2.1. Equivalence of classical Riesz products

Proposition 2.1 ([Reference Kilmer and Saeki9, Theorem 1]). Let $\mu (J, a)$ and $\mu (H, b)$ be classical Riesz products on the unit circle. Assume that

\begin{equation*} \sum _{k\in J\cup H} |a_k - b_k| \lt \infty . \end{equation*}

Then, $\mu (J, a)\approx \mu (H, b)$ .

The main results of the present paper, Theorems1.1 and 4.1, extend the above mutual absolute continuity criterion to the standard and pluriharmonic Riesz products on the unit sphere.

2.2. Basic facts of the harmonic analysis on the sphere $S_n$

Let $\sigma _n$ denote the normalized Lebesgue measure on $S_n$ . Observe that

(2.1) \begin{equation} L^2(\sigma _n) = \mathop {\oplus }\limits _{(\, p,q)\in \mathbb{Z}_+^2} H(p, q). \end{equation}

Specifics of the harmonic analysis on the sphere $S_n$ are suitably illustrated by the multiplication rule for spherical harmonics: if $f \in H(p, q)$ and $g \in H(r, s)$ , then

\begin{equation*} fg\in \sum _{\ell =0}^L H(p + r -\ell , q + s -\ell ), \end{equation*}

where $L = \min (p, s) + \min (q, r)$ , and $f$ and $g$ are restricted to $S_n$ . Proofs of the above facts and further results are presented in Chapter 12 of the monograph [Reference Rudin12].

Let $M(S_n)$ denote the space of all complex Borel measures on the sphere $S_n$ . Let $K_{p,q}(z, \zeta )$ denote the reproducing kernel for the Hilbert space $H(p, q) \subset L^2(S_n)$ . The polynomial

\begin{equation*} \mu _{p,q}(z) = \int _{S_n} K_{p,q} (z, \zeta )\, d\mu (\zeta ), \quad z\in S_n, \end{equation*}

is called the $H(p, q)$ -projection of the measure $\mu \in M(S_n)$ . Given $\mu \in M(S_n)$ , the spectrum $\textrm {spec}(\mu )$ in terms of $H(\, p,q)$ is defined by the equality

\begin{align*} \textrm {spec}(\mu ) = \left \{ (p, q) \in \mathbb{Z}_+^2:\, \mu _{p,q} \neq \mathbf{0} \right \}. \\[-20pt] \end{align*}

2.3. Decomposable measures

For $f\in C(S_n)$ , it is well known that

\begin{equation*} \int _{S_n} f\, d\sigma _n = \int _{S_n} \int _{\mathbb{T}} f_\zeta \, d\sigma _1 d\sigma _n (\zeta ), \end{equation*}

where $f_\zeta (\lambda ) = f(\lambda \zeta )$ , $\lambda \in \mathbb{T}$ and $\zeta \in S_n$ . This property of $\sigma _n$ motivates the following definition.

Definition 3 (cf. [Reference Aleksandrov and Doubtsov3, Definition 2]). A measure $\mu \in M(S_n)$ is called $\text{decomposable}$ if, for $\sigma _n$ -a.e. $\zeta \in S_n$ , there exists a measure $\mu _\zeta \in M(\mathbb{T})$ such that

\begin{equation*} \int _{S_n} \|\mu _\zeta \|\, d\sigma _n(\zeta ) \lt \infty \end{equation*}

and

\begin{equation*} \mu = \int _{S_n} \mu _\zeta \, d\sigma _n(\zeta ) \end{equation*}

in the following weak sense:

(2.2) \begin{equation} \int _{S_n} f\, d\mu = \int _{S_n} \int _{\mathbb{T}} f_\zeta \, d\mu _\zeta \, d\sigma _n(\zeta ) \end{equation}

for all $f\in C(S_n)$ .

2.3.1. Standard Riesz products are decomposable

For $\zeta \in S_n$ , the slice-product

\begin{equation*} \Pi _\zeta (R, J, a)(\lambda ) \, :\!= \, \Pi (R(\lambda \zeta ), J, a),\quad \lambda \in \mathbb{T}, \end{equation*}

is a classical Riesz product defined by the formula:

(2.3) \begin{equation} \prod _{k=1}^\infty \left ( \frac {\overline {a}_k \overline {R}_{j_k} (\zeta )\overline {\lambda }^{\, j_{k}}}{2} + 1 + \frac {{a}_k R_{j_k}(\zeta ) {\lambda }^{\, j_{k}}}{2} \right )\!, \quad \lambda \in \mathbb{T}. \end{equation}

In particular, $\Pi _\zeta (R, J, a)$ is a probability measure.

Proposition 2.2 ([Reference Doubtsov8, Lemma 1]). Let $(R, J, a)$ be a Riesz triple on the sphere $S_n$ , $n\ge 2$ . Then

\begin{equation*} \int _{S_n} f\, d\Pi (R, J, a) = \int _{S_n} \int _{\mathbb{T}} f_\zeta \, d\Pi _\zeta (R, J, a) d\sigma _n(\zeta ) \end{equation*}

for all $f\in C(S_n)$ . Therefore, $\Pi (R, J, a)$ is decomposable in terms of its slice-measures.

2.3.2. Positive pluriharmonic measures are decomposable

Let $B_n = \{z\in \mathbb{C}^n \, : \, |z|\lt 1\}$ . By definition, a measure $\mu \in M(S_n)$ is called pluriharmonic if its Poisson integral

\begin{equation*} P[\mu ](z) = \int _{S_n} \frac {(1-|z|^2)^n}{|1-\langle z, \zeta \rangle |^{2n}} \, d\mu (\zeta ),\quad z\in B_n, \end{equation*}

is a pluriharmonic function. Observe that $\mu$ is pluriharmonic if and only if

\begin{equation*} \mathrm{spec} (\mu ) \subset \{(p, q)\in \mathbb{Z}_+^2\, : \, pq=0\}. \end{equation*}

Now, let $\mu$ be a positive pluriharmonic measure. Put $u = P[\mu ]$ . For $\zeta \in S_n$ , the slice-function $u_\zeta$ is defined by $u_\zeta (z) =u(z\zeta )$ , $z\in \mathbb{D}$ . Since $u$ is pluriharmonic, $u_\zeta$ is a positive harmonic function for all $\zeta \in S_n$ . Hence, for every $\zeta \in S_n$ , there exists a measure $\mu _\zeta \in M(\mathbb{T})$ such that

\begin{equation*} u(z\zeta ) = \int _{\mathbb{T}} \frac {1-|z|^2}{|w-z|^2} \,d\mu _\zeta (w),\quad z\in \mathbb{D}. \end{equation*}

We say that $\mu _\zeta$ is a slice-measure. Also, observe that $\|\mu _\zeta \| = \|\mu \|$ .

Proposition 2.3 ([Reference Aleksandrov and Doubtsov3, Proposition 2.1]). Let $\mu \in M(S_n)$ be a positive pluriharmonic measure. Then

\begin{equation*} \int _{S_n} f\, d\mu = \int _{S_n} \int _{\mathbb{T}} f_\zeta \,d\mu _\zeta d\sigma _n(\zeta ) \end{equation*}

for all $f \in C(S_n)$ . Therefore, $\mu$ is decomposable in terms of its slice-measures.

4.1. Pluriharmonic projection

Let $PLH^2(S_n)$ denote the space of those $f \in L^2(S_n)$ , for which the Poisson integral

\begin{equation*} P[\, f](z) = \int _{S_n} \frac {(1-|z|^2)^n}{|1-\langle z, \zeta \rangle |^{2n}} f(\zeta )\, d\sigma _n(\zeta ),\quad z\in B_n, \end{equation*}

is pluriharmonic in the unit ball $B_n$ . Consider the orthogonal projector

\begin{equation*} K \, : \, L^2(S_n) \to PLH^2(S_n). \end{equation*}

Then every function $\varphi \in L^\infty (S_n)$ generates a Hankel type operator by the formula:

\begin{equation*} H_\varphi [\, f] = \varphi K[\, f] - K[\varphi f], \quad f \in L^2(S_n). \end{equation*}

Now, assume that $\varphi$ is a polynomial. Then the operator $H_\varphi \, : \, C(S_n) \to C(S_n)$ is compact (see, e.g., [Reference Rudin12, Theorem 6.5.4]), hence,

(4.1) \begin{equation} [ \|\, f_j\|_{C(S_n)} \le 1\ \ \ \textrm{and}\ \ \ f_j \to 0 \ \textrm {weakly in}\ L^2(S_n) ] \Rightarrow \|H_\varphi f_j\|_{C(S_n)} \to 0. \end{equation}

The above property is crucial for the transition from the standard Riesz products to the pluriharmonic ones.

4.2. Pluriharmonic Riesz products

Let $(R, a)$ be a Riesz pair, that is, $R$ is a Ryll–Wojtaszczyk sequence and $a = \{a_k\}_{k=1}^\infty$ , $a_k \in \mathbb{D}$ . Note that in this case, the set of indices $J$ is not fixed.

By induction, we construct a sequence of pluriharmonic polynomials $\varphi _k$ , $\varphi _k \gt 0$ on $S_n$ , and an index set $J = \{\, j_k\}_{k=1}^\infty$ , $j_{k+1}/j_k \ge 3$ . As the base of induction, we fix $j_1 \in \mathbb{N}$ and define

\begin{equation*} \varphi _1 = 1 + \mathrm{Re}\, (a_1 R_{j_1}). \end{equation*}

Step $k + 1$

By the induction assumption, we have a strictly positive pluriharmonic polynomial $\varphi _k$ on $S_n$ . Put

\begin{equation*} \varphi _{k+1} = K ( \varphi _k [ 1 + \mathrm{Re}\, ( a_{k+1} R_{j_{k+1}} )]) . \end{equation*}

Using property (4.1), we select $j_{k+1}$ so large that $\varphi _{k+1} \gt 0$ on $S_n$ . Also, the lacunas in the sequence $J$ and the multiplication rule for spherical harmonics guarantee that the new spectrum $\textrm {spec}(\varphi _{k+1} - \varphi _k)$ does not intersect a square $[0,M]^2$ such that $[0,M]^2\supset \textrm {spec}(\varphi _k)$ .

Recall that $\sigma _n$ denotes the normalized Lebesgue measure on the sphere $S_n$ . The above restrictions imply that the sequence $\varphi _k \sigma _n$ weakly^* converges to a probability pluriharmonic measure. This limit measure $\pi = \pi (R, J, a)$ is called a pluriharmonic Riesz product.

4.3. Equivalence of pluriharmonic Riesz products

Theorem 4.1. Let $(R, a^{(d)})$ , $d=1,2$ , be Riesz pairs. Suppose that $a^{(1)}- a^{(2)} \in \ell ^1$ . Then, for all sufficiently lacunary index sets $J=\{\, j_k\}_{k=1}^\infty$ , we have

\begin{equation*} \pi (R, J, a^{(1)}) \approx \pi (R, J, a^{(2)}). \end{equation*}

Proof. Fix a sequence $\{\varepsilon _k\}_{k=1}^\infty \in \ell ^1$ such that $\varepsilon _k \gt 0$ . Below, we simultaneously construct two products $\pi (R, J, a^{(1)})$ and $\pi (R, J, a^{(2)})$ , imposing additional restrictions.

So, on Step $k+1$ of the induction procedure, we assume that positive pluriharmonic polynomials $\varphi ^{(1)}_k$ and $\varphi ^{(2)}_k$ are constructed on $S_n$ .

(1) Conditions to guarantee correct definitions.

For $d=1,2$ , put

\begin{equation*} \varphi _{k+1}^{(d)}(\ell ) = K \left( \varphi _k^{(d)} \left[ 1 + \mathrm{Re}\, \left( a_{k+1}^{(d)} R_{j_{k+1}} \right)\right]\right) . \end{equation*}

By (4.1), we have $\varphi ^{(d)}_{k+1}(\ell ) \ge \mathrm{const}(k) \gt 0$ on $S_n$ , $d=1,2$ , for all sufficiently large $\ell \in \mathbb{N}$ .

(2) Conditions to guarantee equivalence of slices.

For $d = 1, 2$ , we have $\varphi ^{(d)}_{k+1}(\ell ) \ge \mathrm{const}(k) \gt 0$ on $S_n$ ; hence, by (4.1), for all sufficiently large $\ell \in \mathbb{N}$ , we have

(4.2) \begin{eqnarray} \left \|\frac {\varphi _{k+1}^{(d)}(\ell )} {\varphi _k^{(d)}\left ( 1 + \mathrm{Re}\, \left(a_{k+1}^{(d)} R_\ell \right)\right )} -1 \right \|_{C(S_n)} &\lt & \varepsilon _k; \\[-20pt] \nonumber \end{eqnarray}

(4.3) \begin{eqnarray} \left \|\frac {\varphi _k^{(d)}\left ( 1 + \mathrm{Re}\, \left(a_{k+1}^{(d)} R_\ell \right)\right )} {\varphi _{k+1}^{(d)}(\ell )} -1 \right \|_{C(S_n)} &\lt & \varepsilon _k \\[8pt] \nonumber \end{eqnarray}

for $d = 1, 2$ .

(3) Final choice of $j_{k+1}$ .

We select so large $\ell$ that $\ell \ge 3{j_k}$ and all above conditions hold. Put $j_{k+1} = \ell$ . Finally, define $\varphi ^{(d)}_{k+1} = \varphi ^{(d)}_{k+1}(\ell )$ for $d = 1, 2$ . The induction Step $k + 1$ is finished. Observe that the standard products $\Pi (R, J, a^{(d)})$ , $d=1,2$ , are automatically defined.

(4) Verification of the properties for the products $\pi ^{(d)} = \pi (R, J, a^{(d)})$ .

The arguments below do not depend on the index $d$ ; thus, we omit it. We introduce auxiliary functions

\begin{equation*} f_k = 1 + \mathrm{Re}\, (a_k R_{j_k}) \end{equation*}

and consider the partial products $\Pi _p = \prod _{k=1}^p f_k$ . Then, by estimate (4.2), we have

\begin{equation*} \left \|\frac {\varphi _{k+1}}{\Pi _{k+1}}\right \|_{C(S_n)} = \left \| \frac {\prod ^{k}_{j=1} \varphi _{j+1}}{\prod ^k_{j=1} \varphi _{j} \, f_{j+1}} \right \|_{C(S_n)} \le \prod ^k_{j=1} (1 + \varepsilon _j). \end{equation*}

In other words, the sequence $\{(\varphi _k)_\zeta /(\Pi _k)_\zeta \}_{k=1}^\infty$ is bounded in the space $L^\infty (\Pi _\zeta )$ for every point $\zeta \in S_n$ . Observe that

\begin{equation*} \frac {(\varphi _k)_\zeta }{(\Pi _k)_\zeta } \Pi _\zeta \to \pi _\zeta \end{equation*}

weakly^* in the space $M(\mathbb{T})$ as $k \to \infty$ . Hence, we obtain $\pi _\zeta \ll \Pi _\zeta$ . The arguments for the property $\Pi _\zeta \ll \pi _\zeta$ proceed similarly with the help of estimate (4.3). Therefore,

\begin{equation*} \pi _\zeta ^{(d)} \approx \Pi _\zeta ^{(d)}, \quad d=1,2. \end{equation*}

Observe that $\Pi _\zeta (R, J, a^{(1)}) \approx \Pi _\zeta (R, J, a^{(2)})$ for all $\zeta \in S_n$ by Proposition2.1. Hence,

(4.4) \begin{equation} \pi _\zeta (R, J, a^{(1)}) \approx \Pi _\zeta (R, J, a^{(1)})\approx \Pi _\zeta (R, J, a^{(2)}) \approx \pi _\zeta (R, J, a^{(2)}). \end{equation}

Since $\pi (R, J, a^{(d)})$ , $d=1,2$ , are pluriharmonic measures, they are decomposable in terms of their slices $\pi _\zeta (R, J, a^{(d)})$ , $\zeta \in S_n$ , by Proposition2.3. Thus, (4.4) and Lemma3.1 imply the required property $\pi (R, J, a^{(1)}) \approx \pi (R, J, a^{(2)})$ .