbb{B}}^{d}$ and we let ${\mathbb{B}}_{u}({\varepsilon})$
be the subsets in ${\mathbb{B}}^{d}$ so that the collection
$\\{{\mathbb{B}}_{u}({\varepsilon}):u\in\Xi_{\mathbb{B}}({\varepsilon})\\}$ is
a partition of ${\mathbb{B}}^{d}$, and we assume

(4.16)
${\mathbf{c}}_{{\mathbb{B}}}(u,c_{1}{\varepsilon})\subset{\mathbb{B}}_{u}({\varepsilon})\subset{\mathbf{c}}_{{\mathbb{B}}}(u,c_{2}{\varepsilon}),\qquad
u\in\Xi_{{\mathbb{B}}}({\varepsilon}),$

where ${\mathbf{c}}_{{\mathbb{B}}}(u,{\varepsilon})$ denotes the ball centered
at $u$ with radius ${\varepsilon}$ in ${\mathbb{B}}^{d}$, $c_{1}$ and $c_{2}$
depend only on $d$. It is known (see, for example, [19]) that such a
$\Xi_{\mathbb{B}}({\varepsilon})$ exists for all ${\varepsilon}>0$ and its
cardinality satisfies

(4.17)
$c_{d}^{\prime}{\varepsilon}^{-d}\leq\\#\Xi_{{\mathbb{B}}}({\varepsilon})\leq
c_{d}{\varepsilon}^{-d}.$

For the hyperboloid ${\mathbb{X}}^{d+1}$, we denote by
$\Xi_{{\mathbb{X}}}=\Xi_{{\mathbb{X}}}({\varepsilon})$ a maximum
${\varepsilon}$-separated set and, furthermore, denote by
$\\{{\mathbb{X}}(u,t):(tu,t)\in\Xi_{{\mathbb{X}}}\\}$ a partition of
${\mathbb{X}}^{d+1}$. We give one construction of such sets below.

Let ${\varepsilon}>0$ and let
$N=2\lfloor\frac{\pi}{2}{\varepsilon}^{-1}\rfloor$. We define
$t_{j}=\cos{\theta}_{j}$ and $t_{j}^{+}$ and $t_{j}^{-}$, $1\leq j\leq N$, as
in Subsection 3.5. Then ${\mathbb{X}}^{d+1}$ can be partitioned by

${\mathbb{X}}^{(j)}:=\left\\{(x,t)\in{\mathbb{X}}^{d+1}:t_{j}^{-}<t\leq
t_{j}^{+}\right\\},\qquad 1\leq j\leq N.$

Furthermore, the upper and lower hyperboloid ${\mathbb{X}}_{+}^{d+1}$ and
${\mathbb{X}}_{-}^{d+1}$ can be partitioned by

${\mathbb{X}}_{+}^{d+1}=\bigcup_{j=1}^{N/2}{\mathbb{X}}^{(j)}\quad\hbox{and}\quad{\mathbb{X}}_{-}^{d+1}=\bigcup_{j=N/2+1}^{N}{\mathbb{X}}^{(j)}.$

Let ${\varepsilon}_{j}:=\pi{\varepsilon}/(2t_{j})$. Then
$\Xi_{\mathbb{B}}({\varepsilon}_{j})$ is the maximal
${\varepsilon}_{j}$-separated set of ${\mathbb{B}}^{d}$ such that, for each
$j\geq 1$,
$\\{{\mathbb{B}}_{u}({\varepsilon}_{j}):u\in\Xi_{\mathbb{B}}({\varepsilon}_{j})\\}$
is a partition of ${\mathbb{B}}^{d}$ and
$\\#\Xi_{\mathbb{B}}({\varepsilon}_{j})\sim{\varepsilon}_{j}^{-d}.$ For each
$j=1,\ldots,N$, we decompose ${\mathbb{X}}^{(j)}$ by

${\mathbb{X}}^{(j)}=\bigcup_{u\in\Xi_{\mathbb{B}}({\varepsilon}_{j})}{\mathbb{X}}(u,t_{j}),\quad\hbox{where}\quad{\mathbb{X}}(u,t_{j}):=\left\\{(tv,t):t_{j}^{-}<t\leq
t_{j}^{+},\,v\in{\mathbb{B}}_{u}({\varepsilon}_{j})\right\\}.$

Finally, we define the set $\Xi_{{\mathbb{X}}}$ of ${\mathbb{X}}^{d+1}$ by

$\Xi_{{\mathbb{X}}}=\big{\\{}(t_{j}u,t_{j}):\,u\in\Xi_{\mathbb{B}}({\varepsilon}_{j}),\,1\leq
j\leq N\big{\\}}.$

######  Proposition 4.14.

Let ${\varepsilon}>0$ and $N=2\lfloor\frac{\pi}{2}{\varepsilon}^{-1}\rfloor$.
Then $\Xi_{{\mathbb{X}}}$ is a maximal ${\varepsilon}$-separated set of
${\mathbb{X}}^{d+1}$ and
$\\{{\mathbb{X}}(t_{j}u,t_{j}):u\in\Xi_{\mathbb{B}}({\varepsilon}_{j}),\,1\leq
j\leq N\\}$ is a partition

${\mathbb{X}}^{d+1}=\bigcup_{(tu,t)\in\Xi_{\mathbb{X}}}{\mathbb{X}}(u,t)=\bigcup_{j=1}^{N}\bigcup_{u\in\Xi_{\mathbb{B}}({\varepsilon}_{j})}{\mathbb{X}}(u,t_{j}).$

Moreover, there are positive constants $c_{1}$ and $c_{2}$ depending only on
$d$ such that

(4.18)
${\mathbf{c}}_{0}\big{(}(t_{j}u,t_{j}),c_{1}{\varepsilon}\big{)}\subset{\mathbb{X}}(u,t_{j})\subset{\mathbf{c}}_{0}\big{(}(t_{j}u,t_{j}),c_{2}{\varepsilon}\big{)},\quad(t_{j}u,t_{j})\in\Xi_{\mathbb{X}},$

and $c_{d}^{\prime}$ and $c_{d}$ depending only on $d$ such that

$c_{d}^{\prime}{\varepsilon}^{-d-1}\leq\\#\Xi_{{\mathbb{X}}}\leq
c_{d}{\varepsilon}^{-d-1}.$

###### Proof.

The proof is parallel to that of Propositions 3.14 and follows almost
verbatim. We omit the details. ∎

######  Proposition 4.15.

For ${\varepsilon}>0$, let $\Xi_{{\mathbb{X}}}$ be a maximal
${\varepsilon}$-separated set in ${\mathbb{X}}^{d+1}$. Define

$\Xi_{{\mathbb{X}}}^{\varrho}:=\left\\{\left(x,\sqrt{t^{2}-\varrho^{2}}\right):(x,t)\in\Xi_{{\mathbb{X}}}\right\\}.$

Then $\Xi_{{\mathbb{X}}}^{\varrho}$ is a maximal ${\varepsilon}$-separated set
in the solid hyperboloid ${}_{\varrho}{\mathbb{X}}^{d+1}$.

This is an immediate consequence of (3.5). In particular, for
$\Xi_{{\mathbb{X}}}$ defined in Proposition 4.14, both (3.25) and (3.26)
extend to ${}_{\varrho}{\mathbb{X}}^{d+1}$ as well. Moreover, this set is also
evenly symmetric, where the notion of evenly symmetric set on
${\mathbb{X}}^{d+1}$ is defined analogously as in Definition 3.16 with
${\mathbb{X}}_{0}$ replaced by ${\mathbb{X}}$.

With Assertions 1–3 established for $W_{{\gamma},\mu}$ for ${\gamma}\geq 0$
and $\mu\geq 0$, we can now apply [30, Theorem 2.15] to state the
Marcinkiewicz-Zygmund inequality. However, following the consideration in the
case of ${\mathbb{X}}_{0}^{d+1}$, our result holds under symmetry assumption
on the weight $W$ and the set $\Xi_{\mathbb{X}}$, and is restricted to the
subspace of polynomials

$\Pi_{n}^{E}({\mathbb{X}}^{d+1})=\left\\{p\in\Pi_{n}^{d+1}:p(x,t)=p(x,-t),\forall(x,t)\in{\mathbb{X}}^{d+1}\right\\}.$

######  Theorem 4.16.

Let $W$ be a doubling weight on ${\mathbb{X}}^{d+1}$ such that
$W(x,t)=W(x,-t)$ for all $(x,t)\in{\mathbb{R}}^{d+1}$. Let
$\Xi_{{\mathbb{X}}}^{\varrho}$ be a symmetric maximal
$\frac{\delta}{n}$-separated subset of ${\mathbb{X}}^{d+1}$ and $0<\delta\leq
1$.

  1. $(i)$

For all $0<p<\infty$ and $f\in\Pi_{m}^{E}({\mathbb{X}}^{d+1})$ with $n\leq
m\leq cn$,

$\sum_{z\in\Xi_{{\mathbb{X}}}^{\varrho}}\bigg{(}\max_{(x,t)\in{\mathbf{c}}_{\varrho}((z,r),\frac{\delta}{n})}|f(x,t)|^{p}\bigg{)}W\\!\left({\mathbf{c}}_{\varrho}((z,r),\tfrac{\delta}{n})\right)\leq
c_{W,p}\|f\|_{p,W}^{p}$

where $c_{W,p}$ depends on $p$ when $p$ is close to $0$ and on the doubling
constant of $W$.

  2. $(ii)$

For $0<r<1$, there is a $\delta_{r}>0$ such that for $\delta\leq\delta_{r}$,
$r\leq p<\infty$ and $f\in\Pi_{n}^{E}({\mathbb{X}}^{d+1})$,

$\displaystyle\|f\|_{p,W}^{p}\leq
c_{W,r}\sum_{z\in\Xi^{\varrho}}\bigg{(}\min_{(x,t)\in{\mathbf{c}}_{\varrho}\bigl{(}(z,r),\tfrac{\delta}{n}\bigr{)}}|f(x,t)|^{p}\bigg{)}W\bigl{(}{\mathbf{c}}_{\varrho}((z,r),\tfrac{\delta}{n})\bigr{)}$

where $c_{W,r}$ depends only on the doubling constant of $W$ and on $r$ when
$r$ is close to $0$.

###  4.6. Positive cubature rules

We need fast decaying polynomials on the solid hyperboloid, which will verify
Assertion 4.

######  Lemma 4.17.

Let $d\geq 2$ and $\varrho\geq 0$. For each $(x,t)\in{\mathbb{X}}^{d+1}$,
there is a polynomial ${\mathcal{T}}_{x,t}^{\varrho}$ in $\Pi_{n}^{E}$ that
satisfies

  1. (1)

${\mathcal{T}}_{x,t}^{\varrho}(x,t)=1$,
${\mathcal{T}}_{x,t}^{\varrho}(y,s)\geq c>0$ if
$(y,s)\in{\mathbf{c}}((x,t),\frac{\delta}{n})$, and for every ${\kappa}>0$,

$0\leq{\mathcal{T}}_{x,t}^{\varrho}(y,s)\leq
c_{\kappa}\left(1+{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}\big{(}(x,t),(y,s)\big{)}\right)^{-{\kappa}},\quad(y,s)\in{\mathbb{X}}^{d+1}.$

  2. (2)

there is a polynomial $q$ of degree $4n$ such that
$q(x,t){\mathcal{T}}_{x,t}^{\varrho}$ is a polynomial of degree $5n$ in
$(x,t)$ and $1\leq q(x,t)\leq c$.

###### Proof.

We construct our polynomial based on the one given in Lemma 3.18 by following
the approach in Lemma 3.18. For $(x,t),(y,s)\in{\mathbb{X}}^{d+1}$, we
introduce the notation $X=(x,\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}})$ and
$Y=(y,\sqrt{s^{2}-\varrho^{2}-\|y\|^{2}})$. Moreover, denote
$X_{*}=(x,-\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}})$ and
$Y_{*}=(y,-\sqrt{s^{2}-\varrho^{2}-\|y\|^{2}})$. Then $(X,t),(Y,s)$ and
$(X_{*},t)$ and $(Y_{*},s)$ are all elements of ${\mathbb{X}}_{0}^{d+2}$. Let
$T_{(X,t)}^{\varrho}$ denote the polynomial of degree $n$ on
${\mathbb{X}}_{0}^{d+2}$ defined in Lemma 3.18. We now define

${\mathcal{T}}^{\varrho}_{x,t}(y,s):=\frac{T_{(X,t)}^{\varrho}(Y,s)+T_{(X,t)}^{\varrho}(Y_{*},s)}{1+T_{(X,t)}^{\varrho}(X_{*},t)}.$

Since $T_{(X,t)}^{\varrho}(X,t)=1$, it follows that
${\mathcal{T}}_{(x,t)}^{\varrho}(x,t)=1$. Moreover, since

$T_{(X,t)}^{\varrho}(X_{*},t)=\frac{S_{n}(2\|x\|^{2}+1+2\varrho^{2}-2t^{2})+S_{n}(2\|x\|^{2}-1)}{1+S_{n}(2t^{2}-2\varrho^{2}-1)}$

and $0\leq S_{n}(t)\leq c$, we see that $0\leq T_{(X,t)}(X_{*},s)\leq c$. In
particular, it follows that

${\mathcal{T}}_{x,t}^{\varrho}(y,s)\geq c\,T_{(X,t)}^{\varrho}(Y,s)\geq
c>0,\qquad(y,s)\in{\mathbf{c}}\big{(}(x,t),\tfrac{\delta}{n}\big{)},$

since
${\mathsf{d}}_{{\mathbb{X}}}((x,t),(y,s))={\mathsf{d}}_{{\mathbb{X}}_{0}^{d+2}}((X,t),(Y,s))$.
Furthermore, since
$\cos{\mathsf{d}}_{{\mathbb{X}}_{0}}((X,t),(Y,s))\geq\cos{\mathsf{d}}_{{\mathbb{X}}_{0}}((X,t),(Y_{*},s))$,
we obtain

${\mathsf{d}}_{{\mathbb{X}}_{0}}((X,t),(Y_{*},s))\geq{\mathsf{d}}_{{\mathbb{X}}_{0}}((X,t),(Y,s))={\mathsf{d}}_{{\mathbb{X}}}((x,t),(y,s)).$

Hence, using the estimate of $T_{(X,t)}^{\varrho}$ in Lemma 3.18, we conclude
that

$\displaystyle 0\leq{\mathcal{T}}_{x,t}^{\varrho}(y,s)\,$ $\displaystyle\leq
c\left[\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((X,t),(Y,s))\big{)}^{-{\kappa}}+\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}_{0}}((X,t),(Y_{*},s))\big{)}^{-{\kappa}}\right]$
$\displaystyle\leq
c\big{(}1+n{\mathsf{d}}_{{\mathbb{V}}}((x,t),(y,s))\big{)}^{-{\kappa}}.$

Finally, let
$q(x,t)=(1+S_{n}(2t^{2}-2\varrho^{2}-1))T_{(X,t)}^{\varrho}(X_{*},t)$. Then
$q(x,t)$ is a polynomial of degree at most $4n$, so that
$q(x,t){\mathcal{T}}_{x,t}^{\varrho}$ is a polynomial of degree at most $5n$
and $1\leq q(x,t)\leq c$. This completes the proof. ∎

The lemma establishes Assertion 4 with a polynomial in
$\Pi_{n}^{E}({\mathbb{X}}^{d+1})$. Let $W$ be a doubling weight function that
is even in $t$ variable on ${\mathbb{X}}^{d+1}$. We define

(4.19)
$\displaystyle{\lambda}_{n}^{E}(W;x,t):=\inf_{\begin{subarray}{c}g(x,t)=1\\\
g\in\Pi_{n}^{E}({\mathbb{X}}^{d+1})\end{subarray}}\int_{{\mathbb{X}}^{d+1}}|g(x,t)|^{2}W(x,t)\mathrm{d}x\mathrm{d}t,$

which is the Christoffel function for the space
$\Pi_{n}^{E}({\mathbb{X}}^{d+1})$. Just like the case of
${\mathbb{X}}_{0}^{d+1}$, this Christoffel function is related to the kernel
${\mathbf{K}}_{n}^{E}(W)=\sum_{k=0}^{n}{\mathbf{P}}_{k}^{E}(W)$ by

$\displaystyle{\lambda}_{n}^{E}(W;x,t)=\frac{1}{{\mathbf{K}}_{n}^{E}(W;(x,t),(x,t))},\qquad(x,t)\in{\mathbb{X}}^{d+1}.$

Moreover, [30, Propositions 2.17 and 2.18] remain valid for
${\lambda}_{n}^{E}(W)$ on ${\mathbb{X}}^{d+1}$ by using the Lemma 4.17. Hence,
we obtain the following corollary.

######  Corollary 4.18.

Let $W$ be a doubling weight function on ${\mathbb{X}}^{d+1}$ such that
$W(x,t)=W(x,-t)$ for all $(x,t)\in{\mathbb{X}}^{d+1}$. Then

${\lambda}_{n}^{E}\big{(}W;(x,t)\big{)}\leq
c\,W\\!\left({\mathbf{c}}_{\varrho}\left((x,t),\tfrac{1}{n}\right)\right).$

Moreover, for ${\gamma}\geq 0$ and $\mu\geq 0$,

${\lambda}_{n}^{E}\big{(}W_{{\gamma},\mu};(x,t)\big{)}\geq
c\,W_{{\gamma},\mu}\\!\left({\mathbf{c}}_{\varrho}\left((x,t),\tfrac{1}{n}\right)\right)=cn^{-d}W_{{\gamma},\mu}^{\varrho}(n;x,t).$

We are now in position to state the positive cubature rule for solid
hyperboloid, following which holds for polynomials in
$\Pi_{n}^{E}({\mathbb{X}}^{d+1})$ and under the symmetry assumptions.

######  Theorem 4.19.

Let $d\geq 2$ and $\varrho\geq 0$. Let $W$ be a doubling weight on
${\mathbb{X}}^{d+1}$ such that $W(x,t)=W(x,-t)$ for all
$(x,t)\in{\mathbb{X}}^{d+1}$. Let $\Xi^{\varrho}$ be a symmetric maximum
$\frac{\delta}{n}$-separated subset of ${\mathbb{X}}^{d+1}$. There is a
$\delta_{0}>0$ such that for $0<\delta<\delta_{0}$ there exist positive
numbers ${\lambda}_{z,r}$, $(z,r)\in\Xi^{\varrho}$, so that

(4.20)
$\int_{{\mathbb{X}}^{d+1}}f(x,t)W(x,t)\mathrm{d}x\mathrm{d}t=\sum_{(z,r)\in\Xi^{\varrho}}{\lambda}_{z,r}f(z,r),\qquad\forall
f\in\Pi_{n}^{E}({\mathbb{X}}^{d+1}).$

Moreover, ${\lambda}_{z,r}\sim
W\\!\left({\mathbf{c}}_{\varrho}((z,r),\tfrac{\delta}{n})\right)$ for all
$(z,r)\in\Xi^{\varrho}$.

Again, this follows from [30, Theorem 2.20] when the domain becomes
${\mathbb{X}}_{0}^{d+1}$ and it remains valid under symmetry assumptions on
the weight and on the polynomials.

###  4.7. Localized tight frames

The symmetry assumption for the weight and for the ${\varepsilon}$-separated
subset also carries over to the local frame on the solid hyperboloid. Let
$\varrho\geq 0$ and let $W$ be a doubling weight on ${\mathbb{X}}^{d+1}$ that
is even in $t$ variable. Let ${\mathbf{L}}_{n}^{E}(W)*f$ denote the operator
in $\Pi_{2n}^{E}({\mathbb{X}}^{d+1})$ defined by

${\mathbf{L}}_{n}^{E}(W)*f(x):=\int_{{\mathbb{X}}^{d+1}}f(y,s){\mathbf{L}}_{n}^{E}(W;(x,t),(y,s))W(y,s)\mathrm{d}y\mathrm{d}s,$

where ${\mathbf{L}}_{n}^{E}(W;\cdot,\cdot)$ is the highly localized kernel
defined via a cut-off function $\widehat{a}$ that satisfies (3.30). For
$j=0,1,\ldots,$ let $\Xi_{j}^{\varrho}$ be a symmetric maximal
$\frac{\delta}{2^{j}}$-separated subset in ${\mathbb{X}}^{d+1}$, so that

$\int_{{\mathbb{X}}^{d+1}}f(x,t)W(x,t)\mathrm{d}x\mathrm{d}t=\sum_{(z,r)\in\Xi_{j}^{\varrho}}{\lambda}_{(x,r),j}f(z,r),\quad
f\in\Pi_{2^{j}}^{E}({\mathbb{X}}^{d+1}).$

For $j=1,2,\ldots,$ define the operator $F_{j}^{\varrho}(W)$ by

$F_{j}^{\varrho}(W)*f={\mathbf{L}}_{2^{j-1}}^{E}(W)*f$

and define the frame elements $\psi_{(z,r),j}$ for $(z,r)\in\Xi_{j}^{\varrho}$
by

$\psi_{(z,r),j}(x,t):=\sqrt{{\lambda}_{(z,r),j}}F_{j}^{\varrho}((x,t),(z,r)),\qquad(x,t)\in{\mathbb{X}}^{d+1}.$

Then $\Phi=\\{\psi_{(z,r),j}:(z,r)\in\Xi_{j}^{\varrho},\,j=1,2,3,\ldots\\}$ is
a tight frame by [30, Theorem 2.21].

######  Theorem 4.20.

Let $W$ be a doubling weight on ${\mathbb{X}}^{d+1}$ even in its $t$ variable.
If $f\in L^{2}({\mathbb{X}}^{d+1},W)$ and $f$ is even in $t$ variable, then

$f=\sum_{j=0}^{\infty}\sum_{(z,r)\in\Xi_{j}^{\varrho}}\langle
f,\psi_{(z,r),j}\rangle_{\mathsf{w}}\psi_{(z,r),j}\quad\mbox{in
$L^{2}({\mathbb{X}}^{d+1},W)$}$

and

$\|f\|_{2,W}=\Big{(}\sum_{j=0}^{\infty}\sum_{(z,r)\in\Xi_{j}^{\varrho}}|\langle
f,\psi_{(z,r),j}\rangle_{W}|^{2}\Big{)}^{1/2}.$

Furthermore, for ${\gamma}\geq 0$ and $\mu\geq 0$, the frame for
$W_{{\gamma},\mu}^{\varrho}$ is highly localized in the sense that, for every
${\kappa}>0$, there exists a constant $c_{\kappa}>0$ such that

(4.21) $|\psi_{(z,r),j}(x,t)|\leq
c_{\kappa}\frac{2^{j(d+1)/2}}{\sqrt{W_{{\gamma},\mu}^{\varrho}(2^{j};x,t)}(1+2^{j}{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}((x,t),(z,r)))^{\kappa}},\quad(x,t)\in{\mathbb{X}}^{d+1}.$

Like the case on the hyperbolic surface, the frame elements involve only
orthogonal polynomials even in $t$ variable. The localization (4.21) follows
from Theorem 4.9 and ${\lambda}_{(z,r),j}\sim
2^{-j(d+1)}W_{{\gamma},\mu}^{\varrho}(2^{j};x,t)$ which holds for
$W_{{\gamma},\mu}^{\varrho}$ by Corollary 4.18 and (4.9). We note, however,
that the localized tight frame holds for the weight function
$W_{{\gamma},\mu}^{\varrho}$, which does not include the Lebesgue measure on
${\mathbb{X}}^{d+1}$.

###  4.8. Characterization of best appoximation

For $f\in L^{p}({\mathbb{X}}^{d+1},W_{{\gamma},\mu}^{\varrho})$, we denote by
${\mathbf{E}}_{n}(f)_{p,W_{{\gamma},\mu}^{\varrho}}$ the best approximation to
$f$ from $\Pi_{n}^{d+1}({\mathbb{X}}^{d+1})$ in the norm
$\|\cdot\|_{p,W_{{\gamma},\mu}^{\varrho}}$; that is,

${\mathbf{E}}_{n}(f)_{p,W_{{\gamma},\mu}^{\varrho}}:=\inf_{g\in\Pi_{n}({\mathbb{X}}^{d+1})}\|f-g\|_{p,W_{{\gamma},\mu}^{\varrho}},\qquad
1\leq p\leq\infty.$

As in the case of hyperbolic surface, if $f$ is even in $t$ variable, then we
can choose the polynomial of best approximation from
$\Pi_{n}^{E}({\mathbb{X}}^{d+1})$. We can give a characterization of this
quantity for functions that are even in $t$ variable.

For $f\in L^{p}({\mathbb{X}}^{d+1},W_{{\gamma},\mu}^{\varrho})$ and $r>0$, the
modulus of smoothness is defined by

${\omega}_{r}(f;\rho)_{p,W_{{\gamma},\mu}^{\varrho}}=\sup_{0\leq{\theta}\leq\rho}\left\|\left(I-{\mathbf{S}}_{{\theta},W_{{\gamma},\mu}^{\varrho}}\right)^{r/2}f\right\|_{p,W_{{\gamma},\mu}^{\varrho}},\quad
1\leq p\leq\infty,$

where the operator ${\mathbf{S}}_{{\theta},W_{{\gamma},\mu}^{\varrho}}$ is
defined by, for $n=0,1,2,\ldots$ and ${\lambda}={\gamma}+\mu+\frac{d}{2}$,

$\operatorname{proj}_{n}^{E}\Big{(}W_{{\gamma},\mu}^{\varrho};{\mathbf{S}}_{{\theta},W_{{\gamma},\mu}^{\varrho}}f\Big{)}=R_{n}^{({\lambda}-\frac{1}{2},{\lambda}-\frac{1}{2})}(\cos{\theta})\operatorname{proj}_{n}^{E}\Big{(}W_{{\gamma},\mu}^{\varrho};f\Big{)}.$

In terms of the fractional differential operator
$(-{\mathfrak{D}}_{{\gamma},\mu}^{\varrho})^{\frac{r}{2}}$, the $K$-functional
for a doubling weight $W$, even in $t$ variable on ${\mathbb{X}}^{d+1}$, is
defined by

${\mathbf{K}}_{r}(f,t)_{p,W}:=\inf_{g\in
W_{p}^{r}({\mathbb{X}}^{d+1},W)}\left\\{\|f-g\|_{p,W}+t^{r}\left\|(-{\mathfrak{D}}_{{\gamma},\mu}^{\varrho})^{\frac{r}{2}}f\right\|_{p,W}\right\\},$

where the Sobolev space $W_{p}^{r}\big{(}{\mathbb{X}}^{d+1},W\big{)}$ is the
space that consists of $f\in L^{p}({\mathbb{X}}^{d+1},W)$, even in $t$
variable, so that
$\left\|(-{\mathfrak{D}}_{{\gamma},\mu}^{\varrho})^{\frac{r}{2}}f\right\|_{p,W}$
is finite.

For $W_{{\gamma},\mu}^{\varrho}$, Assertions 1 and 3 hold and we now verify
that Assertion 5 holds as well. By (4.12), the kernel $L_{n}^{(r)}(\varpi)$ in
Assertion 5 becomes

${\mathbf{L}}_{n}^{(r)}\big{(}W_{{\gamma},\mu}^{\varrho};(x,t),(y,s)\big{)}=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right)(k(k+2{\gamma}+2\mu+d))^{\frac{r}{2}}{\mathbf{P}}_{k}^{E}\big{(}W_{{\gamma},\mu}^{\varrho};(x,t),(y,s)\big{)}.$

######  Lemma 4.21.

Let ${\gamma},\mu\geq-\frac{1}{2}$ and ${\kappa}>0$. Then, for $r>0$ and
$(x,t),(y,s)\in{\mathbb{X}}^{d+1}$,

$\left|{\mathbf{L}}_{n}^{(r)}\big{(}W_{{\gamma},\mu}^{\varrho};(x,t),(y,s)\big{)}\right|\leq
c_{\kappa}\frac{n^{d+r+1}}{\sqrt{W_{{\gamma},\mu}^{\varrho}(n;t)}\sqrt{W_{{\gamma},\mu}^{\varrho}(n;s)}}\big{(}1+n{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}((x,t),(y,s))\big{)}^{-{\kappa}}.$

###### Proof.

This can be proved similarly as in the proof of Lemma 3.22 and using the
estimate in Theorem 4.9. We omit the details. ∎

We can now state the characterization of the best approximation by polynomials
on the hyperboloid, following [30, Theorem 3.12].

######  Theorem 4.22.

Let $f\in L^{p}({\mathbb{X}}^{d+1},W)$ if $1\leq p<\infty$ and $f\in
C({\mathbb{X}}^{d+1})$ if $p=\infty$. Assume that $f$ satisfies
$f(x,t)=f(x,-t)$. Let $\varrho\geq 0$, $r>0$ and $n=1,2,\ldots$. Then

  1. (i)

for $W=W_{{\gamma},\mu}^{\varrho}$ with ${\gamma},\mu\geq 0$,

${\mathbf{E}}_{n}(f)_{p,W_{{\gamma},\mu}^{\varrho}}\leq
c\,{\mathbf{K}}_{r}(f;n^{-1})_{p,W_{{\gamma},\mu}^{\varrho}};$

  2. (ii)

for every doubling weight $W$ that satisfies $W(x,t)=W(x,-t)$ on
${\mathbb{X}}^{d+1}$,

${\mathbf{K}}_{r}(f;n^{-1})_{p,W}\leq
cn^{-r}\sum_{k=0}^{n}(k+1)^{r-1}{\mathbf{E}}_{k}(f)_{p,W}.$

For $W=W_{{\gamma},\mu}^{\varrho}$, the $K$-functional is equivalent to the
modulus of smoothness.

######  Theorem 4.23.

Let $\varrho\geq 0$, ${\gamma},\mu\geq 0$ and $f\in
L_{p}^{r}({\mathbb{X}}^{d+1},W_{{\gamma},\mu}^{\varrho})$, $1\leq
p\leq\infty$. Then for $0<{\theta}\leq\pi/2$ and $r>0$

$c_{1}{\mathbf{K}}_{r}(f;{\theta})_{p,W_{{\gamma},\mu}^{\varrho}}\leq{\omega}_{r}(f;{\theta})_{p,W_{{\gamma},\mu}^{\varrho}}\leq
c_{2}{\mathbf{K}}_{r}(f;{\theta})_{p,W_{{\gamma},\mu}^{\varrho}}.$

In particular, the characterization in Theorem 4.22 can be stated in terms of
the modulus of smoothness instead.

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# A First Look at COVID-19 Messages on WhatsApp in Pakistan

R. Tallal Javed1, Mirza Elaaf Shuja1, Muhammad Usama1, Junaid Qadir1, Waleed
Iqbal2, Gareth Tyson2, Ignacio Castro2,and Kiran Garimella3

Email: 1(tallal.javed  1Information Technology University, Punjab, Pakistan.
2Queen Mary University of London  3MIT  msds18051  muhammad.usama
<EMAIL_ADDRESS>2(w.iqbal  g.tyson<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The worldwide spread of COVID-19 has prompted extensive online discussions,
creating an ‘infodemic’ on social media platforms such as WhatsApp and
Twitter. However, the information shared on these platforms is prone to be
unreliable and/or misleading. In this paper, we present the first analysis of
COVID-19 discourse on public WhatsApp groups from Pakistan. Building on a
large scale annotation of thousands of messages containing text and images, we
identify the main categories of discussion. We focus on COVID-19 messages and
understand the different types of images/text messages being propagated. By
exploring user behavior related to COVID messages, we inspect how
misinformation is spread. Finally, by quantifying the flow of information
across WhatsApp and Twitter, we show how information spreads across platforms
and how WhatsApp acts as a source for much of the information shared on
Twitter.

###### Keywords:

COVID-19, Misinformation, WhatsApp, Twitter

††publicationid: pubid:  IEEE/ACM ASONAM 2020, December 7-10, 2020
978-1-7281-1056-1/20/$31.00 © 2020 IEEE

##  I Introduction

Social media apps like Facebook, WhatsApp, and Twitter have changed the way we
communicate. The information disseminated through these apps has influenced
our social and cultural norms in an unprecedented way. WhatsApp is one of the
most frequently used and rapidly growing social media apps in the world with
more than 1.5 billion users. WhatsApp is also ranked #1 for the average number
of active users in the world per month [1].

WhatsApp has, therefore, become important for understanding social behavior
and opinion formation. In many cases, the scope of the information shared in
WhatsApp groups, is limited to a community or a country [2]. Recent studies
[3] have shown that WhatsApp (like other social media platforms) is also used
for the dissemination of misinformation. We argue that it is important to
understand how this false information, either spread knowingly
(”disinformation”) or naively (”misinformation”) influences opinion formation
in different societies. This is particularly important in the global south,
where despite users having low digital literacy, WhatsApp is the de facto mode
through which users obtain and share information [3].

With this in mind, we perform a comprehensive analysis of COVID messages being
propagated through WhatsApp in Pakistan. Pakistan is a major developing
country, with approximately 37 million active social media
users111https://datareportal.com/reports/digital-2020-pakistan. Building on
the idea of how political parties around the world [4] are using public
WhatsApp groups to reach their audience, we start by monitoring a large sample
of public WhatsApp groups related to politics in Pakistan. Meanwhile, a major
world event occurred: On March 11, 2020, the World Health Organization (WHO)
declared COVID-19 a pandemic [5]. Our data also holds unique value, as it
encompasses non-covid groups, and gives us insight into how COVID related
content, organically gets propagated across public WhatsApp groups.
Specifically, we explore the following research questions:

  * RQ1:

What kinds of messages, about the pandemic, are being shared, on the publicly
accessible WhatsApp groups of Pakistan?

  * RQ2:

Is there misinformation related to COVID-19, and if so, to what extent and of
which type?

  * RQ3:

What is the general user behavior, and can we detect disinformation from it?

  * RQ4:

What is the interplay between misinformation related to COVID-19 shared on
WhatsApp and Twitter?

To explore these questions, we have collected data from 227 public WhatsApp
groups starting January 10, 2020. To the best of our knowledge, this is the
first dataset and analysis of COVID-19 related conversations from a country in
the global south, involving multiple modalities (text and images) and multiple
platforms (WhatsApp and Twitter). We begin our investigation by analyzing the
content shared in the WhatsApp groups and filtering out the COVID-19 related
content. The filtered content is then further divided into text, images,
videos,222In this study, we focus on text and images, leaving video analysis
for future work. and other related categories. Using this data, we make the
following contributions:

  * •

We offer the first WhatsApp dataset consisting of discussions related to
COVID-19 from Pakistan. The dataset includes texts, images and videos
originating from 227 groups. The (anonymized) dataset will be made publicly
available to the community.

  * •

We show using extensive manual annotation that around 14% of the messages
related to COVID-19 had misinformation about the pandemic.

  * •

We perform a temporal analysis of misinformation related to COVID-19
propagation, across WhatsApp and Twitter, exploring how content is copied
across.

##  II Related Work

###  II-A (Mis)Information on WhatsApp

WhatsApp has been a source of major political misinformation and propaganda
campaigns [6, 7]. Political parties have invested heavily in social media
strategies by creating WhatsApp groups to reach WhatsApp users [8]. Surveys
done in India and Brazil show that at least one in six users are part of one
such public political WhatsApp group [9, 4].

Garimella et al. [10] provide tools to collect and analyze public WhatsApp
group data at scale. Making use of these tools, various studies have shown the
extent of misinformation and manipulation on WhatsApp [11, 12, 13, 14].
Particularly, Resende et al. [12] analyze doctored images to fuel smear
campaigns against political rivals and the dissemination of misinformation
through WhatsApp groups in Brazil. Garimella et al. [14] provide an analysis
of image-based misinformation spread during the 2019 Indian elections and show
that over 13% of the images contained misinformation. Melo et al. [15] provide
a system for gathering analyzing, and visualizing WhatsApp public group data
for identification of misinformation propagated in three countries: India,
Brazil and Indonesia. Maros et al. [16] analyze audio messages shared on
WhatsApp and characterize their propagation dynamics. The analysis is
performed on 20K audio messages from 330 WhatsApp public groups and the
results suggest that the audio messages with misinformation spread further
more than the benign or unchecked audio messages.

###  II-B Health (Mis)Information

A major focus of this paper is understanding the spread of health
misinformation related to COVID-19. WhatsApp has been a major source of health
misinformation especially during the pandemic [17]. This misinformation ranges
from highlighting wrong symptoms to ineffective treatments. Jin et al. [18]
reported a massive wave of misinformation on social media, especially on
Twitter during the Ebola pandemic in Africa. More comprehensive details on how
fake news about Ebola on social media applications is explored in [19].

With the ongoing surge in the COVID-19 pandemic a wealth of misinformation has
already been documented. Sharma et al. [20] provide a dashboard for analyzing
misinformation about COVID-19 on Twitter. They analyze 25 million tweets and
provide a country-wise sentiment analysis of how people are reacting to
COVID-19. Singh et al. [21] analyze Twitter-based misinformation about
COVID-19 and provide insights on how the propagation of misinformation on
social media is connected to the rise in the number of COVID-19 positive
cases. Kouzy et al. [22] analyze Twitter-based misinformation about COVID-19
and report that tweets having the keyword “COVID-19” contains less
misinformation and tweets with keywords “2019-ncov” and “Corona”. Cinelli et
al. [23] provide a comprehensive analysis of the use of different social media
platforms in the COVID-19 pandemic. They analyze Twitter, Instagram, YouTube,
Reddit and Gab, providing a review of how the discourse on these applications
is evolving. They also explore the propagation of misinformation from
different questionable sources in social media.

###  II-C Our Work’s Novelty

To the best of our knowledge, there does not exist any work analyzing
COVID333For brevity, we refer to COVID-19 simply as COVID and use these terms
interchangably related discussions on WhatsApp. Since WhatsApp is arguably the
most frequently used application in the world, it is important to study it to
see how people are using the platform during the pandemic and how the platform
facilitates the spread of COVID-19 misinformation. Although prior work has
focused on misinformation spread via WhatsApp in Brazil and India, we are the
first one to study misinformation on WhatsApp during a major pandemic.
Furthermore, our analysis is focused on Pakistan, which has a thriving Muslim
religious identity, which allows us to see how religion plays a role in the
context of public health. In contrast to the majority of prior work on
misinformation, which focuses on textual analysis, we also provide a detailed
analysis of images related to COVID-19 and study the information spread across
WhatsApp and Twitter both for text messages as well as images.

##  III Methodology

In this section, we delineate our data collection & annotation methodology,
and discuss the related ethical issues.

###  III-A Data Collection

WhatsApp allows its users to create public and private groups. The public
groups can be joined by any user of the platform, typically through an invite
URL of the form chat.whatsapp.com/*. These URLs are frequently shared via
other social web platforms (e.g., Facebook, Twitter) to invite third parties
to join.

Selection of groups. To compile a list of relevant public groups, we looked
for chat.whatsapp.com links on Facebook and Google to find group invite URLs.
We specifically targeted the popular political parties of Pakistan as these
groups tend to be more active and give an idea of the political sphere. Hence,
“WhatsApp” along with political parties’ names and slogans were used to search
for public groups.

Based on the above parameters, we compiled a list of 282 public WhatsApp
groups. In order to ensure the quality of groups, we manually discarded groups
that were unrelated. For instance, if a group’s profile picture, group name or
bio did not contain any relevant information (political aims/motivations) then
it was removed. We further removed groups which were buying and selling
things, and did not have any organic interactions/messages. This left us with
227 public groups, on which the analysis was done.

In order to find these groups, a set of queries, search engines and filters
were used. These queries can be found at https://cutt.ly/8yXhxBd. We also plan
to release our anonymised dataset once the paper is accepted to encourage
further research on WhatsApp data from Pakistan.

WhatsApp data collection. To join and get data from the groups, we used tools
provided by Garimella et al. [10], which uses the Selenium Web Driver to
automate the joining of the groups. WhatsApp stores all message data on the
user’s device in an encrypted SQLite database. We used a rooted Android device
to obtain the decryption key and and extracted the decrypted database every
week. The media content, which is stored as encrypted URLs was downloaded
locally and decrypted using a public tool444https://github.com/ddz/whatsapp-
media-decrypt (slightly modified for our convenience). WhatsApp deletes media
content from their servers after a certain amount of time. As a result, when
decrypting media files, we missed a small subset of the content shared (14%).
The joining of the groups took place over a 1 month period, as new groups were
being identified. The data collection started from 10 January 2020 onward and
continued until 23 February 2020. We have complete data from all groups from
the end of February until the second week of April. The details of the dataset
are summarized in Table I.

Table I: Overview of our WhatsApp dataset. #Groups | 227 |
---|---|---
#Admins | 521 |
#Users | 18,475 |
#Unique users | 16,493 |
Total #messages | 60,202 |
#Text messages | 28,497 | 47%
#Images | 14,633 | 24.5%
#Video | 11,196 | 18.6%
#Audio | 2,688 | 4.5%
Others | 6,740 | 5.4%
#URLs | 3,188 | 2.5%

Twitter data. To compare the data we obtained from the WhatsApp groups to
other open, well studied social media platforms, we also gathered data from
Twitter. Specifically, we obtained historical Twitter data on an extensive
list of hashtags specific to COVID-19 in Pakistan such as #CovidPakistan,
#CoronaFreePakistan555https://cutt.ly/nyXrVyp and other local Twitter trend
variations. This gave us 800,000 tweets.

Ethics note. The groups joined, had been openly propagated on Facebook,
Twitter, and other mediums and can be joined by anyone. The profile bio of our
WhatsApp account declares that we are collecting information for research
purposes. We also anonymized the user data, before analyzing it.

###  III-B Identifying COVID-19 Text Messages

We extract COVID-19 related text messages using a keyword-filtering approach.
We utilize [24], which offers a dictionary of COVID-19 English keywords. We
added small variations and multiple spellings to the dictionary to capture a
wide variety of content related to the pandemic. We translated these keywords
into Urdu and used both the English and Urdu keywords to search our dataset.
The final list includes keywords such as “corona”, “coronavirus”, “covid-19”,
“covid”, “covid19”, and “corona virus”, among others. This keyword based
approach results in a high precision yet low recall method to identify COVID
related messages. Using this approach, we obtained 5,039 COVID related text
messages between March 16, 2020 and April 09, 2020. Figure 1 compares the
number of daily COVID-19 related and Non-COVID-19 related text messages in our
dataset.

Figure 1: COVID vs. Non-COVID texts. Timeline of number of messages and
messages containing COVID related keywords in our WhatsApp dataset

###  III-C Identifying COVID-19 Images

As we see in Table I, around 25% of the content is images. Hence, solely
evaluating text would give a distorted view of the overall information
landscape. Naturally, image content is far harder to automatically categorize.
Therefore, to extract images discussing COVID-19, manual tagging was performed

Two annotators tagged a total of 6,699 images, ranging from 16 March to 9
April, 2020. An image was declared as COVID related if it had any of the
following attributes:

  1. 1.

Contained Coronavirus, COVID-19, or any other related terminology in Urdu or
English.

  2. 2.

Information relating to a lockdown or any restrictions being imposed/relaxed
by the government on business or public/private institutions.

  3. 3.

Sharing of any precautionary measures like prayers for protection from
disease, herbal medications, etc.

  4. 4.

Contained any references to the environmental or economic impact of COVID-19.

  5. 5.

Contained people with personal protective equipment, possible quarantine
centers, and people practicing or encouraging social distancing.

An inter annotator agreement score of 98% was observed. In cases of a
conflict, the annotators were allowed to mutually discuss and agree upon a
label. A total of 2,309 (34.5%) images were identified as COVID related, while
4,390 (65%) were identified as non-COVID images. For context, Figure 2 shows
the percentage of COVID related images over time. We see that, as the pandemic
intensifies, so does the fraction of related images.

Figure 2: Timeline of the percentage of images and images containing COVID
related content in our WhatsApp dataset

##  IV RQ1: Information Sharing on COVID-19

In this section, we answer our first research question, to understand what
types of COVID related information is being shared on WhatsApp. We first
annotate the data into 5 overalapping categories and then use these categories
to understand the types of information.

###  IV-A Message Type Categorization

In order to further characterize the types of COVID-19 related content being
shared within our WhatsApp groups, we categorized the COVID related text and
images manually into the five categories: Information, Misinformation,
Jokes/Satire, Religious, and Ambiguous. The categories were chosen based on a
preliminary manual exploration of the COVID-19 content. The categories are not
mutually exclusive and as a result, a single message (text/image) can belong
to multiple categories. We describe each category below:

  1. 1.

Information: This category consists of WhatsApp content that contains either
factual News or COVID related facts. News reports are fact checked using
Poynter’s COVID Facts database666https://www.poynter.org/ifcn-
covid-19-misinformation/ which contains all of the falsehoods detected by a
large number of fact checking organizations. In addition, AFP Pakistan Fact
Check777https://factcheck.afp.com/afp-pakistan is used to verify news
articles. The contents of the text or image are evaluated against the
falsehoods in the database to verify their validity. Google search was also
used to verify certain claims not present in the Poynter dataset. If the news
is reported by a reputed news source, then it is labelled as “Information”. A
news source is considered reputed if it has a satellite news channel or
newspaper at a national level. COVID related facts are verified using WHO’s
COVID Information and pervalent
myths.888https://www.who.int/emergencies/diseases/novel-
coronavirus-2019/advice-for-public/myth-busters

  2. 2.

Misinformation: This category is the inverse of the above ‘Information’
category. Any content which is either verified to be misinformation or could
not be verified as credible information is placed in this category. Content
was checked using Poynter COVID-Facts and Falsehoods database, AFP Pakistan
and WHO’s COVID Informations and COVID Myths.

  3. 3.

Jokes/Satire: This class consists of content that intends to poke fun at the
COVID-19 pandemic itself or any COVID related government/political actions
using sarcasm, satire or memes. It also contains content that consists of non-
factual opinions/analysis regarding current COVID related events or government
actions.

  4. 4.

Religious: Since Pakistan has a 98% Muslim population, religion plays an
important role in information dissemination. A religious theme in content is
identified by looking for (i) references to spiritual texts, (ii) quotes of
religious scholars (called Maulana, Mufti, or Sheikh), and (iii) mentions of
religious acts such as prayer, fasting etc.

  5. 5.

Ambiguous: If the content does not have enough information to be classified
into one or more of the above categories, it is then assigned to the
‘Ambiguous’ category. This category mainly consists of content where people
are distributing Personnel Protective Equipment (PPE), social media requests
to follow/subscribe, contact information of NGO’s, donation requests, etc.

Figure 3: Percentage of COVID-19 images (Light Green), texts (Medium Green)
and texts + images (Dark Green) for each category. Notably, 14% of the total
messages were labeled to be misinformation.

Note that to maintain consistency in annotation quality between images and
text (images were annotated by two annotators whereas text was done by only
one annotator), we randomly sampled 25% of the 5,039 COVID-19 related texts,
and validated the annotation. We showed the 25% random sample of messages to
one additional annotator and measured the agreement with the original
annotator. We find an 82% agreement between the two annotators, with one or
more common labels counted as an agreement for our non-mutually exclusive
classes. The majority of disagreements were between Information, Jokes/Satire
and Religious classes. This is because a lot of texts contain different
proportions of the three. Very few disagreements were observed when one of the
annotators tagged a text as Misinformation, which were resolved after a
discussion between the two annotators.

###  IV-B Message Type Analysis

We now analyze the different types of COVID-19 related content, in both texts
and images on WhatsApp, based on the above annotation. We have a total of
5,039 texts and 2,309 images which discuss COVID related information between
March 16 and April 9, 2020. The overall distribution of texts and images into
the COVID-19 related content categories are shown in Figure 3. A majority of
the content is simple information (29%), containing news articles, latest
government actions and health information related to COVID-19. This is
followed by religious content (25%). The large amount of religious content
emphasizes the importance of religious sentiment within the society,
especially during times of uncertainty created by the pandemic. Religious
scholars and Holy Verses from religious books were cited in these messages.
Religious content was also event focused. For instance, a ban on
congregational prayers and the rigorous COVID testing of a group of religious
people on a proselytising trip resulted in messages criticising these
government actions.

This was followed by Jokes/Satire representing 17% of the messages. Political
actions by rival parties, and government officials were frequently ridiculed
and mocked, including personally targeted attacks. For instance, many of such
texts were against government action of opening the border with Iran, blaming
the officials for bringing Corona to Pakistan. Interestingly, a non trivial
amount (14%) was Misinformation, which shows that one in seven messages shared
contained some misleading information. This included fake news reporting
deaths of politicians, fake quotes from famous personalities and international
figures, or fake COVID origin stories. We show detailed analysis of
misinformation in Section V.

Finally, a small fraction of messages (12%) were labelled “Ambigious”. Among
these, a significant portion of texts contained Facebook and YouTube
follow/subscribe requests to COVID related pages and channels, donation
requests, or shared contact information for COVID-affected and poverty-
stricken people. We also found some images depicting quarantine centers,
hospitals, doctors, and patients, which did not fit into the above categories
due to the lack of context.

###  IV-C Lifetime of Messages

In this section, we try to understand the impact of the various types of COVID
messages on WhatsApp. To do so, we analyzed the lifetime of various types of
COVID related messages. The lifetime of a message is the difference between
the last and first appearance of a message (in hours) in our dataset. First,
we grouped together perceptually similar images using a popular, state of the
art image hashing tool from Facebook known as PDQ
hashing.999https://github.com/facebook/ThreatExchange The hashes were
generated for all the COVID related images and then instances of similar
images were clustered together by using Hamming distance, with a threshold of
70%, between the hashes. The difference between first and last appearance of a
representative image in each cluster was considered as the lifetime of that
image. For texts, exact string matching was used to find the first and last
appearance of a text. Table II shows the mean lifetime of messages belonging
to the various COVID-19 content categories.

Each category exhibits distinct mean and variance measure for lifetime. The
most short lived messages belong to the “Jokes/Satire” category. This appears
logically coherent since jokes, opinions and satirical texts are generally
dictated by events and die out quickly as the public focus shifts from one
event to another. Interestingly, the lifetime of a message containing
misinformation is quite high, for both text (7 hrs) and images (5.5 hrs),
especially compared to the Information category. This means that
misinformation tends to persist longer compared to information, which supports
existing studies showing similar results [25, 14]. Given that WhatsApp is a
closed platform with no content moderation or third party fact checking, the
fact that misinformation tends to stick around longer might be expected, but
might also be problematic when compared to other social networks, where
eventually corrections can be issued.

Table II: Lifetimes of COVID-19 related texts and images shared on WhatsApp.
_Misinformation tends to have the highest mean lifetime._ Text Messages
---
Label | Num. | Mean | Std Dev
| texts | (hrs) | (hrs)
Information | 1108 | 2.75 | 21.98
Religious | 829 | 6.98 | 29.15
Jokes/Satire | 919 | 1.92 | 9.15
Misinformation | 596 | 7.0 | 28.03
Ambiguous | 313 | 10.05 | 39.2
Images
Label | Num. | Mean | Std Dev
| images | (hrs) | (hrs)
Information | 1069 | 0.55 | 2.87
Religious | 557 | 2.70 | 6.14
Jokes/Satire | 238 | 1.21 | 4.07
Misinformation | 236 | 5.57 | 9.17
Ambiguous | 389 | 1.35 | 4.31
Figure 4: CDFs of Text of COVID-19 categories (Note the broken y-axis: For
better Interpretability).

##  V RQ2: COVID-19 Misinformation

In this section, we specifically look at misinformation posts, and
characterize the types of misinformation shared on WhatsApp. We first
categorize the types of misinformation based on reports from popular fact
checking organizations and then base our characterization on this
categorization.

###  V-A Misinformation Message Analysis

The distribution of the various categories is shown in Figure 5.

Figure 5: Percentage of texts on WhatsApp for each type of COVID-19 related
misinformation.

Fake News. The most frequent form of COVID related misinformation is in the
form of fake news with 45% of misinformation texts. This includes fake news
pertaining to COVID positive tests and COVID related deaths of world figures
such as Ivanka Trump, Prince Williams and even the current Prime Minister of
Pakistan, Imran Khan. Conspiracy theories about Bill Gates intending to place
RFID chips in people to track COVID-19 were also seen. Ironically, fake news
were also observed regarding a doctored government action announcing
‘Punishment for Spreading Fake News on social media’.

Fake Origins. The second most prevalent form of COVID-19 related
misinformation is claiming fake origin stories for the virus with 22% of the
misinformation texts. Fake origin stores include a Corona named lake in
Kazakhstan from which the virus came to being grown in a lab in China or
United States. A few Hollywood movies, namely ‘Contagion’, ‘Resident Evil’ and
‘I am Legend’ along with the book ‘The Eye of Darkness’ were frequently
mentioned stating that COVID-19 had been predicted by them.

Fake Remedies. Making up roughly 20% of the misinformation, this type contains
bogus remedies and treatments such as the 1-minute breath hold test to detect
COVID, and various items like basil seeds, gargling with salt or garlic water,
honey lemon tea and even Hepatitis-C medicine as cures to COVID-19.

Vaccine Myths. The fake origin stories were sometimes accompanied with claims
of the vaccine already being developed and being used as an economic leverage.
Countries such as Israel, China and United States were mentioned with claims
of the vaccine already developed. This category makes up around 6% of the
misinformation texts.

Weather Myths. Four percent of the misinformation claims that the virus can
not survive in winter, summer or rainy seasons and the outbreak would die down
on its own.

Flu Comparison. Only 2% of the misinformation attempted to downplay the
symptoms and severity of the disease by comparison to the common seasonal flu.
Even though this narrative was popular elsewhere (e.g. US), it did not have
much salience in Pakistan, with the general public acknowledging COVID-19 as a
distinct and more dangerous disease as compared to the common flu.

###  V-B Lifetime of Misinformation

The temporal properties of the various categories of misinformation are
analyzed in table III. For each message, we compute the lifetime as the
difference between its last and first occurance. The ‘Fake News’ category has
the shortest lifespan as evidenced by the lowest mean of 4 hrs. This seems to
be consistent with the hypothesis that event triggered content is short-lived,
with similar properties to the ‘Jokes/Satire’ category of COVID-19 related
textual content. The highest lifespan is for the ‘Fake Remedies’ category with
a mean life of 10 hrs, which is significantly larger than other major
categories. This indicates that content that is not tethered to a social event
is more likely to being in circulation on a social media platform like
WhatsApp.

Table III: Lifetime of misinformation texts shared on WhatsApp.  Label | Num. | Mean | Std Dev
---|---|---|---
| texts | (hrs) | (hrs)
Fake News | 307 | 4.06 | 18.7
Fake Origins | 171 | 9.4 | 35.3
Fake Remedies | 125 | 10.6 | 33.8
Weather Myths | 26 | 27.57 | 67.6
Flu Comparison | 16 | 6.68 | 16.66

##  VI RQ3:User Behaviour Analysis

Every WhatsApp group has two types of users: (1) producers and (2) consumers.
Some users share and post messages whereas others silently observe. In
general, producers are few and consumers are many (Table I). In this section,
by examining the user behavior, we hope to understand if there is any
deliberate spread of disinformation.

We use “UpSet”101010For an introduction, see
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4720993/. plots, to visualize
user behavior, where every set is a unique user. The bottom matrix
(combination matrix) of Figure 6,7 shows the intersections of the sets across
COVID categories, while the bars on the top indicate the number of users
(sets) within that intersection. The bars on the left indicate total users
(sets) within a given category.

###  VI-A Text Sharing Trends

The UpSet plot in Figure 6 is plotted against the text messages shared by
individual users. If we observe the combination matrix, we observe that users
are sharing textual content belonging to single category. The most exclusively
shared category is “Ambigous”. It can be attributed to the users that join
WhatsApp groups intending to share advertisements and call-for-donations only.
The second and third highest intersection sets are for “Religious” and
“Misinformation” being shared exclusively. This deviates from the trend
observed for Images. People are more likely to exclusively share “Texts”
containing “Misinformation” as compared to “Images” containing
“Misinformation”. This prompts for need for more research in finding traces of
disinformation within text messages.

Figure 6: UpSet plot for users posting COVID related texts. Top 15
intersection sets are visualized. _More users appear to share texts that
belong to a single category._ Figure 7: UpSet plot for users posting COVID
related images. Only the top 15 intersection sets are visualized. _A lot of
users are sharing information and religious content, whereas some share
misinformation._

###  VI-B Image Sharing Trends

The UpSet plot shown in Figure 7 is made against the type of Images individual
users are sharing. It is good to see that the majority of users are sharing
correct information about the pandemic. An encouraging trend is that users are
not exclusively sharing misinformation, rather a mix of content is being
shared. Only 37 users exclusively shared misinformation, whereas 67 users
shared a mix of content, along with misinformation.

To further understand if there is disinformation, we tried to see if a
specific type of image is being spread more than others. Using the clusters we
already had, made via PDQ hashing and Hamming distance, clusters having
misinformation were identified. The number of images within these clusters are
a good indicator of the impact of given misinformation on the network. 23
unique images were shared more than 1 times and only 8 were shared more than 5
times.

This implies that even if we consider the images shared multiple times to be
disinformation, the quantity of disinformation is very low. Hence we believe,
the misinformation is, rather than being an organized effort, is mainly being
spread due to lack of awareness.

##  VII RQ4: Cross Network Information Spread

Finally, we answer how information flows between WhatsApp and Twitter. Given
that each social network has different properties (closed vs. open),
affordances (e.g.,the ability to see how popular a content is with
retweet/like count) and user bases, such a comparison is interesting.

###  VII-A Methodology

Twitter was chosen as most of Twitter data is public, and it serves as one of
the major information conduits. We obtained more than 0.8 million unique
tweets starting from January 10 to April 9 using an exhaustive list of
hashtags related to COVID-19 in Pakistan. It must be noted, that both Twitter
and WhatsApp datasets only represent a subset of the actual activity and in no
way can be thought to represent the full networks’ behaviour. In order to
understand the flow of information, images and text present within WhatsApp,
from 16 March to 9 April 2020, were compared with tweets in the same time
range. To find similar content across Twitter and WhatsApp, PDQ hashing was
used for images and fuzzy string matching was used for text messages.

###  VII-B Cross Platform Image Spread

Figure 8: CDFs of life of an image, along with content type, as seen on
Twitter. Twitter tends to hold a message alive for a coupe of days. _A healthy
trend is that Information tends to live the longest on Twitter_.

To understand how images were propagated across networks, we isolate the
tweets containing image content from the Twitter dataset. This covers a total
of 67,119 images. We then generate PDQ hashes for both WhatsApp and Twitter
images, and matching two images if their hashes have a Hamming distance of 40
(default value suggested by PDQ). Around 1,500 similar images were found
common to both WhatsApp and Twitter, within the date range of interest (16
March and 9 April). Out of these 1,500 images, 541 were COVID-19 related.

Table IV breaks down the images into the categories previously described,
alongside the average number of retweets, replies and the lifetime of the
image (difference between first and last appearance of an image on Twitter).
Firstly, we observe that the largest category of images shared across both
Twitter and WhatsApp is that of misinformation (29%). We can also see that
misinformation tweets have a high average number of retweets, potentially
reaching tens of thousands of users. This is signal in Twitter (or other
social networks) which does not exist on WhatsApp, where the social popularity
signals like retweet or like counts are shown.

Compared to the other categories, misinformation on Twitter tends to die
quicker. Figure 8 shows a CDF of the lifespan of the various categories of
images on Twitter. We see that most types last at most a day, with most having
a long tail lasting weeks. This is very different from the pattern we observed
on WhatsApp, where on average, most images lasted only a few hours. Images
containing information have a much higher lifetime, which is also in stark
contrast to what we observed on WhatsApp.

The difference in longevity of a message, as seen in Table II and IV, points
towards the nature of interactions. Interactions on WhatsApp are immediate, as
new messages constantly replace old ones. Whereas on Twitter old tweets can be
easily brought back into limelight, using retweets, comments and likes by
influential people. Hence the ability of Twitter to keep conversation, around
a tweet, alive for a long time could be the reason for the overall life of
COVID tweets (in days) compared to WhatsApp (in hours).

Table IV: Characteristics of images mapped between Twitter and WhatsApp. Label | Num. | Retweets | Replies | Life
---|---|---|---|---
| images | (Mean) | (Mean) | (days)
Information | 79 | 64.18 | 75.27 | 5.05
Religious | 108 | 42.66 | 45.82 | 3.25
Jokes/Sarcasm | 104 | 26.96 | 89.71 | 3.0
Misinformation | 183 | 67.72 | 444.28 | 1.6
Ambiguous | 146 | 156.82 | 309.64 | 4.2

A schematic diagram of the temporal flow of images across WhatsApp and Twitter
is shown in Figure 9. The Figure also provides three example case studies of
images which originated on WhatsApp and went on to become widely retweeted on
Twitter. From our analysis of the timelines of images observed on both
platforms, we can conclude that most of the images are seen originating from
WhatsApp and then appearing on Twitter (in our dataset). On average an image
appears on WhatsApp 4 days earlier as compared to Twitter. As a result in
light of the data analyzed, it can be concluded that WhatsApp plays a critical
role in COVID related content dissemination to other networks in Pakistan.
This is especially important in the context of results from Table IV with a
majority (29%) of the content that is common between the two platforms being
misinformation, compared to only 12% being information.

Figure 9: COVID images’ temporal flow across WhatsApp and Twitter (a line’s
thickness depicts the number of images flowing across). Some Observations: a)
a news snippet originates from WhatsApp on 19th March and is seen on Twitter
on 21st; b) religious supplication to fight COVID is observed on WhatsApp 2
days earlier than on Twitter; c) official stats of COVID patients seen on 30th
March on WhatsApp earlier than on Twitter.

##  VIII Conclusions

In this paper, we have provided the first detailed analysis of Pakistani
WhatsApp public groups, focusing on the COVID-19 discourse. We have analyzed
WhatsApp text and image messages collected for more than 6 weeks from 227
public WhatsApp groups to shed light on the salient misinformation
dissemination trends and to share insights on how Pakistani social media users
are experiencing and responding to the COVID-19 pandemic. Our work is unique
as this is the first work to not only study misinformation trends on WhatsApp
but also find a relation between WhatsApp and Twitter. Our analyses showed
that while it is true that the majority of shared information is not
misinformation, misinformation seems to have a longer lifespan on WhatsApp
compared to other types of COVID messages (the lifetime of misinformation is
roughly 4 times that of correct information). On Twitter the inverse was seen,
as COVID misinformation tended to disappear from Twitter 3 times faster than
information. This can potentially be attributed to the open nature of Twitter,
and how a vast number of users can publicly negate such tweets. While
observing user behavior, we found 8 images that could be attributed to
organized disinformation, other than that, we did not find any evidence of
disinformation within images. Whereas more work is required in detecting
disinformation via text messages. We conclude by saying that our dataset has
only scratched the surface of how user interactions happen on WhatsApp. More
work needs to be performed to understand user behavior, and new ways need to
be proposed to detect misinformation in such closed networks.

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