A GENERALIZATION OF SARD’S THEOREM

S.M. BATES AND C.G. MOREIRA

Abstract. Let E be a euclidean space and F any normed vector space. If A ⊂ E is a t-sigmafinite set of rank r for a C^{k} function f : E → F, then f(A) is (r + (t − r)/k)-null for any t > r. We also prove that if α ∈ (0, 1] and A ⊂ E is a t-sigmafinite (resp. t-null) set of rank r for a C^{k+α} mapping f: E → F, then f(A) is (r + (t − r)/k)-sigmafinite (resp. (r + (t − r)/k)-null) for any t > r.

If E is an infinite-dimensional Banach space, then the theorems above are true for k ≤ 2. We give examples of polynomial mappings that show that we cannot improve our results by improving the differentiability hypothesis. In particular, we exhibit a polynomial P: ℓ_{2} → ℝ which maps a 3-finite critical set onto the interval [0, 1].

Introduction

For t ∈ ℤ_{+}, we recall that a set A ⊂ ℝ^{n} is said to be (countably) t-rectifiable provided all but at most a t-null subset of A is contained in a countable union of C^{1}-embedded t-dimensional submanifolds of ℝ^{n}.

Theorem 1. Fix t, s > 0 and r ∈ ℕ with t > r and set d = r + (t − r)/s. Let E be a euclidean space and F any normed vector space. Let A ⊂ E be a t-sigmafinite set of rank r for a differentiable map f: E → F.

1. If s ∈ ℤ_{+} and f ∈ C^{s}, then f(A) is d-null.

2. If s > 1 and f ∈ Λ_{s}, then H^{d}(f(A)) ≤ (2∥f∥_{s})^{d}(1 + r)^{d/2}H^{t}(A).

3. If f ∈ Λ_{s}, t ∈ ℤ_{+} and A is t-rectifiable, then f(A) is d-null.

Note that if k > 1 and t < r, then t < r + (t − r)/k, and the conclusion of the theorem in this case follows trivially from the fact that any C^{1} function maps t-sigmafinite sets to t-sigmafinite sets. If t = r, then the statement of the theorem is false, as illustrated by the identity map on ℝ^{r}. Finally, we note that if F is finite-dimensional and r = dim(F), then the assertion follows from then inequality r < r + (t − r)/k, since F itself is (r + (t − r)/k)-null.

For differentiable mappings from infinite-dimensional domains, similar results hold in the smoothness classes Λ_{s} for s ∈ (1, 3]. When t = dim(E), statement (2) subsumes previous results of [1, 3, 5].

Theorem 2. Fix s ∈ [1, 3] and define d = r + (t − r)/s. Suppose X, Y are Banach spaces and A ⊂ X is a t-sigmafinite rank-r set for a differentiable map f: X → Y.

1. If s ∈ {1, 2} and f is C^{s}, then f(A) is d-null.

2. If f ∈ Λ_{s}, then H^{d}(f(A)) ≤ (2∥f∥_{s})^{d}(1 + r)^{d/2}H^{t}(A).

3. If f ∈ Λ_{s}, t ∈ ℤ_{+} and A is t-rectifiable, then f(A) is d-null.

Supported by an NSF Postdoctoral Fellowship in Mathematics.

Supported by CNPq, Brasil.

In particular, Theorem 2 asserts that for any Banach space X, a C^{3} real-valued function maps any t-dimensional critical set to a set of dimension ≤ 3.

Theorem 3. For any infinite-dimensional Banach space X, there exists a cubic polynomial P: X → ℝ and a 3-finite critical set A ⊂ X such that P (A) = [0, 1].

Notations and Definitions.

For s ∈ (0, 1), we denote by Λ_{s} the Lipschitz space of those continuous functions f: ℝ → ℝ for which

∥f∥_{s} = sup_{x∈ℝ}|f(x)| + sup_{x,h∈ℝ} \frac{|f(x + h) − f(x)|}{|h|^{s}} < ∞.

A function f: ℝ → ℝ belongs to Λ_{1} if f is continuous and

∥f∥_{1} = sup_{x∈ℝ}|f(x)| + sup_{x,h∈ℝ} \frac{|f(x + h) + f(x − h) − 2f(x)|}{|h|^{s}} < ∞.

For s > 1, let [s] denote the greatest integer strictly less than s. The class Λ_{s} is then defined inductively as the space of C^{[s]} functions f: ℝ → ℝ for which

∥f∥_{s} = ∥f∥_{s−1} + ∥f′∥_{s−1} < ∞.

When s > 1 is not an integer, each element of the Hölder class C^{s} coincides locally with some member of Λ_{s}. For k ∈ ℤ_{+}, the Lipschitz space Λ_{k} strictly contains C^{k}. A function f ∈ C^{k−1} coincides locally with some member of Λ_{k} provided that the (k − 1)-st derivative of f satisfies a local Zygmund condition (see [8]).

1. Decompositions of Critical Sets

Lemma 1.1. Fix η > 0. Let E be a euclidean space, F a normed vector space, and A ⊂ E a set of rank-r points for a differentiable map f: E → F.

1. If s ∈ ℤ_{+}, s ≥ 2, and f ∈ C^{s}, then there exists a decomposition {A_{i}} of A such that for each i, there is a constant M_{i} > 0 and a splitting of E such that for any set S ⊂ E,

|f(S ∩ A_{i})| ≤ (∥f∥_{1} + η)|S′| + M_{i}|S′||S′′| + η|S|^{s}.

2. If s > 2 and f ∈ Λ_{s}, then there exists a decomposition {A_{i}} of A such that for each i, there is a constant M_{i} > 0 and a splitting of E such that for any set S ⊂ E,

|f(S ∩ A_{i})| ≤ (∥f∥_{1} + η)|S′| + M_{i}|S′||S′′| + M_{i}|S|^{s}.

3. If A is t-rectifiable for some t ∈ ℤ_{+}, then there exists a decomposition {A_{i}} of A such that A_{0} is t-null, ∑_{i} |A_{i}|^{t} ≤ H^{t}(A_{i})+1, and for each i ∈ ℤ_{+}, there is a constant M_{i} > 0 and a splitting of E such that for any set S ⊂ E,

|f(S ∩ A_{i})| ≤ (∥f∥_{1} + η)|S′| + M_{i}|S′||A_{i}| + M_{i}|A_{i}|^{s}.

To prove the lemma, we will need the following generalization of the Morse decomposition lemma:

Lemma 1.2. Let E be a euclidean space, F any normed vector space, and h: E → F a C^{k} map. If h = 0 on A ⊂ E, then for each η > 0 there exists a decomposition {A_{i}} of A such that A_{0} is countable, and for each i ∈ ℤ_{+}, there is a closed ball B_{i} ⊂ ℝ^{n_{i}} and a C^{1} map ψ_{i}: B_{i} → E such that

1. A_{i} ⊂ ψ_{i}(B_{i}) and |x − y| ≤ |ψ_{i}(x) − ψ_{i}(y)| ≤ (1 + η)|x − y| for all x, y ∈ B_{i}.

2. For all x ∈ ψ_{i}^{−1}(A_{i}) and y ∈ B_{i},

h(ψ_{i}(y)) − h(ψ_{i}(x))| ≤ η|x − y|^{k}.

Proof. Let (n, k) denote the statement of the lemma for all η > 0 and any C^{k} mapping from n-dimensional euclidean space into any Banach space.

We first prove (1, k) for all k ∈ ℤ_{+}. Fix η > 0 and let A_{0} be the set of isolated points of A. If dim(E) = 1 and F is any Banach space, then D^{i}h(x) = 0 for each i = 1, ⋯, k and x ∈ A \ A_{0}. (This follows easily from the mean-value theorem when F = ℝ and can in general be deduced from the equation D^{i}(y^{∗} ∘ f) = y^{∗} ∘ D^{i}f, valid for each y^{∗} ∈ F^{∗}). By continuity of D^{k}f, we can choose a covering of A \ A_{0} by closed intervals B_{i} such that |D^{k}h| < η on each B_{i}. From Taylor’s theorem, it follows that for each x ∈ A_{i} and y ∈ B_{i},

|h(y)| = |h(y) − h(x)| ≤ η|x − y|^{k},

which proves statement (1, k).

To prove case (n, k) of the lemma, we proceed inductively, assuming that statements (n − 1, k) and (n, k − 1) have been proven (if k = 1, we only assume (n − 1, 1)). Given a subset A of an n-dimensional euclidean space E, define

U_{A} = {x ∈ A : every C^{k} function g: E → ℝ with g(A) = 0 satisfies Dg(x) = 0},

V_{A} = A \ U_{A}. Fix η > 0. To decompose A, we find appropriate decompositions of V_{A} and U_{A}.

¿From the definition of V_{A}, it follows that for each η > 0, there exists a countable collection of C^{1} embeddings φ_{i}: B′_{i} → E such that each B′_{i} is a closed ball in ℝ^{n−1}, V_{A} ⊂ ⋃_{i} φ_{i}(B′_{i}), and

|x − y| ≤ |φ_{i}(x) − φ_{i}(y)| ≤ (1 + η)^{1/2}|x − y|.

By applying the (n − 1, k) statement of the lemma to the mapping h ∘ φ_{i}, we obtain a decomposition {A′_{i}} of φ_{i}^{−1}(V_{A}) and a sequence of C^{1} maps ψ_{ij}: B_{ij} → B_{i} such that:

1. A′_{j} ⊂ ψ_{ij}(B_{ij}) and |x − y| ≤ |ψ_{ij}(x) − ψ_{ij}(y)| ≤ (1 + η)^{1/2}|x − y| for all x, y ∈ B_{ij}.

2. For all y ∈ B_{ij},

|h(φ_{i}(ψ_{ij}(y)))| ≤ ηd(y, ψ_{ij}^{−1}(A′_{j}))^{k}.

Setting V_{ij} = φ_{i}(A′_{j}) and ψ̃_{ij} = φ_{i} ∘ ψ_{ij} then gives the desired decomposition of V_{A}.

Next, we note that Dh = 0 on U_{A}. If k = 1, then we can use the continuity of Dh to find a countable covering of U_{A} by balls B_{i} on which ∥Dh∥ ≤ η, so that {B_{i} ∩ U_{A}} gives the desired decomposition of U_{A}. For k > 1, we apply the (n, k − 1) hypothesis to the map Dh: E → L(E, F) in order to find a decomposition {U_{i}} of U_{A} and mappings ψ_{i}: B_{i} → E such that

1. U_{i} ⊂ ψ_{i}(B_{i}) and |x − y| ≤ |ψ_{i}(x) − ψ_{i}(y)| ≤ (1 + η)|x − y| for all x, y ∈ B_{i}.

2. For all x ∈ ψ_{i}^{−1}(U_{i}) and y ∈ B_{i},

|Dh(ψ_{i}(y)) − Dh(ψ_{i}(x))| ≤ η|x − y|^{k−1}.

Given x ∈ ψ_{i}^{−1}(U_{i}) and y ∈ B_{i}, we set γ(t) = ty + (1 − t)x, so that

|h(ψ_{i}(y))| = |∫_{0}^{1} D(h ∘ ψ_{i})(γ(t))γ′(t) dt|
≤ η|x − y|^{k−1}|x − y|.

Thus, |h(ψ_{i}(y))| ≤ ηd(y, ψ_{i}^{−1}(U_{i}))^{k} for all y ∈ B_{i}, and this completes the proof.

□

Lemma 1.3. Let E be a euclidean space, F any normed vector space, and h: E → F a Λ_{s} map with s > 2. If h = 0 on A ⊂ E, then for each η > 0 there exists a decomposition {A_{i}} of A such that A_{0} is countable, and for each i ∈ ℤ_{+}, there is a closed ball B_{i} ⊂ ℝ^{n_{i}} and a C^{1} map ψ_{i}: B_{i} → E such that

1. A_{i} ⊂ ψ_{i}(B_{i}) and |x − y| ≤ |ψ_{i}(x) − ψ_{i}(y)| ≤ (1 + η)|x − y| for all x, y ∈ B_{i}.

2. For all x, x′ ∈ ψ_{i}^{−1}(A_{i}) and y in the line segment xx′,

|h(ψ_{i}(y)) − h(ψ_{i}(x))| ≤ M_{i}|x − y|^{s−1}|x − x′|.

Proof of Lemma 1.1. We first prove the lemma assuming that with respect to some splitting E = E′ ⊕ E′′, the map f is of the form

f(z, y) = (h(z), g(z, y))

for some h, g such that D_{2}g = 0 on A. From Lemma 1.2 we obtain a decomposition {A_{i}} of A and C^{1} mappings ψ_{i}: B_{i} → E such that

1. A_{i} ⊂ ψ_{i}(B_{i}) and |x − y| ≤ |ψ_{i}(x) − ψ_{i}(y)| ≤ (1 + η)|x − y| for all x, y ∈ B_{i}.

2. For all x, x′ ∈ ψ_{i}^{−1}(A_{i}) and y ∈ xx′,

|D_{2}g(ψ_{i}(y))| ≤ η|x − x′|^{s−1}.

Fix i ∈ ℤ_{+} and suppose that (z, y), (z′, y′) ∈ S ∩ A_{i}. Then

|f(z, y) − f(z′, y′)| ≤ |f(z, y) − f(z′, y)| + |f(z′, y) − f(z′, y′)|
≤ ∥f∥_{1}|S′| + |g(z′, y) − g(z′, y′)|.

To bound the term on the right, let γ: [0, 1] → E be the composition of ψ_{i} with the affine path between ψ_{i}^{−1}(z, y) and ψ_{i}^{−1}(z′, y′). If γ̃ : [0, 1] → E is the composition of γ with the projection onto z′ × E′′, then

|g(z, y) − g(z′, y′)| = |∫_{0}^{1} D_{2}g(γ̃(t))γ̃′(t) dt|
≤ ∫_{0}^{1} |D_{2}g(γ̃(t)) − D_{2}g(γ̃(t))||γ̃′(t)| dt + ∫_{0}^{1} |D_{2}g(γ(t))||γ̃′(t)| dt.

Since |γ̃′(t)| ≤ (1 + η)|S′′| for all t, we have

|D_{2}g(γ̃(t)) − D_{2}g(γ̃(t))||γ̃′(t)| ≤ (1 + η)∥f∥_{2}|S′||S′′|.

Finally, if x, y ∈ B_{i} satisfy ψ_{i}(x), ψ_{i}(y) ∈ S ∩ A_{i} and ψ_{i}(y) ∈ Q, then property (1) of the map ψ_{i} implies that |x − y| ≤ |S|. From (2), it follows that ∥D_{2}g∥ is bounded on S ∩ ψ_{i}(B_{i}) by η|S|^{s−1}, and so

∫_{0}^{1} ∥D_{2}g(γ(t))∥|γ̃′(t)| dt ≤ (1 + η)η|S|^{s}.

¿From these estimates, part (1) of the lemma follows.

Part (2) of the lemma follows by exactly the same argument, where Lemma 1.3 is applied in place of Lemma 1.2 to decompose A.

Turning to part (3) of the lemma, we suppose that A is t-rectifiable. By combining Lemma 1.3 with the definition of rectifiability, we then obtain a countable decomposition {A_{i}} of A such that A_{0} is t-null, and for each i ∈ ℤ_{+}, there exists a ball B_{i} ⊂ ℝ^{t} and a C^{1} embedding ψ_{i}: B_{i} → E such that

1. A_{i} ⊂ ψ_{i}(B_{i}) and each point of ψ_{i}^{−1}(A_{i}) is a Lebesgue density point.

2. |x − y| ≤ |ψ_{i}(x) − ψ_{i}(y)| ≤ 2|x − y| for all x, y ∈ B_{i}.

3. For all x, x′ ∈ ψ_{i}^{−1}(A_{i}) and y ∈ xx′,

|D_{2}g(ψ_{i}(y))| ≤ M_{i}|x − x′|^{s−1}.

To obtain the desired decomposition of A, we will further decompose each A_{i}. Given η > 0, we fix a positive integer P such that 2/P < η. Since each x ∈ ψ_{i}^{−1}(A_{i}) is a density point, there exists ε_{x} > 0 such that if Q(x, ε) is a cube of edge ε < ε_{x} centered at x ∈ ψ_{i}^{−1}(A_{i}), then

\frac{H^{t}(Q ∩ ψ_{i}^{−1}(A_{i}))}{H^{t}(Q)} ≥ 1 − P^{−t}.  (∗)

The collection

Q = {Q(x, ε) : x ∈ ψ_{i}^{−1}(A_{i}), ε < ε_{x}}

is a Vitali family for ψ_{i}^{−1}(A_{i}), and so there exists a sequence of pairwise disjoint cubes {Q_{j}} ⊂ Q such that ψ_{i}^{−1}(A_{i}) \ ⋃_{j} Q(x_{j}, ε_{j}) is t-null and ∑_{j} ε_{j}^{t} < H^{t}(ψ_{i}^{−1}(A_{i})) + 1.

Fix j and consider (z, y), (z′, y′) ∈ Q_{j} ∩ ψ_{i}^{−1}(A_{i}). By condition (∗), it follows that there are x_{0}, ⋯, x_{P} in Q ∩ ψ_{i}^{−1}(A_{i}) such that x = x_{0}, y = x_{P}, and |x_{i} − x_{i+1}| < 2|Q|P^{−1} for all i. Let γ: [0, 1] → E be the composition of ψ_{i} with the piecewise linear path connecting the points x_{i}. If γ̃: [0, 1] → E is the composition of γ with the projection onto z′ × E′′, then

|g(z, y) − g(z′, y′)| = |∫_{0}^{1} D_{2}g(γ̃(t))γ̃′(t) dt|
≤ ∫_{0}^{1} |D_{2}g(γ̃(t)) − D_{2}g(γ(t))||γ̃′(t)| dt + ∫_{0}^{1} |D_{2}g(γ(t))||γ̃′(t)| dt.

Since |γ̃(t)| ≤ 2|Q_{j}| for all t, we have

|D_{2}g(γ̃(t)) − D_{2}(γ(t))||γ̃(t)| ≤ 2∥f∥_{2}|S′||Q_{j}|.

Finally, property (1) of the map ψ_{i} implies that for each i, we have |x_{i} − x_{i+1}| ≤ 2|Q|/P, and so from (3) it follows that

∫_{0}^{1} ∥D_{2}g(γ(t))∥|γ̃′(t)| dt ≤ 2|Q_{j}|M_{i} ∑_{i=0}^{P −1} |x_{i} − x_{i+1}|^{s−1} ≤ 2^{s}M_{i}P^{2−s}|Q_{j}|^{s}.

Thus, proceeding as above, and applying the inequality 2^{s}M_{i}P^{2−s} < η, we obtain

|f(S ∩ ψ_{i}(Q_{j}))| ≤ ∥f∥_{1}|S′| + M_{i}|S′||Q_{j}| + η|Q_{j}|^{s}.

For each i ∈ ℤ_{+}, we now choose a collection of cubes Q_{j} associated to A_{i} as above. The decomposition {A_{i}} given by a denumeration of the sets ψ_{i}(Q_{j}) ∩ A, and A_{0} = A \ ⋃ A_{i} then satisfies the lemma.

For the general case, we apply the implicit function theorem locally to find a diffeomorphism φ: E′ ⊕ E′′ → E such that f ∘ φ is of the split form above. If ψ: B → E is a map such that

|f(ψ_{i}(y))| ≤ ηd(y, ψ_{i}^{−1}(A_{i}))^{k}

and φ(x), φ(y) ∈ ψ(B), then....

Note that if we compose the map f in Lemma 1.3 with a C^{1} diffeomorphism φ: E → E, then we obtain the estimate

|(f ∘ φ)(x) − (f ∘ φ)(y)| ≤ (∥f∥_{s} + η)|φ(x) − φ(y)|^{s} ≤ (∥f∥_{s} + η)∥φ∥_{1}^{s}|x − y|^{s}.

In other words, we can compose Morse decompositions with diffeomorphism φ and only increase the constants by the C^{1} norm of φ.

□

Lemma 1.4. Fix η > 0. Let E, F be Banach spaces and A ⊂ E a set of rank-r points for a differentiable map f: E → F.

1. If s ∈ {1, 2} and f ∈ C^{s}, then there exists a decomposition {A_{i}} of A such that for each i, there is a splitting of E such that for any set of the form Q = Q′ × S′′

|f(Q ∩ A_{i})| ≤ ∥f∥_{1}|Q′| + η|Q|^{s}.

2. If s ∈ (1, 3] and f ∈ Λ_{s}, then there exists a decomposition {A_{i}} of A such that for each i, there is a splitting of E such that for any set of the form Q = Q′ × S′′

|f(Q ∩ A_{i})| ≤ ∥f∥_{1}|Q′| + ∥f∥_{s}|Q|^{s}.

Proof. We first prove the lemma assuming that with respect to some fixed splitting E′ ⊕ E′′ of E, the map f assumes the form

f(z, y) = (h(z), g(z, y))

and D_{2}g = 0 on A. Fix (z, y), (z′, y′) ∈ A and define

γ(t) = (t(z′ − z) + z, t(y′ − y) + y).

By the mean-value theorem, there exists a c ∈ (0, 1) such that

|f(z, y) − f(z′, y′)| = |D_{1}f(γ(c))(z′ − z) + D_{2}f(γ(c))(y′ − y)|
≤ ∥f∥_{1}|Q′| + |D_{2}g(γ(c))||Q|,

since D_{2}f = D_{2}g.

1. Let {B_{i}} be a countable covering of A by balls such that |D_{2}g| < η on each B_{i}, and set A_{i} = A ∩ B_{i}. For sets of the form Q = Q′ × S′′, the preceding inequality then implies that

|f(Q ∩ A_{i})| ≤ ∥f∥_{1}|Q′| + η|Q|,

thus proving statement (1) of the lemma for the C^{1} case. If f ∈ C^{2}, then the mean-value theorem implies that for some c′ ∈ (0, c),

|D_{2}g(γ(c)) − D_{2}g(γ(0))| ≤ |D(D_{2}g)(γ(c′))(z − z′, y − y′)|.

Since D_{2}g(γ(0)) = D_{2}g(γ(1)) = 0, there exists c′′ ∈ (0, 1) such that D(D_{2}g)(γ(c′′))(z −z′, y− y′) = 0, so that

|D_{2}g(γ(c))| ≤ |D(D_{2}g)(γ(c′)) − D(D_{2}g)(γ(c′))||(z − z′, y − y′)|.

Let {B_{i}} be a covering of A by balls B_{i} such that |D(D_{2}g)(p) − D(D_{2}g)(p′)| < η for all p, p′ ∈ B_{i}, and set A_{i} = A ∩ B_{i}. Combining (∗) with the preceding inequalities, we find that for any set of the form Q = Q′ × S′′,

|f(Q ∩ A_{i})| ≤ ∥f∥_{1} + η|Q|^{2}.

2. Turning to statement (2) of the lemma, we recall that in the notation above, D_{2}g(γ(0)) = D_{2}g(γ(1)) = 0. If f ∈ Λ_{s} for s ∈ (1, 2], then Norton’s lemma implies that

|D_{2}g(γ(c))| ≤ ∥f∥_{s}|Q|^{s−1},

by Lemma 1.5. Combined with the preceding inequality, this proves the lemma for s ∈ (1, 2].

For s ∈ (2, 3], fix δ > 0 and choose q, q′ ∈ A such that |p − q|, |p′ − q′| < δ. Let \overline{γ} : [−2δ, 2δ] → E be a smooth extension of γ such that ∥γ∥_{1} < |Q| and \overline{γ}(−2δ) = q, \overline{γ}(1 + 2δ) = q′. Set h = D_{2}g ∘ \overline{γ}. Since D_{2}g = 0 on A, it follows that h = 0 at 4 points of [−2δ, 1 + 2δ]. Since h ∈ Λ_{s−1}, Lemma 1.2 implies that

|h(c) − h(0)| ≤ ∥h∥_{s−1}(1 + 4δ)^{s−1}
≤ ∥f∥_{s}|Q|^{s−1}(1 + 4δ)^{s−1}.

Since δ > 0 is arbitrary, this completes the proof.

□

To prove a similar decomposition result for functions of class Λ_{s}, we require the following observation of Norton [5]

Lemma 1.5. (Norton) If g ∈ Λ_{1}(ℝ, ℝ), a < b, and g(a) = g(b) = 0, then for all x ∈ [a, b],

|g(x)| ≤ ∥g∥_{1}|b − a|.

□

2. Proof of the main theorems

Fix t, s > 1, r ∈ ℕ. To prove Theorem 1, we consider a Banach space E, a normed vector space F, and a t-finite set A ⊂ E of rank-r points for a C^{1} map f : E → F. Since Df is continuous, the function x ↦ rank(Df_{x}) is lower semi-continuous on E, and so its level sets are Borel sets. Consequently, there exists a Borel set A′ of rank-r points of f which contains A and satisfies H^{t}(A′) = H(A).

For some μ, η > 0, we suppose for the moment that there is a decomposition {A_{i}} of A′ such that for each i ∈ ℤ_{+} and δ ∈ (0, M_{i}^{−1}), choose a covering {S_{j}} of A_{i} such that |S_{j}| = ε_{j} < δ for each j and

∑_{j} ε_{j}^{t} ≤ H^{t}(A_{i}) + δ.

For each j, let S′_{j}, S′′_{j} denote the orthogonal projections of S_{j} in the subspaces E′_{i} and E′′_{i}, respectively. Evidently, |S′_{j}| ≤ |S_{j}|, and so there exists an r-cube Q′_{j} ⊂ E′ of diameter ≤ (1 + r)^{1/2}ε_{j} which contains S′_{j}. The product Q′_{j} × S′′_{j} then contains S_{j} and has diameter ≤ (1 + r)^{1/2}ε_{j}. Given μ > 0, we partition Q′_{j} into ([μ^{−1}ε_{j}^{1−s}] + 1)^{r} equal subcubes of edge ≤ με_{j}^{s}. For every such subcube Q̃, we have |Q̃| ≤ r^{1/2}ε_{j}, so that by (∗),

|f(A_{i} ∩ (Q̃ × S′′_{j}))| ≤ (P (η, μ) + M_{i}δ)ε_{j}^{s}.

For d = r + (t − r)/s and P = P (η, μ), we then have

H^{d}(f(A_{i})) ≤ ∑_{j}([μ^{−1}ε_{j}^{1−s}] + 1)^{r}((P + M_{i}δ)ε_{j}^{s})^{d} ≤ (P + M_{i}δ)^{d}μ^{−r}(H^{t}(A_{i}) + δ),

so that tending δ to 0 gives H^{d}(f(A_{i})) ≤ P^{d}μ^{−r}H^{t}(A_{i}). Since the A_{i} are disjoint Borel sets, we have

H^{d}(f(A)) ≤ ∑_{i=1}^{∞} H^{d}(f(A_{i})) ≤ P^{d}μ^{−r}H^{t}(A).  (∗∗)

Proof of Theorem 1. To prove statement (1) of Theorem 1, we note that by Lemma 1.1(1), the inequality (∗) above holds for all η, μ > 0 with

P (η, μ) = (∥f∥_{1} + η + 1)μ.

Since d > r, tending η, μ to 0 in the inequality (∗∗) gives H^{d}(f(A)) = 0.

Similarly, Lemma 1.1(2) implies that when f ∈ Λ_{s} for s > 2, then (∗) holds for

P (η, μ) = (∥f∥_{1} + η)μ + ∥f∥_{s} + η).

Thus, we obtain statement (2) of Theorem 1 by setting η = 1 and tending μ to zero in the equality (∗∗).

Finally Lemma 1.1(3) asserts if A is t-rectifiable, then A \ A_{0} admits a decomposition of the form above for all η, μ > 0 and

P (η, μ) = (∥f∥_{1} + η + 1)μ.

Since A_{0} is t-null, Theorem 1(1) implies that f(A_{0}) is d-null, so that the inequality (∗∗) is valid for all η, μ > 0. Since d > r, we conclude that H^{d}(f(A)) = 0.

□

Proof of Theorem 2. To prove statement (1) of Theorem 2, we note that by Lemma 1.1(1), the inequality (∗) above holds for all η, μ > 0 with

P (η, μ) = (∥f∥_{1} + η + 1)μ.

Since d > r, tending η, μ to 0 in the inequality (∗∗) gives H^{d}(f(A)) = 0.

Similarly, Lemma 1.1(2) implies (∗) holds for

P (η, μ) = (∥f∥_{1} + η)μ + ∥f∥_{s} + η).

Thus, we obtain statement (2) of Theorem 1 by setting η = 1 and tending μ to zero in the equality (∗∗).

Finally Lemma 1.1(3) asserts if A is t-rectifiable, then A \ A_{0} admits a decomposition of the form above for all η, μ > 0 and

P (η, μ) = (∥f∥_{1} + η + 1)μ.

Since A_{0} is t-null, Theorem 1(1) implies that f(A_{0}) is d-null, so that the inequality (∗∗) is valid for all η, μ > 0. Since d > r, we conclude that H^{d}(f(A)) = 0.

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Lemma 2.1. Let F be any Banach space. Suppose s > 1 and a < b < c < d are real numbers, and suppose that each of the intervals [a, b], [b, c], and [c, d] contains at least [s] + 1 zeros of a function f ∈ Λ_{s}(ℝ, F). Then

max |f|_{[b,c]}| ≤ ∥f∥_{s} · (d − a)^{s}.

Proof. If F = ℝ, then by repeated application of the mean-value theorem, our hypotheses imply that the derivatives f^{(1)}, ⋯, f^{([s])} of f all have zeros in [a, b], [b, c], and [c, d]. Thus, for k = 1, ⋯, [s] − 1,

max_{[b,c]} |f^{(k)}| ≤ max_{[b,c]} |f^{(k+1)}|(c − b).

Also,

max_{[b,c]} f − min_{[b,c]} f ≤ max_{[b,c]} |f^{(1)}|(c − b).

Hence,

max_{[b,c]} f − min_{[b,c]} f ≤ max_{[b,c]} |f^{([s])}|(c − b)^{[s]}.

For noninteger s, the proof is finished by noting that

max_{[b,c]} |f^{[s]}| ≤ ∥f∥_{s}(c − b)^{(s−[s])},

which follows from the (s − [s])-Hölder condition satisfied by f^{[s]}.

When s is an integer, it suffices to establish that

max_{[b,c]} |f^{[s]}| ≤ ∥f∥_{s}(d − a).

This is implied by Lemma 1.5.

For an arbitrary Banach space F, we can apply the preceding argument to the function u^{∗} ∘ f, where u^{∗} ∈ F^{∗} is any unit covector. Since ∥u^{∗} ∘ f∥_{s} = ∥f∥_{s}, the lemma follows.

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3. Sharpness of the Theorems

Recall that a modulus of continuity 𝔪 is a class of monotone, continuous functions η : [0, 1] → [0, ∞) which are subadditive, i.e.,

η(s + t) ≤ η(s) + η(t)

for all s, t in some subinterval [0, a) of [0, ∞). Any two members η_{1}, η_{2} of 𝔪 are equivalent in the sense that

C^{−1} ≤ \frac{η_{1}(t)}{η_{2}(t)} ≤ C

for some constant C and all t in some subinterval [0, a) of [0, 1]. A partial order ≺ is defined for moduli of continuity by setting 𝔪_{1} ≺ 𝔪2 whenever

lim_{t→0^{+}} \frac{η_{1}(t)}{η_{2}(t)} = ∞

for any η_{i} ∈ 𝔪_{i}, i = 1, 2.

Fix n, m ∈ ℤ_{+} and let X, Y be Banach spaces. For any nonnegative integer k and modulus of continuity η, we define the subset C^{k,𝔪} of C^{k}(X, Y) to consist of all f such that for each x ∈ X, there exists ε_{x} > 0 and η ∈ 𝔪 such that

∥D^{k}f(y) − D^{k}f(z)∥ ≤ η(∥y − z∥)

for all y, z ∈ B(x, ε). Note that if 𝔪_{1} ≺ 𝔪_{2}, then C^{k,𝔪_{2}} ⊂ C^{k,𝔪_{1}}.

Example 3.1. For fixed α ∈ (0, 1], the modulus of continuity 𝔪_{α} containing the function η_{α}(t) = t^{α} gives rise to the familiar spaces C^{k,𝔪_{α}} of C^{k} functions whose k-th derivatives satisfy a local α-Hölder estimate.

Since (− log(t))^{s}t^{ε} → 0 as t → 0 for any fixed ε, s > 0, the modulus of continuity 𝔪 containing the function η(t) = (− log(t))^{s}t^{α} satisfies 𝔪_{β} ≺ 𝔪 ≺ 𝔪_{α} for all β < α.

△

Proposition 3.2. Fix α ∈ (0, 1]. If 𝔪 is a modulus of continuity such that 𝔪 ≺ 𝔪_{α}, then for any k ∈ ℤ_{+} and t ∈ (0, 1), there exists a C^{k,𝔪} function f: ℝ → ℝ and a t-null set A of critical points of f such that f(A) is not (t/(k + α))-sigmafinite.

To prove the proposition, we will use the following extension lemma. Given a compact subset K ⊂ ℝ, we refer to bounded components of ℝ \ K as gaps of K.

Lemma 3.3. Fix k ∈ ℤ_{+} and α ∈ (0, 1], and let η : [0, 1] → ℝ represent a modulus of continuity 𝔪 such that 𝔪 ≺ 𝔪_{α}. If B ⊂ ℝ is a compact set and f : B → ℝ is a monotone embedding such that

|g(J)| ≤ |J|^{k}η(|J|)

for each gap J of B, then there exists a C^{k,𝔪} extension f: ℝ → ℝ for which B is a critical set.

Proof. Let [a, b] denote the convex hull of B and let J_{i} = [a_{i}, b_{i}] denote its gaps. Set l_{i} = |J_{i}| and define t_{i} = |g(b_{i}) − g(a_{i})|.

Let s: [0, 1] → ℝ be any smooth function such that

1. s > 0 on (1/3, 2/3) and s = 0 otherwise, and 2. ∫_{0}^{1} s(x) dx = 1. Denote by M_{n} the C^{n} norm of s. We construct the derivative r of f by defining

r(x) = \{ \begin{matrix}0 & for x ∈ ℝ \ ∪_{i}J_{i} \\ \frac{t_{i}}{l_{i}} s (\frac{x−a_{i}}{l_{i}} ) & for x ∈ J_{i} \end{matrix}

For each x ∈ B, we again let I_{x} denote the set of gaps of B lying to the left of x. By the definitions above, we have

g(x) = g(a) + ∑_{I∈I_{x}} t_{i}.

Define

f(x) = g(a) + ∫_{a}^{x} r(t) dt.

Then

f(x) = g(a) + ∑_{I∈I_{x}} t_{i} ∫_{0}^{1} s(t) dt = g(a) + ∑_{I∈I_{x}} t_{i} = g(x).

Thus, the function f is a monotone extension of the homeomorphism g.

Evidently f is C^{∞} on the complement of B and near isolated points of B. If x ∈ B is not an isolated point, then for any k there exists a neighborhood U of x such that y ∈ U \ B implies that y lies either in a gap with endpoint x, or in a gap of length ℓ less than |U|. In the latter case, property (1) of the function s implies that either f^{(n)}(y) = 0 or |x−y| ≥ ℓ/3, and so

|f^{(n)}(y)| = |r^{(n−1)}(y)| ≤ M_{n}ℓ^{k−n} · η(ℓ)ℓ3^{k+1}|y − x|^{k−n}η(|y − x|).

Thus, the derivative f^{(n)} tends to zero on approach to A for each n ≤ k. We conclude that f is C^{k,𝔪} from the preceding inequality and the following fact, whose proof is left to the reader.

Lemma 3.4. Let K ⊂ ℝ be compact and f: ℝ → ℝ be continuous on ℝ and C^{k} on ℝ \ K. Suppose that

1. for i = 1, …, k, |f^{(i)}(x)| → 0 uniformly as x → K, and

2. |f(x) − f(y)| ≤ o(|x − y|^{k}) uniformly for x, y ∈ K. Then f is C^{k}.

Finally we note that by construction, each connected component of the critical set of f intersects B, and therefore the critical value set of f equals A.

□

Proof of Proposition 3.2. Given a sequence of positive numbers ℓ_{n} such that

∑_{n=0}^{∞} 2^{n}ℓ_{n} ≤ 1,

we let B(𝔰) denote the Cantor set obtained by an analog of the “middle thirds” construction in which 2^{n} intervals of length ℓ_{n} are removed at the n-th stage.

Fix t ∈ (0, 1) and define sequences by

a_{n} = 2^{−n/t} b_{n} = η(2^{−n/t})^{−1/2}2^{−n/t} c_{n} = η(2^{−n/kt})^{−1/2}2^{−n/kt} d_{n} = 2^{−n/kt},

and let A, B, C, D denote the corresponding central Cantor sets. By Lemma 3.3, the obvious homeomorphism B → C extends to a C^{k,𝔪} map f : ℝ → ℝ containing B within its set of critical set.

We claim that B is t-null, whereas C = f(B) is not (t/(k + α))-sigmafinite. Indeed, this can be checked either by a direct computation, or by applying Lemma 3.3 to find C^{1} maps g, h: ℝ → ℝ such that g, h are critical on A, C, respectively, and g(A) = B, h(C) = D. The claim then follows by an application of Theorem 1 to the mappings g, h.

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4. Morse Criticality in Infinite Dimensions

For 1 ≤ p < ∞, we denote by ℓ_{p} the space of real sequences x = (x_{j}) equipped with the norm

∥x∥_{p} = (∑_{j} |x_{j}|^{p})^{1/p} .

By ℓ_{∞} (resp. c_{0}), we denote the space of bounded sequences (resp. sequences tending to 0) with the usual sup-norm. For E = ℓ_{p} (or E = c_{0}), we will say that a sequence of functions f_{j}: ℝ → ℝ satisfies condition (∗)_{E} if (f_{j}(0)) ∈ ℓ_{p} (or (f_{j}(0)) ∈ c_{0}), and the f_{j} have locally uniformly bounded derivatives, i.e., for each k ∈ ℕ, r > 0, there exists a real constant M_{k,r} such that

|f_{j}^{(k)}(x)| ≤ M_{k,r}

whenever |x| < r.

Lemma 4.1. Fix 1 ≤ p < ∞ and consider a sequence of C^{∞} functions f_{j}: ℝ → ℝ satisfying condition (∗)_{ℓ_{p}} (or (∗)_{c_{0}}). Then the map f: ℓ_{p} → ℓ_{p} (or f: c_{0} → c_{0}) given by f(x) = (f_{j}(x_{j})) is C^{∞} Fréchet differentiable.

Proof. From condition (∗)_{ℓ_{p}} and the mean-value theorem, it follows that

∥f(x) − f(y)∥_{p} ≤ M_{1,∥x∥_{p}+∥y∥_{p}},

i.e., f is a well-defined, locally Lipschitz endomorphism of ℓ_{p}. For each k ∈ ℕ, define f^{{k}}: ℓ_{p} → ℓ_{∞} by

f^{{k}}(x) = (f_{j}^{{k}}(x_{j})).

A simple computation shows that each f^{{k}} is differentiable, and )

Df^{{k}}(x)(y) = (f_{j}^{{k}}(x)(y) = (f_{j}^{{k}}(x_{j})y_{j}).

The linear map T: ℓ_{∞} → L(ℓ_{p}, ℓ_{p}) given by T (x)(y) = (x_{i}y_{i}) satisfies

∥T (x)(y)∥_{p} = (∑_{j} |x_{j}y_{j}|^{p})^{1/p} ≤ ∥x∥_{∞}∥y∥_{p}

and is therefore continuous. By the preceding remarks, we have

Df^{{k}} = T ∘ f^{{k+1}}.

¿From this equation, a simple inductive argument shows that each f^{{k}} is C^{∞} Fréchet differentiable. Specifically, each f^{{k}} is continuous. Assuming that each f^{{k}} is C^{n}, the preceding equation implies that Df^{{k}} equals the composition of f^{{k+1}} with the bounded linear map T, and so f^{{k}} is C^{k+1}.

□

By introducing points of infinite singularity, we can improve the target space of f as follows.

Lemma 4.2. Fix 1 ≤ p < ∞ and consider a sequence of C^{∞} functions f_{j} : ℝ → ℝ with locally uniformly bounded derivatives and f_{j}^{{k}}(0) = 0 for all k ∈ ℕ, j ∈ ℤ_{+}. Then the map f: ℓ_{p} → ℓ_{1} given by f(x) = (f_{j}(x_{j})) is C^{∞} Fréchet differentiable.

Proof. For each j ∈ ℤ_{+}, k ∈ ℕ, our hypotheses combined with Taylor’s theorem imply that

|f_{j}(x_{j}) − f_{j}(0)| ≤ M_{[p]+1,|x_{j}|}|x_{j}|^{p},

and so

∥f(x)∥_{1} = ∥f(x) − f(0)∥_{1} ≤ M_{[p]+1,∥x∥_{p}}∥x∥_{p}^{p}

for all x ∈ ℓ_{p}. Thus, f is a well-defined map from ℓ_{p} into ℓ_{1}. A similar argument shows that for each k ∈ ℕ, then map f^{{k}} defined in Lemma 4.1 is also a well-defined map from ℓ_{p} into ℓ_{1}. Moreover, a simple computation again shows that

Df^{{k}}(x)(y) = (f_{j}^{{k+1}}(x_{j})y_{j}).

Evidently the linear map T: ℓ_{1} → L(ℓ_{p}, ℓ_{1}) given by T (x)(y) satisfies

∥T (x)(y)∥_{1} ≤ ∥x∥_{1}∥y∥_{p}

and is therefore continuous. As in Lemma 4.1, we have Df^{{k}} = T ∘f^{{k+1}}, and the assertion follows by the same argument as before.

□

Example 4.3. Consider the polynomial map f: ℓ_{1} → ℓ_{1} given by the component functions

f_{j}(x) = 3 · 2^{−jα}x^{2} − 2x^{3}.

A check of the definitions shows that the critical set of f contains the Cantor set

C = {(β_{j}) : β_{j} ∈ {0, 2^{−jα}}}

of rank-0 points for f, and

f(C) = {(β_{j}) : β_{j} ∈ {0, 2^{−3jα}}}.

The Hausdorff dimensions of C and f(C) are α^{−1} and (3α)^{−1}, respectively, and both have positive, finite measure in these dimensions.

Denoting by s: ℓ_{1} → ℝ the continuous linear map

s(x) = ∑_{j} x_{j},

we note that C is also contained in the critical set of the composition g = s ∘ f. Thus, the critical value set of g contains

g(C) = {∑_{j} β_{j} : β_{j} ∈ {0, 2^{−3jα}}} ,

which has positive, finite Hausdorff measure in dimension (3α)^{−1}. Finally, we note that in the special case of α = 1/3, the set g(C) equals the unit interval [0, 1].

References

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[2] Federer, H., Geometric Measure Theory, Grundlehren Math. Wiss., vol. 153, Springer, New York, 1969.

[3] Moreira, C.G., Hausdorff measures and the Morse-Sard theorem, preprint, IMPA, 1994.

[4] Morse, A., The behavior of a function on its critical set, Ann. Math. 40 (1939) 62–70.

[5] Norton, A., The Zygmund Morse-Sard theorem, J. Geom. Analysis 4 (1994) 403–424.

[6] Sard, A., Images of critical sets, Ann. Math. 68 (1958) 247–259.

[7] Hausdorff measure of critical images on Banach manifolds, Amer. J. Math ************* [8] Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press,

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[9] Yomdin, Y., The geometry of critical and near-critical values of differentiable mappings, Math. Ann. 264 (1983) 495–515.

Department of Mathematics, Columbia University, New York, NY, 10027, USA

IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brasil

E-mail address: smb@math.columbia.edu

E-mail address: gugu@impa.br