3 Proof of the main new results

Schur type inequalities for

multivariate polynomials on convex bodies

András Kroó^a

Communicated by Stefano De Marchi

AMS Subject classification: 41A17, 41A63.

Key words and phrases: convex bodies, multivariate polynomials, Schur type inequalities, Jacobi weights, norms of polynomial factors

Abstract

In this note we give sharp Schur type inequalities for multivariate polynomials with generalized Jacobi weights on arbitrary convex domains. In particular, these results yield estimates for norms of factors of multivariate polynomials.

1 Introduction and main new results

The classical Schur inequality states that for any univariate polynomialpnof degree at mostn−1 we have on the interval I:= [−1, 1]

kpnkI≤nkp

1−x²pnkI.

Throughout this paperkpkK:=sup_x∈K|p(x)|stands for the usual sup norm on a compact setK⊂R^d. The meaning of the above inequality consists in estimating the uniform norm of the polynomial by the uniform norm of the weighted polynomial with the Chebyshev weightp

1−x². This upper bound is known to be sharp as it is attained for the Chebyshev polynomial of the second kind. Schur type inequalities proved to be rather useful in verifying Markov type inequalities for the derivatives of algebraic polynomials. (See[1], p.233 for the basic facts on the classical Schur inequality.) They can be also applied for estimating norms of factors of polynomials. Schur type inequalities have been generalized in two directions: by replacing the uniform norm by a weighted uniform norm and by using instead of Chebyshev weight more general weighted polynomials.

Mastroianni and Totik[6]proved a Schur type inequality with generalized Jacobi weights instead ofp

1−x²in case when the sup norm is endowed with a nonnegative weightwwhich satisfies the so calledA^∗property onI. This means that is there is a constantcw depending only onwsuch that for any intervalJ⊂Iand anyx∈J

w(x)≤ cw

λ(J) Z

J

w(t)d t. (1)

Here and in what followsλstands for the Lebesgue measure.

Then as shown in[6]given a generalized Jacobi type weight h(x):= Y

1≤j≤k

|x−x_j|^γj,γj>0 it follows that for any univariate algebraic polynomialp_nof degree at mostn

kw pnkI≤cn^γkwhpnkI, γ:=maxγ^∗j, (2) wherewis any weight satisfying theA^∗property andγ^∗_j=γjifx_j∈(−1, 1)andγ^∗_j=2γj ifx_j=1 or−1.

In the present note we will give the multivariate analogue of the above Schur type inequality for generalized Jacobi weights. These weights play an important role in the theory of multivariate orthogonal polynomials, see e.g., Y.Xu[7]. It turns out that in the multivariate setting this question is considerably more delicate since the corresponding upper bounds depend on the geometry of the zero sets (algebraic varieties) of the polynomial factors appearing in the weight.

aAlfréd Rényi Institute of Mathematics Hungarian Academy of Sciences and Budapest University of Technology and Economics, Department of Analysis,

Let us introduce now the generalized multivariate Jacobi type weights which will play a central part in our considerations.

As usual,P_n^dwill denote the space of real algebraic polynomials ofdvariables and degree at mostn. Then given any integers m_j∈N, positive real numbersαj >0 and algebraic polynomialsp_j∈P_m^d

j of exact degreem_j, 1≤j≤sthe generalized multivariate Jacobi weight on a convex bodyK⊂R^dis defined by

φ(x):=dist(x,∂K)^α Y

1≤j≤s

|pj(x)|^αj, α≥0. (3)

F. Dai[2]verified a Schur type inequality with generalized Jacobi weights for multivariate polynomials on the unit sphereS^d−1:={x∈R^d:|x|=1}in case whenmj=1, 1≤j≤sandα=0 in the above definition. Then it is verified in[2] that for a certain class of weightswand anyp_n∈P_n^d

kw pnkS^d−1≤cn^ξkwφpnkS^d−1, ξ:= X

1≤j≤s

αj. (4)

If one compares the univariate estimate (2) with the multivariate upper bound (4) then a surprising difference can immediately be noticed: in (2) the maximum of exponents controls the estimate while in (4) the considerably larger sum of the exponents appears in the bound. This raises the natural question: what is the sharp form of the multivariate Schur inequality with generalized Jacobi weights (3) on arbitrary convex bodiesK⊂R^d?

It turns out that in order to give a sharp multivariate Schur inequality for a generalized Jacobi weight (3) one has to take into account possible intersections of the algebraic varieties

H^∗_j :={x∈R^d:p_j(x) =0}, 1≤j≤s

which makes the problem quite delicate. We will need to study their intersection inside the given domain, so we also set Hj:=H^∗_j∩K, 1≤j≤s. In order to handle this difficulty and provide a measure of how fast the weightφgiven by (3) may vanish at a givenx∈K we will introduce now itszero index z(x)as

z(x):= X

{j:x∈Hj}

αjm_j, x∈IntK, z(x):=2α+2 X

{j:x∈Hj}

αjm_j, x∈∂K. (5)

Hence the zero index equals the sum of all properly weighted exponentsαjcorresponding to algebraic varieties intersecting at the given point. In addition, it is important that these exponents are doubled for boundary points. Now we introduce the zero index of the Jacobi type weight as the maximal point wise zero index

z_φ:=max

x∈K z(x). (6)

It turns out that this zero index leads to sharp Schur type inequalities for generalized Jacobi weights on any convex body.

We will verify this for weighted uniform norms withA^∗weightswsatisfying (1) forany segment J⊂K.

Theorem 1.Let K⊂R^d,d≥2be a convex body andφa generalized Jacobi weight (3) with zero index z_φ. Then for any pn∈P_n^dand A^∗weight w we have

kw p_nkK≤cn^zφkwφpnkK (7)

with some constant c>0depending only on K andφ.

When all setsH^∗_j havepair wise empty intersectioninKthen the zero indexz_φwill equal the maximum of corresponding weighted exponentsαjm_j leading to an estimate similar to the univariate case (2). If allH^∗_j-sintersect at a single point inKthenz_φ=P

1≤j≤sαjmjwhich is similar to the estimate (4). In Goetgheluck[3]a multivariate Schur type inequality was given under the condition that the weightφhas rather high order of smoothness. Since a generalized Jacobi weight (3) in general is not smooth the analytic method used in[3]does not work for general Jacobi weights. Another version of multivariate Schur type inequality can be found in[4], where the author gives some general upper bounds. However, the implicit estimates given in[4]are not sharp in general for Jacobi type weights, since they do not distinguish between interior and boundary points of the domain. A crucial point of our approach consists in providing precise bounds which are closely related to the intersection of the algebraic varietiesHj. The approach used in the present paper is a mix of refining some polynomial inequalities combined with geometric considerations.

The Schur type estimate (7) naturally leads to the question wether the zero indexz_φgives the correct rate of increase.

Our next result addresses this question. Letx₀∈Kbe a point in the domain where the zero index is attained, i.e.,z_φ=z(x0).

It will be shown below that ifx₀∈IntKorx₀∈∂Kis a so calledvertexpoint then estimate (7) can be reversed. Recall that x₀∈∂Kis called a vertex point if for some vectorh∈S^d−1normal to∂Katx₀we have with aγ >0 for anyx∈K

|x−x₀| ≤γ|〈x−x₀,h〉|. (8)

We will give the converse to Theorem 1 in the model case whenw=1 and the polynomials in (3) are linear.

Theorem 2. Consider a convex body K⊂R^d,d≥2and Jacobi weight (3) withdegp_j =1,∀j. Assume that one of the following conditions hold:

(i)the zero index z_φis attained at an interior point of K (ii)the zero index is attained at a boundary vertex point.

Then there exist polynomials gn∈P_n^dsuch that with some c>0independent of n kgnkK≥cn^zφkφgnkK, n∈N.

Theorem 2 provides nearly a complete converse to Theorem 1 except for the case when the zero index is attained only at asmooth boundary point. It turns out that there is an essential reason behind this fact because for convex bodies with smooth boundary the upper bound of Theorem 1 can be improved further.

Let us considerC²compact setsDsatisfying the so calleduniform interior ball conditionwhich means that everyx∈Dis contained in a ballB(a,rD)⊂DwithrD>0 depending only on the domain. In what followsB(a,r)will denote the closed ball centered ataand radiusr. Now we consider a generalized Jacobi weight with zero index defined as

ν(x):= Y

1≤j≤s

|p_j(x)|^αj, z_ν=max

x∈D

X

{j:x∈Hj}

αjm_j. (9)

Note that in contrast to (5)-(6) now we do not need to double the exponents at the boundary points. This is related to the fact that forC²compacts with the uniform interior ball condition the boundary points can be treated in the same manner as the interior points. However, we need to modify the definition (1) ofA^∗weights by assuming thatJ⊂Dis any circular arc contained in the domain. With this modifications we have the following analogue of Theorem 1.

Theorem 3.Let D⊂R^d,d≥2be a C²compact satisfying the uniform interior ball condition. Then for any pn∈P_n^dand A^∗ weight w we have

kw p_nkD≤cn^zνkwνpnkD

whereνis a Jacobi type weight (9) and constant c>0depends only on D andφ.

Similarly to Theorem 2 the upper bound of Theorem 3 can be reversed in case whenw=1 and the polynomials in (9) are linear.

Above results also have interesting implications for estimating norms of factors of multivariate polynomials. Let us formulate a typical corollary of this type.

Given a multivariate polynomialqwhich factors into the product of polynomialsq=q₁·...·qswith zero setsHj:={x∈ K:q_j(x) =0}let us consider the zero index ofqrelative to the convex bodyK⊂R^dgiven by

zq:=max

x∈K(τx

X

{j:x∈Hj}

degqj),

whereτx=1 ifx∈IntKandτx=2 ifx∈∂K. It should be noted that the zero indexz_qof polynomialqcan be considerably smaller than its degree. Then for anypn∈P_n^dwe have

kpnkK≤cqn^zqkqpnkK.

In the next section we will verify the auxiliary analytic and geometric results needed in the sequel. First two lemmas provide some refinements of the univariate Schur and Polya inequalities. Lemma 3 presents some crucial information on intersection of algebraic varieties, while Lemma 4 is related to the geometry of convex bodies. Then in Section 3 the proof of the main new results will be given.

2 Auxiliary results

First we will need a refinement of the univariate Schur type inequality. This refinement will involve themonotone rearrange- mentof the weightφwith respect to theChebyshev measuredefined as

µ(E):=

Z

E

pd x 1−x². Letφ(x)≤1 be positive a.e. on[−1, 1]and for anyδ >0 denote

ψ(δ):=sup{c>0 :µ({x∈[−1, 1]:φ(x)≤c})≤δ}.

In the above definition of monotone rearrangement Chebyshev measure is needed in order to handle properly the end points of the interval.

Lemma 1.For any A^∗weight w and pn∈P_n¹

kw pnkI≤ c

ψ(1/n)kwφpnkI.

Proof.Setg(t):=pn(cost). The weightω(t):=w(cost)is also anA^∗weight (see[6], p.68 for details). Furthermore, by the above definition of functionψfor any 0<a< ψ(1/n)

λ(t∈[0,π]:φ(cost)≤a) =µ(x∈[−1, 1]:φ(x)≤a)≤1 n.

We will use now a Remez type inequality forA^∗weights proved in[6], p.60 which states that for any even trigonometric polynomialt_nof degreenand anyE⊂[0,π]with Lebesque measureλ(E)≥π−¹_n, we have

kωtnk_[0,π]≤ckωtnkE, wherec>is an absolute constant.

Hence setting

E:={t∈[0,π]:φ(cost)>a}, λ(E)≥π−1 n yields

kw pk_[−1,1]=kωgk_[0,π]≤ckωgkE≤c

akφ(cost)ω(t)g(t)kE≤c

akwφpk_[−1,1]. ƒ

The next lemma is a refinement of the well known Polya inequality stating that for any univariate monic algebraic polynomialq_mof degreem

λ{t∈[0, 1]:|qm(t)| ≤δ} ≤4δ^m1, 0< δ <1.

We will need an analogue of this inequality for the Chebyshev measure defined above.

Lemma 2.Let q(t) =|t²−1|^αQ

1≤j≤s|t−t_j|^αj,αj>0,α≥0,γ:=α+P

1≤j≤sαj.Then

µ{t∈[−1, 1]:q(t)≤δ} ≤8δ^21γ. (10)

Moreover, if tj∈[−1+a, 1−a]with some0<a<1, 1≤j≤s then settingγ₁:=max{2α,P

1≤j≤sαj}we have

µ{t∈[−1, 1]:q(t)≤δ}=O(δ^1/γ1), (11)

with a constant in the O(.)term depending only on a,α,andαj-s.

Proof.Assume first that eachαjis an integer. Then setting q₁(t):= (t+1)^α Y

1≤j≤s

(t−t_j)^αj, q₂(t):= (t−1)^α Y

1≤j≤s

(t−t_j)^αj

it follows that bothq₁andq₂are monic algebraic polynomials of degreeγ. Thus applying the Polya inequality separately on intervals[−1, 0]and[0, 1]toq₁andq₂, respectively we have

µ{t∈[−1, 1]:q(t)≤δ} ≤2λ{t∈[−1, 1]:q(t)≤δ}¹²

≤2λ{t∈[0, 1]:|q2(t)| ≤δ}¹² +2λ{t∈[−1, 0]:|q1(t)| ≤δ}¹² ≤8δ^2γ1 which is the first estimate of the lemma in case of integer exponents.

Ifαj∈Qwe can considerq^m(t)with a proper integermin order to have only integer exponents. Then using the above estimate withq^m(t)andδ^minstead ofδleads to the same upper bound. Finally, the case of real positive exponents then follows from rational case by continuity.

In order to prove the second estimate we split the interval[−1, 1]into three parts I₁:= [1−a/2, 1],I₂:= [−1,−1+a/2],I₃:= [−1+a/2, 1−a/2], and considerq₃(t):=Q

1≤j≤s(t−tj)^αj,γ2:=P

1≤j≤sαj. As above it suffices to verify the case for integer exponents. Now applying the Polya inequality toq₃on the intervalI₃yields with some constantsc_jdepending only ona,α, andαj-s

µ{t∈[−1, 1]:q(t)≤δ} ≤2µ{t∈I₁:|t−1|^α≤c₁δ}+c₂λ{t∈I₃:|q3(t)| ≤c₃δ} ≤c₄δ^1/γ1. ƒ

Next we present an auxiliary geometric proposition related to intersections of the algebraic varietiesH^∗_j which will be crucial in the proof of Theorem 1.

Lemma 3.Let K andφbe as in Theorem 1. Then there exists aδ0=δ0(K,φ)so that for every0< δ≤δ0,andx∈K such that

Ω(x,δ):={j:|pj(x)|< δ} 6=; we have

∩j∈Ω(x,δ)Hj6=;.

Proof.Assume the contrary. Then∃δk→0⁺andxk∈K such thatΩ(xk,δk)6=;,k∈Nbut at the same time

∩j∈Ω(xk,δk)H_j=;.

SinceKis compact we can assume without the loss of generality thatx_k→x^∗∈K. Furthermore, sinceΩ(x_k,δk)6=;there exist j_k∈Ω(xk,δk), 1≤j_k≤s,k∈N. Again without the loss of generality it can be assumed that for any 1≤j≤r, 1≤r<s index jbelongs only to finitely many ofΩ(xk,δk)-s, while for allr+1≤ j≤sthis index belongs to infinitely many of Ω(x_k,δk)-s. Hence there existqj so that for anyk≥qj

j∈/Ω(xk,δk), 1≤j≤r.

Then settingq:=max{qj, 1≤j≤r}it follows that for anyk≥q

Ω(xk,δk)⊂ {r+1,r+2, ...,s}. (12)

Moreover, using that wheneverr+1≤j≤sthe index jbelongs to infinitely many ofΩ(xk,δk)-s we obtain that relations

|pj(xk)|< δkalso hold for this jwith infinitely manyk∈N. Sinceδk→0 andxk→x^∗∈Kthis means thatpj(x^∗) =0, i.e., x^∗∈Hj,r+1≤j≤s. Thus in view of (12) we have

x^∗∈ ∩r+1≤j≤sHj⊂ ∩_{j∈Ω(xk,δk)}Hj

contradicting our assumption that∩_j∈Ω(x_k,δk)H_j=;.ƒ

Let us denote byS(a,r):={u∈S^d−1:|a−u| ≤r},a∈S^d−1,r>0, the sphere cap inR^dwith centeraand radiusr. Lemma 4. Let K ⊂R^dbe a convex body,B(0,r)⊂K ⊂B(0, 1),r>0. For anyy∈K andu∈S(y₁,r/2),y₁:=y/|y| consider the line l:={y+tu:t∈R}.Then setting l∩K:= [a,b]

|b−a| ≥p

3r, dist(y,∂K)≥ r

4min{|b−y|,|y−a|}. (13)

Proof.Sinceu∈S(y₁,r/2)it follows that ^r₄² ≥ u−y₁

2=2 1− 〈u,y₁〉

. Since|y| ≤1 this yields dist(l,0) =Æ

|y|²− 〈u,y〉²≤Æ

2(|y| − 〈u,y〉)≤r 2.

Clearly this means thatB(0,r/2)∩l6=;. Recalling thatB(0,r)⊂K we easily obtain that|l∩K| ≥p

3r which is the first estimate in (13).

Now we proceed by verifying the second estimate in (13). As shown aboveB(0,r/2)∩l6=;, i.e., we can consider a pointA∈B(0,r/2)∩l. SinceB(0,r)⊂K it follows thatB(A,r/2)⊂K. In addition,A,y∈[a,b], hence we may assume without the loss of generality thaty∈[A,b]. Now setB₀:=B(A,r/2)∩ {w:w−A⊥u}. ThusB₀⊂Kis ad−1 dimensional ball in the hyper plane{w:w−A⊥u}centered atAand of radiusr/2. Then by the convexity ofKthe circular coneQ with vertex atb∈Kand baseB₀is also contained inKand pointybelongs to the axis of this cone. Therefore using that diamK≤2 we have dist(y,∂K)≥dist(y,∂Q) =₄^r|b−y|.ƒ

3 Proof of the main new results

Proof of Theorem 1. We may assume thatB(0,r)⊂K ⊂B(0, 1), 0<r≤1. Since each polynomialpjin (3) is of exact degreem_jit follows that itsm_j-s homogeneous part denoted byh_jis a homogeneous polynomial of degreem_j, 1≤j≤s.

LetH^∗:=∪_1≤j≤s{x∈R^d:hj(x) =0}be the union of all zero sets of these homogeneous polynomials. ClearlyH^∗is nowhere dense inS^d−1. In what follows for any givenε >0 andD⊂R^dletD(ε):=∪x∈DB(x,ε)denote theεenlargement ofD. Then we can choose a propera>0 so thatS(z,r/2)\H^∗(a)6=;, for anyz∈S^d−1and hence

|hj(u)| ≥ξ, ∀u∈S(z,r/2)\H^∗(a), 1≤j≤s (14) with someξdepending only onKandφ.

Now take anyp_n∈P_n^dand lety∈K be such thatkw p_nkK=|w(y)pn(y)|. For anyu∈S(y/|y|,r/2)\H^∗(a)consider the line

l:={y+tu:t∈R}, l∩K:= [y+alu,y+blu],al≤0≤bl.

Thenpj(y+tu), 1≤j≤sare univariate polynomials of variabletof degreemj. Lettk j∈C, 1≤k≤mj, 1≤j≤sdenote all zeros ofp_j(y+tu).

Then for any fixedal≤t≤blwe have by (3), (14) and (13) φ(t):=φ(y+tu) =dist(y+tu,∂K)^α Y

1≤j≤s

|pj(y+tu)|^αj≥c(r,α)|t−al|^α|t−bl|^αξ^Pαjmj Y

1≤j≤s

|gj(t)|^αj, wheregj∈P_m¹

j are themonicunivariate polynomials corresponding topj(y+tu), 1≤j≤s.

Furthermore, denote byMthe maximum of all directional derivatives ofpj, 1≤j≤sinK(1). Now withδ0from Lemma 3 and any fixeda_l≤t≤b_l set

Ωt:={j: min

1≤k≤m_j|t−t_{k j}| ≤M⁻¹δ0}. (15)

.

Forj=1, ...,swe denote byt^∗_j the zero ofgj closest totand we have

|t−t^∗_j|= min

1≤k≤m_j|t−t_{k j}|. Then by the previous estimate

φ(t)≥c₁|t−al|^α|t−bl|^αY

j∈Ωt

|t−t^∗_j|^αjmjY

j∈Ω/ t

|t−t^∗_j|^αjmj.

Here and in the remaining part of the proofcj denote positive constants depending only onKandφ. Evidently, it follows from (15) that for any j∈/Ωtwe have|t−t^∗_j|>M⁻¹δ0. Thus we obtain from the last lower bound withℜt^∗j being the real part of the corresponding zero

φ(t)≥c₂|t−al|^α|t−bl|^αY

j∈Ωt

|t−t^∗_j|^αjmj≥c₂|t−al|^α|t−bl|^αY

j∈Ωt

|t− ℜt^∗j|^αjmj. (16) Clearly, we have by the mean value theorem for everyj∈Ωt

|pj(y+tu)|=|pj(y+tu)−pj(y+t^∗_ju)| ≤M|t−t^∗_j| ≤δ0. This means thatΩt⊂Ω(y+tu,δ0). Thus by Lemma 3

∩j∈ΩtHj6=;.

We will distinguish between the cases when the set∩j∈ΩtHj contains some boundary points ofK, or is completely embedded into IntK.

Case 1.∩j∈ΩtH_j∩∂K6=;, i.e., for somex∈∂Kwe havex∈ ∩j∈ΩtH_j. Then by the definition of the zero indices (5) and (6) it follows that

z_φ≥z(x) =2α+2X

x∈Hj

αj≥2α+2X

j∈Ωt

αj. (17)

Now let us denote byQthe set of all generalized algebraic polynomialsqof the form q(t) =|t−al|^α|t−bl|^α

m

Y

j=1

|t− ℜtj|^αj, m≤s, (18) with eachtj chosen arbitrarily from the set{tk j, 1≤k≤mj}andα,αj-s satisfying (17). ClearlyQis a finite set with some cardinalityNdepending only onmj, 1≤j≤s. Thus in view of (16) we obtain that

µ{t∈[al,bl]:φ(t)≤δ} ≤Nµ{t∈[al,bl]:q(t)≤δ/c2}, (19) whereq(t)is any generalized polynomial of the form (18) with exponents satisfying (17), i.e.,

γ:=α+

m

X

j=1

αj≤z_φ/2.

Now we are going to apply estimate (10) of Lemma 2 to the generalized polynomialq(t)on the interval[al,b_l]instead of [−1, 1]. Note that by Lemma 4 we have 2≥ |a_l−b_l| ≥p

3r, i.e, the transition to this segment can alter the outcome only by a constant factor. Hence by (19) and (10)

µ{t∈[al,b_l]:φ(t)≤δ} ≤Nµ{t∈[al,b_l]:q(t)≤δ/c2} ≤c₃δ^21γ≤c₃δ^z1φ. (20) Now consider the univariate algebraic polynomialgn(t):=pn(y+tu)∈P_n¹,t∈[al,bl]whereal≤0≤bl,|bl−al| ≥p

3r andkw gnk_[a_l,bl]=|w(y)pn(y)|=kw pnkK. Note that by (20) and definition of monotone rearrangementψofφwe have ψ(δ)≥c₄δ^zφ. Now we can apply Lemma 1 togn∈P_n¹(since|bl−al| ≥p

3rwe can obviously transform[al,bl]to[−1, 1]) yielding

kw pnkK=kw gnk_[a_l,b_l]≤ c

ψ(1/n)kwφgnk_[a_l,b_l]≤c(K,φ)n^zφkwφpkK

which is the needed estimate.

Case 2.∩j∈ΩtHj⊂IntK. In this case using the definition of the zero indices (5) and (6) and the obvious relationz_φ>2α it follows that

z_φ≥max{2α,X

j∈Ωt

αj}. (21)

Consider the quantity

Q₁(δ):= min

H_i1∩...∩H_ir∩∂K=;dist(∂K,H_i^∗

1(δ)∩...∩H^∗_i

r(δ)).

We clearly have thatQ₁(0)>0 and hence using the continuity of functions involved it follows thatQ(δ)>0 for anyδ >0 sufficiently small. Let us choose such aδ1>0, where it can be assumed without the loss of generality that 2δ0<Q₁(δ1) withδ0being the quantity from Lemma 3. Then using that∩j∈ΩtHj∩∂K=;we obtain that for anyx∈ ∩j∈ΩtH^∗_j(δ1)

dist(x,∂K)≥Q₁(δ1)>2δ0. (22)

Consider nowy+t^∗_ju,y+t_i^∗u,i,j∈Ωt, where by (15) we have|t−t^∗_j|,|t−t_i^∗| ≤M⁻¹δ0. Then

|pi(y+ℜt^∗ju)|=|pi(y+ℜt^∗ju)−pi(y+t^∗_iu)| ≤M(|ℜt^∗j−t|+|t^∗i−t|)≤2δ0.

This upper bound clearly means thaty+ℜt^∗jumust be in the vicinity ofH_i^∗forδ0small enough, i.e., we may assume without the loss of generality thatδ0is chosen sufficiently small so that y+ℜt^∗_ju ∈ H^∗_i(δ1),∀i,j∈ Ωt. Then clearly y+ℜt^∗_ju∈ ∩_i∈ΩtH_i^∗(δ1), and therefore (22) yields

min{|ℜt^∗_j−al|,|ℜt^∗_j −bl|} ≥dist(y+ℜt^∗_ju,∂K)≥Q₁(δ1)>2δ0. In addition, by (15)|ℜt^∗_j−t| ≤δ0. Sincet∈[al,bl]the last two estimates obviously yield

ℜt^∗_j∈[al+2δ0,bl−2δ0], j∈Ωt. (23) Now similarly to Case 1 we can obtain the upper bound (19) whereqis a generalized algebraic polynomial (18) whose exponents by (21) satisfy relation

max{2α, X

1≤j≤m

αj} ≤z_φ. (24)

Another crucial information concerningqconsists in the fact that in view of relation (23) all its zeros belong to[al+2δ0,bl− 2δ0]. Hence we are in position to apply now the second estimate (11) of Lemma 2 yielding together with (24)

µ{t∈[al,bl]:φ(t)≤δ} ≤Nµ{t∈[al,bl]:q(t)≤cδ}=O(δ^z1φ).

Therefore, similarly to Case 1 this yields for the monotone rearrangementψofφestimateψ(δ)≥cδ^zφ. This means that we can finish the proof now analogously to Case 1.ƒ

Proof of Theorem 2. We may assume that diamK ≤1. Consider first the case when the zero index is attained at a boundary vertex pointx₀∈∂K satisfying (8) for a certain outer normalhatx₀. Then

z_φ=z(x₀) =2α+2 X

x0∈H_j

αj.

Note that 0≤ 〈x₀−x,h〉 ≤diamK≤1,∀x∈K. Moreover, wheneverx₀∈Hj it follows by (8) that

|〈x−c_j,h_j〉|=|〈x−x₀,h_j〉| ≤ |x−x₀| ≤γ|〈x−x₀,h〉|, dist(x,∂K)≤ |x−x₀| ≤γ|〈x−x₀,h〉|.

Now we will need a result from[5](see Proposition 1 on p. 84) according to which given anyρ >0 there exist univariate polynomialsqn∈P_n¹such that

x^ρ|qn(x)| ≤1,x∈[0, 1], |qn(0)| ≥cn^2ρ,n∈N. (25) Setρ:=^z₂^φ. Consider the polynomialp(x):=qn(〈x−x₀,h〉)∈P_n^d. Then using above estimates together with (25) we obtain for anyx∈K

φ(x)|p(x)|=dist(x,∂K)^α Y

1≤j≤s

|〈x−c_j,h_j〉|^αj|p(x)|

≤c₁|〈x−x₀,h〉|^α Y

x0∈H_j

|〈x−x₀,h〉|^αj|p(x)|=c₁|〈x−x₀,h〉|^ρ|qn(〈x−x₀,h〉)| ≤c₁, withc₁>0 independent ofn. On the other hand (25) also yields

|p(x₀)|=|qn(0)| ≥cn^2ρ=cn^zφ.

This is the required lower bound in the case when the zero index is attained at a boundary vertex point.

Now assume that the zero index is attained at an interior pointx₀∈IntK. Hence by (5) z_φ=z(x0) = X

x0∈Hj

αj. Setp(x):=qn(|x−x₀|²)∈P_2n^d. Then using again (25) we have for anyx∈K

φ(x)|p(x)|=dist(x,∂K)^α Y

1≤j≤s

|〈x−x₀,hj〉|^αj|p(x)| ≤ Y

x0∈Hj

|〈x−x₀,hj〉|^αj|p(x)|

≤ |x−x₀|^zφ|p(x)|=|x−x₀|^2ρ|q_n(|x−x₀|²)| ≤1 and

|p(x0)|=|q_n(0)| ≥cn^2ρ=cn^zφ. This completes the proof of Theorem 2.

The proof of Theorem 3 can be given quite similarly to the proof of Theorems 1, in fact it follows the same arguments with considerable simplifications due to the absence of the boundary difficulties. Since for aC²compactDsatisfying uniform interior ball condition any pointx∈Dlies on a sphere of radiusrDimbedded into the domain it suffices to consider the case whenD=S^d−1is the unit sphere. Now we can essentially repeat the arguments of Theorem 1 by modifying Lemma 1 for monotone rearrangements based on circular Lebesgue measure instead of the Chebyshev measure, and considering circular arcs passing throughx, we omit the details.

Acknowledgments: Supported by the OTKA Grant K111742. Written during the author’s visit at the Department of Mathematics of University of Padova.

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