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.

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 $\delta =p-1$. $\square$