March 24, 2026
We establish a precise hierarchy for the maximal growth of the Stein-Wainger oscillatory integral as the regularity of the phase varies over Denjoy-Carleman classes, such as the Gevrey classes and their generalizations. In particular, we resolve a problem posed by Wang–Zhang [1], motivated by eigenfunction restriction estimates on curves, and also provide a new proof of a theorem of Nagel–Wainger [2] on the Hilbert transform along curves. A key ingredient is the sharp estimate on the growth of a phase near a flat point.
Let \(I=[-1,1]\), and let \(\psi\in C^1(I)\) be a real-valued function. For \(\lambda\gg1\), we define the Stein–Wainger oscillatory integral by \[\label{mla} m(\lambda) := \mathrm{p.v.}\int_I e^{i\lambda\psi(t)}\,\frac{dt}{t}.\tag{1}\] The main objective of this paper is to identify the exact relationship between the regularity of \(\psi\) and the maximal possible growth of \(|m(\lambda)|\) as \(\lambda\to\infty\). Throughout this paper, we assume that \(\lambda\) is sufficiently large. We also note that, it is harmless to replace the interval \(I\) by any bounded interval with \(0\) in its interior, since the contribution of the integral outside a fixed neighborhood of \(0\) is \(O(1)\) and is therefore negligible for our purposes.
Our main results establish a new hierarchy of upper bounds, together with matching lower bounds, reflecting the regularity of the phase.
First, for \(C^\alpha\) phases, the maximal growth of \(m(\lambda)\) is logarithmic.
Proposition 1. If \(\psi\in C^\alpha(I)\) for some integer \(\alpha\ge1\), then \(m(\lambda)=O(\log\lambda)\). This bound is sharp: there exists a real-valued \(\psi\in C^\alpha(I)\) such that \(|m(\lambda)|\approx \log\lambda\).
Second, for \(C^\infty\) phases, we may slightly improve the maximal growth to sub-logarithmic.
Theorem 2. If \(\psi\in C^\infty(I)\), then \(m(\lambda)=o(\log\lambda)\). This bound is sharp: for every integer \(k\ge 2\), there exists a real-valued \(\psi\in C^\infty(I)\) such that \[|m(\lambda_n)|\approx \frac{\log \lambda_n}{\log^{(k)}\lambda_n} \quad\text{along a sequence }\lambda_n\to\infty.\]
Third, we may further improve the maximal growth for phases in the Gevrey classes. Recall that for \(s\ge1\), a function \(\psi\in C^\infty(I)\) belongs to the Gevrey class \(G^s(I)\) if there exists a constant \(K>0\) such that for all \(n\ge0\) \[\sup_{t\in I} |\psi^{(n)}(t)| \le K^{n+1} (n!)^s.\] In particular, \(G^1=C^\omega\) is exactly the class of real-analytic functions. Thus Gevrey classes describe a regularity intermediate between smoothness and analyticity. Since their introduction in [3] in connection with regularity properties of the fundamental solution of the heat operator, Gevrey classes have been employed in a variety of contexts within the general theory of linear partial differential operators, including hypoellipticity, local solvability, and the propagation of singularities. For a comprehensive definition and a detailed discussion of Gevrey classes and their applications to linear partial differential operators, we refer the reader to [4].
Theorem 3. Let \(\psi\in G^s(I)\) with \(s>1\). Then \(m(\lambda)=O(\log\log\lambda)\). This bound is sharp: there exists a real-valued \(\psi\in G^s(I)\) such that \[|m(\lambda_n)| \approx \log\log \lambda_n \quad\text{along a sequence }\lambda_n\to\infty.\]
For \(\psi\in G^1(I)\), i.e. the analytic class, it is known that \(m(\lambda)=O(1)\). See Pan [5].
Next, we introduce some refinements between the Gevrey classes and the analytic class. These allow us to see clearly how the maximal growth of \(m(\lambda)\) improves from \(O(\log\log\lambda)\) to \(O(1)\). For \(s\ge 1\) and an integer \(k\ge 1\), a function \(\psi\in C^\infty(I)\) belongs to the refined Gevrey class \(G_k^s(I)\) if there exists a constant \(K>0\) such that for all sufficiently large \(n\) \[\sup_{t\in I} |\psi^{(n)}(t)| \le K^{n+1} n!Q_{k,s}(n)^{n},\] where \[Q_{k,s}(n)=(\log^{(k)} n)^s\prod_{j=1}^{k-1}\log^{(j)} n\] and \(\log^{(k)} n\) is the \(k\)-fold iterated logarithm. For instance, we have \(Q_{1,s}(n)=(\log n)^s\) and \(Q_{2,s}(n)=(\log n)(\log\log n)^s\). Thus these classes are refinements between \(\bigcap_{s>1}G^{s}\) and \(G^1=C^\omega\). Moreover, by the Denjoy–Carleman theorem (see Hörmander [6]), \(G_k^s\) is quasianalytic if and only if \(s=1\), since
\[\sum_{n\gg 1}\frac{1}{n(\log n)\cdots(\log^{(k-1)}n)(\log^{(k)} n)^s}=\infty \iff s=1.\]
The following theorem gives further improvements for the refined Gevrey classes and fills the gap between the Gevrey classes and the analytic class.
Theorem 4. Let \(\psi\in G_k^s(I)\) with \(k\ge1\) and \(s>1\). Then \(m(\lambda)=O(\log^{(k+2)}\lambda).\) This bound is sharp: there exists a real-valued \(\psi\in G_k^s(I)\) such that \[|m(\lambda_n)| \approx \log^{(k+2)} \lambda_n \quad\text{along a sequence }\lambda_n\to\infty.\]
If \(\psi\in G_k^1(I)\), then \(\psi\) is quasianalytic, and it is also known that \(m(\lambda)=O(1)\) by Pan [5]. Furthermore, if \(P(t)\) is a real polynomial of degree \(d\), then a much stronger result holds: \[\label{pbd} \Big|\mathrm{p.v.}\int_{-1}^1\frac{e^{i P(t)}}{t}dt\Big|\le C_d\tag{2}\] where \(C_d\) depends only on \(d\), and the best constant satisfies \(C_d\approx \log d\). This follows from the proof of Parissis [7]. Similarly, if \(R(t):=P(t)/Q(t)\) is a rational function where \(P,Q\) are real polynomials, then \[\label{rbd} \Big|\mathrm{p.v.}\int_{-1}^1\frac{e^{i R(t)}}{t}dt\Big|\le A\tag{3}\] where \(A\) depends only on the degrees of \(P\) and \(Q\) and not on their coefficients. This follows directly from the proof of Folch–Gabayet and Wright [8]. Their results are about the integral on the whole real line, while the proof still works for bounded intervals.
Remark 5. The parameter \(\alpha\) in Proposition 1 and the parameter \(s\) in Theorems 3 and 4 affect only the implicit constants and do not change the maximal growth in an essential way. Suppose that \(\varphi\) is a smooth function on \(\mathbb{R}^n\) and that \(K\) is an odd, \(-n\)-homogeneous function on \(\mathbb{R}^n\) that is integrable on the unit sphere. We have the standard reduction \[\Big|\mathrm{p.v.}\int_{|x|\le 1}e^{i\lambda\varphi(x)}K(x)dx\Big|\le \frac{1}{2}\|K\|_{L^1(S^{n-1})}\sup_{\theta\in S^{n-1}}\Big|\mathrm{p.v.}\int_{-1}^1\frac{e^{i\lambda\varphi(t\theta )}}{t}dt\Big|.\] Thus higher-dimensional analogues also hold whenever the derivatives of the phase \(\varphi(t\theta)\) with respect to \(t\) satisfy estimates uniformly in \(\theta\in S^{n-1}\). For instance, these hold when \(\varphi\) is a real polynomial or rational function by 2 and 3 .
Remark 6. Given the gap between the Gevrey classes and the \(C^\infty\) class, it is also important to examine classes of smooth functions that lie outside every Gevrey class. For instance, Jézéquel [9] established a trace formula conjectured by Dyatlov–Zworski [10] for Anosov flows in dynamical systems that holds for certain intermediate regularity classes. We will discuss them in Section 7 and their extensions in Section 8. We will see how the maximal growth of \(m(\lambda)\) increases to the universal bound \(o(\log\lambda)\) of the \(C^\infty\) class as the classes become larger.
The problem is directly related to the singular oscillatory integral operators \(T_\lambda\) of the form \[\label{Tlambda}T_\lambda f(t)={\rm p.v.}\int e^{i\lambda\phi(t,s)}(t-s)^{-1}a(t,s)f(s)ds,\tag{4}\] where \(\phi\) is smooth, \(\lambda\) is real, and \(a\in C_0^\infty(\mathbb{R}^2)\). These operators and their higher-dimensional analogues have been studied by Stein–Wainger [11], [12], Phong–Stein [13], Ricci–Stein [14], Pan [15], Seeger [16], and Carbery–Pérez [17]. Phong–Stein [13] showed that uniform \(L^2(\mathbb{R})\) estimates for \(T_\lambda\) imply the \(L^2(\mathbb{R}^2)\) boundedness of the Hilbert transform \(\mathcal{H}\) along variable curves: \[\label{ht} \|\mathcal{H}\|_{L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)}\le \sup_{\lambda\in \mathbb{R}}\|T_\lambda\|_{L^2(\mathbb{R})\to L^2(\mathbb{R})}.\tag{5}\] Here \(\mathcal{H}\) is defined initially on functions in \(C_0^\infty(\mathbb{R}^2)\) by \[\mathcal{H}f(x)=\eta(x){\rm p.v.}\int_{-\delta}^{\delta}f(x_1-t,x_2-\phi(x_1,x_1-t))\frac{dt}{t},\] where \(\eta\in C_0^\infty(\mathbb{R}^2)\) and \(\delta>0\) is suitably small. Pan [15] proved that \(T_\lambda\) is uniformly bounded on \(L^2(\mathbb{R})\) if one imposes a weak finite type condition. Later, Seeger [16], Carbery–Pérez [17] investigated certain flat cases in which the finite-type condition fails.
In the translation-invariant case \(\phi(t,s)=\psi(t-s)\), Nagel–Vance–Wainger–Weinberg [18] proved necessary and sufficient conditions under the assumption that \(\psi\) is even (or odd) and convex. Nagel–Wainger [2] constructed an odd smooth function \(\psi\) on \([-1,1]\) such that the Hilbert transform \(\mathcal{H}\) along the curve \((t,\psi(t))\) is unbounded on \(L^2(\mathbb{R}^2)\). Indeed, the function \(\psi\) is flat at \(0\), so the convolution kernel \[\label{nwker} m(x, y)={\rm p . v .} \int_{-1}^{1} e^{i(x t+y \psi(t))} \frac{d t}{t}\tag{6}\] is unbounded on \(\mathbb{R}^2\). Note that \(m(0,\lambda)\) coincides with \(m(\lambda)\) defined in 1 with \(I=[-1,1]\). Thus [2] is also implied by the unboundedness of \(m(\lambda)\). Pan [5] proved boundedness under the assumption that \(\psi\) does not vanish to infinite order at \(0\). Motivated by the study of eigenfunction restriction estimates, Wang–Zhang [1] asked whether \(m(\lambda)\) remains bounded for arbitrary smooth \(\psi\).
Stein–Wainger [11] considered this type of singular oscillatory integrals with \(I=\mathbb{R}\), and they showed that if \(\psi(t)\) is a polynomial of degree \(d\), then \[\Big|{\rm p . v .} \int_{\mathbb{R}} e^{i\lambda\psi(t)} \frac{d t}{t}\Big|\le C_d\] where the constant \(C_d\) depends only on \(d\). See also Parissis [7], Papadimitrakis–Parissis [19], and Al-Qassem, Cheng, and Pan [20]. For rational phases \(\psi\), uniform bounds still hold; see Folch–Gabayet and Wright [8], [21], Wang–Wu [22], and Al-Qassem, Cheng, and Pan [23]. However, this integral may diverge for many real-analytic phases, such as \(\exp(t)\) and \(\arctan(t)\). Nevertheless, our results show that if we replace \(\mathbb{R}\) by a bounded interval, then an interesting new hierarchy emerges for the maximal growth of \(m(\lambda)\) with respect to the regularity of the phase \(\psi\). Moreover, our results give a negative answer to the problem posed by Wang–Zhang [1]. Furthermore, our results imply that 6 is unbounded on the \(y\)-axis, so in particular this gives a new proof of [2] by Nagel–Wainger.
First, we use a van der Corput estimate to handle phases of finite type and reduce the problem to flat phases. Second, we use either Bang’s lemma or the Taylor–Legendre method to prove sharp estimates on the growth of a phase near a flat point. Third, we use these sharp flat-point estimates to establish the oscillatory integral bounds. In particular, we obtain an abstract result (Theorem 7) for general Denjoy–Carleman classes and use it to establish Theorems 3 and 4. Finally, we explicitly construct examples in the relevant classes to show that the estimates are sharp.
In Section 2, we prove some key lemmas on the phase and Proposition 1. In Section 3, we prove Theorem 2 for general smooth phases. In Section 4, we prove Theorem 7 for phases in the Denjoy–Carleman class. In Section 5, we prove Theorem 3 for phases in the Gevrey classes. In Section 6, we prove Theorem 4 for phases in the refined Gevrey classes. In Section 7, we discuss the intermediate regularity classes of smooth functions introduced by Jézéquel [9]. In Section 8, we further discuss larger classes of smooth functions that lie outside every Gevrey class. In Section 9, we prove the derivative bounds for the sharpness examples.
Throughout this paper, \(X\lesssim Y\) means \(X\le CY\) for some positive constant \(C\) independent of \(\lambda\). If \(X\lesssim Y\) and \(Y\lesssim X\), we write \(X\approx Y\). If \(X\ge CY\) for some sufficiently large constant \(C>1\), we write \(X\gg Y\).
The authors would like to thank Shaozhen Xu for helpful discussions and comments. The authors are supported in part by the National Key R&D Program of China 2024YFA1015300. C.Z. is also supported in part by NSFC Grant 12371097. Z.Z. is also supported in part by NSFC Grant 12501065.
We rewrite the principal value integral as \[\label{eq:pv} m(\lambda) = \int_0^1 \frac{e^{i\lambda\psi(t)}-e^{i\lambda\psi(-t)}}{t}\,dt.\tag{7}\] Let \[\label{phidef} \phi(t)=\psi(t)-\psi(-t).\tag{8}\] If \(\phi\equiv 0\), then \(\psi\) is even and \(m(\lambda)=0\) identically. If \(\phi(t)\) has a finite order of vanishing at \(0\), then standard oscillatory integral estimates imply that \(m(\lambda)\) is bounded. See e.g. Pan [5]. We give a direct proof for completeness.
Lemma 1. Suppose that \(\phi\) has a finite order of vanishing at 0. Then \(m(\lambda)=O(1)\).
Proof. Assume \(\phi\not\equiv 0\) and let \(k\) be the smallest odd integer with \(\phi^{(k)}(0)\neq 0\). Then Taylor’s theorem gives \[\phi(t)=c\,t^k+O(t^{k+2}),\qquad c=\frac{\phi^{(k)}(0)}{k!}\neq 0.\] Hence there exists \(\delta\in(0,1)\) and constants \(c_0,C_0>0\) such that for \(0<t\le\delta\), \[\label{eq:finite-type} c_0 t^k\le |\phi(t)|\le C_0 t^k, \qquad |\phi^{(k)}(t)|\ge c_0.\tag{9}\]
Split \[m(\lambda)=\int_0^\delta \frac{e^{i\lambda\psi(t)}-e^{i\lambda\psi(-t)}}{t}\,dt +\int_\delta^1 \frac{e^{i\lambda\psi(t)}-e^{i\lambda\psi(-t)}}{t}\,dt.\] The second integral is \(O(1)\), so it remains to bound the first one, denoted by \(m_0(\lambda)\), uniformly in \(\lambda\).
Let \(I_j=[2^{-j-1}\delta,2^{-j}\delta]\), \(t_j=2^{-j}\delta\) (\(j\ge 0\)). Dyadically decompose the integral and rescale \(t=t_js\), \(s\in[1/2,1]\). We obtain \[m_0(\lambda)=\sum_{j\ge 0} J_j(\lambda), \qquad J_j(\lambda):=\int_{1/2}^1 \frac{e^{i\lambda\psi(t_js)}-e^{i\lambda\psi(-t_js)}}{s}\,ds.\] Define the scale parameter \(\Lambda_j:=\lambda\,t_j^k=\lambda\,\delta^k\,2^{-jk}.\) We claim that \[\label{clm} |J_j|\lesssim\min \{\Lambda_j,\;\Lambda_j^{-1/k}\}.\tag{10}\] We postpone the proof of this claim and use it to obtain a uniform bound for \(m_0(\lambda)\).
Let \(j_*\) be such that \(\Lambda_{j_*}\ge 1>\Lambda_{j_*+1}\), i.e. \(2^{j_*}\approx (\lambda\delta^k)^{1/k}\). Then \[\sum_{j\le j_*}|J_j(\lambda)| \lesssim \sum_{j\le j_*}\Lambda_j^{-1/k} =\sum_{j\le j_*}(\lambda\delta^k)^{-1/k}2^{j} \lesssim (\lambda\delta^k)^{-1/k}2^{j_*} \lesssim 1,\] and \[\sum_{j> j_*}|J_j(\lambda)| \lesssim \sum_{j>j_*}\Lambda_j =\sum_{j>j_*}\lambda\delta^k\,2^{-jk} \lesssim \lambda\delta^k\,2^{-k(j_*+1)} \lesssim 1.\]
Now it remains to prove the claim 10 . On the one hand, using \(|e^{ix}-e^{iy}|\le |x-y|\) and 9 , we obtain \[\big|e^{i\lambda\psi(t)}-e^{i\lambda\psi(-t)}\big| \le \lambda\,|\psi(t)-\psi(-t)| =\lambda\,|\phi(t)| \lesssim \lambda\,t^k \quad (0<t\le\delta).\] Hence for \(s\in[1/2,1]\), \[\big|e^{i\lambda\psi(t_js)}-e^{i\lambda\psi(-t_js)}\big| \lesssim \lambda\,(t_js)^k\lesssim \lambda\,t_j^k=\Lambda_j.\] Therefore \[\label{eq:small} |J_j(\lambda)| \lesssim\int_{1/2}^1 \frac{\Lambda_j}{s}\,ds \lesssim \Lambda_j.\tag{11}\]
On the other hand, write \(J_j=A_j-B_j\) where \[A_j(\lambda):=\int_{1/2}^1 \frac{e^{i\lambda\psi(t_js)}}{s}\,ds, \qquad B_j(\lambda):=\int_{1/2}^1 \frac{e^{i\lambda\psi(-t_js)}}{s}\,ds.\] Let \(\Phi_j(s)=\psi(t_js)\). Since \(k\) is odd and \[\phi^{(k)}(t)=\psi^{(k)}(t)+\psi^{(k)}(-t)=2\psi^{(k)}(0)+O(t),\] by 9 we have \(|\psi^{(k)}(u)|\gtrsim 1\) for \(|u|\le\delta\) (up to adjusting \(\delta\)), hence \(|\Phi_j^{(k)}(s)|\gtrsim t_j^k\) on \([1/2,1]\). Then a standard van der Corput estimate for phases with a nonvanishing \(k\)th derivative on a fixed interval yields \[|A_j(\lambda)|\lesssim (\lambda t_j^k)^{-1/k}=\Lambda_j^{-1/k}.\] Similarly \(|B_j(\lambda)|\lesssim \Lambda_j^{-1/k}\), hence \[\label{eq:large} |J_j(\lambda)|\le |A_j(\lambda)|+|B_j(\lambda)|\lesssim \Lambda_j^{-1/k}.\tag{12}\] This proves the claim 10 and completes the proof. ◻
Next, we need the following one-dimensional Bang’s lemma (Bang [24], Bruna [25]), which allows us to control the rate of increase of a function near a flat point. For completeness, we give a direct proof of the version we use.
Lemma 2 (Bang). Let \(r>0\) and let \(g\in C^\infty((-r,r))\) satisfy \(g(x)=0\;(x\le 0).\) Let \((A_n)_{n\ge 0}\) be a sequence of positive numbers such that \[\sup_{x\in (-r,r)}|g^{(n)}(x)|\le A_n\qquad (n\ge 0),\] and such that the quotient sequence \(\eta_n:=A_n/A_{n-1}\;(n\ge 1)\) is nondecreasing. Fix \(x\in(0,r)\) and an integer \(\ell\ge 1\). If \[\sum_{j=\ell}^{\infty}\frac{1}{\eta_j}> 4x,\] then \[|g(x)|\le A_0 2^{-\ell}.\]
Proof. The idea is to connect \(x\) to \(0\) by a chain of short intervals whose lengths are comparable to \(\eta_j^{-1}\) and then to iterate the fundamental theorem of calculus from high derivatives down to lower ones. The monotonicity of (\(\eta_j\)) is exactly what allows each step to close.
Choose \(n\ge \ell\) so that \[\sum_{j=\ell}^n \frac{1}{\eta_j}\ge 4x.\] Set \[a:=\frac{x}{\sum_{j=\ell}^n \eta_j^{-1}}\le \frac{1}{4}.\] Choose points \[t_\ell=x>t_{\ell+1}>\cdots > t_{n+1}=0\] so that \[t_j-t_{j+1}=\frac{a}{\eta_j}\qquad (\ell\le j\le n).\] For \(\ell\le j\le n\) set \(I_j=[t_{j+1},t_j]\) and let \(I_{n+1}=(-r,0]\). For integers \(0\le p\le q\le n+1\), define \[F(p,q):=\sup_{x\in I_q}|g^{(p)}(x)|.\] We claim that \[\label{eq:claim} F(p,q)\le (2a)^{q-p}A_p \qquad (\ell\le q\le n+1,\; 0\le p\le q).\tag{13}\] This is clear when \(p=q\), and also when \(q=n+1\), because \(g\) and all its derivatives vanish on \(I_{n+1}=(-r,0]\).
Now let \(\ell\le q\le n\) and \(0\le p<q\). For every \(y\in I_q\), the fundamental theorem of calculus gives \[|g^{(p)}(y)| \le |g^{(p)}(t_{q+1})| + (t_q-t_{q+1})\sup_{x\in I_q}|g^{(p+1)}(x)|.\] Hence \[F(p,q)\le F(p,q+1)+\frac{a}{\eta_q}F(p+1,q).\] Assuming 13 already known for \((p,q+1)\) and \((p+1,q)\), we get \[F(p,q) \le (2a)^{q+1-p}A_p+\frac{a}{\eta_q}(2a)^{q-p-1}A_{p+1}.\] Because \((\eta_j)\) is nondecreasing and \(p+1\le q\), we have \(A_{p+1}=A_p\eta_{p+1}\le A_p\eta_q\), so \[F(p,q) \le (2a)^{q-p}A_p\Bigl(2a+\frac{1}{2}\Bigr) \le (2a)^{q-p}A_p,\] since \(a\le 1/4\). This proves 13 . Finally, since \(x=t_\ell\in I_\ell\), 13 with \((p,q)=(0,\ell)\) gives \[|g(x)|\le F(0,\ell)\le A_0(2a)^\ell\le A_0 2^{-\ell}.\] ◻
Suppose that \(\psi\in C^\alpha([-1,1])\) for some integer \(\alpha\ge1\). Using \(|e^{ix}-e^{iy}|\le \min\{2,|x-y|\}\), 7 , and \(|\phi(t)|\lesssim|t|\), we obtain \[|m(\lambda)| \lesssim \int_0^1 \frac{\min\{1,\lambda t\}}{t}\,dt\lesssim\log\lambda.\] We remark that one may obtain a more precise estimate, namely \(|m(\lambda)|\le \frac{2}{\alpha}\log\lambda+O(1)\), by using a refined argument similar to the proof of Theorem 2.
Next, we choose \(\psi(t)=\max(t^{\alpha+1},0)\in C^{\alpha}([-1,1])\) and then \[\begin{align} m(\lambda) &= \int_0^1 \frac{e^{i\lambda t^{\alpha+1}}-1}{t}\,dt=\frac{1}{\alpha+1}\int_0^\lambda\frac{e^{iu}-1}{u}\,du\\ &= -\frac{1}{\alpha+1}\log\lambda + O(1). \end{align}\] This proves the sharpness of the upper bound. For large \(\alpha\), this also shows that the coefficient \(\approx\frac{1}{\alpha}\) is essentially sharp.
We prove Theorem 2 in this section. Let \(\psi\in C^\infty([-1,1])\) and \(\phi(t)=\psi(t)-\psi(-t)\). By Lemma 1, it remains to handle the case in which \(\phi\) is flat at \(0\). Then for every \(N\ge1\) there exist constants \(C_N,\delta_N>0\) such that \[|\phi(t)|\le C_N |t|^N \qquad (|t|\le\delta_N).\] Hence \[|e^{i\lambda\psi(t)}-e^{i\lambda\psi(-t)}| \le \lambda C_N |t|^N.\]
Split 7 at \[r:=\min\bigl(\delta_N,(\lambda C_N)^{-1/N}\bigr).\] Then \[|m(\lambda)| \le \int_0^r \lambda C_N t^{N-1}\,dt + 2\int_r^1 \frac{dt}{t} \le \frac{1}{N} + \frac{2}{N}\log\lambda + O(1).\] Dividing by \(\log\lambda\) and then letting \(\lambda\to\infty\), followed by \(N\to\infty\), yields \(m(\lambda)=o(\log\lambda)\) as claimed.
We construct examples to show that the sub-logarithmic growth \(o(\log\lambda)\) is sharp. The construction uses a “dyadic plateau” phase for which \(m(\lambda)\) grows linearly in an index \(n\) along a chosen frequency subsequence \(\lambda_n\), and then we tune the subsequence so that \(n\approx \log\lambda_n/\log^{(k)}\lambda_n\).
Fix \(\eta\in C_c^\infty((1/2,1))\) with \(0\le \eta\le 1\) and \[\eta\equiv 1\;\text{ on } [3/5,4/5].\] For \(j\ge 1\) set \[\eta_j(t):=\eta(2^j t), \qquad J_j:=\Big[\frac{3}{5\cdot 2^j},\frac{4}{5\cdot 2^j}\Big].\] Then \(\eta_j\equiv 1\) on \(J_j\) and \(\operatorname{supp}\eta_j\subset(2^{-(j+1)},2^{-j})\), so the supports are pairwise disjoint.
Let \(\{q_j\}_{j\ge 1}\) be an increasing sequence of odd integers with \(q_j\ge 3\). Define \[Q_j:=\prod_{\ell=1}^j q_\ell, \qquad a_j:=\frac{\pi}{Q_j}.\] Define the phase \[\label{eq:psi-def} \psi(t):= \begin{cases} 0, & t\le 0,\\[4pt] \displaystyle \sum_{j=1}^\infty a_j\,\eta_j(t), & t>0. \end{cases}\tag{14}\] Because the supports of \(\eta_j\) are disjoint, the sum is locally finite on \((0,1)\).
Now we choose \(q_j\) to force a given iterated-log profile. For \(k\ge 2\), fix a large integer \(j_0\) so that \(\log^{(k-2)}(j+j_0)\ge 2\) for all \(j\ge 1\), and define \[\label{eq:qj-choice} q_j:=2\Big\lfloor \log^{(k-2)}(j+j_0)\Big\rfloor+1.\tag{15}\] Then \(q_j\) is odd, \(q_j\ge 5\), and \(\log q_j \approx \log^{(k-1)} j\) and \(\log Q_n \approx n\,\log^{(k-1)} n\).
Lemma 3. With the choice of \(q_j\) above, the function \(\psi\) defined by 14 lies in \(C^\infty(\mathbb{R})\).
Proof. Fix an integer \(m\ge 0\). On \(\operatorname{supp}\eta_j\) we have \(\psi(t)=a_j\eta_j(t)\) and \[\psi^{(m)}(t)=a_j\,2^{jm}\,\eta^{(m)}(2^j t).\] Since \(\log Q_j\approx j\log^{(k-1)}j\), we obtain \[\sup_{t\in(0,2^{-\ell})}|\psi^{(m)}(t)| \lesssim \sup_{j\ge \ell}\frac{2^{jm}}{Q_j}\to 0\quad (\ell \to\infty).\] Therefore \(\psi^{(m)}(t)\to 0\) as \(t\to 0^+\) for every \(m\). Since \(\psi\equiv 0\) on \((-\infty,0]\), all derivatives match at \(0\) and \(\psi\in C^\infty(\mathbb{R})\). ◻
Let \(\lambda_n=Q_n\). Since \(\log \lambda_n\approx n\log^{(k-1)}n\), we have \(n\approx \log \lambda_n/\log^{(k)}\lambda_n\) for all large \(n\).
Lemma 4 (Lower bound). There exists \(c_0>0\) such that for all \(n\ge 1\), \[|m(\lambda_n)|\ge c_0\,n.\]
Proof. Since \(\psi(-t)=0\) for \(t>0\) by construction, we have \[- {\rm Re}\,m(\lambda)=\int_0^1 \frac{1-\cos(\lambda\psi(t))}{t}\,dt\ge 0.\] Fix \(n\) and take \(t\in J_j\) with \(1\le j\le n\). Then \(\eta_j(t)=1\) and \(\eta_\ell(t)=0\) for \(\ell\neq j\), so \(\psi(t)=a_j=\pi/Q_j\). Therefore \[\lambda_n\psi(t)=Q_n\cdot\frac{\pi}{Q_j} =\pi\prod_{\ell=j+1}^n q_\ell.\] Since each \(q_\ell\) is odd, we have \(\cos(\lambda_n\psi(t))=-1\) on \(J_j\). Thus \[- {\rm Re}\,m(\lambda_n) \ge \sum_{j=1}^n \int_{J_j}\frac{2}{t}\,dt =2\sum_{j=1}^n \log\!\left(\frac{ \frac{4}{5\cdot 2^j} }{ \frac{3}{5\cdot 2^j} }\right) =2n\log\!\left(\frac{4}{3}\right).\] So \(|m(\lambda_n)|\ge - {\rm Re}\,m(\lambda_n)\ge c_0 n\) with \(c_0:=2\log(4/3)\). ◻
Lemma 5 (Upper bound). There exists \(C_0>0\) such that for all \(n\ge 1\), \[|m(\lambda_n)|\le C_0\,n.\]
Proof. Write \(m(\lambda_n)=\int_0^1 \frac{e^{i\lambda_n\psi(t)}-1}{t}\,dt\). Split \((0,1)\) into dyadic shells \(I_j:=(2^{-(j+1)},2^{-j})\). On each \(I_j\) we have \(|e^{i\lambda_n\psi(t)}-1|\le 2\) and \[\int_{I_j}\frac{dt}{t}=\log 2,\] so \[\int_{I_j}\frac{|e^{i\lambda_n\psi(t)}-1|}{t}\,dt \le 2\log 2.\] Summing this over \(j=1,\dots,n\) gives a bound \(\lesssim n\).
For the tail \(j>n\), we use \(|e^{ix}-1|\le |x|\) and the fact that on \(I_j\) the phase equals \(a_j\eta_j\) (or \(0\)), hence \(|\psi(t)|\le |a_j|=\pi/Q_j\). Thus for \(j>n\), \[\int_{I_j}\frac{|e^{i\lambda_n\psi(t)}-1|}{t}\,dt \le \int_{I_j}\frac{|\lambda_n|\,|\psi(t)|}{t}\,dt \le \frac{\pi Q_n}{Q_j}\int_{I_j}\frac{dt}{t} = \pi\log 2\cdot \frac{Q_n}{Q_j}.\] Since \(q_\ell\ge 3\), we have \(Q_j\ge Q_n\,3^{j-n}\) for \(j>n\), hence \(\sum_{j>n}Q_n/Q_j\le \sum_{r\ge 1}3^{-r}\lesssim 1\). Therefore the tail contributes \(O(1)\) uniformly in \(n\). Combining, \(|m(\lambda_n)|\le C_0 n\). ◻
Thus, by Lemmas 4 and 5, we obtain \[|m(\lambda_n)|\approx n \approx \frac{\log \lambda_n}{\log^{(k)}\lambda_n}.\]
In this section, we prove general upper bounds for \(m(\lambda)\) when the phase belongs to a Denjoy–Carleman class associated with a log-convex sequence.
Let \(M=(M_n)_{n\ge 0}\) be a positive sequence with \(M_0=1\). We say that \(M\) is log-convex if \[M_n^2\le M_{n-1}M_{n+1}\qquad (n\ge 1).\] For such a sequence, define the quotient sequence \[\mu_n:=\frac{M_n}{M_{n-1}}\qquad (n\ge 1).\] Log-convexity is equivalent to the monotonicity of \((\mu_n)_{n\ge 1}\).
For a log-convex sequence \(M\), a function \(\psi\in C^\infty([-1,1])\) belongs to the Denjoy–Carleman class \(\mathcal{C}_M([-1,1])\) if there exists a constant \(K>0\) such that for all \(n\ge0\) \[\label{defdc} \sup_{t\in[-1,1]}|\psi^{(n)}(t)|\le K^{n+1}M_n.\tag{16}\] The Gevrey class \(G^s\) and its refinements \(G_k^s\) are important special cases of the Denjoy–Carleman class. We first establish a general theorem for the Denjoy-Carleman class in this section, and then discuss concrete estimates and sharpness examples for the Gevrey class \(G^s\) and its refinements \(G_k^s\) in the next two sections.
The central quantity below is the tail function \[\label{eq:TM} T_M(N):=\sum_{j\ge N}\frac{1}{\mu_j},\tag{17}\] with the convention \(T_M(N)=\infty\) if the series diverges. By the Denjoy–Carleman theorem (see Hörmander [6]), the class is quasianalytic exactly when \(T_M(1)=\infty\). Specifically, for the Gevrey class \(G^s\) with \(s>1\), we have \(T_M(N)\approx N^{1-s}\), and for the refined Gevrey class \(G_k^s\) with \(s>1\), we have \(T_M(N)\approx (\log^{(k)}N)^{1-s}\).
Theorem 7. Let \(M=(M_n)_{n\ge 0}\) be a log-convex sequence, and let \(\mu_n=M_n/M_{n-1}\) and \(T_M\) be as in 17 .
(i) If \(T_M(1)=\infty\), then \(m(\lambda)=O(1).\)
(ii) Assume now that \(T_M(1)<\infty\) and \(\mu_NT_M(N)\ge C_0\) for some \(C_0>1\) and all large \(N\). Then there exist constants \(C,c>0\) such that \[\label{ub3} |m(\lambda)|\le C\Bigl(1+\log \frac{1}{T_M(c\log \lambda)}\Bigr) \qquad (\lambda\ge 2).\qquad{(1)}\]
The condition “\(\mu_NT_M(N)\ge C_0\)” roughly means that the series \(\sum_{j\ge N}1/\mu_j\) cannot converge faster than a geometric series. It can be relaxed, but it is enough for our purposes, since it holds for the Gevrey class \(G^s\) \((s>1)\) and the refined Gevrey class \(G_k^s\) \((s>1)\).
To prove the theorem, we need to control the rate of growth of \(\phi\) near 0. We achieve this by using Bang’s Lemma 2.
Proposition 8 (Flat-point estimate). Let \(M=(M_n)_{n\ge 0}\) be a log-convex sequence. Let \[N_M(r):=\sup\bigl\{N\ge 1:\;T_M(N)\ge r\bigr\}\in \mathbb{N}\cup\{0,\infty\}.\] Let \(\psi\in \mathcal{C}_M([-1,1])\) and \[\phi(t):=\psi(t)-\psi(-t).\] Assume that \(\phi\) is flat at \(0\). Then there exist constants \(C,c>0\) such that for all sufficiently small \(|t|\) \[\label{eq:flat-bound} |\phi(t)|\le C\,2^{-N_M(c|t|)}.\qquad{(2)}\] If in addition \(T_M(1)=\infty\), then \(\phi\equiv 0\) on \([-1,1]\).
Note that \(N_M(r)\) is roughly the “inverse” of the function \(r=T_M(N)\).
Proof. Extend \(\phi\) by zero to the left: \[F(x):= \begin{cases} 0,& x\le 0,\\ \phi(x),& 0<x<1. \end{cases}\] Since \(\phi\) is flat at \(0\), the extension \(F\) belongs to \(C^\infty([-1,1])\) and obeys \[\sup_{x\in[-1,1]}|F^{(n)}(x)|\le 2K^{n+1}M_n.\] Apply Lemma 2 with \(A_n:=2K^{n+1}M_n\). Then \[\eta_n=\frac{A_n}{A_{n-1}}=\frac{KM_n}{M_{n-1}}=K\mu_n,\] which is nondecreasing by log-convexity. Therefore, if \(x\in(0,1)\) and \(T_M(\ell)>4Kx\), then \[|F(x)|\le 2\cdot 2^{-\ell}=2^{1-\ell}.\] Equivalently, there are constants \(C,c>0\) such that \[|\phi(x)|\le C\,2^{-N_M(cx)} \qquad (0<x\ll 1).\] Applying the same argument to \(x\mapsto \phi(-x)\) gives ?? for both signs of \(t\).
If \(T_M(1)=\infty\), then \(T_M(\ell)=\infty\) for every \(\ell\). Fix \(x\in(0,1)\) and let \(\ell\to\infty\) in Lemma 2. We obtain \(F(x)=0\). Thus \(\phi(x)=0\) for \(x>0\), and by symmetry also for \(x<0\). ◻
Remark 9. It is natural to estimate the flat functions by Taylor’s theorem and Legendre transform. Indeed, since \(\phi\) is flat at 0, we have \[\label{taylor} |\phi(t)|\le \inf_{n\ge1}\frac{K^{n+1}M_n |t|^n}{n!}=K\exp\Big(-\Phi^*\Big(\log\frac{1}{K|t|}\Big)\Big)\qquad{(3)}\] where \(\Phi(n)=\log(M_n/n!)\) and \[\label{leg} \Phi^*(y) := \sup_{n \geq 1}\{ny - \Phi(n)\}\qquad{(4)}\] is the Legendre transform of \(\Phi\). This Taylor–Legendre method works for the Gevrey classes and yields Proposition 10, but it cannot produce the correct estimates for the refined Gevrey classes in Proposition 11. For example, if \(\phi\in G_2^2\), the Taylor–Legendre method only gives, for some constant \(c>0\), \[|\phi(t)|\lesssim\exp\Big(-\exp\Big(\frac{c}{|t|(\log\frac{1}{|t|})^2}\Big)\Big),\;\;|t|\ll1,\] while the correct estimate is given by Proposition 11: for some constants \(B,\gamma>0\) \[|\phi(t)|\lesssim\exp\Big(-B\exp\Big(\exp\Big(\gamma |t|^{-1}\Big)\Big)\Big),\;\;|t|\ll1.\] Nevertheless, we observe that this Taylor–Legendre method can still produce correct estimates for the intermediate regularity classes of smooth functions that lie outside every Gevrey class. See Remark 14.
By the Denjoy–Carleman theorem (see Hörmander [6]) and Lemma 1, we only need to prove part (ii) of Theorem 7 when \(\phi\) is flat at \(0\). By Proposition 8, after shrinking constants if needed, there exist \(C_1,c_1>0\) such that \[|\phi(t)|\le C_1 2^{-N} \qquad \text{whenever } 0<t\le t_N:=c_1 T_M(N).\] Since \(T_M(N)\downarrow 0\), the intervals \(I_N:=(t_{N+1},t_N]\;(N\gg 1)\) form a shell decomposition near the origin. We write \[|m(\lambda)|\lesssim 1+\sum_{N\ge N_0}\int_{I_N}\min\{2,\lambda|\phi(t)|\}\,\frac{dt}{t}\] for some fixed large \(N_0\). On \(I_N\) we have \(|\phi(t)|\le C_1 2^{-N}\), hence \[\label{eq:shellsum} |m(\lambda)| \lesssim 1+\sum_{N\ge N_0}\min\{1,\lambda 2^{-N}\}\log\frac{T_M(N)}{T_M(N+1)}.\tag{18}\] For \(N\lesssim\log\lambda\), we obtain for some \(c>0\) \[\begin{align} \sum_{N_0\le N\lesssim\log\lambda}\log\frac{T_M(N)}{T_M(N+1)} \lesssim 1+\log\frac{1}{T_M(c\log\lambda+1)}. \end{align}\]
For \(N\gg \log\lambda\), by the assumption that \(T_M(N)\ge C_0/\mu_N\) for some \(C_0>1\) and all large \(N\), we have for all large \(N\) \[T_M(N+1)=T_M(N)\Bigl(1-\frac{1}{\mu_NT_M(N)}\Bigr)\approx T_M(N).\] Hence \[\sum_{N\gg \log\lambda}\lambda 2^{-N}\log\frac{T_M(N)}{T_M(N+1)} \lesssim \sum_{N\gg \log\lambda}\lambda2^{-N} \lesssim 1.\] Combining the two ranges yields \[|m(\lambda)|\lesssim 1+\log\frac{1}{T_M(c\log\lambda)}.\]
We prove Theorem 3 in this section. For \(s\ge1\), fix \(\psi\in G^s([-1,1])\) and define \(\phi(t)=\psi(t)-\psi(-t)\). By Lemma 1, it remains to handle the case in which \(\phi\) is flat at \(0\). We need to estimate the rate of growth of \(\phi\) near \(0\). Since \(M_n=(n!)^s\), we can apply Proposition 8 with \[T_M(N)\approx N^{1-s}\quad \text{and}\quad N_M(r)\approx r^{-1/(s-1)}\] when \(s>1\). Note that \(T_M(1)=\infty\) when \(s=1\).
Proposition 10 (Flat-point estimate). Let \(s\ge1\), and let \(\phi\in G^s([-1,1])\) be flat at \(0\), i.e. \(\phi^{(n)}(0)=0\;(n\ge 0).\) Then the following statements hold.
(i) If \(s=1\), then \(\phi\equiv 0\) on \([-1,1]\).
(ii) If \(s>1\), then there exist constants \(A,B>0\) such that for all sufficiently small \(|t|\), \[\label{ub2} |\phi(t)|\le A\exp\bigl(-B |t|^{-1/(s-1)}\bigr).\qquad{(5)}\]
The estimate ?? is essentially sharp, since it is classical that \(\exp(-|t|^{-1/(s-1)})\) is flat at \(0\) and belongs to \(G^s([-1,1])\). By Theorem 7 we have \(m(\lambda)=O(1)\) when \(s=1\) and \(m(\lambda)=O(\log\log\lambda)\) when \(s>1\). This proves the upper bound in Theorem 3.
Fix \(s>1\) and set \(a=\frac{1}{s-1}\). Define \[\psi(t)= \begin{cases} 0,& t\le 0,\\[2pt] e^{-t^{-a}},& t>0. \end{cases}\] It is classical that \(\psi\in G^s([-1,1])\) and is flat at \(0\).
For this choice, \[m(\lambda) = \int_0^1 \frac{e^{i\lambda\psi(t)}-1}{t}\,dt.\] With the substitution \(u=\psi(t)=e^{-t^{-a}}\), one computes \[\frac{dt}{t}=(s-1)\frac{du}{u\log(1/u)},\] hence \[m(\lambda) = (s-1)\int_0^{e^{-1}} \frac{e^{i\lambda u}-1}{u\log(1/u)}\,du.\] Let \(\lambda_n=2\pi n\) and \[I_j:=\Bigl[\frac{j+1/4}{n},\,\frac{j+3/4}{n}\Bigr],\quad j=1,2,3,\dots.\] On \(I_j\), we have \(1-\cos(2\pi n u)\ge 1\). Since the weight \[w(u):= \frac{1}{u\,\log(1/u)}\] is decreasing for small \(u\), summing over \(1\le j\lesssim n\) yields, for some fixed \(c_0\in (0,1/e)\), \[- {\rm Re}\,m(\lambda_n) \gtrsim (s-1)\int_{1/n}^{c_0} w(u)\,du\gtrsim(s-1)\log\log n.\] Therefore \[|m(\lambda_n)|\ge - {\rm Re}\,m(\lambda_n) \gtrsim (s-1)\log\log \lambda_n.\] This shows that the upper bound \(O(\log\log\lambda)\) is sharp.
We prove Theorem 4 in this section. For \(s\ge1\) and an integer \(k\ge1\), fix \(\psi\in G_k^s([-1,1])\) and define \(\phi(t)=\psi(t)-\psi(-t)\). By Lemma 1, we only need to consider the case in which \(\phi\) is flat at \(0\). Since \(M_n=(\log^{(k)} n)^s\prod_{j=1}^{k-1}\log^{(j)} n\) for large \(n\), we can apply Proposition 8 with \[T_M(N)\approx (\log^{(k)}N)^{1-s}\quad \text{and}\quad N_M(r)\approx \exp^{(k)}(r^{-1/(s-1)})\] when \(s>1\). Here \(\exp^{(k)}\) is the \(k\)-fold iterated exponential. Note that \(T_M(1)=\infty\) when \(s=1\).
Proposition 11 (Flat-point estimate). For \(s\ge1\) and integer \(k\ge1\), let \(\phi\in G_k^s([-1,1])\) be flat at \(0\), i.e. \(\phi^{(n)}(0)=0\;(n\ge 0).\) Then the following statements hold.
(i) If \(s=1\), then \(\phi\equiv 0\) on \([-1,1]\).
(ii) If \(s>1\), then there exist constants \(A,B,\gamma>0\) such that for all sufficiently small \(|t|\), \[\label{ub1} |\phi(t)|\le A\exp\bigl(-B \exp^{(k)}(\gamma |t|^{-1/(s-1)})\bigr).\qquad{(6)}\]
The estimate ?? is essentially sharp, since \(\exp(-\exp^{(k)}(|t|^{-1/(s-1)}))\) is flat at \(0\) and belongs to \(G_k^s([-1,1])\) (see Section 9.1). By Theorem 7 we have \(m(\lambda)=O(1)\) when \(s=1\) and \(m(\lambda)=O(\log^{(k+2)}\lambda)\) when \(s>1\). This proves the upper bound in Theorem 4.
For convenience, we denote \(E_1(x)=e^x,\;E_{j+1}(x)=\exp(E_j(x)),\;j\ge1\). Let \(k\ge 1\) and \(s>1\) and \[a:=\frac{1}{s-1}, \qquad \psi(t):= \begin{cases} 0,& t\le 0,\\[1mm] \exp\!\bigl(-E_k(t^{-a})\bigr),& t>0. \end{cases}\] We will prove that \(\psi\in G_k^s([-1,1])\) in Section 9.1 and that \(|m(\lambda_n)| \approx \log^{(k+2)} \lambda_n\) along a sequence \(\lambda_n\to\infty\) in the following.
Since \(\psi(-t)=0\) for \(t>0\), \[m(\lambda)=\int_0^1 \frac{e^{i\lambda \psi(t)}-1}{t}\,dt.\] Set \[u=\psi(t)=\exp\!\bigl(-E_k(t^{-a})\bigr).\] Write \(x=t^{-a}\). Then \[\frac{du}{u}=-E_k'(x)\,dx = -\Bigl(\prod_{j=1}^k \log^{(j)}(1/u)\Bigr)\,dx\] and \[dx=-a\,x\,\frac{dt}{t} = -a\,\log^{(k+1)}(1/u)\,\frac{dt}{t}.\] Hence \[\frac{dt}{t} = (s-1)w(u)du,\] where \[w(u)= \frac{1}{u\,\log(1/u)\,\log^{(2)}(1/u)\cdots \log^{(k+1)}(1/u)}.\] Therefore \[m(\lambda) = (s-1)\int_0^{u_0} (e^{i\lambda u}-1)w(u) du\] for some fixed \(u_0\in(0,1)\).
Now take \(\lambda_n:=2\pi n\). Then \[- {\rm Re}\,m(\lambda_n) = (s-1)\int_0^{u_0} (1-\cos(2\pi n u))w(u)du.\] Let \[I_j:=\Bigl[\frac{j+1/4}{n},\,\frac{j+3/4}{n}\Bigr],\qquad j=1,2,3,\dots.\] On \(I_j\), we have \(1-\cos(2\pi n u)\ge 1\). Since the weight \(w(u)\) is decreasing for small \(u\), summing over \(1\le j\lesssim n\) yields for some fixed \(c_0\in(0,u_0)\) \[- {\rm Re}\,m(\lambda_n) \gtrsim (s-1) \int_{1/n}^{c_0} w(u)\,du\gtrsim(s-1)\log^{(k+2)}n.\]
Thus \[|m(\lambda_n)| \ge - {\rm Re}\,m(\lambda_n) \gtrsim (s-1)\log^{(k+2)}\lambda_n.\] This proves the sharpness of the upper bound \(O(\log^{(k+2)}\lambda)\).
In this section, we investigate classes of smooth functions that lie outside every Gevrey class. Jézéquel [9] proved a trace formula conjectured by Dyatlov–Zworski [10] for Anosov flows in dynamical systems that holds for certain intermediate regularity classes. See also [26]–[28].
Let \(I=[-1,1]\). Let \(M=(M_n)_{n\ge 0}\) be a positive sequence with \(M_n=\exp(cn^\alpha)\) (\(\alpha>1\), \(c>0\)). It is log-convex. We consider the Denjoy–Carleman class \(\mathcal{C}_M\) associated with this sequence. If \(\psi\in \mathcal{C}_M(I)\), then there exists a constant \(K>0\) such that for all \(n\ge 0\) \[\label{defdc1} \sup_{t\in I}|\psi^{(n)}(t)|\le K^{n+1}\exp(cn^\alpha).\tag{19}\] It is essentially the class used by Jézéquel [9], and it is larger than any Gevrey class since \(\exp(cn^\alpha)\gg (n!)^s\) for any \(\alpha,s>1\) and \(c>0\).
Theorem 12. Let \(\psi\in \mathcal{C}_M(I)\). Then we have \(m(\lambda)=O((\log\lambda)^{1-\frac{1}{\alpha}})\). This bound is sharp: there exists a real-valued \(\psi\in \mathcal{C}_M(I)\) such that \[|m(\lambda_n)| \approx (\log\lambda_n)^{1-\frac{1}{\alpha}} \quad\text{along a sequence }\lambda_n\to\infty.\]
The following flat-point estimate is the key to the theorem.
Proposition 13 (Flat-point estimate). Let \(\phi\in \mathcal{C}_M(I)\) be flat at \(0\), i.e. \(\phi^{(n)}(0)=0\;(n\ge 0).\) Then there exist constants \(A,B>0\) such that for all sufficiently small \(|t|\), \[\label{ub4} |\phi(t)|\le A\exp\Bigl(-B \Big(\log\frac{1}{|t|}\Big)^{\frac{\alpha{\alpha-1}}{}}\Bigr).}\qquad{(7)}\]
The estimate ?? is essentially sharp, since \(\exp\Bigl(- \Big(\log\frac{1}{|t|}\Big)^{\frac{\alpha{\alpha-1}}{}}\Bigr)}\) belongs to \(\mathcal{C}_M(I)\) (see Section 7.2). It can be proved by the Taylor–Legendre method in Remark 9. Indeed, by Taylor’s theorem and minimization, we obtain for some constant \(B>0\) \[|\phi(t)|\le \inf_{n\ge1}\frac{K^{n+1}\exp(cn^\alpha)|t|^n}{n!}\approx \exp\Bigl(-B \Big(\log\frac{1}{|t|}\Big)^{\frac{\alpha{\alpha-1}}{}}\Bigr),\quad |t|\ll1,}\] where the minimum is achieved at \(n_*\approx (y/\alpha)^{\frac{1}{\alpha-1}}\) with \(y=\log\frac{1}{K|t|}\).
Remark 14. Applying Proposition 8 gives only a weaker upper bound: \[|\phi(t)|\lesssim\exp\Bigl(-B \Big(\log\frac{1}{|t|}\Big)^{\frac{1}{\alpha-1}}\Bigr),\quad |t|\ll1.\] Roughly speaking, the Taylor–Legendre method works well for the Gevrey classes and other larger classes, while Proposition 8 (Bang’s Lemma 2) works well for the Gevrey classes and the refined Gevrey classes. See also Remark 9 and Bang [24]. From this perspective, the Gevrey class is exactly the borderline between these two methods.
By the reduction above, it suffices to consider the case in which \(\phi\) is flat at \(0\). Let \(\beta=\alpha/(\alpha-1)\). Then Proposition 13 gives constants \(A,B>0\) and \(\delta\in(0,1)\) such that \[\label{eq:flat-small} |\phi(t)|\le A\exp\!\Bigl(-B\bigl(\log(1/t)\bigr)^{\beta}\Bigr) \qquad (0<t\le \delta).\tag{20}\] Using \(|e^{ix}-e^{iy}|\le \min\{2,|x-y|\}\), we obtain \[|m(\lambda)| \lesssim 1+\int_0^\delta \min\{1,\lambda|\phi(t)|\}\,\frac{dt}{t}.\] Let \(u_\lambda>0\) solve \(\lambda\exp(-Bu^\beta)=1\). Then \(u_\lambda\approx (\log\lambda)^{1/\beta}\). We split the interval and change variables to obtain \[\begin{align} |m(\lambda)| &\lesssim u_\lambda+ \int_{u_\lambda}^\infty \lambda e^{-Bu^\beta}du\\ &\lesssim u_\lambda+\int_{u_\lambda}^\infty \lambda e^{-Bu_\lambda^{\beta-1}u}du\\ &\lesssim u_\lambda+u_\lambda^{1-\beta}\lesssim u_\lambda. \end{align}\] This proves the upper bound in Theorem 12.
Let \(\alpha>1\) and \(\beta=\alpha/(\alpha-1)\) and \[\psi(t)=\begin{cases} 0,& t\le 0,\\[1mm] \exp(-(\log\frac{1}{t})^{\beta}),& 0<t\le1/e. \end{cases}\] We may take \(I=[-1/e,1/e]\). It is harmless and simplifies the calculation. We show that \(\psi\in \mathcal{C}_M(I)\) in Section 9.2 and that \(|m(\lambda_n)| \approx (\log\lambda_n)^{1-\frac{1}{\alpha}}\) along a sequence \(\lambda_n\to\infty\) in the following.
Since \(\psi(-t)=0\) for \(t>0\), \[m(\lambda)=\int_0^{e^{-1}}\frac{e^{i\lambda\psi(t)}-1}{t}\,dt.\] Set \[s:=\psi(t)=e^{-(\log(1/t))^\beta}.\]
Differentiating gives \[\frac{dt}{t} = \frac{1}{\beta} \frac{ds}{s(\log(1/s))^{1-\frac{1}{\beta}}} = \frac{1}{\beta} \frac{ds}{s(\log(1/s))^{1/\alpha}}.\] Write \[w(s):=\frac{1}{s(\log(1/s))^{1/\alpha}}, \qquad 0<s\le e^{-1}.\]
Take \(\lambda_n:=2\pi n.\) Then \[- {\rm Re}\,m(\lambda_n) = \frac{1}{\beta} \int_0^{e^{-1}} (1-\cos(2\pi n s))\,w(s)\,ds.\] Fix \[c_0:=\frac{1}{4e}, \qquad N_n:=\lfloor c_0 n\rfloor.\] For \(1\le j\le N_n\) set \[I_{j,n}:=\Bigl[\frac{j+1/4}{n},\,\frac{j+3/4}{n}\Bigr].\] Since \((j+3/4)/n\le c_0+1/n<e^{-1}\) for all large \(n\), these intervals lie inside \((0,e^{-1})\). Moreover, on \(I_{j,n}\) one has \[1-\cos(2\pi n s)\ge 1.\] Therefore \[- {\rm Re}\,m(\lambda_n)\ge \frac{1}{\beta} \sum_{j=1}^{N_n}\int_{I_{j,n}} w(s)\,ds.\] Because \(w\) is decreasing, \[\int_{I_{j,n}} w(s)\,ds \ge \frac{1}{2n}w\Bigl(\frac{j+1}{n}\Bigr) \ge \frac{1}{2}\int_{(j+1)/n}^{(j+2)/n} w(s)\,ds.\] Summing in \(j\) gives \[- {\rm Re}\,m(\lambda_n) \ge \frac{1}{2\beta}\int_{2/n}^{(N_n+2)/n} w(s)\,ds \gtrsim \int_{2/n}^{c_0} \frac{ds}{s(\log(1/s))^{1/\alpha}}\approx (\log n)^{1-\frac{1}{\alpha}}.\] Since \(\lambda_n=2\pi n,\) \[|m(\lambda_n)|\ge - {\rm Re}\,m(\lambda_n)\gtrsim (\log \lambda_n)^{1-\frac{1}{\alpha}}.\] This completes the proof.
In this section, we consider the Denjoy–Carleman classes that are substantially larger than the Gevrey and intermediate classes discussed earlier. Let \(I=[-1,1]\), and let \(M=(M_n)_{n\ge0}\) be a positive sequence of the form \(M_n=\exp^{(k)}(c n^\alpha)\), where \(c,\alpha>0\) and \(k\ge2\) is an integer. For large \(n\), this sequence is log-convex, so it defines a Denjoy–Carleman class \(\mathcal{C}_M\). If \(\psi\in\mathcal{C}_M(I)\), then there exists a constant \(K>0\) such that \[\label{defdc2} \sup_{t\in I} |\psi^{(n)}(t)| \le K^{n+1}\exp^{(k)}(c n^\alpha), \qquad n\ge0.\tag{21}\] Our goal is to understand how the oscillatory integral bound changes in this case. In what follows, we show that as the function classes become larger, the maximal growth of \(m(\lambda)\) increases to the universal \(C^\infty\) bound \(o(\log\lambda)\).
Theorem 15. Let \(\psi\in \mathcal{C}_M(I)\). Then \(m(\lambda)=O((\log\lambda)/(\log^{(k)}\lambda)^{\frac{1}{\alpha}})\). This bound is sharp: there exists a real-valued \(\psi\in \mathcal{C}_M(I)\) such that \[|m(\lambda_n)| \approx (\log\lambda_n)/(\log^{(k)}\lambda_n)^{\frac{1}{\alpha}} \quad\text{along a sequence }\lambda_n\to\infty.\]
As before, the key ingredient is a flat-point estimate.
Proposition 16 (Flat-point estimate). Let \(\phi\in \mathcal{C}_M(I)\) be flat at \(0\), i.e. \(\phi^{(n)}(0)=0\;(n\ge 0).\) Then there exist constants \(A,B>0\) such that for all sufficiently small \(|t|\), \[\label{ub5} |\phi(t)|\le A\exp\Bigl(-B \Big(\log\frac{1}{|t|}\Big)\Big(\log^{(k)}\frac{1}{|t|}\Big)^{\frac{1}{\alpha}}\Bigr).\qquad{(8)}\]
The estimate ?? is essentially sharp, since \(\exp\Bigl(- \Big(\log\frac{1}{|t|}\Big)\Big(\log^{(k)}\frac{1}{|t|}\Big)^{\frac{1}{\alpha}}\Bigr)\) belongs to \(\mathcal{C}_M(I)\) (see Section 8.2). It can be proved by the Taylor–Legendre method in Remark 9. Indeed, by Taylor’s theorem and minimization, we obtain for some constant \(B>0\) \[|\phi(t)|\le \inf_{n\ge1}\frac{K^{n+1}\exp^{(k)}(cn^\alpha)|t|^n}{n!}\approx \exp\Bigl(-B \Big(\log\frac{1}{|t|}\Big)\Big(\log^{(k)}\frac{1}{|t|}\Big)^{\frac{1}{\alpha}}\Bigr),\quad |t|\ll1,\] where the minimum is achieved at \(n_*\approx (\log^{(k-1)} y) ^{\frac{1}{\alpha}}\) with \(y=\log\frac{1}{K|t|}\).
By the reduction above, it suffices to consider the case in which \(\phi\) is flat at \(0\). Then Proposition 16 gives constants \(A,B>0\) and \(\delta\in(0,1)\) such that \[|\phi(t)|\le A\exp\Bigl(-B \Big(\log\frac{1}{t}\Big)\Big(\log^{(k)}\frac{1}{t}\Big)^{\frac{1}{\alpha}}\Big) \qquad (0<t\le \delta).\] Using \(|e^{ix}-e^{iy}|\le \min\{2,|x-y|\}\), we obtain \[|m(\lambda)| \lesssim 1+\int_0^\delta \min\{1,\lambda|\phi(t)|\}\,\frac{dt}{t}.\] Let \(u_\lambda>0\) solve \(\lambda\exp(-Bu(\log^{(k-1)}u)^{\frac{1}{\alpha}})=1\). Then \(u_\lambda\approx \frac{\log\lambda}{(\log^{(k)}\lambda)^{\frac{1}{\alpha}}}\).
We split the interval and change variables to obtain \[\begin{align} |m(\lambda)| &\lesssim u_\lambda+ \int_{u_\lambda}^\infty \lambda\exp(-B u(\log^{(k-1)}u)^{\frac{1}{\alpha}})du\\ &\lesssim u_\lambda+\int_{u_\lambda}^\infty \lambda\exp(-B(\log^{(k-1)}u_\lambda)^{\frac{1}{\alpha}}u)du\\ &\lesssim u_\lambda+(\log^{(k)}\lambda)^{-1/\alpha}\lesssim u_\lambda. \end{align}\] This proves the upper bound in Theorem 15.
For \(x\) sufficiently large we write \[L_1(x):=\log x, \qquad L_{j+1}(x):=\log L_j(x)\quad (j\ge 1),\] and \[E_1(x):=e^x, \qquad E_{j+1}(x):=\exp(E_j(x))\quad (j\ge 1).\] Fix \(k\ge 2\), \(\alpha>0\), and \(0<\delta\ll1\). Let \(I=(-\delta,\delta)\) and define \[\psi(t):= \begin{cases} 0, & -\delta<t\le 0,\\[4pt] \exp\!\Bigl(-\bigl(\log(1/t)\bigr)\,L_k(1/t)^{1/\alpha}\Bigr), & 0<t<\delta. \end{cases}\] We will show that \(\psi\in \mathcal{C}_M(I)\) in Section 9.3 and that \(|m(\lambda_n)| \approx (\log\lambda_n)/(\log^{(k)}\lambda_n)^{\frac{1}{\alpha}}\) along a sequence \(\lambda_n\to\infty\) in the following.
Since \(k\ge 2\), for \(u:=\log(1/t)\) this may be rewritten as \[\psi(t)=e^{-Q(u)}, \qquad Q(u):=uL_{k-1}(u)^{1/\alpha}.\]
Since \(\psi(-t)=0\) for \(t>0\), \[m(\lambda)=\int_0^{\delta}\frac{e^{i\lambda\psi(t)}-1}{t}\,dt.\] Set \[s:=\psi(t)=e^{-Q(u)}\in (0,s_0], \qquad s_0:=\psi(\delta)>0.\] Hence \[\label{eq:m-omega} m(\lambda)=\int_0^{s_0}(e^{i\lambda s}-1)\,\omega(s)\,ds, \qquad \omega(s):=\frac{1}{sQ'(u(s))}.\tag{22}\]
For \(0<s\le s_0\ll1\), \[\label{eq:weight-comparison} \omega(s)\approx \frac{1}{sL_k(1/s)^{1/\alpha}}.\tag{23}\]
Take \(\lambda_n:=2\pi n.\) By 22 , \[- {\rm Re}\,m(\lambda_n)=\int_0^{s_0}(1-\cos(2\pi ns))\,\omega(s)\,ds.\] Using the lower bound in 23 , \[- {\rm Re}\,m(\lambda_n) \gtrsim \int_0^{s_0}(1-\cos(2\pi ns))\,\frac{ds}{sL_k(1/s)^{1/\alpha}}.\] Choose a fixed number \(0<c_*<s_0/4\) and let \(N_n:=\lfloor c_*n\rfloor\). For \(1\le j\le N_n\) set \[I_{j,n}:=\Bigl[\frac{j+1/4}{n},\,\frac{j+3/4}{n}\Bigr].\] For large \(n\), all these intervals lie in \((0,s_0]\), and on each \(I_{j,n}\) one has \[1-\cos(2\pi ns)\ge 1.\] Therefore \[- {\rm Re}\,m(\lambda_n) \gtrsim \sum_{j=1}^{N_n}\int_{I_{j,n}}\frac{ds}{sL_k(1/s)^{1/\alpha}}.\] The weight \(s\mapsto \bigl(sL_k(1/s)^{1/\alpha}\bigr)^{-1}\) is decreasing on \((0,s_0]\), so \[\int_{I_{j,n}}\frac{ds}{sL_k(1/s)^{1/\alpha}} \ge \frac{1}{2}\int_{(j+1)/n}^{(j+2)/n}\frac{ds}{sL_k(1/s)^{1/\alpha}}.\] Summing in \(j\) gives \[- {\rm Re}\,m(\lambda_n) \gtrsim \int_{2/n}^{c_*}\frac{ds}{sL_k(1/s)^{1/\alpha}}\approx \frac{\log n}{L_k(n)^{1/\alpha}}.\]
Hence \[\label{eq:lower-seq} |m(\lambda_n)|\ge - {\rm Re}\,m(\lambda_n) \gtrsim \frac{\log \lambda_n}{L_k(\lambda_n)^{1/\alpha}}.\tag{24}\]
Let \(E_1(x)=e^x,\;E_{j+1}(x)=\exp(E_j(x)),\;j\ge1\). Let \(k\ge 1\) and \(s>1\) and \[a:=\frac{1}{s-1}, \qquad \psi(t):= \begin{cases} 0,& t\le 0,\\[1mm] \exp\!\bigl(-E_k(t^{-a})\bigr),& t>0. \end{cases}\] We will prove that \(\psi\in G_k^s([-1,1])\). For \(x\ge 1\), write \[M_k(x):=\prod_{j=1}^{k-1}E_j(x), \qquad \Omega_k(x):=x\,M_k(x)\] (with the convention \(M_1(x)=1\), so \(\Omega_1(x)=x\)).
Lemma 6. For every \(k\ge 1\) there exists \(A_k\ge 1\) such that for all \(r\ge 0\) and all \(x\ge 1\), \[E_k^{(r)}(x)\le r!\,A_k^r\,M_k(x)^r\,E_k(x).\]
Proof. We argue by induction on \(k\).
For \(k=1\), \(E_1^{(r)}(x)=e^x=E_1(x)\), so the claim is immediate.
Assume it holds for \(k-1\). Since \(E_k=e^{E_{k-1}}\), the Taylor expansion of \(E_k(x+h)\) at \(x\) gives \[\sum_{r=0}^\infty \frac{E_k^{(r)}(x)}{r!}h^r = E_k(x)\exp\!\bigl(E_{k-1}(x+h)-E_{k-1}(x)\bigr).\] Taking absolute values and using the induction hypothesis, \[\sum_{r=0}^\infty \frac{|E_k^{(r)}(x)|}{r!}|h|^r \le E_k(x)\exp\!\left( \sum_{m=1}^\infty A_{k-1}^m M_{k-1}(x)^m E_{k-1}(x)\,|h|^m \right).\] Since \(M_k(x)=E_{k-1}(x)M_{k-1}(x)\), choosing \[|h|=\frac{1}{2A_{k-1}M_k(x)}\] makes the exponent bounded by \(1\). Hence \[\sum_{r=0}^\infty \frac{|E_k^{(r)}(x)|}{r!}|h|^r \le e\,E_k(x).\] Comparing coefficients gives \[E_k^{(r)}(x)\le r!\,(2eA_{k-1})^r M_k(x)^r E_k(x).\] So the induction closes with \(A_k:=2eA_{k-1}\). ◻
Now put \[x(t):=t^{-a}\qquad (0<t\le 1), \qquad g(t):=E_k(x(t)), \qquad e^{-g(t)}=\psi(t)\quad (t>0).\] We first estimate derivatives of \(x(t)\). Since \[x^{(m)}(t)=(-1)^m a(a+1)\cdots (a+m-1)\,t^{-a-m},\] there is a constant \(B_a\ge 1\) such that for all \(m\ge 1\), \[|x^{(m)}(t)|\le m!\,B_a^m\,t^{-m}x(t).\]
Lemma 7. There exists \(C_1\ge 1\) such that for all \(n\ge 0\) and all \(0<t\le 1\), \[|g^{(n)}(t)|\le n!\,C_1^n\,t^{-n}\Omega_k(x(t))^n\,g(t).\]
Proof. Using the Taylor expansion of \(E_k(x(t+h))\) around \(x(t)\), \[\sum_{n=0}^\infty \frac{g^{(n)}(t)}{n!}h^n = \sum_{r=0}^\infty \frac{E_k^{(r)}(x(t))}{r!}\, \bigl(x(t+h)-x(t)\bigr)^r .\] Taking absolute values and using the previous lemma, \[\sum_{n=0}^\infty \frac{|g^{(n)}(t)|}{n!}|h|^n \le g(t)\sum_{r=0}^\infty \left( A_k M_k(x(t)) \sum_{m=1}^\infty \frac{|x^{(m)}(t)|}{m!}|h|^m \right)^r .\] Also, \[\sum_{m=1}^\infty \frac{|x^{(m)}(t)|}{m!}|h|^m \le x(t)\sum_{m=1}^\infty (B_a t^{-1}|h|)^m = x(t)\frac{B_a t^{-1}|h|}{1-B_a t^{-1}|h|}.\] Choose \[|h|=\frac{1}{4A_kB_a\,t^{-1}\Omega_k(x(t))}.\] Then \(B_a t^{-1}|h|\le 1/4\), hence the geometric factor above is at most \(2x(t)B_a t^{-1}|h|\), and therefore \[A_kM_k(x(t)) \sum_{m=1}^\infty \frac{|x^{(m)}(t)|}{m!}|h|^m \le \frac{1}{2} .\] So \[\sum_{n=0}^\infty \frac{|g^{(n)}(t)|}{n!}|h|^n \le 2g(t).\] Comparing coefficients yields the claim. ◻
Lemma 8. There exists \(C_2\ge 1\) such that for all \(n\ge 0\) and all \(0<t\le 1\), \[|\psi^{(n)}(t)| \le n!\,C_2^n\,t^{-n}\Omega_k(x(t))^n\,e^{-g(t)/2}.\]
Proof. Since \(\psi=e^{-g}\), \[\sum_{n=0}^\infty \frac{\psi^{(n)}(t)}{n!}h^n = e^{-g(t)}\exp\!\bigl(-(g(t+h)-g(t))\bigr).\] Taking absolute values and using the previous lemma, \[\sum_{n=0}^\infty \frac{|\psi^{(n)}(t)|}{n!}|h|^n \le e^{-g(t)} \exp\!\left( \sum_{m=1}^\infty C_1^m t^{-m}\Omega_k(x(t))^m g(t)\,|h|^m \right).\] Choose \[|h|=\frac{1}{4C_1\,t^{-1}\Omega_k(x(t))}.\] Then the series in the exponent is at most \(g(t)/2\), so \[\sum_{n=0}^\infty \frac{|\psi^{(n)}(t)|}{n!}|h|^n \le e^{-g(t)/2}.\] Comparing coefficients gives \[|\psi^{(n)}(t)| \le n!\,(4C_1)^n\,t^{-n}\Omega_k(x(t))^n\,e^{-g(t)/2}.\] ◻
Because \(a=1/(s-1)\), we have \(t^{-1}=x(t)^{\,s-1}\). Hence \[t^{-1}\Omega_k(x(t)) = x(t)^s \prod_{j=1}^{k-1}E_j(x(t)).\] Therefore \[\label{eq:star} |\psi^{(n)}(t)| \le n!\,C_2^n \left( x(t)^s\prod_{j=1}^{k-1}E_j(x(t)) \right)^n e^{-E_k(x(t))/2}.\tag{25}\] Now choose \(n_0\) large and define \[x_n:=\log^{(k)}\bigl((n+n_0)^2\bigr).\] Then \[E_k(x_n)=(n+n_0)^2, \qquad x_n\approx \log^{(k)}n, \qquad E_j(x_n)=\log^{(k-j)}\bigl((n+n_0)^2\bigr)\approx \log^{(k-j)}n\] for \(1\le j\le k-1\). Hence \[x_n^s\prod_{j=1}^{k-1}E_j(x_n)\lesssim Q_{k,s}(n).\]
If \(1\le x\le x_n\), then by monotonicity, \[x^s\prod_{j=1}^{k-1}E_j(x)\le x_n^s\prod_{j=1}^{k-1}E_j(x_n)\lesssim Q_{k,s}(n).\]
If \(x\ge x_n\), set \[R(x):=x^s\prod_{j=1}^{k-1}E_j(x).\] Since \[\log R(x)=s\log x+x+E_1(x)+\cdots+E_{k-2}(x)=o(E_{k-1}(x)),\] we have \(R(x)\le E_k(x)^{1/2}\) for all large \(x\). Therefore, writing \(y=E_k(x)\), \[R(x)^n e^{-E_k(x)/2}\le y^{n/2}e^{-y/2}.\] As \(y\ge E_k(x_n)=(n+n_0)^2\ge n\), the function \(y^{n/2}e^{-y/2}\) is decreasing, so \[y^{n/2}e^{-y/2}\le (n+n_0)^n e^{-(n+n_0)^2/2}\lesssim 1.\] Combining this with 25 , we obtain \[\sup_{0<t\le 1}|\psi^{(n)}(t)|\le K^{n+1} n!Q_{k,s}(n)^n\] for some constant \(K>0\) and all large \(n\). Since \(\psi^{(n)}(t)\to 0\) as \(t\downarrow 0\) by 25 , the extension by \(0\) to \((-\infty,0]\) is \(C^\infty\) and flat at \(0\). Therefore \(\psi\in G_k^s([-1,1])\).
Let \(\alpha>1\) and \(\beta=\alpha/(\alpha-1)\) and \[\psi(t)=\begin{cases} 0,& t\le 0,\\[1mm] \exp(-(\log\frac{1}{t})^{\beta}),& 0<t\le1/e. \end{cases}\] We take \(I=[-1/e,1/e]\) and show that \(\psi\in \mathcal{C}_M(I)\) with \(M_n=\exp(c n^\alpha)\) for some \(c>0\).
For \(0<t<e^{-1}\) set \[u:=\log(1/t)\ge 1, \qquad g(u):=e^{-u^\beta},\] so that \(\psi(t)=g(u)\).
Lemma 9. For every \(n\ge 1\) there exist real numbers \(a_{n,k}\), \(1\le k\le n\), such that \[\frac{d^n}{dt^n}g(\log(1/t)) = t^{-n}\sum_{k=1}^n a_{n,k}\,g^{(k)}(u),\] and \[\sum_{k=1}^n |a_{n,k}|\le n!.\]
Proof. For \(n=1\) this is immediate: \[\frac{d}{dt}g(\log(1/t))=-t^{-1}g'(u),\] so we may take \(a_{1,1}=-1\).
Assume the statement true for some \(n\ge 1\). Differentiate \[\frac{d^n}{dt^n}g(\log(1/t)) = t^{-n}\sum_{k=1}^n a_{n,k}\,g^{(k)}(u).\] Since \(u'=-t^{-1}\), we get \[\frac{d}{dt}\Bigl(t^{-n}F(u)\Bigr) = -t^{-n-1}\bigl(nF(u)+F'(u)\bigr).\] Applying this with \(F(u)=\sum_{k=1}^n a_{n,k}g^{(k)}(u)\) yields \[\frac{d^{n+1}}{dt^{n+1}}g(\log(1/t)) = t^{-n-1}\sum_{k=1}^{n+1} a_{n+1,k}\,g^{(k)}(u),\] where, with the convention \(a_{n,0}=a_{n,n+1}=0\), \[a_{n+1,k}=-(n a_{n,k}+a_{n,k-1}).\] Hence \[\sum_{k=1}^{n+1}|a_{n+1,k}| \le n\sum_{k=1}^n|a_{n,k}|+\sum_{k=1}^n|a_{n,k}| \le (n+1)n!=(n+1)!.\] This closes the induction. ◻
We next estimate \(g^{(k)}\).
Lemma 10. There exists a constant \(C_0\ge 1\) such that for all integers \(k\ge 1\) and all \(u\ge 1\), \[|g^{(k)}(u)| \le C_0^k\,k!\,k^k\,u^{(\beta-1)k}e^{-u^\beta}.\]
Proof. Write \[h(u):=-u^\beta, \qquad g(u)=e^{h(u)}.\] For \(m\ge 1\), \[h^{(m)}(u)=-(\beta)_m\,u^{\beta-m},\] where \((\beta)_m=\beta(\beta-1)\cdots (\beta-m+1)\). Since \[|\,\beta-j\,|\le |\beta|+j\le (1+|\beta|)(j+1) \qquad (j\ge 0),\] we obtain \[|(\beta)_m| \le (1+|\beta|)^m m!.\] Therefore, after enlarging the constant if needed, \[\label{eq:hm} |h^{(m)}(u)|\le C_0^m m!\,u^{\beta-m} \qquad (m\ge 1,\;u\ge 1).\tag{26}\]
Now apply Faà di Bruno in the partition form: \[g^{(k)}(u) = e^{h(u)} \sum_{\pi\in \varPi_k}\prod_{B\in\pi} h^{(|B|)}(u),\] where \(\varPi_k\) is the set of partitions of \(\{1,\dots,k\}\). Fix \(\pi\in\varPi_k\), let \(r:=|\pi|\) (the number of blocks in \(\pi\)) and let \(m_1,\dots,m_r\) be its block sizes, so \(m_1+\cdots+m_r=k\). Using 26 , \[\prod_{\ell=1}^r |h^{(m_\ell)}(u)| \le C_0^k\Bigl(\prod_{\ell=1}^r m_\ell!\Bigr)\,u^{r\beta-k}.\] Since \(r\le k\) and \(\prod_{\ell=1}^r m_\ell!\le k!\), we obtain \[\prod_{\ell=1}^r |h^{(m_\ell)}(u)| \le C_0^k k!\,u^{(\beta-1)k}.\] The number of partitions of a \(k\)-element set is the Bell number \(B_k\), and \(B_k\le k^k\). Hence \[|g^{(k)}(u)| \le C_0^k\,k!\,k^k\,u^{(\beta-1)k}e^{-u^\beta}.\] ◻
Combining Lemmas 9 and 10, for \(n\ge 1\) and \(0<t<e^{-1}\) we get \[|\psi^{(n)}(t)| \le t^{-n}\sum_{k=1}^n |a_{n,k}|\,|g^{(k)}(u)| \le t^{-n} e^{-u^\beta}\Bigl(\sum_{k=1}^n |a_{n,k}|\Bigr) \max_{1\le k\le n}\bigl(C_0^k k!\,k^k\,u^{(\beta-1)k}\bigr).\] By Lemma 9, \[\sum_{k=1}^n|a_{n,k}|\le n!,\] and since \(k\le n\) and \(u\ge 1\), \[C_0^k k!\,k^k\,u^{(\beta-1)k} \le C_0^n n!\,n^n\,u^{(\beta-1)n}.\] Therefore \[|\psi^{(n)}(t)| \le C_0^n (n!)^2 n^n\,e^{nu}\,u^{(\beta-1)n}e^{-u^\beta}.\] Taking logarithms and using Stirling in the crude form \(\log(n!)\lesssim n\log(n+1)\), we obtain \[\label{eq:rough-bound} |\psi^{(n)}(t)| \le \exp\!\Bigl(C_1 n\log(n+1)+nu+(\beta-1)n\log u-u^\beta\Bigr)\tag{27}\] for a constant \(C_1>0\).
For fixed \(n\), the exponent on the right-hand side tends to \(-\infty\) as \(u\to\infty\). Hence \(\psi^{(n)}(t)\to 0\) as \(t\downarrow 0\) for every \(n\), so the extension by \(0\) to \(t\le 0\) is \(C^\infty\) at the origin.
It remains to optimize 27 . Since \(u\ge 1\), we have \(\log u\le u\), hence \[nu+(\beta-1)n\log u \le \beta n u.\] Because \(\alpha\) and \(\beta\) are conjugate exponents, Young’s inequality gives, for every \(\varepsilon\in(0,1)\), \[\beta n u \le \varepsilon u^\beta + C_\varepsilon n^\alpha.\] Choosing \(\varepsilon=\frac{1}{2}\), \[nu+(\beta-1)n\log u-u^\beta \le -\frac{1}{2} u^\beta + C_2 n^\alpha \le C_2 n^\alpha.\] Thus 27 implies \[|\psi^{(n)}(t)| \le \exp\!\bigl(C_1 n\log(n+1)+C_2 n^\alpha\bigr) \qquad (0<t<e^{-1}).\] Since \(\alpha>1\), we have \(n\log(n+1)\le \varepsilon n^\alpha + C_\varepsilon n\). Absorbing the linear term into \(K^{n+1}\), we obtain \[\sup_{t\in I}|\psi^{(n)}(t)|\le K^{n+1}e^{c n^\alpha}\] for suitable constants \(K,c>0\) and all \(n\ge 0\). This proves \(\psi\in \mathcal{C}_M(I)\) in Section 7.
For \(x\) sufficiently large we write \[L_1(x):=\log x, \qquad L_{j+1}(x):=\log L_j(x)\quad (j\ge 1),\] and \[E_1(x):=e^x, \qquad E_{j+1}(x):=\exp(E_j(x))\quad (j\ge 1).\] Fix \(k\ge 2\), \(\alpha>0\), and choose \(\delta>0\) so small that \[L_j(1/t)\ge 2 \qquad (0<t\le \delta,\;1\le j\le k).\] Let \(I:=(-\delta,\delta)\) and define \[\psi(t):= \begin{cases} 0, & -\delta<t\le 0,\\[4pt] \exp\!\Bigl(-\bigl(\log(1/t)\bigr)\,L_k(1/t)^{1/\alpha}\Bigr), & 0<t<\delta. \end{cases}\] Since \(k\ge 2\), for \(u:=\log(1/t)\) this may be rewritten as \[\psi(t)=e^{-Q(u)}, \qquad Q(u):=uA(u), \qquad A(u):=L_{k-1}(u)^{1/\alpha}.\] For \(0<t<\delta\) set \(g(u):=e^{-Q(u)}\). To estimate \(g^{(r)}\) we first record a direct bound on the derivatives of \(A\) and \(Q\). Introduce the auxiliary functions \[\eta_0(u):=u^{-1}, \qquad \eta_j(u):=L_j(u)^{-1}\quad (1\le j\le k-1).\] Observe that \[\eta_0'(u)=-\eta_0(u)^2, \qquad \eta_j'(u)=-\eta_0(u)\eta_1(u)\cdots \eta_{j-1}(u)\eta_j(u)^2 \quad (j\ge 1).\] In particular, each derivative introduces at least one extra factor of \(\eta_0(u)=u^{-1}\).
Lemma 11. There exists a constant \(C\ge 1\) such that, for every integer \(r\ge 1\) and every sufficiently large \(u\), \[|A^{(r)}(u)|\le C^r r!\,A(u)u^{-r}, \qquad |Q^{(r)}(u)|\le C^r r!\,A(u)u^{1-r}.\] Moreover, \[Q'(u)=A(u)\Bigl(1+\frac{1}{\alpha L_1(u)\cdots L_{k-2}(u)L_{k-1}(u)}\Bigr),\] so in particular \[Q'(u)\approx A(u) \qquad (u\to\infty).\]
Proof. Since \[A(u)=L_{k-1}(u)^{1/\alpha}=\eta_{k-1}(u)^{-1/\alpha},\] a straightforward induction shows that for every \(r\ge 0\), \[A^{(r)}(u)=A(u)u^{-r}P_r\bigl(\eta_1(u),\dots,\eta_{k-1}(u)\bigr),\] where \(P_r\) is a polynomial with real coefficients and where the \(\ell^1\)-norm of the coefficient vector of \(P_r\) is bounded by \(C^rr!\) for a suitable constant \(C\) independent of \(r\). Indeed, differentiating the identity above produces three kinds of terms: one from differentiating \(A\), one from differentiating \(u^{-r}\), and one from differentiating the polynomial in the variables \(\eta_j\). Each differentiation contributes one extra factor of \(u^{-1}=\eta_0\), and the coefficient growth is at most linear in \(r\) at each step; hence the total coefficient growth is bounded by \(C^rr!\). Since \(0<\eta_j(u)\le 1\) for large \(u\), this gives \[|A^{(r)}(u)|\le C^rr!\,A(u)u^{-r}.\]
Now \(Q(u)=uA(u)\), so Leibniz’ rule gives, for \(r\ge 1\), \[Q^{(r)}(u)=uA^{(r)}(u)+rA^{(r-1)}(u).\] Using the bound just proved, \[|Q^{(r)}(u)| \le C^rr!A(u)u^{1-r}+rC^{r-1}(r-1)!A(u)u^{1-r} \le (2C)^rr!A(u)u^{1-r}.\] This proves the stated estimate after enlarging \(C\). Finally, \[A'(u)=\frac{A(u)}{\alpha uL_1(u)\cdots L_{k-2}(u)L_{k-1}(u)},\] so \[Q'(u)=A(u)+uA'(u) =A(u)\Bigl(1+\frac{1}{\alpha L_1(u)\cdots L_{k-2}(u)L_{k-1}(u)}\Bigr),\] which implies \(Q'(u)\approx A(u)\) for large \(u\). ◻
Lemma 12. There exists a constant \(C\ge 1\) such that for every integer \(r\ge 1\) and every sufficiently large \(u\), \[|g^{(r)}(u)|\le C^r (r!)^2 A(u)^r e^{-Q(u)}.\]
Proof. Apply Faà di Bruno in the partition form: \[g^{(r)}(u)=e^{-Q(u)}\sum_{\pi\in\varPi_r}\prod_{B\in\pi}(-Q^{(|B|)}(u)),\] where \(\varPi_r\) is the set of partitions of \(\{1,\dots,r\}\). For a partition \(\pi\in\varPi_r\), let \(m_B:=|B|\). By Lemma 11, \[\prod_{B\in\pi}|Q^{(m_B)}(u)| \le C^r\Bigl(\prod_{B\in\pi}m_B!\Bigr)A(u)^{|\pi|} \le C^r r!\,A(u)^r,\] because \(|\pi|\le r\) and \(\prod_B m_B!\le r!\). Summing over partitions and using the Bell-number bound \(|\varPi_r|\le e^r r!\), we obtain \[|g^{(r)}(u)|\le (Ce)^r(r!)^2A(u)^re^{-Q(u)}.\] Absorb \(e\) into the constant. ◻
We may now estimate \(\psi^{(n)}\) directly. By Lemmas 9 and 12, for \(n\ge 1\) and \(0<t<\delta\), \[|\psi^{(n)}(t)| \le t^{-n}\sum_{r=1}^n |a_{n,r}|\,|g^{(r)}(u)| \le t^{-n}n!\max_{1\le r\le n}|g^{(r)}(u)|.\] Hence \[|\psi^{(n)}(t)| \le C^n (n!)^3 e^{nu}A(u)^n e^{-uA(u)}.\] Taking logarithms and using \(\log(n!)\le C_0n\log(n+1)\), we get \[\label{eq:basic-exponent} |\psi^{(n)}(t)| \le \exp\!\Bigl(C_1n\log(n+1)+nu+n\log A(u)-uA(u)\Bigr).\tag{28}\] We now optimize the exponent.
Set \[F_n(u):=nu+n\log A(u)-uA(u).\] There are two cases.
Case 1: \(A(u)\le 2n\). Then \(L_{k-1}(u)=A(u)^\alpha\le (2n)^\alpha\), hence \[u\le E_{k-1}((2n)^\alpha).\] Therefore \[F_n(u)\le nE_{k-1}((2n)^\alpha)+n\log(2n).\] Since \(k\ge 2\), the function \(E_{k-1}(x)\) dominates polynomial factors; thus, after enlarging the constant, \[F_n(u)\le E_{k-1}(C_2n^\alpha).\]
Case 2: \(A(u)\ge 2n\). Since \(\log A(u)=\frac{1}{\alpha} L_k(u)=o(u)\), for large \(u\), \(\log A(u)\le \frac{u}{2}\). If \(A(u)\ge 2n\), then \[F_n(u) =nu+n\log A(u)-uA(u) \le nu+\frac{n}{2}u-uA(u) \le \frac{3}{2} nu-2nu = -\frac{1}{2} nu\le 0.\]
Thus in this case as well, \[F_n(u)\le E_{k-1}(C_2n^\alpha).\]
Substituting into 28 , we obtain \[|\psi^{(n)}(t)|\le \exp\!\bigl(C_1n\log(n+1)+E_{k-1}(C_2n^\alpha)\bigr).\] Because \(k\ge 2\), one has \(n\log(n+1)\le E_{k-1}(C_3n^\alpha)\) for a larger constant \(C_3\), and therefore \[|\psi^{(n)}(t)|\le \exp\!\bigl(E_{k-1}(cn^\alpha)\bigr)=E_k(cn^\alpha)\] for some \(c>0\). After increasing the constants to account for finitely many small values of \(n\), we obtain \[\sup_{0<t<\delta}|\psi^{(n)}(t)|\le K^{n+1}E_k(cn^\alpha).\] Since the exponent in 28 tends to \(-\infty\) as \(u\to\infty\) for each fixed \(n\), all derivatives tend to \(0\) as \(t\downarrow 0\); thus the extension by \(0\) to \(t\le 0\) is \(C^\infty\) at the origin. This proves \(\psi\in \mathcal{C}_M(I)\) in Section 8.