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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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 \(p\in [2,\infty )\),
Proof
If \(\mathcal {P}^+\) is the projection of \(L^2({\mathbb {T}})\) onto \(H^2({\mathbb {T}})\) (analytic Hardy space), then
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
Proof
Consider the following operator-valued function:
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
understood in a weak sense. By induction, one can then easily show the well-known formula
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,
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
The Hunt–Muckenhoupt–Wheeden theorem ([20], p.205) asserts that
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,
Proof
Set \(\Vert w^{-1}\Vert _1=M\). Then, by the Cauchy–Schwarz inequality,
On the other hand, by the John–Nirenberg estimate for \(w^{-1}\),
Choosing \(\lambda =(4\pi )^{-1}M\), we get
Then, \(\Vert w\Vert _1=1\), and therefore
By the John–Nirenberg inequality, we have
We choose \(p=2\) in the last estimate and use the Cauchy–Schwarz inequality in (18) to get
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
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
We will interpolate between two bounds: the standard Coifman–Rochberg–Weiss theorem for \(p=2\) ([7, 20]),
and the following estimate:
(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,
Let \(A=t\). From the Chebyshev inequality and (21), we get
For \(I_2\), we use (22) (notice that \(g_A\) is continuous and piece-wise smooth)
We have
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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DOI: https://doi.org/10.1007/s00365-016-9350-6