Markov's inequality on Koornwinder's domain in $L^p$ norms

TL;DR

This paper establishes the optimal exponent in Markov's inequality for a specific domain in the plane within $L^p$ norms, advancing understanding of polynomial inequalities on complex domains.

Contribution

It determines the exact exponent value (4) for Markov's inequality on Koornwinder's domain in $L^p$ norms, which was previously unknown.

Findings

Optimal exponent in Markov's inequality is 4.

The result applies to the domain defined by $ ext{Omega}$ in the plane.

Provides a precise mathematical characterization of polynomial bounds.

Abstract

Let $\Omega=\{(x,y) \in \mathbb{R}^2 : |x|<y+1, \, x^2>4y\}$. We prove that the optimal exponent in Markov's inequality on $\Omega$ in $L^p$ norms is 4.

Equations

$${\Vert \frac{\partial P}{\partial x_j}\Vert }_E\le M(\mathrm{deg}P{)}^r\Vert P{\Vert }_E$$

$${\Vert \frac{\partial P}{\partial x_i}\Vert }_{L^p(S,w)}\le C{n}^2\Vert P{\Vert }_{L^p(S,w)}(i=1,2).$$

$$\max \left\{{\Vert w\frac{\partial P}{\partial x_1}-P\Vert }_{L^{\infty }(S)},{\Vert w\frac{\partial P}{\partial x_2}+P\Vert }_{L^{\infty }(S)}\right\}\le \frac{2(n+1{)}^2}{r_S}\Vert wP{\Vert }_{L^{\infty }(S)},$$

$${\Vert w\frac{\partial P}{\partial x_i}\Vert }_{L^{\infty }(S)}\le \frac{2(\kappa {\delta }_S+1)(n+1{)}^2}{{\delta }_S}{\Vert P\Vert }_{L^{\infty }(S,w)}(i=1,2).$$

$${\Vert \frac{\partial P}{\partial x_i}\Vert }_{L^p(S,w)}\le \sum _{j=0}^2{\left({\int }_{D_j}{\left|\frac{\partial P}{\partial x_i}(x_1,x_2)\right|}^p(x_2-x_1)d{x}_{1}d{x}_2\right)}^{1/p}$$

$$D_0=\{(x_1,x_2)\in {\mathbb{R}}^2:-1<x_1<0,x_1+1<x_2<1\},$$

$$D_1=\{(x_1,x_2)\in {\mathbb{R}}^2:-1<x_1<0,x_1<x_2<x_1+1\},$$

$$D_2=\{(x_1,x_2)\in {\mathbb{R}}^2:0<x_2<1,x_2-1<x_1<x_2\}.$$

$${\Vert \frac{\partial P}{\partial x_i}\Vert }_{L^p(D_j,w)}\le \tilde{C}(\mathrm{deg}P{)}^2{\Vert P\Vert }_{L^p(S,w)},j=0,1,2.$$

$${\left({\int }_{D_0}{\left|\frac{\partial P}{\partial x_i}(x_1,x_2)\right|}^p(x_2-x_1)d{x}_{1}d{x}_2\right)}^{1/p}$$

$$\le C_0(\mathrm{deg}P{)}^2\Vert P{\Vert }_{L^p(S,w)}.$$

$${\left({\int }_{D_1}{\left|\frac{\partial P}{\partial x_i}(x_1,x_2)\right|}^p(x_2-x_1)d{x}_{1}d{x}_2\right)}^{1/p}.$$

$${\left({\int }_{-1}^0{\int }_0^1{\left|\frac{\partial P}{\partial x_i}(t,s+t)\right|}^{p}sdsdt\right)}^{1/p}.$$

$$\frac{\partial Q}{\partial t}(t,s)-\frac{\partial Q}{\partial s}(t,s)=\frac{\partial P}{\partial x_1}(t,s+t),\frac{\partial Q}{\partial s}(t,s)=\frac{\partial P}{\partial x_2}(t,s+t).$$

$${\int }_0^1{\left|\frac{\partial P}{\partial x_2}(t,s+t)\right|}^{p}sds\le C_1^p(\mathrm{deg}Q{)}^{2p}{\int }_0^1{\left|Q(t,s)\right|}^{p}sds.$$

$${\Vert \frac{\partial P}{\partial x_2}\Vert }_{L^p(D_1,w)}$$

$$=C_1(\mathrm{deg}P{)}^2{\left({\int }_{-1}^0{\int }_0^1{\left|P(t,s+t)\right|}^{p}sdsdt\right)}^{1/p}$$

$$\le C_1(\mathrm{deg}P{)}^2\Vert P{\Vert }_{L^p(S,w)}.$$

$${\Vert \frac{\partial P}{\partial x_1}\Vert }_{L^p(D_1,w)}\le$$

$$+{\left({\int }_{-1}^0{\int }_0^1{\left|\frac{\partial Q}{\partial s}(t,s)\right|}^{p}sdsdt\right)}^{1/p}.$$

$${\int }_{-1}^0{\left|\frac{\partial Q}{\partial t}(t,s)\right|}^{p}dt\le {\hat{C}}_1^p(\mathrm{deg}Q{)}^{2p}{\int }_{-1}^0{\left|Q(t,s)\right|}^{p}dt,$$

$${\int }_0^1{\left|\frac{\partial Q}{\partial s}(t,s)\right|}^{p}sds\le C_1^p(\mathrm{deg}Q{)}^{2p}{\int }_0^1{\left|Q(t,s)\right|}^{p}sds.$$

$${\Vert \frac{\partial P}{\partial x_1}\Vert }_{L^p(D_1,w)}\le$$

$$+{\left({\int }_{-1}^0\left[C_1^p(\mathrm{deg}P{)}^{2p}{\int }_0^1{\left|Q(t,s)\right|}^{p}sds\right]dt\right)}^{1/p}.$$

$${\Vert \frac{\partial P}{\partial x_1}\Vert }_{L^p(D_1,w)}$$

$$\le ({\hat{C}}_1+C_1)(\mathrm{deg}P{)}^2\Vert P{\Vert }_{L^p(S,w)}.$$

$$\{\text{there exists }\alpha \in {\mathbb{N}}^2\text{ such that }{\alpha }_1+{\alpha }_2\le d\text{ and }|R^{(\alpha )}(x)|\ge m>0(x\in A)\}$$

$$\Vert P{\Vert }_{L^p(A,w)}\le C_d{m}^{-1}{ϵ}^{-1}(n+k{)}^{2d}\Vert PR{\Vert }_{L^p(S,w)}+ϵ\Vert P{\Vert }_{L^p(S,w)}.$$

$$|P(x)|\le m^{-1}|P(x)R(x)|\text{for}x\in A.$$

$$\Vert P{\Vert }_{L^p(A,w)}\le m^{-1}\Vert PR{\Vert }_{L^p(A,w)}\le m^{-1}\Vert PR{\Vert }_{L^p(S,w)}+ϵ\Vert P{\Vert }_{L^p(S,w)}.$$

$$I=\left\{({\beta }_1,{\beta }_2)\in {\mathbb{N}}^2:0<|\beta |,0\le {\beta }_1\le {\alpha }_1,0\le {\beta }_2\le {\alpha }_2\right\}.$$

$$|P(x)|\le m^{-1}\left[|(PR{)}^{(\alpha )}(x)|+\sum _{\beta \in I}(\genfrac{}{}{0ex}{}{\alpha }{\beta })|R^{(\alpha -\beta )}(x)||P^{(\beta )}(x)|\right].$$

$$B_0=\{x\in A:|R^{(\alpha -\beta )}(x)|\le \frac{mϵ}{{\eta }^2}{\left(\genfrac{}{}{0ex}{}{\alpha }{\beta }\right)}^{-1}(C{n}^2{)}^{-|\beta |},\beta \in I\},$$

$$|P(x)|\le m^{-1}|(PR{)}^{(\alpha )}(x)|+\frac{ϵ}{{\eta }^2}\sum _{\beta \in I}(C{n}^2{)}^{-|\beta |}|P^{(\beta )}(x)|.$$

$$\Vert P{\Vert }_{L(B_0,w)}$$

Full text

$L^{p}$-norm estimate for the Bergman projection on Hartogs triangle

Tomasz Beberok

Faculty of Mathematics and Computer Science, Jagiellonian University,

Lojasiewicza 6, 30-048 Krakow, Poland

**Markov’s inequality on Koornwinder’s domain in $L^{p}$ norms **

Tomasz Beberok

Abstract. Let $\Omega=\{(x,y)\in\mathbb{R}^{2}\colon|x|<y+1,\,x^{2}>4y\}$. We prove that the optimal exponent in Markov’s inequality on $\Omega$ in $L^{p}$ norms is 4.

Keywords: Markov inequality; $L^{p}$ norms; Markov exponent AMS Subject Classifications: primary 41A17, secondary 41A44

1 Introduction

Throughout this paper $\mathcal{P}(\mathbb{R}^{N})$ ($\mathcal{P}_{n}(\mathbb{R}^{N})$, respectively) denotes the set of algebraic polynomials of $N$ variables with real coefficients (with total degree at most $n$). We begin with the definition of multivariate Markov’s inequality.

Definition 1.1

Let $E\subset\mathbb{R}^{N}$ be a compact set. We say that $E$ admit Markov’s inequality if there exist constants $M,r>0$ such that for every polynomial $P\in\mathcal{P}(\mathbb{R}^{N})$ and $j\in\{1,2,\ldots,N\}$

[TABLE]

where $\|\cdot\|_{E}$ is the supremum norm on $E$.

A compact set $E$ with this property is called a Markov set. The inequality (1) is a generalization of the classical inequality proved by A. A. Markov in 1889, which gives such estimate on $[-1,1]$. The theory of Markov inequality and it’s generalizations is still the active and fruitful area of approximation theory (see, for instance, [5, 7, 17]). For a given compact set $E$, an important problem is to determine the minimal constant $r$ in (1). This can be used to minimize the loss of regularity in problems concerning the linear extension of classes of $\mathrm{C}^{\infty}$ functions with restricted growth of derivatives (see [24, 25]). Such an $r$ is so-called Markov’s exponent of $E$ (see [4] for more detail on this matter). In the case of supremum norm various information about Markov’s exponent is known (see, e.g., [8, 11, 21, 26, 27]). The Markov type inequalities were also studied in $L^{p}$ norms (see [1, 6, 12, 13, 14, 15]). In this case the question of Markov’s exponent problem is much more complex. In particular, to the best of our knowledge, there is no example of a compact set in $\mathbb{R}^{N}$ with cusps for which Markov’s exponent (with respect to the Lebesgue measure) is known. The attempts to solve this problem led, among others, to a so-called Milówka–Ozorka identity (see [3, 22, 23] for discussion). The aim of this note is to give such an example. More precisely, we show that, in the notation above, Markov’s exponent of $\Omega$ in $L^{p}$ norms is 4. Here $\Omega=\{(x,y)\in\mathbb{R}^{2}\colon|x|<y+1,\,x^{2}>4y\}$ which is depicted in the Figure 1. Since $\Omega$ is the region for Koornwinder orthogonal polynomials (first type), see [18, 19], we call this set Koornwinder’s domain.

2 Some weighted polynomial inequalities on simplex

The following lemma will be particularly useful in the proof of our main result.

Lemma 2.1

Let $S=\{(x_{1},x_{2})\in\mathbb{R}^{2}\colon-1<x_{1}<x_{2}<1\}$, $w(x_{1},x_{2})=x_{2}-x_{1}$ and $1\leq p\leq\infty$. Then there exists a positive constant $C(S,w)$ such that, for every $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$, we have

[TABLE]

**Proof. ** We start with $p=\infty$. Since $\overline{S}$ is a convex body in $\mathbb{R}^{2}$, the result of Wilhelmsen [30] gives

[TABLE]

where $\delta_{S}$ is the width of the convex body (the minimal distance between parallel supporting hyperplanes). Therefore by Lemma 3 from [13], there is a constant $\kappa>0$ such that, for all $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$,

[TABLE]

Thus we conclude that (2) holds when $p=\infty$. Now, for each $1\leq p<\infty$, it is clear that

[TABLE]

where

[TABLE]

We shall show that there is a constant $\tilde{C}>0$ such that, for all $P\in\mathcal{P}(\mathbb{R}^{2})$,

[TABLE]

Since $D_{0}$ is a bounded convex set and $w$ is bounded away from zero on $D_{0}$, we have (see [9, 12, 13, 20])

[TABLE]

Now consider the case $j=1$. The integral is then

[TABLE]

We perform the change of variables $t=x_{1}$, $s=x_{2}-x_{1}$. The integral becomes

[TABLE]

Define $Q(t,s)=P(t,s+t)$. Then

[TABLE]

Hence, (using Goetgheluck’s result–see [10])

[TABLE]

Therefore

[TABLE]

On the other hand,

[TABLE]

We have, arguing as before, that there exists constants $\hat{C}_{1},C_{1}$ such that for every polynomial $Q\in\mathcal{P}(\mathbb{R}^{2})$

[TABLE]

Therefore we see immediately that

[TABLE]

Thus we finally have

[TABLE]

A similar result for $D_{2}$ obtains if one considers the substitution $t=x_{2}$, $s=x_{2}-x_{1}$ and polynomial $\tilde{Q}(t,s)=P(t-s,t)$. We omit the details. Thus we have shown that, if $\tilde{C}=2\max\{C_{0},\hat{C}_{1},C_{1},\hat{C}_{2},C_{2}\}$, then (4) holds. That completes the proof.

Now we shall prove the following weighted Schur-type inequality.

Lemma 2.2

(with previous notation). Let $d$ be a natural number. Then, for every $A\subset\bar{S}$ and $R\in\mathcal{P}_{k}(\mathbb{R}^{2})$, satisfying the condition

[TABLE]

one can find a constant $C_{d}$ such that, for any $\epsilon>0$ and every $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$, we have

[TABLE]

**Proof. ** The idea of the proof comes from [13]. Thus we proceed by induction on the length of $\alpha$. If $\alpha_{1}=\alpha_{2}=0$, then

[TABLE]

Therefore

[TABLE]

Now, for $d_{0}\leq d$, assume that (5) holds when $\alpha_{1}+\alpha_{2}<d_{0}$. We shall show that (5) is still valid for $\alpha$ such that $|\alpha|=d_{0}$. Here $|\alpha|$ denotes the length of $\alpha$. Let

[TABLE]

Notice that the set $I$ contains at most $\frac{(d_{0}+1)(d_{0}+2)}{2}-1$ elements. By Leibniz’s rule, if $x\in A$, then

[TABLE]

Let $C$ be a constant so that (2) holds. We set

[TABLE]

where $\eta=\frac{(d_{0}+1)(d_{0}+2)}{2}$. Then, for each $x\in B_{0}$, we have

[TABLE]

This yields

[TABLE]

Therefore by the preceding lemma,

[TABLE]

On the other hand, if $x\in A\setminus B_{0}$, then there exists $\beta\in I$ such that

[TABLE]

Thus we can share $x\in A\setminus B_{0}$ into at most $\eta-1$ disjoint subsets $B_{j}$ such that, for every $x\in B_{j}$, there exists an index $\beta$ for which (6) holds. Therefore, since $|\beta|>0$, on each $B_{j}$, replacing $\epsilon$ by $\frac{\epsilon}{\eta}$, we conclude by induction that

[TABLE]

Since $A=\bigcup_{j}B_{j}$ we see that

[TABLE]

with

[TABLE]

which completes the induction and the proof.

3 Main result

Our main result reads as follows:

Theorem 3.1

Let $p\geq 1$. Then there exists constant $C=C(\Omega,p)$ such that for every polynomial $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$ we have

[TABLE]

**Proof. ** Let us first prove the inequality (7) with respect to the second variable. Let $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$. Then the integrals

[TABLE]

become, under a change of variables $x=u+v$, $y=uv$,

[TABLE]

where $S=\{(u,v)\in\mathbb{R}^{2}\colon-1<u<v<1\}$. Let us define polynomial $Q(u,v)=P(u+v,uv)$. Then

[TABLE]

We now see, using Lemma 2.1, that

[TABLE]

Lemma 2.2 tells us that

[TABLE]

Hence

[TABLE]

That completes the proof of (7) for the derivative of $P$ with respect to $y$. To prove the remaining part we need to consider the polynomials $uQ$ and $vQ$. Then

[TABLE]

Hence

[TABLE]

Thus using an argument similar to the one that we carry out in detail in the previous case, one can obtain the desired estimate.

Remark 3.1

In the same fashion, we may prove that there exists a positive constant $C_{l}$ such that for every $P\in\mathcal{P}_{n}(\mathbb{R}^{2})$ we have

[TABLE]

where $\Delta_{l}=\left\{(x,y)\in\mathbb{R}^{2}\colon|x|^{1/l}+|y|^{1/l}\leq 1\right\}$ and $l$ is a positive odd number.

4 Sharpness of the exponents

In fact, according to [2], it is enough to prove sharpness in the supremum norm. The discussion here is based on unpublished work of M. Baran. Let us consider following sequence of polynomials

[TABLE]

where $T_{k}$ is the $k$th Chebyshev polynomial of the first kind. Note that the degree of a polynomial $P_{k}$ is equal $5k-4$. Since

[TABLE]

we may conclude that

[TABLE]

for any $(x,y)\in\Omega$. On the other hand,

[TABLE]

A similar calculation shows that, for $Q_{k}=\left[\frac{1}{k}T_{k}^{\prime}\left(\frac{1+y}{2}\right)\right]^{5}\left(\frac{x^{2}}{4}-y\right)$,

[TABLE]

Let $P_{n}^{(\alpha,\beta)}$ denote the Jacobi polynomials. In order to prove sharpness of (8), we consider the sequence of polynomials $W_{n}(x,y)=yP_{n}^{(\alpha,\alpha)}(x)$. Thus

[TABLE]

By the well known symmetry relation (see [29], Chap. IV)

[TABLE]

we find that

[TABLE]

Now Bernoulli’s inequality, for each positive integer $l$ and $x\in[0,1]$, implies that

[TABLE]

Hence, if $n\rightarrow\infty$, then

[TABLE]

We may now apply the result of Szegö (see [29], Chap. VII) to get

[TABLE]

where $\mu_{\alpha,p}=\alpha p-2+p/2$. If $2(p+1)l<\mu_{\alpha,p}$, we can combine (9) and (10) to see that

[TABLE]

That is what we wished to prove.

Acknowledgment

The author deeply thanks Mirosław Baran and Leokadia Białas-Cież who pointed out some important remarks, corrections and shared their unpublished notes.

The author was supported by the Polish National Science Centre (NCN) Opus grant no. 2017/25/B/ST1/00906.