Generalized Area Operators on Hardy Spaces

Generalized Area Operators on Hardy Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO. 216, 112]121 Ž1997. AY975663 Generalized Area Operators on Hardy Spaces William S. Co...

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

216, 112]121 Ž1997.

AY975663

Generalized Area Operators on Hardy Spaces William S. Cohn Department of Mathematics, Wayne State Uni¨ ersity, Detroit, Michigan 48202 Submitted by Brian S. Thomson Received August 23, 1995

We show that if 0 - p - ` then the operator Gf Ž z . s HGŽ z . < f Ž z .< d mrŽ1 y < z <. maps the Hardy space H p to L p Ž< d z <. if and only if m is a Carleson measure. Here G Ž z . is the usual nontangential approach region with vertex z on the unit circle G Ž z . s  z g D : <1 y z < F 1 y < z < 2 4, and < d z < is arclength measure on the circle. We also show that if 0 - p F 1, b ) 0, and 1 y b p ) 0 then the operator Gf maps the Hardy]Sobolev space Hbp into L p Ž< d z <. if and only if the function GmŽ z . s HGŽ z . d mrŽ1 y < z <. belongs to the Morrey space L p, 1y b p. In case p s 1, this condition is equivalent to the condition that m ŽT Ž I .. F C < I < 1y b for all arcs I contained in the circle, where T Ž I . is the tent over I contained in the unit disk. Q 1997 Academic Press

Let D denote the unit disk in the complex plane and T its boundary, the unit circle. For 0 - p F `, H p will be the usual Hardy space of functions holomorphic on the unit disk, i.e., a holomorphic function f belongs to H p if its nontangential maximal function belongs to L p ŽT .. For b ) 0, Hbp will denote the Hardy]Sobolev space of functions f whose radial fractional derivative of order b , D b f, belongs to H p ; see w1x. It is well known Ža proof can be found in w2x. that if a ) 0, a function f belongs to H p if and only if

ž

HT HG Ž z . Ž 1 y r .

2a

a

D f Ž z.

2

pr2

dA

Ž 1 y < z <.

2

/

< d z < - `.

Here, dA is the Lebesgue measure on the plane. Therefore the operator Aa f Ž z . s

žH

GŽ z .

Ž1 y r .

2a

112 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

Da f Ž z.

2

1r2

dA

Ž 1 y < z <.

2

/

OPERATORS ON HARDY SPACES

113

maps H p to L p. If a s 0 it is clear Žconsider the function f Ž z . s 1. that Aa is not a bounded operator from H p to L p. In this note we investigate dArŽ1 y < z <. must be replaced by in order to obtain a bounded operator for the case where a s 0. Replacing f 2 by f and dArŽ1 y < z <. by a positive measure m on D yields the question of characterizing measures m such that

ž

HT HG Ž z .

f Ž z.

dm Ž z . 1 y < z<

p

/

F C 5 f 5 pp

Ž 1.

for a constant C independent of f. Let Gf Ž z . s

HG Ž z .

dm

f Ž z.

1 y < z<

.

The operator G is therefore a generalization of the area integral. Our first result is the following theorem. THEOREM 1. Let 0 - p - `. Necessary and sufficient that Ž1. holds for all f in H p is that m is a Carleson measure. See w6x for basic facts about Carleson measures. Notice that dArŽ1 y < z <. ‘‘just misses’’ being a Carleson measure; this is the motivation for the formulation of the problem we have chosen. The proof of sufficiency in Theorem 1 is based on the same idea as the proof of Lemma 1 in w5x. The necessity is easy if p G 1. The real interest of Theorem 1, from the point of view of this note, is the necessity in the case 0 - p - 1, where the proof relies on the argument of John and Nirenberg in conjunction with the Calderon]Zygmund decomposition. It is also worth pointing out that the class of measurable functions with nontangential maximal function in L p Ž< d z <. is characterized by verifying the imbedding Ž1.. We can also look at the operator Gf Ž z . s

HG Ž z .

f Ž z.

dm 1 y < z<

in the context of Hardy]Sobolev spaces. The situation is easiest to understand for 0 - p F 1, and can be described in terms of the function Gm Ž z . s

dm

HG Ž z . 1 y < z < .

114

WILLIAM S. COHN

THEOREM 2. Let 0 - p F 1. Suppose b ) 0 and 1 y b p ) 0. A necessary and sufficient condition that there is a constant C independent of f such that

ž

HT HG Ž z .

f Ž z.

dm 1 y < z<

p

F C 5 f 5 Hp bp

/

Ž 2.

is that Gm belongs to the Morrey space L p, 1y b p. If p s 1 then this is equi¨ alent to the condition that

m Ž T Ž I . . F C < I < 1y b for all arcs I contained in the circle T. Here T Ž I . is the tent over I in the unit disk Ža convenient substitute for the usual Carleson square over I; see w9, p. 372x. and L p, e is the Morrey space of functions g satisfying the condition that HI < g < p F C < I < e for all arcs I contained in T ; see w4; 9, p. 215x. Since the condition on m in Theorem 1 is independent of p, it is natural to ask if the conditions appearing in Theorem 2 are also really independent of p. We give an example to show that this is not the case. The problem of determining which measures m verify Ž2. is also of interest for p ) 1. This problem seems to be more difficult. We give some partial results ŽTheorem 3. at the end of the note. Proof of Theorem 1. Necessity for 1 F p - ` follows easily by applying Ž1. to the test functions Ž1 y z I z .ym , where m is a positive integer and I is an arc contained in the circle. Here z I is the point in the disk z I s re i u with e i u in the center of I and 1 y r s < I <. Testing this function yields

ž

dm

HI HT Ž I .lG Ž z . 1 y < z <

p

/

< dz <
FC

for a constant C independent of I. Since 1 F p - `, Jensen’s inequality shows that dm

HH I T Ž I .lG Ž z . 1 y < z <

< dz <
FC

and interchanging the order of integration shows that m is a Carleson measure. Necessity for the case 0 - p - 1 seems to lie deeper. Denote the measure of a subset I of the circle by < I <. Let D denote the collection of Žclosed. dyadic subarcs of T. If Q is such an arc and k a

OPERATORS ON HARDY SPACES

115

positive integer, let R k Ž Q . be the ‘‘Carleson rectangle’’ R k Ž Q . s  re i u : 1 y r F 2 k < Q < , e i u g Q 4 and let R k Ž Q . t denote the closed set t R k Ž Q . s  re i u : 2 ky1 < Q < F 1 y r F 2 k < Q < , e i u g Q 4 .

Let x E denote the characteristic function of a set E. If we choose k large enough and for simplicity write Q t in place of R k Ž Q . t , then Ž1. implies that

HJ

ž

Ý Q;J , Qg D

m Ž Qt . xQ Ž z . < Q<

p

/

< dz < F C < J < .

Without loss of generality we may assume C s 1. Let ÝQ ; J, Q g D Ž m Ž Q t .r< Q <. x QŽ z .. Then Ž2.X may be stated as 1
HJ F Ž z . J

p

< dz < F 1

Ž 2.

X

Fj Ž z . s

Ž 3.

for all dyadic arcs J. The idea now is to use the Calderon]Zygmund decomposition at level a in the manner of John and Nirenberg Žsee w6, p. 230x or w9, p. 202x. to obtain the distribution inequality

 z g J : FJ Ž z . ) t 4

F Ceyc t < J < , p

where the constants c and C are independent of J. If a ) 1 then from Ž3. we may find dyadic arcs I j1 , j s 1, . . . , N1 contained in J such that if G1 s D I j1 then Ža. FJp F a on J y G1; Žb. Ž1r< I j1 <.HI 1 FJp < d z < F 2 a , for j s 1, . . . , N1; and j Žc. < G1 < F Ž1ra .< J <. p

Notice that the subset of J where FJ is larger than a is contained in G . Now for each j, 1 F j F N1 , use Ž3. to apply the Calderon decomposip tion Žat the same level a . described above to the function FI j1 on the 1 1 interval I j . For each j, we obtain dyadic subarcs of I j , which we denote as Il2, j, such that, with Gj2 s Dl Il2, j, 1

Ža 1 . FIp1 F a on I j1 y Gj2 ; j Žb 1 . Ž1r< Il2, j <.HI 2, j FIp1 < d z < F 2 a for all l; and l j Žc 1 . < Gj2 < F Ž1ra .< I j1 <, for all j.

116

WILLIAM S. COHN

Use the fact that 0 - p - 1 to deduce that on I j1 y Gj2 FJ s < FI j1 y FI j1 q FJ < p p

F FI j1 q < FI j1 y FJ < p p

F a q < FI j1 y FJ < p . Since < FI j1 y FJ < p is constant on I j1 we may estimate it by averaging to obtain 1

< FI 1 y FJ < p s

< I j1 <

j

1

F

< I j1 <

HI < F 1 j

I j1

p I j1

HI ž F 1 j

y FJ < p < d z < p q FJ < d z <

/

F 1 q 2a. p

It follows therefore that the subset of J where FJ G 1 q 3 a is contained in the union G 2 s Dj Gj2 . Combine estimates Žc. and Žc 1 . above to estimate that < G2 < F

1

Ý < Gj2 < F Ý a < Ij1 < j

F

1

a2

j

< J <.

Iterate this process n times to obtain subsets G n of J satisfying the inequality < G n < F Ž1ra n .< J < and which contain all points of J where p FJ G nŽ1 q 3 a .. It follows that

 z g J : FJ Ž z . p ) l4

F Ceyc l < J < ,

where the constants c and C are independent of J. Thus

HJ Ý

Q;J

m Ž Qt . xQ Ž z . d z F C < Q<

`

yc t p

H0 e

dt < J < .

Since the left hand side is larger than m ŽT Ž J .. the proof of necessity is complete. We now turn to sufficiency. Suppose first that m is a Carleson measure and that 1 F p - `. Duality shows that it is enough to prove the estimate

HTg Ž z . Gf Ž z . < dz < F C 5 g 5

p

X

5 f 5 p,

117

OPERATORS ON HARDY SPACES

where pX is conjugate to p and g ) 0. Interchanging the order of integration yields the equivalent inequality

HD f Ž z . G*g Ž z . d m Ž z . F C 5 g 5

p

X

5 f 5 p,

where G*g Ž z . s

1 1 y < z<

HTx

GŽ z .

Ž z . g Ž z . < dz < .

It is clear that < G*g < is dominated by the Poisson integral of g. Holder’s inequality shows then that

HDG*g Ž z . f Ž z . d m Ž z .

F

ž

HD< G*g <

p

X

1rp

dm

X

1rp

/ ž

HD< f <

p

dm

/

F 5 f 5 p 5 g 5 pX since m is a Carleson measure. Suppose next that 0 - p - 1. Let f *Ž z . s sup z g GŽ z . < f Ž z .< be the nontangential maximal function of f. Estimate that p

dm

HT HG Ž z .< f Ž z . < 1 y < z <

ž

F

< dz <

/

H Ž f *. p 1yp Ž z . Ž

.

T

žH / žH žH GŽ z .

1 y < z<

1yp

F

žH

T

p

Ž f *. < dz <

T

GŽ z .

p

dm

p

f Ž z.

f Ž z.

/

< dz <

dm

p

1 y < z<

p

< dz <

/ /

p

F C 5 f 5 ppŽ1yp.

žH

f Ž z.

D

p

dm Ž z .

/

F C 5 f 5 pp , since m is a Carleson measure. This completes the proof of Theorem 1. Proof of Theorem 2. We consider first the matter of sufficiency. Following Ahern w1x, if f g Hbp then f Ž z . F C Ý l j d jby1r pA j Ž z . , j

118

WILLIAM S. COHN

where A j F 1 is supported on a set of the form T Ž I j . where I j is an arc on T of length d j and Ý j l jp F C 5 f 5 Hp bp . Since 0 - p F 1, if we let 4 I denote an arc whose length is 4 times the length of I and whose center is the same as that of I then Minkowski’s inequality yields that

ž

f Ž z.

HT HG Ž z .

1 y < z<

ž

FC

p

dm

HT HG Ž z . Ý

< dz <

/

l j d jby1r pA j

Ž z.

j

F C Ý l jp d jb py1 j

H4 I G Ž z . m

p

dm 1 y < z<

p

/

< dz <

< dz <

j

F C Ý l jp . j

This proves sufficiency. Necessity in Theorem 2 follows from two observations. First, by looking at test functions of the form Ž1 y zz Q .ym where m is a large integer, a necessary condition that Ž2. hold is that

m Ž Q t . F < Q < 1y b for all arcs Q contained in the circle. From this it follows easily that

ž

dm

HQ HG Ž z .yT ŽQ . 1 y < z <

p

/

< d z < F C < Q < 1y b p .

Ž 4.

The same test functions also show that the condition

ž

dm

HQ HG Ž z .lT ŽQ . 1 y < z <

p

/

< d z < F C < Q < 1y b p

is necessary that Ž2. holds. Together the two inequalities yield the result. If p s 1 then the second estimate shows that

m Ž T Ž I . . F C < I < 1y b

Ž 5.

for all arcs I : T, and it is easy to see that this latter condition implies the estimate Ž4.. Therefore condition Ž5. implies that Gm g L1, 1y b . This completes the proof of necessity.

OPERATORS ON HARDY SPACES

119

EXAMPLE. It is natural to ask whether or not the condition that Gm g L p, 1y b p Žwith b ) 0 fixed. occurring in Theorem 2 really depends on p; after all the characterization of the ‘‘b s 0’’ case given in Theorem 1 is independent of p. The following example shows that the condition does depend on p. Let I denote a subinterval of T of length 1. Fix an integer n and assume a s 1 y b p ) 0 and 0 - p - 1. Let  In, j : j s 1 . . . 2 n4 be 2 n intervals each of length Ž1r2.1r a chosen according to the standard construction of a singular Lebesgue function for a symmetric Cantor set. Žsee w7, Chapt. 1x, for details.. For each j let z j s re i u j where e i u j is the center of In, j and 1 y r s Ž1r2. n r a . Let m n s Ý j Ž1r2. nŽŽ1y b .r a .d z j , where d z denotes a unit mass at z. Notice that 5 m n 5 s 2 nŽ1yŽ1y b .r a . and that 1 y Ž1 y b .ra ) 0. It can be verified that Gmpn s < In, j
HTu Ž h .
1y b

< dh < .

Non-linear potential theory shows that a condition necessary that Ž2. hold is that

ž

dm

HE HT Ž E .lG Ž z . 1 y < z <

p

/

< d z < F C Cbp Ž E . ,

Ž 6.

where E is a finite union of disjoint arcs and Cbp denotes the Bessel capacity associated with the potential space Hbp ; see w3x. It is natural to ask whether or not condition Ž6. is sufficient that Ž2. hold. We are able to prove this in a special case. THEOREM 3. Suppose  z k 4 is an interpolation sequence and m s Ý k w k d k where d k denotes unit mass at z k . Then Ž6. is necessary and sufficient that Ž2. hold.

120

WILLIAM S. COHN

Proof. As above, let G*g Ž z . s

1 1 y < z<

HTx

GŽ z .

Ž z . g Ž z . < dz < .

If 1 - p - ` then Ž2. is by duality equivalent to to

HDG*g Ž z . f Ž z . d m Ž z . F C 5 g 5

p

X

5 f 5 Hbp .

Ž 7.

Let GŽ z . s G*g Ž z . and d k s 1 y < z k <. Then Ž7. is equivalent to X

p f Ž zk . Ý G Ž z k . d1r k

k

wk

ž / p d 1r k

X

F C 5 g 5 pX 5 f 5 Hbp .

Ž 8.

Since  z k 4 is an interpolation sequence, the operator sending g to the X X X p4 p p Ž sequence  GŽ z k 4 d 1r maps L onto l . The easiest way to see this is to k verify that the conjugate operator which sends a sequence  a k 4 in l p to the function Ý k Ž a kr< Ik < 1r p . x I k, where z k s z I k, is bounded from below. This, in turn is an easy consequence of the Carleson embedding theorem.. Therefore Ž8. is equivalent to

Ý f Ž zk . k

p

wk

p

ž / dk

d k F C 5 f 5 Hp bp .

Ž 9.

It follows from w8x that Ž9. holds if and only if

Ý z kgT Ž E .

wk

ž / dk

p

d k F C Cbp Ž E . ,

Ž 10 .

where E is a union of disjoint subarcs of T. The proof will be complete if we show that Ž6. implies Ž10.. For this, we again use duality: condition Ž6. is equivalent to:

Ý z kgT Ž E .

G Ž z k . w k F C 5 g 5 pX Ž Cbp Ž E . .

1rp

,

Ž 11 .

where g is supposed on E. Using again the fact that the operator sending X X p functions g in L p supported on E to the sequence  GŽ z k . d 1r : zk g k X p T Ž E .4 maps onto l , duality shows that Ž11. is equivalent to Ž10.. This completes the proof.

OPERATORS ON HARDY SPACES

121

REFERENCES 1. P. Ahern, Exceptional sets for homorphic Sobolev functions, Michigan Math. J. 35 Ž1988., 29]41. 2. P. Ahern and J. Bruna, Maximal and area integral characterizations of Hardy]Sobolev spaces in the unit ball of C n , Re¨ . Mat. Iberoamericana 4, No. 1 Ž1988., 123]153. 3. P. Ahern and W. S. Cohn, Exceptional sets for holomorphic Hardy]Sobolev functions, p ) 1, Indiana Uni¨ . Math. J 38, No. 2 Ž1989., 417]453. 4. S. Campanato, Proprieta di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa 18 Ž1964., 137]160. 5. W. Cohn, Weighted Bergman projections and operators on Hardy spaces, preprint. 6. J. Garnett, ‘‘Bounded Analytic functions,’’ Academic Press, New YorkrLondon, 1982. 7. J. P. Kahane and R. Salem, ‘‘Ensembles parfaits et series trigomometriques,’’ Hermann, Paris, 1962. 8. D. Stegenga, Multipliers of the Dirichlet spaces, Illinois J. Math. 24, No. 1 Ž1980., 113]140. 9. Torchinsky, ‘‘Real Varible Methods in Harmonic Analysis,’’ Academic Press, Orlando, 1986.