Here, the open set $\mathcal{U}$ and the involved constants are uniform with respect to $\alpha$, $\nu \ge 1$ and $\delta \in (0, 1/\nu)$. Moreover, the real numbers $d_\alpha$ and $e_\alpha$ satisfy

$$d_{\alpha} \ll \sum_{j=2}^{d} \delta + \sum_{j=2}^{d} \left( \left( \frac{1}{\delta} \right)^{1-\frac{1}{k_j}} \right)^{k_j} \ll \delta \quad \text{and} \quad e_{\alpha} \ll \delta \sum_{j=2}^{d} \left( \frac{1}{\delta} \right)^{1-\frac{1}{k_j}} \ll \delta^{\frac{1}{k_1}}.$$

We now apply the inverse function theorem to solve the system (26). Provided $\nu$ is large enough, we get a solution $(\rho_1(\alpha), u_1(\alpha))$ satisfying $\sup(\rho_1(\alpha), |u_1(\alpha)|) \ll \delta^{1/k_1}$. In this case, the involved constant depends on $\nu$, but it is uniform with respect to $\alpha$ and $\delta$.

Now, a look at the first equation of (26) shows that we in fact have the more precise inequality $\delta \ll \rho_1(\alpha) \ll \delta$, provided $\nu$ is sufficiently large, and this is independent of $\alpha$ and $\delta \in (0, 1)$. Looking now at the second equation of (26), if $\alpha \ne \beta \in \Lambda$ satisfy $\alpha_j = \beta_j$ for $j \ge 2$, so that $e_\alpha = e_\beta$ and $h_\alpha = h_\beta$, then $|u_1(\alpha) - u_1(\beta)| \gg \delta$.

Hence, we have obtained $\prod_{j=2}^{d} \lfloor (1/\delta)^{1-\frac{1}{k_j}} \rfloor$ solutions to the equation $\Phi(Z(\alpha)) = s_m$, and they satisfy the conclusions of Lemma 25 since the inequalities on $u_j(\alpha)$ are also valid for $\theta_j(\alpha)$ up to a constant depending only of $\Phi$. $\square$

*Final part in the proof of the lower bound of Theorem 16.* We apply Lemma 21 to $S_\delta$ and $Z_\delta$ given by the previous lemma, for

$$\delta = \left(\frac{1}{n}\right)^{\frac{k_1}{k_1-1}},$$

so that $S_\delta$ has cardinal number equal to $n$. Since

$$M_{\mathfrak{J}^2}(S_\delta) \ll 1 \quad \text{and} \quad \|\mu_{Z,H^2(\mathbb{T}^d)}\|_{\epsilon, H^2(\mathbb{T}^d)} \ll 1$$

by Lemma 19 and Lemma 20, we are left to estimate the sum $N_\Phi(s_m; Z)\zeta(2\operatorname{Re} s_m)$ for any $m$. Using the fact that $\rho_j(\alpha) \gg \delta$ for any $j = 1, \dots, d$ and any $\alpha$, we obtain

$$N_{\Phi}(s_m; Z)\zeta(2\operatorname{Re} s_m) \gg \left(\frac{1}{\delta}\right)^{\sum_{j=2}^{d} \left(1-\frac{1}{k_j}\right)} \cdot \delta^d \cdot \delta^{-1} \gg \delta^{\sum_{j=2}^{d} \frac{1}{k_j}} \gg \left(\frac{1}{n}\right)^{\left(\sum_{j=2}^{d} \frac{1}{k_j}\right) \times \frac{k_1}{k_1-1}},$$

and we are done. $\square$

<figure><img src="image_10.png" /></figure>

**Remark.** We may modify the proof of Theorem 16 so that we do not assume that there exists a unique $w \in \mathbb{T}^d$ such that $\operatorname{Re} \Phi(w) = 0$. Suppose that $\varphi$ is boundary regular at any $w \in \mathbb{T}^d$ such that $\operatorname{Re} \Phi(w) = 0$. Define now the *compactness index* of $\varphi$ as the real number

$$\eta_{\varphi}(s) = \inf\{\eta_{\varphi,w} : \operatorname{Re} \Phi(w) = 0\}.$$

It should be observed that this infimum is in fact a minimum. Indeed, our assumptions imply that the points $w \in \mathbb{T}^d$ such that $\operatorname{Re} \Phi(w) = 0$ are isolated. Theorem 16 remains true with this new definition of $\eta_\varphi$.
References

[Bayart 2002] F. Bayart, “Hardy spaces of Dirichlet series and their composition operators”, *Monatsh. Math.* **136**:3 (2002), 203–236. MR Zbl

[Bayart 2003] F. Bayart, “Compact composition operators on a Hilbert space of Dirichlet series”,
*Illinois J. Math.* **47**:3 (2003), 725–743. MR Zbl

[Bayart and Brevig 2015] F. Bayart and O. F. Brevig, Xcas file for Lemma 11, 2015, available at http://bayart.perso.math.cnrs.fr/nonlin.xws.

[Berndtsson et al. 1987] B. Berndtsson, S.-Y. A. Chang, and K.-C. Lin, “Interpolating sequences in the polydisc”, *Trans. Amer. Math. Soc.* **302**:1 (1987), 161–169. MR Zbl

[Bruce and Giblin 1992] J. W. Bruce and P. J. Giblin, *Curves and singularities: a geometrical introduction to singularity theory*, 2nd ed., Cambridge University Press, 1992. MR Zbl

[Chang 1979] S.-Y. A. Chang, “Carleson measure on the bi-disc”, *Ann. of Math.* (2) **109**:3 (1979), 613–620. MR Zbl

[Finet and Queffélec 2004] C. Finet and H. Queffélec, “Numerical range of composition operators on a Hilbert space of Dirichlet series”, *Linear Algebra Appl.* **377** (2004), 1–10. MR Zbl

[Finet et al. 2004] C. Finet, H. Queffélec, and A. Volberg, “Compactness of composition operators on a Hilbert space of Dirichlet series”, *J. Funct. Anal.* **211**:2 (2004), 271–287. MR

[Gordon and Hedenmalm 1999] J. Gordon and H. Hedenmalm, “The composition operators on the space of Dirichlet series with square summable coefficients”, *Michigan Math. J.* **46**:2 (1999), 313–329. MR Zbl

[Hardy and Wright 1979] G. H. Hardy and E. M. Wright, *An introduction to the theory of numbers*, 5th ed., Oxford University Press, New York, 1979. MR Zbl

[Queffélec and Seip 2015a] H. Queffélec and K. Seip, “Approximation numbers of composition operators on the $H^2$ space of Dirichlet series”, *J. Funct. Anal.* **268**:6 (2015), 1612–1648. MR Zbl

[Queffélec and Seip 2015b] H. Queffélec and K. Seip, “Decay rates for approximation numbers of composition operators”, *J. Anal. Math.* **125** (2015), 371–399. MR Zbl

Received November 17, 2015.

FRÉDÉRIC BAYART
LABORATOIRE DE MATHÉMATIQUES, UMR 6620, CNRS
CLERMONT UNIVERSITÉ, UNIVERSITÉ BLAISE PASCAL, CAMPUS DE CÉZEAUX
3, PLACE VASARELY, BP 10448
TSA 60026, CS 60026
63178 AUDIÈRE CEDEX
FRANCE

bayart@math.univ-bpclermont.fr

OLE FREDRIK BREVIG
DEPARTMENT OF MATHEMATICAL SCIENCES
NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY (NTNU)
7491 TRONDHEIM
NORWAY

ole.brevig@math.ntnu.no
References

[Bayart 2002] F. Bayart, “Hardy spaces of Dirichlet series and their composition operators”, *Monatsh. Math.* **136**:3 (2002), 203–236. MR Zbl

[Bayart 2003] F. Bayart, “Compact composition operators on a Hilbert space of Dirichlet series”,
*Illinois J. Math.* **47**:3 (2003), 725–743. MR Zbl

[Bayart and Brevig 2015] F. Bayart and O. F. Brevig, Xcas file for Lemma 11, 2015, available at http://bayart.perso.math.cnrs.fr/nonlin.xws.

[Berndtsson et al. 1987] B. Berndtsson, S.-Y. A. Chang, and K.-C. Lin, “Interpolating sequences in the polydisc”, *Trans. Amer. Math. Soc.* **302**:1 (1987), 161–169. MR Zbl

[Bruce and Giblin 1992] J. W. Bruce and P. J. Giblin, *Curves and singularities: a geometrical introduction to singularity theory*, 2nd ed., Cambridge University Press, 1992. MR Zbl

[Chang 1979] S.-Y. A. Chang, “Carleson measure on the bi-disc”, *Ann. of Math.* (2) **109**:3 (1979), 613–620. MR Zbl

[Finet and Queffélec 2004] C. Finet and H. Queffélec, “Numerical range of composition operators on a Hilbert space of Dirichlet series”, *Linear Algebra Appl.* **377** (2004), 1–10. MR Zbl

[Finet et al. 2004] C. Finet, H. Queffélec, and A. Volberg, “Compactness of composition operators on a Hilbert space of Dirichlet series”, *J. Funct. Anal.* **211**:2 (2004), 271–287. MR

[Gordon and Hedenmalm 1999] J. Gordon and H. Hedenmalm, “The composition operators on the space of Dirichlet series with square summable coefficients”, *Michigan Math. J.* **46**:2 (1999), 313–329. MR Zbl

[Hardy and Wright 1979] G. H. Hardy and E. M. Wright, *An introduction to the theory of numbers*, 5th ed., Oxford University Press, New York, 1979. MR Zbl

[Queffélec and Seip 2015a] H. Queffélec and K. Seip, “Approximation numbers of composition operators on the $H^2$ space of Dirichlet series”, *J. Funct. Anal.* **268**:6 (2015), 1612–1648. MR Zbl

[Queffélec and Seip 2015b] H. Queffélec and K. Seip, “Decay rates for approximation numbers of composition operators”, *J. Anal. Math.* **125** (2015), 371–399. MR Zbl

Received November 17, 2015.

FRÉDÉRIC BAYART
LABORATOIRE DE MATHÉMATIQUES, UMR 6620, CNRS
CLERMONT UNIVERSITÉ, UNIVERSITÉ BLAISE PASCAL, CAMPUS DE CÉZEAUX
3, PLACE VASARELY, BP 10448
TSA 60026, CS 60026
63178 AUDIÈRE CEDEX
FRANCE

bayart@math.univ-bpclermont.fr

OLE FREDRIK BREVIG
DEPARTMENT OF MATHEMATICAL SCIENCES
NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY (NTNU)
7491 TRONDHEIM
NORWAY

ole.brevig@math.ntnu.no
PACIFIC JOURNAL OF MATHEMATICS

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<figure><img src="image_11.png" /></figure>

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©2017 Mathematical Sciences Publishers
PACIFIC JOURNAL OF MATHEMATICS

Founded in 1951 by E. F. Beckenbach (1906–1982) and F. Wolf (1904–1989)

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EDITORS

Don Blasius (Managing Editor)
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Los Angeles, CA 90095-1555
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Department of Mathematics
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Los Angeles, CA 90095-1555
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Vyjayanthi Chari
Department of Mathematics
University of California
Riverside, CA 92521-0135
chari@math.ucr.edu

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Department of Mathematics
University of California
Santa Barbara, CA 93106-3080
cooper@math.ucsb.edu

Robert Finn
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
finn@math.stanford.edu

Kefeng Liu
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
liu@math.ucla.edu

Jiang-Hua Lu
Department of Mathematics
The University of Hong Kong
Pokfulam Rd., Hong Kong
jhlu@maths.hku.hk

Sorin Popa
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
popa@math.ucla.edu

Igor Pak
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
pak.pjm@gmail.com

Jie Qing
Department of Mathematics
University of California
Santa Cruz, CA 95064
qing@cats.ucsc.edu

Paul Yang
Department of Mathematics
Princeton University
Princeton NJ 08544-1000
yang@math.princeton.edu

PRODUCTION

Silvio Levy, Scientific Editor, production@msp.org

SUPPORTING INSTITUTIONS

ACADEMIA SINICA, TAIPEI
CALIFORNIA INST. OF TECHNOLOGY
INST. DE MATEMÁTICA PURA E APLICADA
KEIO UNIVERSITY
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NEW MEXICO STATE UNIV.
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These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no
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See inside back cover or msp.org/pjm for submission instructions.

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Subscriptions, requests for back issues and changes of subscriber address should be sent to Pacific Journal of Mathematics, P.O. Box
4163, Berkeley, CA 94704-0163, U.S.A. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH,
PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and Web of Knowledge (Science Citation Index).

The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 798 Evans Hall
#3840, Berkeley, CA 94720-3840, is published twelve times a year. Periodical rate postage paid at Berkeley, CA 94704, and additional
mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163.

PJM peer review and production are managed by *EditFLOW®* from Mathematical Sciences Publishers.

PUBLISHED BY

<figure><img src="image_11.png" /></figure>

mathematical sciences publishers

nonprofit scientific publishing

http://msp.org/

©2017 Mathematical Sciences Publishers
PACIFIC JOURNAL OF MATHEMATICS

Volume 291 No. 1 November 2017

<table><tr><td>Chain transitive homeomorphisms on a space: all or none<br>ETHAN AKIN and JUHO RAUTIO</td><td>1</td></tr><tr><td>Spinorial representation of submanifolds in Riemannian space forms<br>PIERRE BAYARD, MARIE-AMÉLIE LAWN and JULIEN ROTH</td><td>51</td></tr><tr><td>Compact composition operators with nonlinear symbols on the H<sup>2</sup><br>space of Dirichlet series<br>FRÉDÉRIC BAYART and OLE FREDRIK BREVIG</td><td>81</td></tr><tr><td>A local relative trace formula for PGL(2)<br>PATRICK DELORME and PASCALE HARINCK</td><td>121</td></tr><tr><td>Regularity of the analytic torsion form on families of normal coverings<br>BING KWAN SO and GUANGXIANG SU</td><td>149</td></tr><tr><td>Thick subcategories over isolated singularities<br>RYO TAKAHASHI</td><td>183</td></tr><tr><td>Projections in the curve complex arising from covering maps<br>ROBERT TANG</td><td>213</td></tr><tr><td>The local Ginzburg-Rallis model over the complex field<br>CHEN WAN</td><td>241</td></tr></table>

<figure><img src="image_12.png" /></figure>

0030-8730(201711)291:1;1-Y
PACIFIC JOURNAL OF MATHEMATICS

Volume 291 No. 1 November 2017

<table><tr><td>Chain transitive homeomorphisms on a space: all or none<br>ETHAN AKIN and JUHO RAUTIO</td><td>1</td></tr><tr><td>Spinorial representation of submanifolds in Riemannian space forms<br>PIERRE BAYARD, MARIE-AMÉLIE LAWN and JULIEN ROTH</td><td>51</td></tr><tr><td>Compact composition operators with nonlinear symbols on the H<sup>2</sup><br>space of Dirichlet series<br>FRÉDÉRIC BAYART and OLE FREDRIK BREVIG</td><td>81</td></tr><tr><td>A local relative trace formula for PGL(2)<br>PATRICK DELORME and PASCALE HARINCK</td><td>121</td></tr><tr><td>Regularity of the analytic torsion form on families of normal coverings<br>BING KWAN SO and GUANGXIANG SU</td><td>149</td></tr><tr><td>Thick subcategories over isolated singularities<br>RYO TAKAHASHI</td><td>183</td></tr><tr><td>Projections in the curve complex arising from covering maps<br>ROBERT TANG</td><td>213</td></tr><tr><td>The local Ginzburg-Rallis model over the complex field<br>CHEN WAN</td><td>241</td></tr></table>

<figure><img src="image_12.png" /></figure>

0030-8730(201711)291:1;1-Y