Estimation of the maximal operator M when the coordinates are

We now turn to the proof of our main result, Theorem 1.2. As observed in the introduc-tion, we may assume thatS is locally the graph S={(x₁, x₂,1+φ(x₁, x₂)):(x₁, x₂)∈Ω}

of a function 1+φ. Here and in the subsequent sections, φ∈C^∞(Ω) will be a smooth real-valued function of finite type defined on an open neighborhood Ω of the origin inR² and satisfying

φ(0,0) = 0 and ∇φ(0,0) = 0.

In this section we shall consider the easiest cases where the coordinatesx are adapted toφ,or whereh(φ)<2.

Recall thatA_t,t>0, denotes the corresponding family of averaging operators A_tf(y) :=

Z

S

f(y−tx)%(x)dσ(x),

where dσ denotes the surface measure on S and %∈C₀^∞(S) is a non-negative cut-off function. We shall assume that%is supported in an open neighborhoodU of the point (0,0,1) which will be chosen sufficiently small. The associated maximal operator is given by

Mf(y) := sup

t>0

|Atf(y)|, y∈R³. (7.1)

The averaging operatorA_tcan be rewritten in the form Atf(y) :=

Z

R²

f(y1−tx1, y2−tx2, y3−t(1+φ(x1, x2)))η(x1, x2)dx,

where η is a smooth function supported in Ω. If χ is any integrable function defined on Ω, we shall denote byA^χ_t the correspondingly localized averaging operator

A^χ_tf(y) :=

Z

R²

f(y1−tx1, y2−tx2, y3−t(1+φ(x1, x2)))χ(x)η(x)dx, and byM^χ the associated maximal operator

M^χf(y) := sup

t>0

|A^χ_tf(y)|, y∈R³.

Proposition7.1. Let φbe as above,and assume that =(1,2)is a given weight such that 0<162<1. As in (2.2),we decompose

φ=φ+φr

into its -principal part φ and the remainder term φ_r consisting of terms of -degree greater than 1. Then, if the neighborhood Ω of the point (0,0) is chosen sufficiently small, the maximal operator Mis bounded on L^p(R³)for every p>max{2, h(φ)}.

Proof. Let us modify our notation slightly and write points inR³in the form (x, x3), withx∈R² andx3∈R. Recall from Corollary 2.2 the crucial fact that

h(φ) = max{m(φ), d_h(φ)}.

In particular, the multiplicity of every real root of the-homogeneous polynomialφ is bounded byh(φ).

Choosing a dyadic decomposition

∞

X

k=k0

χk(x) = 1

for 06=x∈Ω as in§6, withχ_k(x):=χ(δ₂kx), where theδ_rdenote the dilations associated with the weight, and where χis supported in an annulus 16|x|6R, we can write At

as a sum of averaging operators

Atf(y, y3) =

∞

X

k=k₀

A^k_tf(y, y3),

whereA^k_t:=A^χ_t^k. Notice that by choosing Ω small, we can choose k0∈Nto be large.

If we apply the change of variablesx7!δ₂−k(x) in the integral above, we obtain A^k_tf(y, y3) = 2^−k||

Z

R²

f(y−tδ₂−k(x), y3−t(1+2^−kφ^k(x)))η(δ₂−k(x))χ(x)dx,

where

φ^k(x) :=φ(x)+2^kφr(δ₂−k(x))

and where the perturbation term 2^kφr(δ₂−k(·)) is of orderO(2^−εk) for someε>0 in any C^M-norm. To express this fact, we shall in the sequel again use the short-hand notation

2^kφ_r(δ₂−k(·)) =O(2^−εk).

ByM^k we shall denote the maximal operatorM^χk associated with the averaging oper-atorsA^k_t.

Assume now thatp>max{2, h(φ)}. We define the scaling operatorT^k by T^kf(y, y₃) := 2^k||/pf(δ₂k(y), y₃).

ThenT^k acts isometrically onL^p(R³), and (T^−kA^k_tT^k)f(y, y3) = 2^−k||

Z

R²

f(y−tx, y3−t(1+2^−kφ^k(x)))η(δ₂−k(x))χ(x)dx.

Assuming that Ω is a sufficiently small neighborhood of the origin, we need to consider only the case wherekis sufficiently large.

Letx⁰∈D be a fixed point.

If ∇φ(x⁰)6=0, then by Euler’s homogeneity relation one can easily derives that rank(D²φ(x⁰))>1 (see [18, Lemma 3.3]). Therefore, we can find a unit vector e∈R² such that∂_e²φ(x⁰)6=0, where ∂e denotes the partial derivative in direction ofe.

If ∇φ(x⁰)=0, then by Euler’s homogeneity relation we have φ(x⁰)=0 as well.

Thus the function φ vanishes in x⁰ at least of order 2, so that m(φ)>2, and hence h(φ)>2. On the other hand, by what we remarked earlier, it vanishes along the circle passing throughx⁰ and centered at the origin at most of orderh(φ). Therefore, we can find a unit vectore∈R² such that∂^m_e φ(x⁰)6=0, for somemwith 26m6h(φ).

Thus, in both cases, after rotating coordinates so that e=(0,1), we may apply Proposition 4.5 to conclude that forp>max{2, h(φ)}and sufficiently large k,

kT^−kM^kT^kfkp6C2^k(1/p−||)kfkp, f∈ S(R³),

if we replaceχin the definition ofA^k_t byχη, whereηis a bump function supported in a sufficiently small neighborhood ofx⁰. This is equivalent to

kM^kfkp6C2^k(1/p−||)kfkp.

Decomposingχ and correspondingly A^k_t by means of a suitable partition of unity into a finite number of such pieces, we see that the same estimate holds for the original operatorsM^k.

Since 1/||=d_h(φ)6h(φ)<p, we can sum over all k>k₀ and obtain the desired estimate forM.

Let us apply this result first to the case where the coordinatesxare adapted toφ, possibly after a rotation of the coordinate system (x1, x2). Observe first that a linear change of the coordinates (x1, x2) induces a corresponding linear change of coordinates in R³ which fixes the coordinate x3. This linear transformation is an automorphism of R³, so that it preserves the convolution product onR³ (up to a fixed factor), and hence the norm of the maximal operator M. We may thus assume that the coordinates are adapted toφ.

If we choose the weightas in Lemma 2.4 of§2.4, then Proposition 7.1 immediately implies the following result.

Corollary 7.2. Let φ be as above, and assume that, possibly after a rotation of the coordinate system,the coordinates xare adapted to φ,i.e.,that h(φ)=d(φ). Then,if the neighborhood Ωof the point (0,0) is chosen sufficiently small, the maximal operator Mis bounded on L^p(R³)for every p>max{2, h(φ)}.

Remark 7.3. One can easily extend Corollary 7.2 as follows: If the neighborhood Ω of the point (0,0) is chosen sufficiently small, then the maximal operatorMis bounded onL^p(R³) for everyp>max{2, h(φ_pr)}, no matter if the coordinates are adapted toφor not.

Proof. Indeed, if the coordinates are adapted, thenh(φ_pr)=h(φ). So, assume that the coordinates (x₁, x₂) are not adapted to φ. Then the principal face of the Newton polyhedron is a compact edge, so that the principal part φ_pr of φ is -homogeneous, where satisfies the assumptions of Proposition 7.1. Since φ_pr=φ, the result then follows from this proposition.

The result above holds even when the coordinates are not adapted, but then it will in general not be sharp, since we haveh(φpr)>h(φ) (see [19, Corollary 4.3]), and in general strict inequality holds.

For example, letφ(x1, x2):=(x2−x²₁)²+x⁵₁. Then we haveφpr(x)=(x2−x²₁)². The coordinate system is not adapted to φ, becaused(φ)=⁴₃<2, where 2 is the multiplicity of the root of φpr. A coordinate system which is adapted to φ is given by y2:=x2−x²₁ andy1:=x1. It is then easy to see thath(φpr)=2>¹⁰₇=h(φ).

Corollary7.4. If h(φ)<2,and if the neighborhood Ωof the point (0,0)is chosen sufficiently small, then the maximal operator M is bounded on L^p(R³) for any p>2, also when the coordinates are not adapted to φ.

Proof. IfD²φ(0,0)6=0 then we have at least one non-vanishing principal curvature at the origin, so that the result follows from C. D. Sogge’s main theorem in [36].

Next, we consider the case whereD²φ(0,0)=0. Then necessarily D³φ(0,0)6=0, for otherwiseh(φ)>d(φ)>2. In particular,h(φ)>1. Denote byP3 the Taylor polynomial of degree 3 with base point 0 of the functionφ, so thatP3=φ, if we choose:= ¹₃,¹₃

. If h(P3)62, then we obtain the desired estimate from Proposition 7.1. Assume therefore that h(P3)>2. Then, by formula (2.6), P3 must have a root of order 3. Thus, possibly after rotating the coordinate system, we may assume thatP3(x1, x2)=cx³₂ withc6=0.

Now, we consider the Taylor supportT(φ) ofφ. As T(φ)⊂

(t1, t2):¹₃t1+¹₃t2>1 , one easily checks that the subset

(t1, t2) :¹₆t1+¹₃t2<1 ∩T(φ)

ofT(φ) contains at most three points, namely (4,0), (5,0) and (3,1), all of them lying below the bisectrixt1=t2.

Moreover, any line passing through the point (0,3)∈T(φ) corresponding toP3=cx³₂ contains at most one of these points. Thus, if

(t1, t2) :¹₆t1+¹₃t2<1}∩T(φ)6=∅,

then the principal partφpr ofφcontains only two monomials, one corresponding to the point (0,3) above the bisectrix and the other one corresponding to one of the points listed above which lie below the bisectrix, i.e., φpr is of the form dx⁴₁+cx³₂, dx⁵₁+cx³₂ or dx³₁x2+cx³₂, withd6=0 (note that these all satisfyd(φpr)<2). We remark that these correspond to the singularities of type Ek with k=6,7,8 in Arnold’s classification (see [2] and [11]). Therefore on the unit circleφpr has no root of multiplicity bigger than 1, so that the coordinate system is adapted toφ, and thus h(φ)<2. The desired estimate forMfollows therefore in this case from Corollary 7.2.

Finally assume that

(t1, t2) :¹₆t1+¹₃t2<1}∩T(φ) =∅. ThenT(φ)⊂

(t1, t2):¹₆t1+¹₃t2>1}, and henceh(φ)>d(φ)>2, which contradicts our as-sumption.

8. Non-adapted coordinates: Estimation of the maximal operator M