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Journal of Approximation Theory 267 (2021) 105593
www.elsevier.com/locate/jat
Full Length Article
Sharp L p Bernstein type inequality for cuspidal domains in R d
András Kroó
1,∗Alfréd Rényi Institute of Mathematics, Hungary
Budapest University of Technology and Economics, Department of Analysis, Budapest, Hungary Received 14 February 2021; received in revised form 9 April 2021; accepted 4 May 2021
Available online 7 May 2021 Communicated by V. Totik
Abstract
In this paper we verify a sharp Bernstein type inequality for multivariate algebraic polynomials inLp norm on general cuspidal domains which is based on a proper measure of the distance to the boundary of the domain. In particular, in case of Lipγ cuspidal graph domains the exact rate is given by n
2γ−1 . c
⃝2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
MSC:41A17; 41A63
Keywords:Multivariate polynomials; Cuspidal sets;Lpnorm; Bernstein–Markov inequality
Introduction
The main goal of this paper is to provide a new sharp Bernstein type inequality in Lp norm for general cuspidal domains in Rd. The univariate Lp,1 ≤ p ≤ ∞ Markov type inequality for algebraic polynomialsq of degree at mostn states that
∥q′∥Lp[a,b]≤ cpn2
b−a∥q∥Lp[a,b].
Above upper bound was first verified by A.A. Markov when p= ∞with the sharp numerical constant c∞ = 2. This inequality and its various extensions play a central role in various
∗ Correspondence to: Alfr´ed R´enyi Institute of Mathematics, Hungary.
E-mail address: kroo.andras@renyi.mta.hu.
1 Supported by the NKFIH-OTKA, Hungary Grant K128922.
https://doi.org/10.1016/j.jat.2021.105593
0021-9045/ c⃝ 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
approximation theoretical problems. Subsequently Markov’s classical estimate was extended for any p>0 and weighted Lp norms.
Given any function f :Rd →Rk we denote by
∥f∥Lp w(K):=
(∫
K
|f|pw )1/p
its weightedLp norm on K ⊂Rd with weightw. Whenw=1 the notation ∥f∥Lp(K) is used below. Let us recall now certain weighted Markov and Bernstein type inequalities given in [3], p. 407 for Jacobi type weights w of degreem (seeDefinition 2). It is shown therein that for any such weight and algebraic polynomialq of degree≤n with we have
∥q′∥Lp
w[−1,1]≤cp,mn2∥q∥Lp
w[−1,1]. (1)
On the other hand for trigonometric polynomials t of degree ≤ n the next Bernstein type inequality holds:
∥t′∥Lp
w[−π,π]≤cp,mn∥t∥Lp
w[−π,π], p>0. (2)
The constants cp,m above depend only on p and m. It is remarkable, that the order n2 of derivatives in (1) in algebraic case reduces to n in trigonometric case (2). The standard trigonometric substitution x=cost transforms(2) into Bernstein type inequality
∥√
1−x2q′∥Lp
w[−1,1]≤cp,mn∥q∥Lp
w[−1,1], p>0 (3)
withq being any algebraic polynomials of degree≤n. Thus introduction of a weight
√ 1−x2 into the derivative norms reduces their size by a factor of n. This phenomenon plays a significant role in numerous approximation theory results and applications. The upper bounds (1) and (3) will be used simultaneously on multivariate domains so it will be convenient for us to combine (1) and (3) into a single Bernstein–Markov type inequality for algebraic polynomialsq of degreen and p>0
∥(n−1+√
1−x2)q′∥Lp
w[−1,1]≤cp,mn∥q∥Lp
w[−1,1]. (4)
Here as abovewis a Jacobi type weight of degreem. It should be noted that in [9] Bernstein–
Markov type inequality(4)is shown to hold for a more general class of the so called doubling weights. However for multivariate extensions it will be more suitable to use Jacobi type weights.
Lp Bernstein type inequality for general cuspidal domains
Now we turn our attention to the multivariate case and the space Pnd of real algebraic polynomials ofd variables and degree at most n. It is well known that for Lip1 and thus, in particular convex domains the upper bound of ordern2 forLp,0< p≤ ∞Markov inequality extends for multivariate polynomials, as well, see e.g., [6,10]. Similarly, order n is preserved inLp, 0<p≤ ∞Bernstein type inequalities on convex and Lip1 domains if analogously to (3)we insert the square root of the Hausdorff distance to the boundary of the domain into the derivative norm. However, when the underlying domain is cuspidal the order of magnitude can change drastically. In particular, for Lipγ, 0< γ <1 cuspidal domains inRd the sharp order inL∞Markov inequality is known to benγ2 (see for instance, [1,4,8]). Typically multivariate Markov-type inequalities inL∞norm are proved by inscribing suitable polynomial curves into the domain and reducing the problem to the univariate setting on these curves. In case of
2
Lp,0< p<∞norm this reduction of dimension technique is not as straightforward and the corresponding problem becomes more complex.
The multivariate Bernstein problem will be considered below on the so called graph domains. For anyr >0,a ∈Rd andu∈ Sd−1:= {x∈Rd : |x| =1}the cylinder La(r,u) of radiusr >0, centera and axisu is given by
Lr(a,u):= {x∈Rd : |x−a|2<r2+ ⟨x−a,u⟩2}.
Furthermore,lx(u) will denote the line inRd in directionu∈Sd−1through point x∈Rd. Definition 1. K is called agraph domain with respect to the cylinder Lr(a,u) if for every x ∈ Lr(a,u) we have that lx(u)∩K =[d1(x),d2(x)] is a finite segment withdi(x),i =1,2 being continuous forx∈Lr(a,u) and
δr(a,u):= inf
x∈Lr(a,u)
|d1(x)−d2(x)|>0. (5) Moreover, K ⊂ Rd is a piecewise graph domain if it can be covered by finite number of cylinders so that K is a graph domain with respect to each of them.
When K ⊂ Rd is a piecewise graph domain the smoothness of its boundary denoted by BdK is related to the moduli of continuity of all functionsdi(x) specified above. Recall that the modulus of continuity of F ∈C(D) is given by
ωF(t):=max{|F(x)−F(y)| :x,y∈ D, |x−y| ≤t}, t >0.
Let us denote byωK(t) the modulus of continuity of the boundary of piecewise graph domain K which is defined as the maximum of moduli of continuity of all functionsdi(x) involved in the corresponding finite covering by cylinders. Forcuspidal domains we have that ωKt(t) →
∞,t →0. If ωK(t)=O(tγ),0 < γ ≤1 then the piecewise graph domain K is called Lipγ. A model example of Lipγ piecewise graph domain is provided by thelγ ball inRd defined by
Bγd := {x∈Rd : |x1|γ + · · · + |xd|γ ≤1}.
In order to avoid superfluous technical details we will always assume below that ωK(t) is concave and strictly increasing. That is we replace it if needed by its smallest concave increasing upper bound. It is well known that this does not alter the rate of convergence to zero of the modulus of continuity.
Bernstein type inequalities involve bounds for weighted derivatives of polynomials with weights vanishing at the boundary of the domain. As can be seen from (3) in the classical univariate case this decreases the size of the derivatives for polynomials of degreenby a factor ofn. In order to achieve similar phenomena in the multivariate case it is crucial to find a proper measure of thedistance to the boundaryof the domain. A natural candidate for measuring the distance to the boundary of K is the quantity
hK(x):=inf{|x−y| :y∈Bd K}, x∈K.
However it turned out that this standard measure of the distance to the boundary does not lead to sharp upper bounds in case of cuspidal domains. Therefore we will introduce now an alternate radial distance to the boundary for cuspidal graph domains.
If K is a graph domain with respect to the cylinder L := Lr(a,u) with corresponding continuous boundary functionsdi(x),i =1,2 then for anyx∈ K∩Lr(a,u) we set
ρL(x):=min{|x−d1(x)|,|x−d2(x)|}.
3
Furthermore when K ⊂Rd is a piecewise graph domain covered by the cylindersLj,1≤ j ≤ m then we can define the distance to the boundary of K by
ρK(x):= min
1≤j≤mρLj(x).
EvidentlyρK(x)≥hK(x),x∈K. It can be easily seen for instance in case oflγ balls Bγd that the standard distancehK(x) can be substantially smaller thanρK(x) in the vicinity of a cusp. It turns out that for cuspidal domainsρK(x) is the proper measure of the distance to the boundary which yields sharp Bernstein type inequalities.
Letϵ:=ϵn(K) be the solution of the equation 2n2ωK
(ϵ n2
)
=1, n∈N. (6)
It is easy to see thatϵn(K)→0,n → ∞for cuspidal piecewise graph domains.
Let ∂f stand for the gradient of f : Rd → R. Furthermore, Duf = ⟨∂f,u⟩ denotes the derivative of f in directionu∈Rd. We will use the notationc...for various positive constants depending only on the quantities specified in the subscript. Recently in [7] a sharp Markov type inequality for algebraic polynomials in Lp norm on piecewise graph cuspidal domains was verified. Namely, as shown in [7] for any cuspidal piecewise graph domain K ⊂Rd and anyq ∈ Pnd
∥∂q∥Lp(K)≤cK,p
n2
ϵn(K)∥q∥Lp(K), n∈N.
In particular, if K is Lipγ, 0< γ <1 this yields
∥∂q∥Lp(K)≤cK,pnγ2∥q∥Lp(K).
It was also shown in [7] that for a wide family of Lipγ, 0< γ <1 graph domains the above upper boundnγ2 is attained. (Note that in a recent paper [2] the sharp boundn4 is verified for the specific Lip12 domain{|x| ≤y+1,x2≥4y} ⊂R2.)
The main result of the present paper establishes a corresponding sharp Bernstein type inequality for algebraic polynomials in Lp metric on piecewise graph cuspidal domains. This Bernstein type inequality will be based on applying the distance function ρK(x) introduced above. In addition, our results will be derived in a more general setting ofLp norms endowed with multivariate Jacobi type weights. In order to introduce Jacobi type weights we will need generalized algebraic polynomials of degreemwhich are defined as functions of the form
g(x)=c ∏
1≤j≤k
|x−xj|rj, rj >0,c∈R, m:= ∑
1≤j≤k
rj.
Definition 2.LetK ⊂Rd,d ≥1 be any compact set. Given a nonnegative weightwonK and m >0 we will say thatw is Jacobi type weight of degree m if there exist positive constants a,b depending only on K so that for every line l(t) = ut +a ⊂ Rd,t ∈ R there exists a generalized algebraic polynomialg(t) of degree at mostm such that
ag(t)≤w(l(t))≤bg(t), ∀l(t)∈ K. (7)
Now let us present the main theorem of this paper providing a sharp Lp Bernstein type inequality for general cuspidal domains.
4
Theorem 1. Let K ⊂ Rd be a cuspidal piecewise graph domain and consider a Jacobi type weight w ∈ Lp(K),0 < p < ∞. Then for any q ∈ Pnd,n ∈ N we have with some c=cK,p,w>0
∥√
ρK(x)∂q∥Lp
w(K) ≤ cn ϵn(K)∥q∥Lp
w(K). In particular, if K is Lipγ, 0< γ <1 then
∥√
ρK(x)∂q∥Lp
w(K) ≤cnγ2−1∥q∥Lp w(K).
Thus similarly to the classical univariate results theLp Bernstein type estimates for general cuspidal domains exhibit a decrease by a factor ofnin comparison to the corresponding Markov type bounds given above.
Theorem 1gives in general, the best possible Lp Bernstein type upper bound for cuspidal graph domains. Indeed, as shown below it is attained for a large family of cuspidal domains. But first we should note that a certain natural restriction on the magnitude of modulus of continuity wKis needed in order to verify lower bounds. Consider for instance the modulusw(t)= 1
log(1/t). Then clearly the solution of Eq.(6)given byϵn =n2e−2n2 will converge to zero exponentially fast asn → ∞. On the other hand it is known (see [5] for details) that for compact domains with analytic parametrization (which include in particular all graph domains)Lp Markov type derivative estimates for multivariate polynomials have sub exponential magnitude. Thus we need to exclude from the consideration moduli wK(t) for which solutions of(6) tend to zero exponentially. This will be accomplished by assuming thatt−swK(t) is an increasing function for somes>0.
So let us consider a model cuspidal cylinder given by
D:= {x=(x1, . . . ,xd): |x1| +w∗(|x2|)≤1,0≤xj ≤1,3≤ j≤d} ⊂Rd (8) where 0 < w∗(t) < 1,t > 0 is a given positive concave continuous function such that t−sw∗(t),t>0 is an increasing function for somes>0. Evidently,wD(t)=w∗(t).
Theorem 2. Let D⊂Rd be the cuspidal graph domain given by (8)withw∗(t),t >0being a positive concave continuous function such that t−sw∗(t),t >0is an increasing function for some s >0. Then there exists qn ∈Pnd,n∈Nsatisfying
∥√
ρD(x)∂qn∥Lp(D) > cn
ϵn(D)∥qn∥Lp(D), 0< p<∞ whereϵn(D)is the solution of(6) with abovew∗ and c=cd,p.
Thus above theorem provides a matching lower bound toTheorem 1in case of the cuspidal cylinderDandw∗(t) such thatt−sw∗(t),t>0 is increasing for somes>0. It should be noted that the proof ofTheorem 2given below does not work if we use the standard distance function hK(x) instead ofρK(x). This is related to the fact that for cuspidal domains (for instance D given by (8)) the quantity hD(x) can be substantially smaller thanρD(x) in the vicinity of a cusp. On the other hand for Lip 1 domains with wK(t) = O(t) the two distances hK(x) and ρK(x) are asymptotically equal.
Proofs
First we need to verify a crucial lemma stating that for cuspidal graph domains Bernstein type estimates similar to (3) remain valid after certain small perturbations of lines. It should
5
be noted that lines may intersect cuspidal domains along disconnected linear sets making the application of univariate Bernstein–Markov inequalities problematic. But as shown in [7]
appropriate small perturbations of lines make it possible to apply Remez type inequalities and thus reduce the considerations to line segments. In addition, since we are dealing with Bernstein type problem we also need to provide a delicate analysis of how these small perturbations affect the size of the distance functionρK.
Lemma 1. Let K ⊂Rd be a graph domain relative to the cylinder L :=Lr(a,u)so that(5) holds. Consider a Jacobi type weight w on K . Given any 0 < ρ < r and w ∈ Sd−1,w⊥u setun =u+ϵnw, withϵn :=ϵn(K)being a solution of(6). Then for any line ln :=l(un)⊂ Lρ(a,un)
∫
K∩ln
(√
ρL(x)|Dunq|)pw≤cK,p,wnp
∫
K∩ln
|q|pw, q∈ Pnd. (9) Proof. The proof can be conducted in the 2-dimensional plane containingl(un) and w. Thus we may assume that u = (0,1),w = (1,0) and hence un = (ϵn,1). Let [an,bn] ⊂ ln be the smallest segment on this line so that K ∩ln ⊂ [an,bn] (an,bn ∈ Bd K). Since ln ⊂ Lρ(a,un) for some 0 < ρ < r and ϵn → 0,n → ∞ it follows that for n large enoughan,bn ∈ Bd K∩Lr(a,u). Using(5)this implies that|an−bn| ≥δr(a,u)
2 for sufficiently large n. Therefore we may assume without loss of generality that an = −bn = (ϵn,1). Set ln = {z(t):=tun,t ∈R}, In :=[z(t),|t| ≤1−n−2] and Jn:=[z(t),|t| ≤1+n−2]. Then it is verified in [7], p. 4 that
In ⊂(K ∩ln)⊂[−an,an]⊂Jn, n ≥n0. (10) This shows that the set K ∩ ln contained in the segment [−an,an] differs from it by an exceptional set of linear measure at most 4n−2√
1+ϵn2 =O(n−2). Furthermore letm be the degree of the Jacobi type weightw and consider the generalized algebraic polynomialg(t) of degree at mostm for which relations(7) hold. Then it follows that
ag(t)|q(z(t))|p≤w(z(t))|q(z(t))|p ≤bg(t)|q(z(t))|p, ∀z(t)∈ K. (11) Now we need to recall an Lp Remez type inequality for generalized algebraic polynomials given in [3], Theorem A.4.10. which states that for every generalized algebraic polynomial f of degree N and every set A ⊂[−1,1] of Lebesgue measure at mosts ≤1/2 we have with an absolute constantc
∫
[−1,1]
|f|p≤(1+ecp N
√s)
∫
[−1,1]\A
|f|p, p>0. (12) Applying the above estimate for the generalized algebraic polynomial f(t):=g(t)1/p|q(z(t))|
of degree N ≤ n+m/p ≤cp,mn on the interval [−an,an], |an| ≥1 with A:= [−an,an]\ (K∩ln) ands=O(n−2) we obtain
∫
[−an,an]
|q(z(t))|pg(t)dt ≤cK,w,p
∫
K∩ln
|q(z(t))|pg(t)dt. (13) Let us now estimate the size ofρL(z(t)) wheneverz(t)∈K∩ln,|t| ≤1+n−2. SinceK ⊂R2 is a graph domain relative to Lr(0,u),u=(0,1) it follows that for every (x,y)∈K we have d2(x) ≤ y ≤ d1(x),|x| ≤ x0 where d1(x),d2(x) are the corresponding continuous functions specified in Definition 1. Furthermore, for any z(t) =(xt,yt)∈ K ∩ln,d2(xt) ≤ yt ≤ d1(xt)
6
denote byz1:=(xt,d1(xt)),z2:=(xt,d2(xt))∈BdK the points on the boundary of the domain corresponding toz(t). Then by the definition of the distance function ρL it follows that
ρL(z(t))=min{|z(t)−z1|,|z(t)−z2|}.
We may assume that ρL(z(t)) = |z(t)−z1|, the case ρL(z(t)) = |z(t)−z2| can be treated analogously.
Let us verify now the crucial estimate ρL(z(t))
2 =|z(t)−z1|
2 ≤ 1
n2 + |z(t)−an|, z(t)∈ K∩ln. (14) Assume first thatd1(xt)≤1. Then recalling thatan=(ϵn,1)∈ Bd K∩ln yields
|z(t)−an| ≥1−yt≥d1(xt)−yt = |z(t)−z1|,
and thus (14)follows immediately. So we may assume that 1 < d1(xt). Then setting a∗ :=
(xt,1) yields a∗ ∈ [z(t),z1]. Recalling that z(t),an = (ϵn,1) ∈ ln = l(un),un = (ϵn,1) we evidently have that
|xt−ϵn| = |a∗−an| =ϵn|z(t)−a∗|.
Furthermore using that an =(ϵn,1) =(ϵn,d1(ϵn)),z1 =(xt,d1(xt)) ∈ Bd K we have by the definition of the modulus of continuity wK
|z1−a∗| = |d1(ϵn)−d1(xt)| ≤wK(|ϵn−xt|)=wK(|a∗−an|)=wK(ϵn|z(t)−a∗|). Thus by the obvious relation|z(t)−z1| = |z(t)−a∗| + |z1−a∗|,a∗ ∈[z(t),z1] we obtain
|z(t)−z1| ≤ |z(t)−a∗| +wK(ϵn|z(t)−a∗|). (15) Now consider first the case when|z(t)−a∗| ≤ 1
n2. Recalling that 2n2ωK
(ϵn n2
)
=1 it follows from(15)that
ρL(z(t))= |z(t)−z1| ≤ 1 n2 +wK
(ϵn
n2 )
= 3 2n2, i.e.,(14)holds in this case.
So we may assume that|z(t)−a∗|> n12. Since equation 2ωK(ϵnt)=thas a unique solution in variable t for concave modulus wK and this solution is n12 we clearly have that whenever
|z(t)−a∗|>n12 this implies
2wK(ϵn|z(t)−a∗|)≤ |z(t)−a∗|. This together with(15)yields
ρL(z(t))= |z(t)−z1| ≤ 3
2|z(t)−a∗| ≤3
2|z(t)−an|.
Hence relation(14)holds again with all the possible cases being covered.
We can repeat the above argument for−an replacingan in which case(14)yields ρL(z(t))
2 ≤ 1
n2 + |z(t)+an|, z(t)∈ K∩ln. Since|an|>1 combining the last bound with(14)gives
ρL(z(t))
2 ≤ 1
n2 + |z(t)−an ∥z(t)+an|, z(t)∈ K∩ln. (16)
7
Last upper bound together with(10)yields
∫
K∩ln
(√
ρL(z(t))|Dunq(z(t))|)pw(z(t))dt
≤cK,p,w
∫
[−an,an]
((1 n +√
|z(t)−an ∥z(t)+an| )
|Dunq(z(t))| )p
g(t)dt.
Applying now the univariate Lp Bernstein–Markov inequality(4)on the interval [−an,an] in directionun together with(13)and(11)we obtain for the last integral
∫
[−an,an]
((1 n +√
|z(t)−an∥z(t)+an| )
|Dunq(z(t))|
)p
g(t)dt
≤cK,p,wnp
∫
[−an,an]
|q(z(t))|pg(t)dt
≤cK,p,wnp
∫
K∩ln
|q(z(t))|pw(z(t))dt.
Clearly the last two upper bounds yield(9) and hence the proof of the lemma is complete.
Proof ofTheorem 1.Letej:=(δi j,1≤i ≤d),1≤ j ≤d be the standard basis inRd, where δi j is the Kronecker delta. Since K ⊂Rd is a piecewise graph domain it can be covered bym open cylindersLrk(uk) of radiusrk>0 and axisuk, withK being a graph domain with respect to each Lrk(uk),1≤k≤m. Clearly we can chooserk∗ <rk sufficiently close tork so thatK is covered byLr∗
k(uk),1≤k≤m, as well. Now we will establish the needed estimates for the derivatives in each of the cylinders Lr∗
k(uk),1≤k≤m. Let for instancek=1. Without loss of generality assume thatu1=e1. Then by(5)every line in Lr1(e1) in directione1 intersects K along a single line segment [z1,z2],z1,z2∈BdK of lengths≥cK >0. Thus evidently
ρK(z)≤min{|z−z1|,|z−z2|} ≤ 2 cK
|z−z1||z−z2|, z∈Rd.
Thus applying the univariate Lp Bernstein inequality (3) along each of these segments evidently implies that
∫
K∩Lr∗ 1(e1)
(√ ρK
⏐
⏐
⏐
⏐
∂q
∂x1
⏐
⏐
⏐
⏐ )p
w≤cK,p,wnp
∫
K
|q(x)|pw, q ∈ Pnd. (17) Furthermore choosing any r1∗ < ρ < r1 and using above Lemma 1 with w := ej,un,j = e1+ϵnej,2≤ j ≤d for every lineln,j :=l(un,j)⊂L(ρ,un,j) we obtain by Fubini theorem
∫
K∩Lρ(un,j)
(√ ρK
⏐
⏐
⏐
⏐
∂q
∂x1
+ϵn
∂q
∂xj
⏐
⏐
⏐
⏐ )p
w
≤cK,p,wnp
∫
K∩Lρ(un,j)
|q(x)|pw, 2≤ j ≤d, q ∈ Pnd. (18) Sincer1∗< ρ andun,j →e1,n→ ∞we have that K∩L(r1∗,e1)⊂(K∩L(ρ,un,j)) for every 2≤ j≤d andn large enough. Hence it follows by(18)
∫
K∩Lr∗ 1(e1)
(√ ρK
⏐
⏐
⏐
⏐
∂q
∂x1
+ϵn
∂q
∂xj
⏐
⏐
⏐
⏐ )p
w≤cK,p,wnp
∫
K
|q(x)|pw, 2≤ j≤d, q ∈ Pnd. (19)
8
Now we combine(17)and(19)with the next upper bound for the gradient
|∂q| ≤
√ d max
1≤j≤d
⏐
⏐
⏐
⏐
∂q
∂xj
⏐
⏐
⏐
⏐
≤ 2
√ d ϵn
max {⏐
⏐
⏐
⏐
∂q
∂x1
⏐
⏐
⏐
⏐,
⏐
⏐
⏐
⏐
∂q
∂x1
+ϵn
∂q
∂xj
⏐
⏐
⏐
⏐,2≤ j ≤d }
, x∈Rd. This evidently yields
∫
K∩Lr∗ 1(e1)
(√
ρK(x)|∂q|)pw≤cK,p,wnp ϵnp
∫
K
|q|pw, q ∈Pnd. Since the last upper bound holds for each cylinder Lr∗
k(uk),1≤k≤mand the union of these cylinders contains K the statement of the theorem follows.
Proof ofTheorem 2.Given the strictly increasing modulus of continuity w∗(t) let us denote by W(t) the inverse function ofw∗(t). Since t−sw∗(t) is an increasing function of t >0 for somes>0 it easily follows thatt−1/sW(t) is decreasing. We will prove the lower bound when d =2, the general case can be treated analogously. Clearly
D= {|x| +w∗(|y|)≤1} = {|x| ≤1,|y| ≤W(1− |x|)} ⊂R2. (20) Now similarly to [7] we will apply certain properties of Jacobi polynomialsPn(α,β)(x), α, β >
−1 verified in [11], (7.34.1), p. 173. It is essentially shown there that
∫ 1 0
(1−x)µ⏐
⏐Pn(α,β)(x)⏐
⏐
pd x∼n−2µ−2+αp, 2µ < αp−3
2. (21)
Moreover, the lower bound in the above asymptotic relation holds for integration over any interval [a,1],0 <a <1, see [11], (7.34.4). (In fact, in [11] these asymptotic relations are verified for p = 1 but they follow analogously for any p ≥ 1.) Now consider the bivariate polynomial qn(x,y) := y Pn(α,α)(x) ∈ Pn+1d ,n ∈ N with α to be specified below. Since the function|y Pn(α,α)(x)|is even in both variables it clearly follows from(20)that
∫
D
|qn|p≤4
∫ 1 0
∫ W(1−x)
0
yp⏐
⏐Pn(α,α)(x)⏐
⏐
pd yd x ≤4
∫ 1 0
W(1−x)p+1⏐
⏐Pn(α,α)(x)⏐
⏐
pd x. (22) Now recalling thatt−1/sW(t) is decreasing i.e., (1−t)−1/sW(1−t) is increasing we have by the upper bound in(21)settingµ=(p+1)/sand choosingα > 2µp+2
∫ 1−n−2 0
W(1−x)p+1⏐
⏐Pn(α,α)(x)⏐
⏐
pd x
=
∫ 1−n−2 0
(1−x)−(p+1)/sW(1−x)p+1(1−x)(p+1)/s⏐
⏐Pn(α,α)(x)⏐
⏐
pd x
≤n2(p+1)/sW(n−2)p+1
∫ 1−n−2 0
(1−x)(p+1)/s⏐
⏐Pn(α,α)(x)⏐
⏐
pd x≤cW(n−2)p+1n−2+αp. Furthermore using the well known estimate ∥Pn(α,α)(x)∥C[−1,1] = O(nα), see [11], Theorem 7.32.1 it follows that
∫ 1 1−n−2
W(1−x)p+1⏐
⏐Pn(α,α)(x)⏐
⏐
pd x≤n−2W(n−2)p+1∥Pn(α,α)(x)∥C[−1p ,1]
≤cW(n−2)p+1n−2+αp.
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Hence the last two upper bounds combined with(22)yield the estimate
∫
D
|qn|p≤cW(n−2)p+1n−2+αp. (23) Now we need to give a lower bound for the weighted norm of the gradient ofqn on D. The distance functionρD(x,y) can be easily seen to satisfy the lower bound
ρD(x,y)≥1−x−w(y), 0≤x≤1,y≥0,(x,y)∈ D. Hence it follows that
∫
D
(√
ρD|∂qn|)p≥
∫
D
ρDp/2
⏐
⏐
⏐
⏐
∂qn
∂y
⏐
⏐
⏐
⏐
p
≥
∫ 1 0
∫ W((1−x)/2) 0
(1−x−w(y))p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd yd x
≥2−p/2
∫ 1 0
∫ W((1−x)/2) 0
(1−x)p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd yd x
≥cW(n−2)
∫ 1−n−2 0
(1−x)p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd x.
Furthermore using again the Lp Remez type inequality (12) for the generalized algebraic polynomial f(x):= |1−x|1/2|Pn(α,α)(x)|of degreen+1/2 implies that
∫ 1 0
(1−x)p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd x≤c
∫ 1−n−2 0
(1−x)p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd x.
Thus combining the last two estimates and using (21) withµ = p/2 we arrive at the next lower bound for the gradient
∫
D
(√
ρD|∂qn|)p≥cW(n−2)
∫ 1 0
(1−x)p/2⏐
⏐Pn(α,α)(x)⏐
⏐
pd x≥cW(n−2)n−p−2+αp. Recalling also the upper bound(23)we get
∫
D
(√
ρD|∂qn|)p≥cW(n−2)−pn−p
∫
D
|qn|p, i.e.,
∥√
ρD∂qn∥Lp(D) ≥ c nW(n12)
∥qn∥Lp(D). (24)
SinceW(t) is the inverse function ofw∗(t) andϵn(D) is the solution of(6)with this modulus w∗(t) it follows that ϵn(D) = n2W
( 1 2n2
)
. Moreover, using that t−1/sW(t) is a decreasing function it is easy to see that 21/sW(
1 2n2
)
≥W(
1 n2
) . Hence W
(1 n2
)
≤21/sW ( 1
2n2 )
=21/sn−2ϵn(D). Thus(24)clearly yields
∥√
ρD(x)∂qn∥Lp(D) > cn
ϵn(D)∥qn∥Lp(D). □
Remark. Our main emphasis in this paper was to provide Lp Bernstein type inequalities on cuspidal domains which are sharp with respect to the degreenof the multivariate polynomials.
On the other hand estimates of the main Theorem 1 can be made more explicit in order to
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exhibit their dependence on the degree m of the Jacobi type weightw used in the Lp norm.
Indeed, the univariate Bernstein–Markov type inequality (4) is known to hold with n being replaced by (n +m)m and a constant cp depending only on p, see [3], p. 407 for details.
Likewise, since the Remez type inequality (12)is used inLemma 1for generalized algebraic polynomials of degree N ≤cp(n+m) throughout the proof ofTheorem 1ϵn can be replaced byϵn+m withm being the degree of the Jacobi type weightw. This will result in the estimate
∥√
ρK(x)∂q∥Lp
w(K) ≤cK,p
(n+m)m ϵn+m(K) ∥q∥Lp
w(K), q ∈Pnd
with some cK,p > 0 depending only on K and p. In particular, if graph domain K is Lipγ, 0< γ <1 then
∥√
ρK(x)∂q∥
Lwp(K) ≤cK,pm(n+m)γ2−1∥q∥
Lwp(K), q ∈Pnd.
The sharpness of the above upper bounds with respect to the degree m of the Jacobi type weightwis not known even in the univariate case.
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