bernstein’s inequality and the resolution of spaces of analytic functions

BERNSTEIN’S INEQUALITY AND THE RESOLUTIONOF SPACES OF ANALYTIC FUNCTIONS

CHARLES FEFFERMAN AND RAGHAVAN NARASIMHAN

0. Introduction. The growth and smoothness properties of polynomials F onare controlled by standard inequalities of the following type:

Csup IVFI -- sup Ifl (0.1)

sup IFI < C sup IFI (0.2)Btx, p) B(x,p/2)

sup IFI <- IF(y)I dyl(x,p) pn (x,p)

(0.3)

sup IFI < C sup IFI. (0.4)Bc(X, t) B(x, t)

Here, C denotes a constant depending only on the degree of F, while B(x, p),Be(x, p) denote the ball, with center x and radius p in R and C", respectively.We call inequalities (0.1)-(0.4) the Bernstein inequalities.In this article, we shall show that if Vx is a finite-dimensional vector space of

real analytic functions of n variables depending real-analytically on a parameter2 Rm, then the Bernstein inequalities (0.1)-(0.4) continue to hold for F e V,locally uniformly with respect to 2. Thus, our first main result is as follows.

BERNSTEIN THEOREM. Let F,, Fs, z be holomorphic functions on the com-plex ball Bc(O, 1 + e), e > O, in C dependin# real-analytically on 2 U R (U anopen set). Let Vz be the linear span of the Fk, z, 1 < k < N.

Then, for any compact set K U, there is a constant C > 0 such that the Bern-stein inequalities (0.1)-(0.4) hold for any F Vz, 2 K, and B(x, p) B(O, 1).

For example, if 2 (1, N) (R")N and V span{e<,’x>, e<N’x>}, theabove theorem asserts that (1)-(4) hold for F V with a constant C dependingonly on upper bounds for Ixl, p, Ixl, I1.The difficulty in proving these uniform estimates comes from the fact that V

may degenerate (i.e., that its dimension may drop). This difficulty arises already ifwe attempt to prove (0.1)-(0.4) for a single vector space Vo because the estimates

Received 5 August 1994.Fefferman supported in part by a grant from the National Science Foundation.

77

78 FEFFERMAN AND NARASIMHAN

involve the parameters x, p, which play the same role as 2 in our proof (and,except in trivial cases, the spaces involved certainly degenerate at p 0). To over-come this difficulty, we prove a result which "resolves the degeneracies of V."This theorem, stated below, is of independent interest.

Before stating the theorem, we give some definitions and a simple example.With {Fk, a} as above, let Gl,x, GM, x be holomorphic functions on Be(0, 1 + s)depending real-analytically on 2 U. We say that the family {G,x, GM,z}resolves {F,x,..., FN, x} if it has the following properties.

(a) For each 2 U, G,x, Gu, x are linearly independent.(b) span{F,, FN,} c span{Gl,, G,} for all 2 U.(c) There is a nowhere-dense analytic subset A of U such that, for 2 U\A, the

{Fk,} and the {Gj, x} span the same vector space.For example, let Fx,x(x) 1, F2,x(x) e<z’x>, x, 2 Rn, and let Vx span{Fx,x,

F2,x }. Then dim Vx is constant for 2 :/: 0, but drops when 2 0.If n 1, we can resolve {Fx,x, F2,x} simply by taking G,x 1, G2,(x)=

(1/2)(ex 1) for 2 - 0, G2,0(x) x.However, when n > 1, this construction makes no sense, and, in fact, it is not

hard to see that {F,, F2,} cannot be resolved in the above sense. One can getover this difficulty by performing a blow-up in the A-space. More precisely, write2 in polar coordinates 2 z09 with z R1, 09 S"-1. Pulling back F,x, F2,; weget functions

F,,(,,) =- 1, F,(,,o,)(x) e’<’’’>,

and we can now resolve this system by taking the functions

G,t,,,o 1, G2,(r,o)(x _1 (e,<,,x 1) for z -: 0, G,o,)(x) <o, x>.

Our main theorem on resolution asserts, in effect, that this is the case in gen-eral, i.e, that a general system {F,,a, Fs, a } can be resolved after a succession ofblowings-up of the 2-space. To state it, we introduce one more definition.

Let U c R= be open and let 2o U. A dominating family at 2o is a finite collec-tion (U, K, n) < .<o where

(i) U R is open, K U is compact;(ii) zr- U ---} U is real analytic and the jacobian det rr’ does not vanish on any

nonempty open set;(iii) the images n(K) cover a neighborhood of 20.Given such a dominating family and a system {Fk, x}<.k<. of functions param-

etrized by 2 U, the pull-back of the system to U is the system {Fk,%(0) <k<Nparametrized by ( U.

1Fefferman has found applications of this resolution theorem to the study of the training of neuralnetworks, which he will treat in a later paper.

SPACES OF ANALYTIC FUNCTIONS 79

Our second main result is the following.

RESOLUTION THEOREM. Given a finite system {Fl,x, FN,x} of holomorphicfunctions on a ball in C parametrized real-analytically by 2 U c Rm, U open, anda point 2o e U, there exists a dominating family (U, K, 7r) <<o at 2o such that,for each , the pull-back of the system to U can be resolved.

We shall obtain both these results as special cases of results in which we regardfunctions of two sets of variables (z, 2) as Banach-space-valued functions of 2.These are stated precisely in Section 1 and Section 6, respectively.Uniform Bernstein inequalities for certain families of algebraic functions played

an important role in Parmeggiani’s work [P] on subunit balls for the symbol of apseudodifferential operator. They were first proved in our papers [FN1], [FN2],in which they were obtained as consequences of an extension theorem. The exten-sion theorem of [FN1], [FN2] asserts that polynomials can be extended fromcertain algebraic varieties in R" to rational functions with bounds uniform in thevariety (while they cannot always be extended to polynomials with such uniformbounds). The Bernstein inequalities proved in the present paper contain the Bern-stein inequalities proved in [FN1], [FN2] as special cases, although the resultsproved here do not imply the extension theorem mentioned above.

1. Definitions and statement of the Main Theorem. We give here the defini-tions used in the rest of the paper and state the Main Theorem on resolving thedegeneracies of spaces of analytic functions. The terminology is slightly differentfrom that used in the introduction.

Definition 1.1 Let n > 1. Suppose that U is an open set in C and that XoU c R. A dominating family at Xo is a finite collection (U, K, n)l<<o where,for each v, the following conditions hold:

(a) U is an open set in C.(b) K is a compact subset of U c R.(c) n is a holomorphic map of U into U.(d) The jacobian det n’ does not vanish identically on any (nonempty) ball in

u.(e) ) <<o n(K) is a neighborhood of Xo in Rn.

Note. We are not assuming that the maps n are generically injective (as onedoes with blowing up) but only that they are generally finite-to-one (on any com-pact subset of U).

Definition 1.2. Let H be a (complex) Banach space; let U c C" be open. Sup-pose that f, fN: U H are holomorphic maps. A resolution of {fl, fs} isa finite set of holomorphic maps g, /M: U H with the following properties:

(a) For each z U, the values (z), gM(z) are linearly independent vectorsin H.


(b) For each z U, we have

span {fl (z),..., fN(z)} c span {g (z), gM(z)}

(c) There is a nowhere-dense analytic subset V U such that, for each zU\ V, we have

span {g (z),..., gM(z)} c span{f (z),..., fN(z)}.

We also say that the system {gx, gu} resolves {fx,..., f}.Note. The condition on the analytic set V is equivalent to the condition that V

have codimension > 1 at each of its points.

Definition 1.3. (a) Let U, H, f, fs be as in Definition 1.2. We say that{f,...,f} is resolvable (on U) if there exists a resolution {g,..., gu} of{f, f).

(b) If xo e U r R", we say that {fl, f} is potentially resolvable at xo if thrxists a dominating family (U, K, rt).<.<o at xo such that, for azh , thfamily of maps {f, o rt, f o r} of U into H is resolvabl (on U).

Remark 1.1. By convention, the empty family of vectors has span {0}, so thatth singl map 0 is resolved by the empty family.

reso ve {f,,..., v ,< tat,Definition 1.2(b) and (z), span{g (z),..., g(z)} span {f, (z),..., f(z)} for somz e U, while, by (a), span {g (z),..., g(z)} has dimension M for all z e U.)

We can now state our Main Theorem.

MAIN THEOREM. Let H be a complex Banach space, let U be open in C, and letf,, f: U -- tt be holomorphic maps. Then, for any xo U r R, {f,, fN} ispotentially resolvable at Xo.

Note that we recover the Resolution Theorem as stated in the introductionfrom this result. If F: f x U- C is a real-analytic function, (f Cs, U R)such that, for fixed 2 U, the function w - F(w, 2) is holomorphic, then, forfo f and 2o 6 U, there is a neighborhood Uo of 2o in C" such that FIfo x(Uo c R) extends to a holomorphic map of fo x Uo into C. Thus, to obtainthe Resolution Theorem of the introduction, we can take H to be the space ofbounded holomorphic functions on the unit ball.

2. Elementary remarks

LEMMA 2.1. Let H be a complex Banach space, let U C be an open set,and let f, fs: U H be holomorphic. Suppose that (U, K, rr)t<<o is adominating family at Xo U R such that, for each and each y K, the family{ft o r, f o rc} is potentially resolvable at y. Then {ft,..., fs} is potentiallyresolvable at Xo.


Proof. For 1 < v < Vo, Y e K, we can find a dominating family (U’y), K’y),’)t<r<rot, at y, such that

("Y) } is a resolvable family of maps of U’y) to H.{f o n:, o 7r’y) f o n, o

(2.1)

(’Y)" U(’Y) U. isNow U’y) is open in C’, K’Y)c U’Y) R" is compact,holomorphic and

det(u’Y)) 0 on any nonempty open set (2.2)

while

J (’Y) (-,y)) W(V,y) W(V,y)r (K where is an open neighborhood of y in Rn

Thus, {W(r’Y)}yr is an open covering of the compact set K.

Let Y c K be a finite subset such that n(’Y)(K(’Y)),y Y, 1 < < o(, y)cover K.

(2.3)

(2.4)

We claim that

{U’y), K(’y), n o n,y): 1 < v < Vo, y Y, 1 < , < o(v, Y)}is a dominating family at Xo. (2.5)

Properties (a), (b), and (c) in the definition of dominating family (Definition 1.1)are obvious, while (e) follows from (2.4) above. Property (d) is a consequence ofthe following simple remark:

If ft, "2, "3 are open in Cn, and 1" fl "2, 2" ’2 f3 are holomorphicmaps such that det 0 on any nonempty open subset of fl, 1, 2, thendet(u2 o )’ 0 on any nonempty open subset of f. (2.6)

In fact, if # Wx = f is an open set and B1 c W, Bx - tZI is an open setwith det (z) # 0 for all z B, then r(B1) Wz = fl. is a nonempty open setin fz, so that it contains a point w (z), z e B. for which det n(w) 0. Hence,

det(z2 o u)’(z) det r[(w). det ri(z) - 0.

From (2.5) and (2.1) above, we see that {f, fN} is potentially resolvable atXo, proving the lemma.

LEMMA 2.2. Let H be a complex Banach space, and let U C be open. Sup-pose that fl, f2" U -- H are holomorphic maps such that IIA(z)ll < 1/2llf (z)ll for allz U. Then, if {f ) is resolvable, so is {f: + f2 ).


Proof. We may assume that ft 0 on U. Hence, by Remark 1.2, a resolutionof {ft } consists of one map o:U- H. The conditions (a)-(c) of Definition 1.2become: t: U H is holomorphic and 9t (z) # 0 for each z U; f (z) 0(z)g (z),tp(z) C for z U; and (z) # 0 for z U\ V, V being a nowhere-dense analyticsubset of U. It is clear that z - o(z) is holomorphic on U. Now, the map U\VH given by

1a() --(h (z) + f(z))

tpz)

is holomorphic. If z z U\ V, we have

1 211fx(z)llI[B(z)ll < lift(z) + f2(z)[I < 2llgl(Z)ll,

I(z)l I(z)l

and [[gt is bounded on any U’ c c U, so that g extends to a holomorphic mapof U to H, which we again denote by g. Clearly,

fl+f2=qgg onU.

To check that fl + f2 is resolvable, it is sufficient to show that O(z) q: 0 for allz U. If z U\ V, we have

1 IIf(z)ll 1 II0(z)x(z)ll 1II(z)ll -Io(z)l IIA(z) + A(z)ll .]0ii I(z)l Elle,(z)ll.

Since , #t are holomorphic on U, it follows that

1II(z)ll IIx(z)ll > 0 for all z e U.

COROLLARY 2.1. Let U, H be as in Lemma 2.2, and let ft, f2" U H be holo-morphic maps with IIA(z)ll < (1/2)llft(z)ll for all z U. If {ft } is potentially resolv-able at Xo U c R, then so is {f + f2 }.

Proof. If (U, K, n)l <<o is a dominating family at Xo such that {fto rr}is resolvable for each v, then so also is {(ft + f2)o r}, since IIf2(o(z))ll <(1/2)llf((z))ll for z U.LEMMA 2.3. Let H be a complex Banach space, U C, an open set. Let ft,

fs and Ft, F be holomorphic mapsfrom U to H. Let Xo U c R be given.Assume that

span{fl(z), fs(z)} span{F(z), F.(z)} for all z U, (2.1)


and

span {Fx (z), Fz(z)} = span{fx(z), fs(z)} for z U\ V, (2.2)

where

V U is a nowhere dense analytic subset. (2.3)

Then, if {Fx, F.} is potentially resolvable at Xo, so also is {fa,..., fN}.

Proof. By assumption, we can find a dominating family (U, K, rr)x <<o atXo, and, for each v, a finite family of holomorphic maps Oj: U H, j 1, J,such that {O,}.<s resolves {Fx o r, Ft. o z} on U (for each v). We shallcheck that {Oj}<s also resolves {f o r, fN o n}.By the definition of resolution and condition (2.1), we have

span{f, o n,(z),

= span{o(z)}<s for all z U. (2.4)

There is a nowhere dense analytic set V = U such thatspan{o,(z)}<.s span{F o rr(z), Fz o n(z)} for z e U\ V. (2.5)

Hence, from (2.2) and (2.5), we see that

span{o(z)}j span{f o n(z), fu o n(z)} for z U\W, (2.6)

where

W, V nS(V). (2.7)

Clearly, W is an analytic subset of U,. We have only to check that W isnowhere dense in U. Since V is nowhere dense, it is sufficient to check thatnx(V) is nowhere dense. Now, by condition (d) in the definition of a dominatingfamily (Definition 1.1), the set

A {z Uldet r’(z)= 0}

is nowhere dense. Moreover, the map zrl U\A U is open, so that, since V isnowhere dense in U, n-x(V)(U\A) is nowhere dense in U\A. Since A is no-where dense in U, it follows that n-x(v) A w (n-x(V) c (U\A)) is also nowheredense.

Finally, by condition (d) of Definition 1.2, {0(z)}<s are linearly independentfor each z e U. Thus, (2.4), (2.6), and the fact that W is nowhere dense in U


show that {gj}j<s is a resolution of {fl o n, f o n}, completing the proofof the lemma.

3. Blowing up to normal crossings. Our proof of the Main Theorem uses thefollowing theorem in an essential way. The theorem is contained in Hironaka’sresults on desingularization. A direct proof of the result is contained in the articleof Bierstone-Milman IBM]. (See also the remark following the statement.)

If (,..., ), fl (fl,..., fl) are multi-indices (j, flj > 0 being integers),we write <fl (or fl>) to mean that <fl for all j. As usual, if z--(z,..., z) C and (,..., ) is a multi-index, z denotes z...z,.THEOREM 3.1. Let U c C" be open, let Xo U R, and let f:, fN: U -, C

be holomorphic functions. We assume that no f vanishes identically in a neighbor-hood of xo. Then there exists a dominating family (U, K, 7r)1 <<o at xo and, foreach z,, there exist multi-indices , such that the following hold:

f o n(z) O,(z)z’o for z u, 1 < j < N, (3.1)

where

0, is holomorphic and nowhere zero on U. (3.2)

Moreover,

for fixed , the multi-indices 1, are totally ordered(i.e., for 1 < i, j < N, we either have < xi or < ). (3.3)

In particular, ifjo is a value ofj for which is minimal, (3.1) and (3.2) imply

for 1 < j < N, the function f o r/f o rc is holomorphic on U(so that the ideal 9enerated by the f o r is principal). (3.4)

Remark 3.1. When N 1, this theorem is contained in Theorem 4.4 of Bier-stone-Milman IBM]. Implicit inl their proof is a derivation of the general casefrom the case N 1. In view of IBM, Lemma 4.7-1, it is sufficient to apply thecase N 1 to the function

fl...f H (f,

the product being taken over those pairs (i, j) for which f f near Xo.

4. Reduction to the case of a single map into H. We now begin the proof ofthe Main Theorem. We shall use induction on the dimension n. When n 0,U c C consists of a single point, and the Main Theorem simply asserts the fol-


lowing: If fl, fN H, there exist linearly independent vectors 91,..., 9M Hsuch that span {0, Ou} span {f, fs}.

This being true, the Main Theorem holds in dimension zero, and we can startthe induction. From now on, we fix n, and assume that the following is true.

INDUCTION HYPOTHESIS 4.1. The Main Theorem holds for all complex Banachspaces H, and all holomorphic maps of any open set in C"-1 into H.

Our task is to prove the Main Theorem in dimension n. Most of the work goesinto proving the following.

MAIN LEMMA. Let H be a complex Banach space, U c C", an open set. Letf: U - H be a holomorphic map. Then, under the Induction Hypothesis 4.1, {f} ispotentially resolvable at any point Xo U c R".

In this section, we shall show that the Main Lemma implies the Main Theo-rem; in the next section, we shall prove the Main Lemma itself.To reduce the Main Theorem to the Main Lemma, we use induction on N, the

number of functions in our family. When N 1, the two statements coincide.Now fix N > 2 and assume the following.

INDUCTION HYPOTHESIS 4.2. The Main Theorem in dimension n holds forfamilies of at most (N 1) functions fl,

We shall prove the Main Theorem for families fl,..., fu: U H. By our as-sumption in the Main Lemma, there is a dominating family (U, K,at Xo such that, for each v, the singleton {fu o } is resolvable on U. If{gl, g,M,} resolves {fN o }, we have M 0 if f, o/1: 0 on U, andM. 1 otherwise.

If M. 0, then {fl o ,..., fN o } is potentially resolvable at any point ofU R" because of the Induction Hypothesis 4.2 (since fu o . 0).

Suppose, therefore, that M # 0. Then {fu o } is resolved by a singleton{F.}, F: U H being holomorphic and nowhere zero. By definition, we havespan{f o n} c span{F} on U while span(F)c span{f o rr} on U\ V, Vbeing a nowhere dense analytic subset of U.We want to show the following.

Statement 4.1.U c R".

{f o n, fs o r} is potentially resolvable at any point of

In view of Lemma 2.3, this is equivalent to the next statement.

Statement 4.2. {fl o r,...,fN_l o n, F} is potentially resolvable at any pointy U c R" (i.e., we may assume that the last function in the family is nowherezero).

We proceed to the proof of Statement 4.2. Given y U R", we can pick aclosed subspace H = H (of eodimension 1) complementary_to F(y) (si_nce F(y) 4:0). It follows that H is complementary to F(z) for all z U, where U is a small


open neighborhood of y in U. Thus, we can write

f o o(z) q(z)F(z) + f(z), zeUs, 1 <j < N- 1,

where o(z)e C and (z)e H. Moreover, j .is holomorphic on t. Since

span_{f o(z), f_: oz,(z), F(z)} span{f,(z), f_,(z), F(z)} forz e U, we see that the following is equivalent to Statement 4.2.

Statement 4.3. {fl, fN-,, F} is potentially resolvable at y.

To prove Statement 4.3, we use the Induction Hypothesis 4.2 for the functions{fx,..., f_x,} (co_nsidered as mappings from to^H).. Thus, for a given(and the given y e U), there is a dominating family (Ur, Kr, r)-.<r<ro at y suchthat {fl o r fN-l, r} is resolvable on r" In other words, for 1 < y < Yo,we can find holomorphic maps G,r" Or H, 1 <j < J(v), such thatis a resolution of {j, o r}We now claim the following.

Statement 4.4. The family {G,r}<<s(r)w{F o r} is a resolution of {fa o r,fv_,,, o , F, o tr} on 0r"

Clearly, Statement 4.4 implies Statement 4.3 which, as remarked above, isequivalent to Statement 4.1.

First, for fixed v, y and z s 0r, the Gr(z are linearly independent vectors inH, which is complementary to F o r2r(z), so that the vectors Gr(z), 1 < j < J(),F o tr(z in H are linearly independent, thus verifying condition (a) in Definition1.2.

Next, by condition (b) in the definition of resolution (Definition 1.2), we have,for any z r, span{j, o r(z), 1 < k < N 1} c span{Gr(z), 1 <j < J()}, sothat

span {f, o (z),..., fu_,,, o (z), F o (z)= span{G(z), F, o (z), 1 <j< J(y) },

which is condition (b) of Definition 1.2, for our systems of functions.Finally, there is a nowhere-dense analytic subset c 0 such that, for z e

Ur\ V, we have

span{G#(z), 1 <j < J(y)} c span{j, o r(z), 1 < k < N 1},

so that, for z Ur\ Vr,

span {Gr(z), F, o r(z), 1 < j < J(V)}

span{f,. o err(z), 3-, o r(z), F, o err(z)},

verifying Definition 1.1 (c) for our systems of functions.


This proves Statement 4.4, and hence also Statement 4.1. Thus, for each v

(whether M 0 or 1), {fl o , f o } is potentially resolvable at everypoint of U,, c R". An application of Lemma 2.1 now shows that {fl, f} ispotentially resolvable at Xo, which is the Main Theorem in dimension n.

This completes the induction, reducing the Main Theorem to the Main Lemma(i.e., the case of a single map U H).

5. Proof of the Main Lemma. We fix the dimension n > 1. Our inductionhypothesis throughout this section is the following:

INDUCTION HYPOTHESIS 5.1. The Main Theorem holds in dimension n 1.

Our goal is to prove the Main Lemma. Thus, let H be a complex Banach space,and U c C" an open set. We are given a holomorphic map f: U H and a pointXo e U R". Our aim is to prove the following.

Statement 5.1. {f} is potentially resolvable at Xo.We may assume that f is not identically zero in any neighborhood of Xo.

Let H* be the dual of H. For each continuous linear functional h* e H*, letV(h*) {z Ulh* o f(z) 0}. Clearly, V(h*) is an analytic set in U and, if V(z Ulf(z) 0}, we have V (’]h*n* V(h*). Now, if F is any family of analyticsets in U and U’= = U, there is a finite subset Fo = F so that (’]aAo U’=0a lo Ac U’. Hence, shrinking U if necessary, we may assume that:

We have finitely many linear functionals h’, h H* such that if z U,then f(z) 0 if and only if h*;(f(z)) hy(f(z)) 0. (5.1)

We may also drop any h’ for which h* o f vanishes identically in a neighbor-hood of Xo. Thus, we may assume:

In (1), h o f has a nonzero germ at Xo for each j 1, J. Moreover,IIh’ 1 for each j (norm in n*). (5.1’)

Since f 0 near Xo, we clearly have J > 1. We apply the blowing-up theorem,Theorem 3.1, to hj o f, j 1, J. Thus, there exists a dominating family(U, K, rr)l <<o at Xo such that, for each , the following holds.

Statement 5.2. We can find multi-indices 1, s and holomorphic (scalar)functions 01, Os which are nowhere zero on U and have the followingproperties.

and

h o f o n(z) Oj(z)z’J for all z e U, (5.2)

the multi-indices x, s are totally ordered. (5.3)


By shrinking U slightly (so as to still contain K), we may assume that there isa c > 0 so that 10(z)l > c for z U, 1 < j < J. Fixing v, let t be the smallest ofthe 09, say 0 o" Then, by (5.2) and (5.3), h o f o n,(z) 0 for 1 < j < ./if andonly if z 0(z U). Together with (5.1) this gives:

for z U, we have f o n(z) 0 if and only if z 0; (5.4)

IIf o rr(z)ll Iho o f o n(z)l > clzl, z U (and c > 0, c < inf 10o(Z)l.) (5.5)

We shall show that f o z is potentially resolvable at any point y e U c R".This, and Lemma 2.1, imply the Main Lemma, which is our goal.Dropping the subscript v and the notation used so far, we see that the Main

Lemma is thus reduced to proving the following statement.

Statement 5.3. Let H be a complex Banach space, and let U c C" be an openset. Suppose given a holomorphic map f: U H having the following properties:

There is a multi-index 0 and a constant c > 0 such that IIf(z)ll c Izlfor all z U. (5.6)

If z U and z 0, then f(z) O.

Under these hypotheses, f is potentially resolvable at any point Xo U c R".

Fix Xo U c R", and let (Xox, Xo,) be its coordinates. Let (x, 0,).By a permutation of the coordinates, we may assume that, for a certain integer m,0 < rn < n, we have the following:

Xo=0 and >0 forl<j<m (5.8)

while

eitherxo-0 or 09=0 form<j<n. (5.9)

If we replace U by a smaller neighborhood of Xo, by (01, 0,, 0, 0),and c by a smaller constant > 0, (5.6) and (5.7) still hold with in place of 0 sincez’= I-I<,,,z]JI-I>,nZ]J, and the second product is bounded away from 0 in aneighborhood of Xo. Thus, in place of (5.9), we may assume the following:

for j > m. (5.10)

Now, if m 0, then (5.6) and (5.10) imply that f(z) # 0 for all z U and {f}itself is a resolution of (f}. We may therefore assume that rn > 1.


Let S =Sm be the set of permutations a: {1,..., m} {1,..., m}. Given e > 0and tr S, we define the following objects.

u, {(1, ,) C=lljl < forj > m, Ijl < 2 for 1 <j < m}. (5.11)

K,, {(1, .) 6 R"III < /2 forj > m, Il < 1 for i <j < m}. (5.12)

r,: U,, --, C" is the map r,(, .) Xo + (z, z.), (5.13)

where

z for j > m, (5.14)

Also, we.set

z,t)= I-I (i for l <j<m. (5.15)< <

E,,, {Xo + (Zl, z,) C’[Izl < 2"5 for allj and

for 1 <j < m}, (5.16)

Fo, {Xo + (xx, x.) R"I Ixl < 5/2 for all j and

Ixl < Ixj/l)l for 1 < j < m}. (5.17)

As is easily verified, if is small enough, we have

(5.18)

Clearly, U,., is open in C", K,. c U,. c R" is compact, and n, is holomorphic.Moreover, if e is small enough, then E,. c U (since U is open), so that, by (5.18),r, maps U,., into U. Also, (5.18) and the definition of F,. imply that r,(K,.)contains a small open neighborhood of Xo in R". Furthermore, det r(() 0 if# 0,..., m #0.Thus, we have

If e > 0 is small, then (U,., K,.,, n,),s is a dominating family at Xo e U. (5.19)

By Lemma 2.1, the Statement 5.3 above is reduced to proving that

for each tr S, {f o n,} is potentially resolvable at each point y K,,(if e is small enough). (5.20)

Fix a S, let N ]a], and let P,(z) be the Nth-order Taylor expansion of f(z)


in the variable go-(l) about z,a) 0; thus,

N

P(z) fk(Z)Za) (5.21)k=O

where

fk is a holomorphic map of a neighborhood of Xo into H anddepends only on the coordinates z with j a(1). (5.22)

Moreover,

IIf(z) e(z)[I Clztx)l+x in a neighborhood of Xo where C > 0 is a constant.(5.23)

Let z e E,,; then z Xo + w with Iwl < 2he for all j and Iwox)l 2mlW[ for allj < m. Since, by (5.8) and (5.10), Xo 0 for j < m and 09 0 for j > m, we have

z= w and IZotl)l IWotl)l.

Hence, since N I1, there is a constant Ca > 0 so that

Izx)lTM CllZl. (5.24)

Using (5.23), (5.24) and our hypothesis (5.6) (that IIf(z)ll c lz]), we concludethat, if e is small enough, then

1IIf(z)- P(z)ll IIP(z)ll for z e E,,.

Hence, because of (5.18) we obtain

IIf o zr=(() e o r=(()ll 1/2 IIe o r(()ll for all ( e U,,(if e > 0 is sufficiently small). (5.25)

We now claim that Statement 5.3, and hence also the Main Lemma, are both aconsequence of the following.

Statement 5.4., {po o ro} is potentially resolvable at every point of Uo, c R".

In fact, this statement and (5.25) enable us to use Lemma 2.2 to concludethat (f o no } is potentially resolvable at every point of Uo, R". Since (U,,, Ko.,zr,)os is a dominating family at Xo U, Lemma 2.1 then shows that {f} is poten-tially resolvable, which is exactly Statement 5.3.


Thus, to prove the Main Lemma we have only to prove Statement 5.4 above;we now turn to this.From the definitions (5.13)-(5.15), we see that each zj with j a(1) depends

only on (2,..., (N, (not on (1), while z,tl) (... (,,. Hence (5.21) and (5.22) showthat P o n, is of the form

N

P, o n,(() Fk((2, (U)(k, ( U,., (5.26)k=0

where, if U’= ((2, (s) c- I11 < 2, 2 <j < m, I1 < e for m <j < n,j > 2},

Fk is a holomorphic map of U’ into H. (5.27)

In view of the form of P, o r, given by (5.26) and (5.27), we see that Statement5.4, and hence also the Main Lemma, are both consequences of the followingresult.

Statement 5.5. Let U {(gl,..., g.) 6 C"llz l < 1, 1 < j < n} and U’ {(Zl,Z_l) c-lllzl < 1, 1 <j < n- 1} be the unit polydiscs in C", C"- respec-

tively. Let N > 0 be an integer, and let fo, fs" U’ H be holomorphic maps.Set

N

f(z) A(z, z_)(z,,)k, Z (Z1,... Zn) 6-. U.

Then {f} is potentially resolvable at any point of U c R (under the inductivehypothesis that the Main Theorem holds in dimension (n 1)).

To prove Statement 5.5, fix Xo U c R, Xo (x, x,) R-1 x R. By theinductive hypothesis (applied to the coefficients fo, fN above), there is adominating family (U’, K’, ’)l<<o at x (relative to U’) so that {fo o ’,fu o n’ } is resolvable on U’. Set

u u’ {z CIIzl < 1}, K K’ -, r n’ x (identity): U U.

Clearly, (U, K,n)<<o is a dominating family at Xo (relative to U). By Lemma2.1, it is sufficient to prove that

Now

{f o n} is potentially resolvable at every point of U c R". (5.28)

N

f o r(z,..., z,) fk(Z’)zk,, Z’ (Z,..., Z,_I), (5.29)k=O


where

A A o (5.30)

Moreover, the family {j, fn} is resolvable on U’, so that we can find holo-morphic maps G, Gu: U’ H such that

G (z’), Gu(z’) are linearly independent, for any z’ U’, (5.31)

and

Mfk(Z’) , tPk, m(Z’)Gm(Z’), Z’ U’, k 0,...,N; (5.32)

m=l

here, the tp.,(z’) are scalar-valued functions on U’. They are uniquely determinedholomorphic functions on U’ because of (5.31). We shall abuse notation and re-gard G,,, qg, also as holomorphic functions on U (which do not depend on thelast coordinate z).From (5.29) and (5.32), we have

M

f o 7,(z)= m(z)Gm(z), (5.33)

where

I]m(Zl, Z,) Z q)km(Zt)Zkn’ Z (z, z._x). (5.34)k=0

Thus, to prove Statement 5.5, and hence also the Main Lemma, we have onlyto prove the following.

Statement 5.6. Let H be a complex Banach space, let U = C" be open, and letGx, Gu: U--, H be holomorphic maps such that, for any z U, Gx(z),G(z) are linearly independent.Then, if kx, qt: U C are any (scalar) holomorphic functions, the single-

ton{q,G,} is potentially resolvable at any point of U c R".

We drop the notation used so far and prove Statement 5.6. By the blowing-uptheorem, Theorem 3.1, part (4), there exists a dominating family (U, K, n)x <<oat Xo such that, for each z,, the functions q/m o 7r have the following property.

There is an index too, 1 < mo < M, such that if we set O(z) ’mo o Try(z), thenfor each m, we have

m o 7r, O" q)m, q9m holomorphic on U (5.35)


and

Pmo 1. (5.36)

Clearly, we may assume that mo 0 on any nonempty open set in U, so that

V {z e UlO(z) 0} is a nowhere dense analytic subset of U. (5.37)

Thus, if we set

M

G(z) tpm(z)Gm o n(z), z U, (5.38)m=l

we have

O(z)G(z) (,: @,G,) o z(z), z U. (5.39)

Further, since the {Gin(z)} are linearly independent and qgmo 1, we see thatG(z) v 0 for all z U. Hence, (5.37) and (5.39) show that {G} is a resolution of

This proves Statement 5.6 above. As remarked earlier, this also completes theproof of the Main Lemma and, consequently, (as shown in Section 4) proves theMain Theorem stated in Section 1.

6. Abstract Bernstein inequalities. Let H1 and H2 be complex Banach spaces.Suppose that we are given a linear, continuous, injective map

j: H1 - H2

We keep these data fixed throughout this section.

Definition 6.1. Let f, fN: U H1 be holomorphic maps defined on anopen set U c C". We say that {fl, fN} has the Bernstein property if the follow-ing holds:For any compact set K c U c R", there exists a constant Cr > 0 such that

whenever z K and v e span {fx (z),..., fs(z)}, we have

vii c Iljvll.

(Of course, Ilvll is the norm in H, Iljoll that in H2.)

We shall use the Main Theorem to prove the following.

ABSTRACT BERNSTEIN TI-mOREM. For any open U C" and holomorphic mapsf, f: U H, the system {fl, fs} has the Bernstein property.


We begin with the following preliminary result.

LEMMA 6.1. Let U C be open. Let f, fs: U --, H be holomorphic maps.Suppose that {f(z), fs(z)} are linearly independent for each z U. Then(f, fN} has the Bernstein property.

Proof. Let K c U c R be compact, and let Xo K. Since Jfl(xo), jfs(xo)are linearly independent, there exists a constant C1 > 0 such that

N

Ial Cxk=l

(All norms on C are equivalent.) Now, fl,-.., fs are continuous at Xo and j iscontinuous by hypothesis. Hence, there is a neighborhood Uxo of Xo in U suchthat

1Ilj(fk(z)) --j(fk(X0))ll -- for z e U,o k 1,..., N (6.2)

and

Ilf(z)ll c2 for z Uxo C2 > 0 being a constant. (6.3)

From (6.1) and (6.2), we get, if A 1,..., As C and z e Uxo,

k=l+ Ihkl Ilfk(z) fk(Xo)ll

k=l

1 N

k=l

so that

IAkl < 2Cl j Akfk(Z) for all A1,..., As e C, 2 6 Uxo. (6.4)k=l k=l

From (6.3) and (6.4), we obtain

N [INx hA(z) C Z Ihl

< 2C1C2 for A, As C, 7,

_Uxo. (6.5)


Covering K by finitely many Uxo, Xo e K, as above, we conclude that there isCr > 0 so that

N

Akfk(Z)k=l

for all A1,..., AN e C and all z K. This is the Bernstein property for fl,

COROLLARY 6.1. Let U c C" be open and let fl, fN: U H1 be holo-morphic. If {f1,..., fN} is resolvable, then it has the Bernstein property.

Proof. If {gl, gM} resolves {f, fN}, then, by definition, span{f (z),fN(z)} c span{gl(z), gM(Z)} for all z U. Since {gl, gM} has the Bernsteinproperty (because gl (z),..., gM(z) are linearly independent by the definition of aresolution), so also does {f, fN}.

Proof of the Abstract Bernstein Theorem. Let U, H1, H2, j be as in the state-ment of the theorem, and let fl, fN: U H1 be holomorphic. Fix a compactset K U Rn, and let Xo e K. By the Main Theorem, there exists a dominatingfamily (U, K, n)l <<o at Xo such that (fl o n, fN o r) is resolvable on Ufor each v. Corollary 6.1 above shows that for each v, there is C > 0 such that

Ilvll Cllj(v)ll whenever v span{f1 o n(z), fN o n(z)}, zeK. (6.6)

Since the images n(K) cover a neighborhood of Xo in U c Rn, this impliesthat there is a neighborhood W,o of Xo in U c R such that if z e Wo andv span {fl (z),..., fN(z)}, we have

Ilvll CtllJ(v)ll, with CtX= max C.< < vo

(6.7)

Covering the compact set K U c R" by finitely Wo, and taking Cr to be themaximum of the corresponding Ctxo), we obtain the statement of the theorem.

7. Bernstein inequalities for families of analytic functions. In this section, wedenote by B(x, r) the open ball in 11 with centre x and radius r, and by (z, r),the open ball in C with centre z and radius r.

Let F1, FN: B,(O,r)x B(0, r’)C be holomorphic functions, where r,r’ > 1. For fixed B,(0, r’), let H’ be the linear span of the functions z Fk(Z, t),1 < k < N (z B(0, r)). We shall prove that the usual Bernstein-type inequalitiesfor polynomials hold also for functions in Ht, uniformly in t for t Bin(O, 1).

THEOREM 7.1. There exists a constant C. > 0 such that if e Bin(O, 1), /fBn(x, p) Bn(O, 1), and if F e H, then the following inequalities hold:

sup IFI < C, sup Ill. (7.1)Cntx, p) Bntx,/2


C,sup [VF[ < sup IFI (where VF is the [lradient of F).n(X,) p (x,)

(7.2)

C_,sup [FI |a(x,p) pn d BntX,)

Ig(y)l ,y. (7.3)

This theorem is clearly equivalent to the Bernstein Theorem stated in theintroduction.

Proof of Theorem 7.1. We shall simply apply the Abstract Bernstein Theoremto a suitable system {f1,..., fN} and suitable Banach spaces H1, H2. Let U{(w, p, t) C C Cmllwl / IPl < 1 + e, Itl < 1 + e}, where e > 0 is small. For(w, p, t) U, let fk(W, p, t) be the function z Fk(Pz + w, t), k 1,..., N. Clearly,if e is small enough, this is defined and holomorphic on the ball (0, 1 + e). IfK {(x, p, t) R" [0, ) Rmllxl + p < 1, Itl < 1}, then K is a compact subsetof U.

Proof of (7.1). Let H be the space of bounded holomorphic functions onB(0, 1), H2 the space of bounded functions on B.(0, 1/2) (both with sup norm),and j: H1 "-" H2 the obvious restriction map. We can look upon fk(w, P, t)l(0, 1)as defining a holomorphic map (which we denote again by fk) of U into H1.The Abstract Bernstein Theorem applied to this choice of fk, U, K, H1, H2, and

j yields (7.1).

Proof of (7.2). Take H1 to be the space of functions tp (Bn(O, 1)) whichare bounded and have bounded first derivatives with norm sup Iol / sup IVql.Let H2 be the space of bounded functions on Bn(0, 1) (with sup norm), and letj: Ht ---, H2 be the obvious inclusion. Applying the Abstract Bernstein Theorem tothe fk and these choices, we obtain the following:

sup IVF(px + w)l < C. sup IF(px + w)lX C Bn(O, 1) X C Bn(O, 1)

whenever F Ht, t Bin(0, 1), p > 0, w R", and Iwl + p 1. (Of course, V, de-notes the gradient as a function of x.) This inequality is obviously equivalent to(7.2).

Proof of (7.3). Let H1 be the space of bounded continuous functions onB(0, 1) with the sup norm, and let Hz L(B(O, 1)). We take for j the obviousinclusion. The Abstract Bernstein Theorem gives, with these choices,

sup IF(px + w)l < C. | IF(px + w)l dx,x Bn(O, 1) )Bn(O, 1)

whenever F Ht, t Bin(0, 1), p > 0, w e R and Iwl + p 1. This is equivalentto (7.3).


8. Some remarks on the Bernstein constants. It is obviously of interest to esti-mate the size of the constant C occurring in the Bernstein inequalities (0.1)-(0.4)stated in the introduction. We call the best constant occurring in (0.1) (estimating/ in terms of F) the Bernstein constant, and that occurring in (0.2) (estimating F onB(x, p) in terms of its values on B(x,/9/2)) the doublin9 constant.

Neither the methods of this paper, nor those in [FN2], are effective; theywould, in any case, require knowledge of the best constants for a single vectorspace of functions. We make a few remarks on the problem of these constants inthe following case.

Let X be a smooth analytic curve, 0 6 X, in some neighborhood U of 0 in R2.Let d > 0 be an integer, and let

V ((PIX){P 6 R[x, y] being a polynomial of degree < d}.

Remark 8.1. Given any sequence of integers (kd)a> 1, there is a curve X suchthat the Bernstein constant and the doubling constant for V are at least kd forinfinitely many d.

In fact, consider a rapidly increasing sequence of integers {m.} and a sequence{a.} of positive real numbers tending rapidly to zero (depending on the {m,}).Consider

X= x,y) eR2 y= anxm.n=l

Then y- .,=1 a=xm"lX e Vm. It has a zero of order m,+l 1 at (0, 0)s X. It iseasily checked that if the {h,} tend to zero sufficiently rapidly, the Bernsteinconstant for this function is at least c’m,+l, and the doubling constant is at leastc. 2m’/1 (c > 0 being an absolute constant).

Note. The curve X is not compact. The example can be modified to make itcompact by considering y2 + 2x2 + ,oo__ 2 a,x2m, 1, which is a small perturba-tion of the ellipse y2 + 2x2 1.

Remark 8.2. If X is a smooth compact algebraic curve in R2 and F is a poly-nomial of degree < d in two variables, Bos, Levenberg, and Taylor [BLT] haveshown that

sup IVxFI C" d sup IFIx x

where Vx is the gradient along X and C > 0 is a constant depending only on X.When X is the unit circle, one form of the classical Bernstein inequality is that wemay take C 1. Note that the suprema are taken globally on X.

However, one does not have such a result if one asks for estimates for thesefunctions on a small smooth piece of an algebraic variety in R; the classical


Bernstein inequality for the interval [-1, 1] c R2 asserts that the Bernstein con-stant is d 2. It would be of great interest if one could obtain good estimates forthese constants in this context (i.e., for restrictions of polynomials of degree < d inxl, xN to a fixed smooth piece of a fixed algebraic variety in RN).Remark 8.3. If X {(x, y) R2ly e 1}, then it is easy to construct f V,

f 0, having a zero of order cd2 (c > 0 an absolute constant) at zero. In thiscase, the doubling constant has to be at least 2cd2. It seems very likely that onecan prove the doubling inequality with a constant (Cd)’, C an absolute constant.

REFERENCES

IBM] E. BIERSTONE AND P. MILMAN, Semianalytic and sub-analytic sets, Inst. Hautes ttudes Sci.Publ. Math. 67 (1988), 5-42.

[BLT] L. Bos, N. LEVENBERG, AND B. A. TAYLOR, Characterization of smooth, compact aloebraiccurves R2, preprint.

[FN1] C. FEFFERMAN AND R. NARASlMHAN, Bernstein’s inequality on algebraic curves, to appear inAnn. Inst. Fourier.

I-FN2] ,On the polynomial-like behavior of certain algebraic functions, to appear in Ann. Inst.Fourier.

[P] A. PARMEGGIANI, Subunit balls for the symbol of a pseudo differential operator, to appear inAdv. Math.

FEFFERMAN: DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY 08544,USANARASIMHAN: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO, CmCAGO, ILLINOIS 60637,

USA

bernstein’s inequality and the resolution of spaces of analytic functions

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