Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions \(w,w^{-1}\in \mathrm{BMO}\)

Abstract

For the weight w satisfying \(w,w^{-1}\in \mathrm{BMO}({\mathbb {T}})\), we prove the asymptotics of \(\{\Phi _n(e^{i\theta },w)\}\) in \(L^p[-\pi ,\pi ], 2\leqslant p<p_0\), where \(\{\Phi _n(z,w)\}\) are monic polynomials orthogonal with respect to w on the unit circle \({\mathbb {T}}\). Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial entropies. The estimates on higher-order commutators between the Calderon–Zygmund operators and BMO functions play the key role in the proofs of main results.

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Acknowledgments

The work of SD done in the second part of the paper was supported by RSF-14-21-00025, and his research on the rest of the paper was supported by the Grant NSF-DMS-1464479. The research of KR was supported by the RTG Grant NSF-DMS-1147523.

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Correspondence to Sergey Denisov.

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Communicated by Percy A. Deift.

Appendix

Appendix

In this Appendix, we collect some auxiliary results used in the main text.

Lemma 3.1

For every \(p\in [2,\infty )\),

$$\begin{aligned} \Vert P_{[1,n]}\Vert _{p,p}\leqslant \cot \left( \frac{\pi }{2p}\right) . \end{aligned}$$
(14)

Proof

If \(\mathcal {P}^+\) is the projection of \(L^2({\mathbb {T}})\) onto \(H^2({\mathbb {T}})\) (analytic Hardy space), then

$$\begin{aligned} P_{[1,n]}=z\mathcal {P}^+z^{-1}-z^{n+1} \mathcal {P}^+ z^{-(n+1)} = 0.5 \left( zHz^{-1}-z^{n+1} H z^{-(n+1)}\right) , \end{aligned}$$
(15)

where H is the Hilbert transform. Since \(\Vert H\Vert _{p,p}=\cot (\pi /(2p))\) [16], we have the lemma by the triangle inequality. \(\square \)

The proof of the following lemma uses some standard results of harmonic analysis.

Lemma 3.2

If \(\Vert w\Vert _{\mathrm{BMO}}=t\) and \(p\in [2,3]\), then we have

$$\begin{aligned} \Vert \mathbf{C}_j\Vert _{p,p}\leqslant (Cjt)^j. \end{aligned}$$

Proof

Consider the following operator-valued function:

$$\begin{aligned} F(z)=e^{zw}P_{[1,n]}e^{-zw}. \end{aligned}$$

If we can prove that F(z) is weakly analytic around the origin (i.e., analyticity of the scalar function \(\langle F(z)f_1,f_2\rangle \) with fixed \(f_{1(2)}\in C^\infty \)), then

$$\begin{aligned} F(z)=\frac{1}{2\pi i}\int _{|\xi |=\epsilon } \frac{F(\xi )}{\xi -z}d\xi , \quad z\in B_\epsilon (0), \end{aligned}$$

understood in a weak sense. By induction, one can then easily show the well-known formula

$$\begin{aligned} \mathbf{C}_j=\partial ^{j}F(0)=\frac{j!}{2\pi i}\int _{|\xi |=\epsilon } \frac{F(\xi )}{\xi ^{j+1}}d\xi , \end{aligned}$$

which explains that we can control \(\Vert \mathbf{C}_j\Vert _{p,p}\) by the size of \(\Vert F(\xi )\Vert _{p,p}\) on the circle of radius \(\epsilon \). Indeed,

$$\begin{aligned} \Vert \mathbf{C}_j\Vert _{p,p}= & {} \sup _{f_{1(2)}\in C^\infty , \Vert f_1\Vert _p\leqslant 1, \Vert f_2\Vert _{p'}\leqslant 1} |\langle \mathbf{C}_j f_1,f_2\rangle | \\\le & {} \frac{j!}{2\pi }\sup _{f_{1(2)}\in C^\infty , \Vert f_1\Vert _p\leqslant 1, \Vert f_2\Vert _{p'}\leqslant 1}\left| \int _{|\xi |=\epsilon } \frac{\langle F(\xi )f_1,f_2\rangle }{\xi ^{j+1}}d\xi \right| \\\leqslant & {} \frac{j!}{\epsilon ^j}\max _{|\xi |=\epsilon }\Vert F(\xi )\Vert _{p,p}. \end{aligned}$$

The weak analyticity of F(z) around the origin follows immediately from, e.g., the John–Nirenberg estimate ([20], p.144). To bound \(\Vert F\Vert _{p,p}\), we use the following well-known result (which is again an immediate corollary from the John–Nirenberg inequality, see, e.g., [20], p.218).

There is \(\epsilon _0\) such that

$$\begin{aligned} \Vert \widetilde{w}\Vert _{\mathrm{BMO}}<\epsilon _0 \implies [e^{\widetilde{w}}]_{A_p}\leqslant [e^{\widetilde{w}}]_{A_2}<C,\quad p>2. \end{aligned}$$

The Hunt–Muckenhoupt–Wheeden theorem ([20], p.205) asserts that

$$\begin{aligned} \sup _{[\widehat{w}]_{A_p}\leqslant C} \Vert H\Vert _{(L^p_{\widehat{w}}({\mathbb {T}}),L^p_{\widehat{w}}({\mathbb {T}}))}= & {} \sup _{[\widehat{w}]_{A_p}\leqslant C} \Vert \widehat{w}^{1/p}H\widehat{w}^{-1/p}\Vert _{p,p} \nonumber \\= & {} C(p)<\infty , \quad p\in [2,\infty ). \end{aligned}$$
(16)

Taking \(\epsilon \ll t^{-1}\), we get the statement. \(\square \)

The following lemma provides an estimate that is not optimal, but it is good enough for our purposes.

Lemma 3.3

Suppose \(w\geqslant 0,\Vert w\Vert _{\mathrm{BMO}}=t, \Vert w^{-1}\Vert _{\mathrm{BMO}}=s\), and \(\Vert w\Vert _1=1\). Then,

$$\begin{aligned} (2\pi )^2\leqslant \Vert w^{-1}\Vert _1\lesssim 1+(1+t)s. \end{aligned}$$

Proof

Set \(\Vert w^{-1}\Vert _1=M\). Then, by the Cauchy–Schwarz inequality,

$$\begin{aligned} 2\pi \leqslant \Vert w\Vert _1^{1/2}\Vert w^{-1}\Vert ^{1/2}_1=M^{1/2}. \end{aligned}$$

On the other hand, by the John–Nirenberg estimate for \(w^{-1}\),

$$\begin{aligned} |\{\theta : |w^{-1}-(2\pi )^{-1}M|>\lambda \}|\lesssim \exp \left( -\frac{C\lambda }{s}\right) . \end{aligned}$$

Choosing \(\lambda =(4\pi )^{-1}M\), we get

$$\begin{aligned} |\Omega ^c|\lesssim \exp \left( -\frac{CM}{s}\right) \lesssim \left( \frac{s}{M}\right) ^2,\quad \mathrm{where}\quad \,\Omega =\Bigl \{\theta :\frac{4\pi }{3M}\leqslant w\leqslant \frac{4\pi }{M}\Bigr \}. \end{aligned}$$
(17)

Then, \(\Vert w\Vert _1=1\), and therefore

$$\begin{aligned} 1= & {} \int _{w\leqslant (4\pi )/M}wd\theta +\int _{w>(4\pi )/M}wd\theta \nonumber \\ \int _{w>(4\pi )/M}wd\theta\geqslant & {} 1-8\pi ^2 M^{-1}. \end{aligned}$$
(18)

By the John–Nirenberg inequality, we have

$$\begin{aligned} \Vert w-(2\pi )^{-1}\Vert _p<Ctp,\quad p<\infty . \end{aligned}$$
(19)

We choose \(p=2\) in the last estimate and use the Cauchy–Schwarz inequality in (18) to get

$$\begin{aligned} 1-8\pi ^2 M^{-1}\leqslant & {} \int _{w>(4\pi )/M}wd\theta \leqslant \Vert w\Vert _2 \cdot |\{\theta : w>4\pi /M\}|^{1/2} \\\leqslant & {} \Vert w\Vert _2 \cdot |\Omega _c|^{1/2} \lesssim \frac{(1+t)s}{M}, \end{aligned}$$

where we used (17) and (19) for the last bound. So, \( M\lesssim (1+t)s+1\). \(\square \)

Lemma 3.4

For \(p\in [2,\infty )\), we have

$$\begin{aligned} \Vert [w,P_{[1,n]}]\Vert _{p,p}\lesssim p^2\Vert w\Vert _{\mathrm{BMO}}. \end{aligned}$$

Proof

The proof is standard, but we give it here for completeness. Assume \(\Vert w\Vert _{\mathrm{BMO}}=1\). By duality and formula (15), it is sufficient to show that

$$\begin{aligned} \Vert [w,H]\Vert _{p,p}\leqslant C(p-1)^{-2}, \quad p\in (1,2]. \end{aligned}$$
(20)

We will interpolate between two bounds: the standard Coifman–Rochberg–Weiss theorem for \(p=2\) ([7, 20]),

$$\begin{aligned} \Vert [H,w]\Vert _{2,2}\leqslant C, \end{aligned}$$
(21)

and the following estimate:

$$\begin{aligned} |\{x:|([H,w]f)(x)|>\alpha \}|\leqslant C \int _{{\mathbb {T}}} \frac{|f(t)|}{\alpha }\left( 1+\log ^+\left( \frac{|f(t)|}{\alpha }\right) \right) dt. \end{aligned}$$
(22)

(See [15]; the estimate was obtained on \({\mathbb {R}}\) for smooth f with compact support. The proof, however, is valid for \({\mathbb {T}}\) as well and, e.g., piece-wise smooth continuous f). Assume a smooth f is given and set \(\lambda _f(t)=|\{x: |f(x)|>t\}|, t\geqslant 0\). Take \(A>0\), and consider \(f_A=f\cdot \chi _{|f|\leqslant A}+A\cdot \mathrm{sgn}f\cdot \chi _{|f|>A}\), \(g_A=f-f_A\). Let \(T=[H,w]\). Then,

$$\begin{aligned} \Vert Tf\Vert _p^p= & {} p\int _0^\infty t^{p-1}\lambda _{Tf}(t)dt\leqslant p\int _0^\infty t^{p-1}\lambda _{Tf_A}(t/2)dt \\&+ p\int _0^\infty t^{p-1}\lambda _{Tg_A}(t/2)dt=I_1+I_2. \end{aligned}$$

Let \(A=t\). From the Chebyshev inequality and (21), we get

$$\begin{aligned} I_1\lesssim & {} \int _0^\infty t^{p-3}\Vert f_A\Vert _2^2dt=2\int _0^\infty t^{p-3}\int _0^A \xi \lambda _f(\xi )d\xi dt \\\lesssim & {} (2-p)^{-1}\int _0^\infty \xi ^{p-1}\lambda _f(\xi )d\xi \lesssim (2-p)^{-1}\Vert f\Vert _p^p. \end{aligned}$$

For \(I_2\), we use (22) (notice that \(g_A\) is continuous and piece-wise smooth)

$$\begin{aligned} I_2\lesssim & {} -\int _0^\infty t^{p-1}\int _0^\infty \frac{\xi }{t}\left( 1+\log ^+\frac{\xi }{t}\right) d\lambda _{g_A}(\xi )\\\lesssim & {} \Vert f\Vert _p^p+\int _0^\infty t^{p-1}\int _{2t}^\infty t^{-1}\Bigl (1+\log ^+((\tau -t)/t)\Bigr )\lambda _{f}(\tau )d\tau \\\lesssim & {} \Vert f\Vert _p^p \int _0^1 \xi ^{p-2}\left( 1+\log ^+\frac{1-\xi }{\xi }\right) d\xi . \end{aligned}$$

We have

$$\begin{aligned} \int _0^{1/2} \xi ^{p-2}\left( 1+\log ^+\frac{1-\xi }{\xi }\right) d\xi \lesssim \int _2^\infty u^{-p}\log u du\lesssim \int _0^\infty e^{-\delta t}tdt\lesssim \delta ^{-2} \end{aligned}$$

with \(\delta =p-1\). \(\square \)

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Denisov, S., Rush, K. Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions \(w,w^{-1}\in \mathrm{BMO}\) . Constr Approx 46, 285–303 (2017). https://doi.org/10.1007/s00365-016-9350-6

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Keywords

  • Orthogonal polynomials
  • Weight
  • Bounded mean oscillation

Mathematics Subject Classification

  • 42C05
  • 33D45