An extension of Hedberg's convolution inequality and applications

An Extension of Hedberg’s Convolution Inequality

and Applications

Andrea Cianchi*

Istituto di Matematica, Facolta di Architettura, Uni` ¨ersita di Firenze, Via dell’ Agnolo`

14, 50122, Firenze, Italy

and Bianca Stroffolini†

Dipartimento di Matematica e Applicazioni ‘‘R. Caccioppoli,’’ Uni¨ersita di Napoli`

Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, 80126, Napoli, Italy Submitted by N. H. Bingham

Received November 26, 1997

1. INTRODUCTION AND FIRST RESULTS

Given any measurable nonnegative function f on⺢n_{, we denote by Mf}

the Hardy᎐Littlewood maximal function of f, defined by 1 n Mf x

Ž .

s sup

_H

f y dy

Ž .

for xg ⺢ ,

Ž

1.1

.

<B r

Ž .

< B rŽ . r)0 x x Ž . < <

where B r stands for the ball centered at x and having radius r, andx ⭈

Ž . < <␣yn is Lebesgue measure. Moreover, for 0-␣ - n, we set I x s x_␣ , the Riesz kernel, and denote by) the convolution product, so that

f y

Ž .

_n

I␣) f x s

Ž .

_H

_{n<xy y<}ny␣ dy for xg ⺢ ,

Ž

1.2

.

⺢

the Riesz potential of f. * E-mail: cianchi@cesit1.unifi.it.

†

E-mail: stroffol@matna2.dma.unina.it 166 0022-247Xr98 $25.00

Copyright䊚 1998 by Academic Press All rights of reproduction in any form reserved.

w x

Hedberg 11 proved the following pointwise inequality between I_␣) f and Mf: 1y␣ prn ␣ pr n_{p n} n 5 5 I␣) f x F C f

Ž .

L Ž⺢ .Mf x

Ž .

for xg ⺢ .

Ž

1.3

.

5 5 p n

Here 1F p - nr␣, and ⭈ L Ž⺢ .is the usual norm in the Lebesgue space

p_Ž n_.

L ⺢ . Henceforth, C will denote a positive constant, not necessarily the

same in different occurrences. In particular, C depends only on ␣, p, and Ž .

n in 1.3 .

Ž .

One of the interesting features of estimate 1.3 is that it allows one to reduce certain problems concerning I_␣) f to analogous problems for Mf, which are often easier to deal with. This is the case, for instance, when interpolation techniques are involved. Actually, since the operator M is of

Ž . Ž . Ž

type ⬁, ⬁ and of weak type 1, 1 , interpolation theorems e.g., of .

Marcinkiewicz type in diagonal form, usually simpler than those off diagonal, can be applied. Thus, for example, the Sobolev inequality for potentials

5I_␣) f5_LŽ n prŽ ny␣ p.._{Ž⺢ .}n F C f5 5_{L Ž}p_{⺢ .}n , 1- p - nr␣ ,

_Ž

1.4

_.

where C is a constant independent of f, immediately follows from

inequal-Ž . pŽ n.

ity 1.3 , thanks to the boundedness of the operator M on L ⺢ , the latter being a consequence of the Marcinkiewicz interpolation theorem.

Ž . Our basic result is an optimal Orlicz-space version of inequality 1.3 . Recall that, given any measurable subset G of⺢n and any Young function

A

Ž .

A, the Orlicz space L G is the Banach function space of those functions f for which the Luxemburg norm

<f x

Ž .

< A 5 5f L ŽG.s inf

½

␭ ) 0:

H

A

ž

/

dxF 1

5

Ž

1.5

.

␭ G Ž . s Ž .

is finite. A is called a Young function if A s s H a r dr for s G 0,₀

w . w x

where a: 0,⬁ ª 0, ⬁ is left-continuous and nondecreasing. Plainly,

A_{Ž .} p_{Ž . Ž .} p _{Ž .}

L G s L G if either 1 F p - ⬁ and A s s s , or p s ⬁ and A s ' 0

Ž .

for 0F s F 1, A s ' ⬁ otherwise.

THEOREM1. Let 0-␣ - n and let A be a Young function such that the

function H , defined by_␣ Žny␣ rn. Ž . ␣r ny␣ s r H_␣

Ž .

s s

ž

_H

ž

/

dr

/

for sG 0,

Ž

1.6

.

A r

Ž .

0

is finite. Then a constant C, depending only on ␣ and n, exists such that Mf x

Ž .

_n A n 5 5 I␣) f x F C f

Ž .

L Ž⺢ .H␣

ž

_{5 5f A n}

/

for xg ⺢ ,

Ž

1.7

.

L Ž⺢ . A_Ž n_{. Ž .}

for all nonnegati¨e fg L ⺢ . Moreo¨er, inequality 1.7 is sharp, in the

Ž .

sense that if 1.7 holds with H replaced by some nondecreasing continuous_␣

w . w . Ž . Ž .

function H: 0,⬁ ª 0, ⬁ , then a constant c exists such that H s F cH s_␣

for sG 0.

w x Let us mention that a result in the same direction is contained in 14 ; however, such a result requires additional assumptions on A and is not optimal.

Theorem 1 will be proved in the next section. As a first application, we show here how it can be combined with an extension of the Hardy᎐

Ž

Littlewood maximal theorem to give simplified proofs under slightly more .

restrictive assumptions on A of some recent results concerning Sobolev inequalities in Orlicz spaces. The extension of the maximal theorem we

A_Ž n_{. Ž .}

need states that the operator M is bounded on L ⺢ if and only if the

˜

Young conjugate A of A, defined by

˜

A s

Ž .

s sup rs y A r : r G 0



Ž .

4

for sG 0,

Ž

1.8

.

Ž

belongs to the class ⌬ this is, e.g., a consequence of Theorem 5.17 of₂ w3, Chap. 3 or Theorem 1.2.1 of 12 . Recall that a function Ax w x. g ⌬ if a₂

Ž . Ž .

positive constant c exists such that A 2 s F cA s for s G 0. Thus, if we

˜

5 5

assume that Ag ⌬ and denote by M the norm of the operator M on₂

A_Ž n_{. Ž .}

L ⺢ , then we deduce from 1.7 that

<I␣) f x

Ž .

< 1 I␣) f x<

Ž .

< y1 y1 A H dxF A H dx

H

_⺢n

ž

␣

ž

C M5 5 5 5f _{L Ž}A_{⺢ .}n

/

/

H

_⺢n

ž

5 5M ␣

ž

C f5 5_{L Ž}A_{⺢ .}n

/

/

< < M f

Ž .

x F

_H

A dxF 1 1.9

Ž

.

n

ž

5 5 5 5M f A n

/

⺢ L Ž⺢ . A Ž n. Ž .

for every fg L ⺢ . Observe that the first inequality in 1.9 is due to the

5 5 y1

fact that M G 1 and that the left-continuous inverse H_␣ of H_␣ is a Ž .

Young function. Inequality 1.9 yields the following

COROLLARY 1. Under the same assumptions as Theorem 1, suppose in

˜

addition that Ag ⌬ . Let A be the Young function defined by_{2 ␣}

Then a constant C, depending only on ␣, n, and A exists such that

5I_␣) f5LA␣Ž⺢ .n F C f5 5L ŽA⺢ .n

_Ž

1.11

_.

A_Ž n_.

for all fg L ⺢ .

Corollary 1 can be used to derive, in a standard way, Sobolev᎐ Poincare

_´

᎐type inequalities for functions from the Orlicz᎐Sobolev space

k, A_Ž n_.

W ⺢ , defined for a positive integer k as

Wk , A _⺢n _{s u g L}A _⺢n _{: u has weak derivatives D}k_u



Ž

.

Ž

.

of order k and Dk_{ug L}A _⺢n _.

4

Ž

.

Indeed, the estimate

<_{u x} <_{F CI ) D u x}< k < _{for a.e. xg ⺢}n _1.12

Ž .

k

Ž .

Ž

.

holds, with C depending only on k and n, for all compactly supported

n

Ž functions u on⺢ which are weakly differentiable up to the order k see w18, Remark 2.8.6 . Inequalities 1.11x. Ž . Ž᎐ 1.12 imply.

˜

COROLLARY2. Let A be a Young function such that Ag ⌬ and let k be₂

a positi¨e integer - n. Assume that the function H is finite. Then a constant_k C, depending only on k, n, and A exists such that

5 5u LAkŽ⺢ .n F C D u5 k 5L ŽA⺢ .n

_Ž

1.13

_.

k, A_Ž n_.

for all functions ug W ⺢ ha¨ing compact support.

Ž .

Remark 1. We emphasize that inequality 1.11 is sharp, in the sense

A_␣

Ž n.

that L ⺢ cannot be replaced by any smaller Orlicz space. This can be shown by an argument similar to that which proves the optimality of

Ž . Ž . w x

inequality 1.7 Section 2 . See also 7 , where a complete characterization of norm inequalities between I_␣) f and f in Orlicz spaces is established.

Ž . w x

Notice that, in the case where ks 1, inequality 1.13 was proved in 6 to hold and to be sharp, via rearrangement and interpolation techniques,

˜

even without the assumption Ag ⌬ .₂

Let us mention that earlier results about convolutions and about Sobolev

w x w x

inequalities in Orlicz spaces are contained in 15 and 8 , respectively.

Remark 2. Consider the situation when functions f whose support

Ž .

sprt f has finite measure are taken into account. Then inequalities

Ž . Ž .

analogous to 1.7 and 1.11 can be shown to hold, with constants C < Ž .<

depending also on A and sprt f , even if the integral on the right-hand Ž .

side of 1.6 diverges. If this is the case, one has just to replace A in the definition of H_␣ and A_␣ by a Young function A0 which makes the

Ž .

Ž . Ž . Ž .

that A c s_{0 1} F A s F A c s for some fixed c , c ) 0 and for suffi-_{0 2 1 2} ciently large s. This follows from the same arguments as in the proofs of Theorem 1, Section 2, and of Corollary 1, above, and from the fact that, in an Orlicz space over a set of finite measure, replacing the defining Young function with a Young function equivalent near infinity results in an equivalent Luxemburg norm. Such an equivalence of norms allows one

˜

also to weaken the assumption that Ag ⌬ in Corollary 1. Actually, for2

˜

Ž1.11 to hold with C depending on sprt f. Ž < Ž .<.it suffices that Ag ⌬ near2

˜

infinity. This amounts to requiring that A be finite and that the inequality

˜

in the definition of the class ⌬ be satisfied by A for large values of the2 argument. An analogous remark applies to Corollary 2.

Ž . p

EXAMPLES. When A s s s with 1 F p - nr␣, then

H

_{Ž .}

s s c sŽ ny␣ p.r n and A

_{Ž .}

s s c sn prŽ ny␣ p.

␣ 1 ␣ 2

for suitable constants c and c . Thus, in particular, Theorem 1 includes1 2 Ž .

Hedberg’s inequality 1.3 and Corollaries 1 and 2 include Sobolev’s theorem.

Ž . nr␣ Ž .

In the borderline case where A s s s , H s_␣ is equivalent to

1y␣ r n_{Ž . Ž . Ž} nrŽny␣ ._.

log 1q s near infinity and A s is equivalent to exp s_␣ y 1 near infinity. Therefore, Theorem 1 and Remark 2 yield that

Mf x

Ž .

1y␣ r n n nr␣ n 5 5 I␣) f x F C f

Ž .

L Ž⺢ .log

ž

1q

/

for xg ⺢ , nr␣ n 5 5f L Ž⺢ . 1.14

Ž

.

for nonnegative functions f whose support has finite measure, with C < Ž .<

depending on ␣, n and sprt f . Moreover, Corollaries 1 and 2 reproduce

w x w x

the limiting inequalities by Trudinger 17 and Strichartz 16 . Let us notice

Ž . Ž . w x

that inequality 1.14 is very close to inequality 4 of 11 , which was used in that paper to prove a sharper version of Trudinger’s and Strichartz’s results.

Ž .

Sobolev inequalities in the borderline situation where A s is equivalent

nr␣ ␤_{Ž .}

to s log 1q s near infinity for some ␤ g ⺢, which have been

re-w x w x

cently proved in 9 and 10 , can be recovered from Corollaries 1 and 2 as well; for instance, the double exponential integrability result contained in w x9 is reproduced, since A_␣Ž .s is equivalent to exp exp sŽ Ž nrŽ ny␣ ...y e near

Ž .

infinity when ␤ s n y ␣ r␣. Further examples can be easily worked out on choosing special Young functions. In particular, Theorem 1 tells us that

A

Ž n. the Riesz potential of order ␣ is a bounded operator from L ⺢ into

⬁_Ž n

. ⬁Ž Ž ..␣rŽny␣ .

Ž .

Theorem 1 or rather Corollary 1 is also a fundamental tool for an extension of a well-known result about the capacity of the Lebesgue set of

p_Ž n_.

Riesz potentials. Such result states that if f is any function from L ⺢ and gs I ) f with 1 - p F nr␣, then there exists a function g such that_␣

1 lim

_H

g y dy

Ž .

s g x

Ž .

q <_{B r}

_{Ž .}

< rª0 x B rxŽ . and 1 q 5 5 lim _{1r q} g

Ž .

⭈ y g x

Ž .

L Ž B Ž r .._x s 0

Ž

1.15

.

q < < rª0 B rx

Ž .

Ž . n Ž .

for ␣, p q.e. x g ⺢ . Here, q equals npr n y ␣ p , the Sobolev conju-gate of p, if p- nr␣, and is any number G 1 if p s nr␣. Moreover, the

Ž . Ž .

convergence in 1.14 and 1.15 is uniform outside an open set of

arbitrar-Ž . Ž .

ily small ␣, p capacity, g is an ␣, p quasi-continuous representative for

Ž . Ž w x

g and gs I ) f_␣ ␣, p q.e. see, e.g., 1, Theorem 6.2.1 for a proof in the

. analogous case where I is replaced by the Bessel kernel ._␣

Our extension will be stated in Theorem 3, Section 3. In Theorem 2 below we limit ourselves to presenting a refinement, in the limiting situation where ps nr␣, of the classical result we just recalled. Theorem 2 is a straightforward consequence of Theorem 3 and of the subsequent Remark 4.

THEOREM2. Assume that ps nr␣ and that f has compact support in the

Ž .

statement abo¨e. Then Eq. 1.15 can be replaced by

1

1y␣ r n 5 5 nrŽ ny␣ .

lim log_q

ž

1q <_{B r} <

/

g

Ž .

⭈ y g x

Ž .

ExpŽ L .Ž B Ž r ..x s 0, 1.16

Ž

.

Ž .

rª0 x

Ž nrŽ ny␣ ..

where Exp L is the Orlicz space associated with the Young function

Ž nrŽny␣ ..

exp s y 1.

2. PROOF OF THEOREM 1 Ž .

Inequality 1.7 is a consequence of Lemmas 1 and 2 below. Lemma 1 is an abstract version, of possible independent interest, of Hedberg’s inequal-ity in the general framework of rearrangement invariant Banach function

Ž . w x

spaces briefly, r.i. spaces . We refer to 3, Chap. 2 for an exhaustive treatment of r.i. spaces. In view of our purposes, we limit ourselves to

Ž n.

recalling here the following facts. An r.i. space X ⺢ is a Banach function

n 5 5 _n

space on⺢ endowed with a norm ⭈ X Ž⺢ . such that

whenever f *s g*. Here f * stands for the nonincreasing rearrangement of

w .

f, i.e., the nonincreasing, right-continuous function on 0,⬁

equimeasur-able with f.

Ž n. Two more r.i. spaces can be associated with every r.i. space X ⺢ : the

n

Ž . Ž . Ž .

representation space X 0,⬁ and the associate space X⬘ ⺢ . X 0, ⬁ is

Ž .

the unique r.i. space on 0,⬁ such that

n

5 5f X Ž⺢ .s f *5 5X Ž0 ,⬁.

Ž

2.2

.

Ž n.

for all fg X ⺢ . Notice that, for customary r.i. spaces, such as Lebesgue, 5 5

Lorentz, and Orlicz spaces, the norm ⭈ X Ž0,⬁. can be immediately com-Ž n.

puted from the norm in the original space X ⺢ ; for a general formula,

w x

see 3, proof of Theorem 4.10, Chap. 2 . Ž n.

The norm in the associate space X⬘ ⺢ is defined by

n n

5 5f X⬘Ž⺢ .s sup

½

H

_n f x g x

Ž .

Ž .

dx : g5 5X Ž⺢ .F 1 .

5

Ž

2.3

.

⺢

The following Holder-type inequality is an obvious consequence of defini-

¨

Ž . tion 2.3 : n n 5 5 5 5 f x g x

Ž .

Ž .

dxF f g .

Ž

2.4

.

H

_n X Ž⺢ . X⬘Ž⺢ . ⺢ Ž n. pŽ n. Ž n. p⬘Ž n.

Observe that, if X ⺢ s L ⺢ , then X⬘ ⺢ s L ⺢ with p⬘ s

˜

n A n n A n

Ž . Ž . Ž . Ž . Ž .

pr p y 1 . When X ⺢ s L ⺢ , one can show that X⬘ ⺢ s L ⺢

and that

˜ ˜

A n A n A n

5 5f L Ž⺢ .F f5 5Ž L .⬘Ž⺢ .F 2 f5 5L Ž⺢ ..

Ž

2.5

.

Ž n.

LEMMA1. Let X ⺢ be any r.i. space. Set

5 5

␾ s s ␹X

Ž .

Ž0 , s . X Ž0 ,⬁.,

Ž

2.6

.

Ž n.

the fundamental function of X ⺢ , and

y1q␣rn

␺_{␣ , X}

Ž .

s s ⭈

Ž .

␹_{Ž s ,⬁.}

Ž .

⭈ _{X⬘ 0, ⬁Ž .}

Ž

2.7

.

for sG 0. Here ␹ denotes the characteristic function of the set E. AssumeE

Ž .

that␺_{␣, X} s is finite for s) 0 and define

␻_{␣ , X}

Ž .

s s␺_{␣ , X}(␾y1X

Ž

1rs

.

for s) 0,

Ž

2.8

.

where␾y1_X is the right-continuous in¨erse of␾ . Then a constant C, depend-_X

ing only on ␣ and n, exists such that

Mf x

Ž .

_n n 5 5 I␣) f x F C f

Ž .

X Ž⺢ .␻␣ , X

ž

_{5 5f n}

/

for xg ⺢ ,

Ž

2.9

.

X Ž⺢ . Ž n.

Proof. It is not difficult to see that, for every xg ⺢n _{and ␦ ) 0,}

f y

Ž .

n _␣

dyF ␦ Mf x

Ž .

Ž

2.10

.

H

_{<xy y<}ny␣ _␣

y : x< yy -␦< 4

Žsee 1, inequality 3.1.1 . Moreover, by 2.5 and 2.7 ,w Ž .x. Ž . Ž .

f y

Ž .

_␣yn n 5 5 < < dyF f xy ⭈ ␹

Ž .

⭈ n

H

_{ < < 4<xy y<}ny␣ X Ž⺢ .  y :< xyy < G␦ 4 X⬘ ⺢Ž . y : xyy G␦ Ž ny␣ .r n_{5 5 n} n s Cn f X Ž⺢ .␺␣ , X

Ž

Cn␦ ,

.

Ž

2.11

.

n nr2 _{Ž .}

for xg ⺢ , where C s_n ␲ r⌫ 1 q nr2 , the measure of the

n-dimen-Ž . Ž .

sional unit ball. Combining 2.10 and 2.11 and choosing 1rn 1 _y1 n 5 5 ␦ s

ž

␾X

Ž

f X Ž⺢ .rMf x

Ž .

.

/

Cn yield ␣rn n 5 5 n 1 _y1 f X Ž⺢ . I_␣) f x F

Ž .

ž

␾X

ž

/

/

Mf x

Ž .

␣ Cn Mf x

Ž .

Mf x

Ž .

Ž ny␣ .r n5 5 n q Cn f X Ž⺢ .␻␣ , X

ž

/

.

Ž

2.12

.

n 5 5f X Ž⺢ . Ž .

Consequently, inequality 2.9 will follow if we show that ␣rn

y1 1y␣ r n

s

_Ž

␾X

Ž

1rs

.

_.

F

Ž

␣rn q 1 2

.

␻␣ , X

Ž .

s for s) 0. 2.13

Ž

.

Ž .

Inequality 2.13 is a consequence of the inequality

␣r n 1y␣ r n_{5 5}

s F

Ž

␣rn q 1 2

.

␹Ž0 , s . X Ž0 ,⬁. y1q␣rn

= ⭈

Ž .

␹_{Ž s ,⬁.}

Ž .

⭈ _{X⬘ 0, ⬁Ž .} for s) 0, which, in turn, follows from

⬁ ␣r n ␣ r ␣r n s s

ž

q 1

/

_H

␹Ž0 , s .

Ž .

r dr n 0 s ␣rn ␣

Ž .

⭈ ␹Ž0 , s .

Ž .

⭈ 5 5 F

ž

q 1

/

␹Ž0 , s . X Ž0 ,⬁. for s) 0, Ž . n s X⬘ 0, ⬁

since U ␣rn ⭈

Ž .

␹Ž0 , s .

Ž .

⭈

Ž .

r

ž

s

/

␣rn ␣rny1 sy r ␹ r rq s

Ž

.

Ž .

Ž

.

Ž

Ž0 , s .

.

_{␣ r ny1} s F␹Ž0 , s .

Ž .

r s F ␣ r ny1 s 2 U ␣rny1 1y␣ r n s 2

_ž

Ž .

⭈ ␹Ž s ,⬁.

Ž .

⭈

_/

Ž .

r for r) 0.

LEMMA2. Under the same assumptions as Theorem 1, positi_¨e constants

k and k exist such that_{1 2}

k H_{1 ␣}

Ž .

s F␻_{␣ , L}A

Ž .

s F k H s_{2 ␣}

Ž .

for s) 0.

Ž

2.14

.

Ž . Ž .

Proof. Definitions 2.6 and 1.5 yield

1

A

␾L

Ž .

s s _y1 for s) 0,

Ž

2.15

.

A

Ž

1rs

.

y1 _{Ž .}

where A is the right-continuous inverse of A. On the other hand, 1.5 and a change of variable show that

y1q␣ r n ⬁ r y1q␣rn

_˜

⭈ ␹ ⭈ s inf ␭ ) 0: A drF 1

Ž .

_{Ž s ,⬁.}

Ž .

_LA˜_{Ž0 ,⬁.}

½

H

ž

/

5

␭ s sy1q␣ r n s _y1 for s) 0, D

Ž

1rs

.

nrŽ ny␣ . s

_˜

y1yn rŽ ny␣ .

Ž . Ž . Ž . Ž .

where D s s s E s and E s s H A t t₀ dt for sG 0.

Thus, on setting Ž . nr ny␣ y1 nrŽ ny␣ .

ˆ

A␣

Ž .

s s sE

Ž

Ž

s

.

.

,

Ž

2.16

.

we have y1q␣rn

_ˆ

_y1 ⭈ ␹ ⭈ s A 1rs for s) 0, 2.17

Ž .

Ž s ,⬁.

Ž .

LA˜Ž0 ,⬁. ␣

Ž

.

Ž

.

y1

_ˆ

y1

where E is the left-continuous inverse of E and A_␣ is the

right-ˆ

Ž .

if we show that positive constants c and c exist such that_{1 2}

ˆ

A␣

Ž

c s1

.

F A s F A c s␣

Ž .

␣

Ž

2

.

for sG 0.

Ž

2.18

.

Ž . Ž . y1Ž . Ž . Ž .

In order to prove 2.18 , let us set L s s 2 srA s and B s s A s rs.

y1_{Ž .}

_˜

y1_{Ž .}

_˜

y1_{Ž . Ž .}

Since A s A s F 2 s, we have A s F L s for s G 0. Denote by Ly1 and By1 the left-continuous inverses of L and B, respectively. Then,

y1_{Ž .}

_˜

_{Ž . Ž . Ž .}

since L s F A s and A s rs F a s , the following chain of

inequali-ties is easily verified to hold:

s Ly1

Ž .

t E s

Ž .

G

_H

_{1qn rŽ ny␣ .} dt t 0 Ž . nr ny␣ y1 ny␣ Ly1Ž .s A

Ž .

r _{y1 nrŽ␣yn.} s

ž

_H

ž

/

dry L

Ž .

s s

/

n 0 2 r ny␣ G n Ž . nr ny␣ ␶ y1Ž . B sr2 nrŽ␣yn. y1 nrŽ␣yn. = 2

ž

_H

ž

/

a

Ž .

␶ d␶ y L

Ž .

s s

/

A

Ž .

␶ 0 Ž . ␣r ny␣ ny␣ nrŽ␣yn. By1Žsr2. ␶

_˜

nrŽ␣yn. G

ž

2

H

ž

/

d␶ y A s s

Ž .

/

n 0 A

Ž .

␶ 2.19

Ž

.

˜

_{Ž .} nrŽ␣yn. _{Ž .}

for s) 0. Hence, since A s s F E 2 s , we deduce that a positive Ž .Ž ny␣ .r n Ž y1Ž ..

constant c exists such that cE cs G H B_␣ s . The last

inequal-ity implies that there exists a positive constant c such that

ˆ

A_␣

Ž .

s F A cs_␣

Ž

.

for s) 0,

Ž

2.20

.

y1 nrŽ ny␣ .

Ž . Ž Ž Ž ...

where A_␣ s s sB H_␣ s . On making use of inequalities

sA_␣

Ž .

r 2 sA_␣

Ž .

r

drF A s F

Ž .

dr ,

Ž

2.21

.

H

_r ␣

H

_r

0 0

performing a change of variable in the last integral, taking into account the

y1_{Ž .} y1_{Ž . Ž .}

fact that 2 H_␣ s F H_␣ 2 s for sG 0, and observing that 2.21 also Ž .

holds with A_␣ r replaced by A we get

ny␣

A_␣

Ž

sr2 F A s F A 2 s

.

_␣

Ž .

_␣

Ž

.

for s) 0.

Ž

2.22

.

n

Ž . Ž . Ž .

Ž .

As far as the first inequality in 2.18 is concerned, owing to inequalities

˜

_{Ž .} y1_{Ž .} y1_{Ž .} y1

A s rs F a s F B s , where a is the left-continuous inverse of a, one has s ay1

Ž .

t ny␣ ay1Ž .s ␣r ␣ynŽ . _{␣ rŽ␣yn.} E s

Ž .

F

_H

_{nrŽny␣ .} dtF

ž

_H

a r

Ž .

y s dr

/

␣ t 0 0 Ž . ␣r ny␣ ny␣ _By1_{Ž .s} ␶ F

ž

_H

_ž

_/

d␶

/

␣ 0 A

Ž .

␶ ny␣ _y1 nr ny␣Ž . s _␣

Ž

H␣

Ž

B

Ž .

s

.

.

for s) 0, y1_Ž nrŽ ny␣ ._{. Ž} y1_{Ž ..}

whence E r G B H_␣ cr for rG 0 and for some positive

Ž . Ž .

constant c. Thus, by 2.22 , also the first of inequalities 2.18 follows. Let us now prove the second part of Theorem 1. Assume that an

Ž .

inequality of type 1.7 holds with H_␣ replaced by H. On taking nonin-creasing rearrangements of both sides, we get

I ) f * s Mf * s

Ž

␣

. Ž .

Ž

. Ž .

F CH

ž

/

for s) 0.

Ž

2.23

.

A n A n

5 5f L Ž⺢ . 5 5f L Ž⺢ .

Consider radially symmetric functions f, namely, functions having the form

Ž . Ž < <n. w . w .

f x s␾ C xn for some␾: 0, ⬁ ª 0, ⬁ . It is easily verified that

f y

Ž .

I␣) f x G

Ž .

_H

_{<xy y<}ny␣ dy y : y< < < <4) x

⬁

1y␣ r n ␣yn y1q␣ r n n

G Cn 2

_H

_n␾ r r

Ž .

dr for xg ⺢ ,

< <

C xn

whence

⬁

1y␣ r n ␣yn y1q␣ r n

I ) f * s G C 2 ␾ r r dr for s) 0. 2.24

Ž

_␣

. Ž .

n

_H

Ž .

Ž

.

s

w x

Moreover, by 3, Theorem 3.8, Chap. 3 , a constant C, depending only on

n, exists such that

s s

C C

Mf * s F f * r drs ␾* r dr for s) 0. 2.25

Ž

. Ž .

_H

Ž .

_H

Ž .

Ž

.

Ž . Ž . Ž . Inequalities 2.23 , 2.24 , and 2.25 yield

Hs⬁␾ r r

Ž .

y1q␣ r ndr

Ž

Crs H

.

0s␾* r dr

Ž .

F CH

ž

/

for s) 0, 2.26

Ž

.

A A

5 5␾ L Ž0 ,⬁. 5 5␾ L Ž0 ,⬁.

for some constant C independent of ␾. For fixed s, we have

H0s␾* r dr

Ž .

A 5 5␾ L Ž0 ,⬁. H0⬁␾* r ␹

Ž .

Ž0 , s .

Ž .

r dr s A 5 5␾ L Ž0 ,⬁. 2 1 y1 ˜ A 5 5 F 2 ␹Ž0 , s . L Ž0 ,⬁.s

_˜

_y1 F 2 sA

ž /

.

Ž

2.27

.

s A

Ž

1rs

.

Ž . Ž .

Notice that the first inequality is due to 2.2 and 2.5 , and the last y1_{Ž .}

_˜

y1_{Ž .}

inequality holds because rF A r A r for rG 0. On the other hand,

H0⬁␾ r r

Ž .

y1q␣ r n␹Ž s ,⬁.

Ž .

r dr y1q␣rn ˜ A 5 5 sup G ⭈

Ž .

␹Ž s ,⬁.

Ž .

⭈ L Ž0 ,⬁. A 5 5␾ AŽ . L Ž0 ,⬁. ␾gL 0, ⬁ 1 1 y1 y1

ˆ

s A_␣

ž /

G CA_␣

ž /

Ž

2.28

.

s s

for some constant C independent of s. The first inequality, the equation,

Ž . Ž . Ž .

and the last inequality in 2.28 are consequences of 2.5 , 2.17 , and Ž2.18 , respectively..

Ž . Ž . y1Ž .

From 2.26᎐ 2.28 we deduce that a constant C exists such that A_␣ s

Ž y1Ž .. Ž . Ž .

F CH A Cs for s) 0. Hence, CH Cs F H s for some positive C_␣ Ž . and, since H is concave and vanishes at 0, we can conclude that CH s_{␣ ␣}

Ž .

F H s for some positive C and for all s ) 0. The proof of Theorem 1 is complete.

3. CAPACITY AND LEBESGUE POINTS

The present section deals with capacitary estimates for the Lebesgue set

A_Ž n_.

of Riesz potentials of functions from an Orlicz space L ⺢ . Our results

Ž .

DEFINITION. Let 0-␣ - n and let A be a Young function. For any

E: ⺢n _{the quantity}

5 5 A n A n

C␣ , A

Ž

E

.

s inf f



L Ž⺢ .: fg L ⺢ and I ) f x G 1 for x g E

Ž

.

␣

Ž .

4

3.1

Ž

.

Ž .

will be called the ␣, A capacity of E.

C_{␣, A} satisfies the customary properties of a capacity, namely:

C_{␣ , A}

Ž

⭋ s 0;

.

Ž

3.2

.

E: F implies C_{␣ , A}

Ž

E

.

F C_{␣ , A}

Ž

F ;

.

Ž

3.3

.



4

C_{␣ , A}

ž

D

Ei

/

F

_Ý

C␣ , A

Ž

Ei

.

for every countable family of sets E .i

i i

3.4

Ž

.

Ž . Ž . Ž .

Properties 3.2 and 3.3 are straightforward; 3.4 is a special case of w x

Proposition 2 below. We refer to 2 for a more extensive study of C_{␣, A}.

Ž . p w .

In the case where A s s s for some p g 1, ⬁ , C_{␣, A} will be simply

w x Ž

denoted by C_{␣, p}. Note that, by Proposition 2.3.13 of 1 , C_{␣ , p}agrees up to

. Ž . Ž .

a multiplicative constant with the 1rp th power of the classical ␣, p capacity.

Ž .

In what follows, we shall say that some property holds for ␣, A q.e.

n _{Ž .}

xg ⺢ if it holds outside a set of zero ␣, A capacity. Furthermore, a

Ž .

function f will be said to be ␣, A quasi-continuous if for every ⑀ ) 0 Ž .

there exists an open set ⍀ such that C_{␣, A} ⍀ -⑀ and f, restricted to ⺢n_{_ ⍀, is continuous.}

Notions of a geometric nature which will play a role in our discussion Ž w x. are those of upper p-estimate and lower q-estimate for norms see 13 .

A

Ž n.

Recall that an Orlicz space L ⺢ is said to satisfy an upper p-estimate or a lower q-estimate if there exists a constant N or N such that, forp q

 4

every sequence f of functions with disjoint supports, we havei p p A n 5 5 f F N f ,

Ž

3.5

.

Ý

i p

Ý

i L Ž⺢ . A_Ž n_. i L ⺢ i or q q A n 5 5 f G N f ,

Ž

3.6

.

Ý

i q

Ý

i L Ž⺢ . A_Ž n_. i L ⺢ i p_Ž n_.

respectively. For instance, L ⺢ simultaneously satisfies an upper and a lower p-estimate with Nps 1. Notice that every Orlicz space satisfies an

Ž .

upper 1-estimate, with N₁s 1 in 3.5 , by the triangle inequality. A

Ž . Ž .

characterization of those ps or qs for which 3.5 or 3.6 holds is known in terms of the Matuszewska᎐Orlicz indices of Ay1, defined by

log inf

Ž

s) 0

Ž

Ay1

Ž

␭s rA

.

y1

Ž .

s

.

.

is lim and log␭ ␭ªq⬁ 3.7

Ž

.

log sup

Ž

s) 0

Ž

Ay1

Ž

␭s rA

.

y1

Ž .

s

.

.

Is lim , log␭ ␭ªq⬁ w x

and satisfying 0F i F I F 1 4 . Actually, from Remark 2 after Proposition

w x w x

2.b.5 of 13 and from the Theorem of 5 , we get

1rI s sup p: LA _⺢n _{satisfies an upper p-estimate , 3.8}



Ž

.

4

Ž

.

1ri s inf q: L



A

Ž

⺢n

.

satisfies a lower q-estimate .

4

Ž

3.9

.

˜

In particular, inasmuch as Ag ⌬ if and only if i ) 0 and A g ⌬ if and2 2

w x A

Ž n. only if I- 1, then there exists q - ⬁, resp. p ) 1 such that L ⺢

w x

satisfies a lower q-estimate upper p-estimate if and only if Ag ⌬2

˜

A n

wAg ⌬ . Moreover, if L ⺢ satisfies an upper p-estimate and a lower₂x Ž .

q-estimate, then pF q.

We are now in a position to state the main result of this section. THEOREM 3. Let 0-␣ - n. Let A be a Young function such that

˜

A_Ž n_.

A, Ag ⌬ and let p and q be numbers such that L ⺢₂ satisfies an upper p-estimate and a lower q-estimate. Assume that

Ž . ␣r ny␣ r dr- ⬁

Ž

3.10

.

H

_ž

_{A r}

_/

Ž .

0 and Ž . ␣r ny␣ ⬁ r drs ⬁.

Ž

3.11

.

H

ž

_{A r}

/

Ž .

A_Ž n_.

Gi¨en any fg L ⺢ , set g s I ) f. Then a function g exists such that_␣

1

lim

_H

g y dy

Ž .

s g x

Ž .

Ž

3.12

.

q <_{B r}

_{Ž .}

<

and prq 1 y1 _{5 5 A} lim_q

ž

A␣

ž

<_{B r} <

/

/

g

Ž .

⭈ y g x

Ž .

L ␣Ž B Ž r ..x s 0

Ž

3.13

.

Ž .

rª0 x Ž . n Ž . Ž .

for ␣, A q.e. x g ⺢ . Moreo¨er, the con¨ergence in 3.12 and 3.13 is

Ž .

uniform outside an open set of arbitrarily small ␣, A capacity, g is an

Ž␣, A quasi-continuous representati. ¨e for g, and gs g Ž␣, A q.e.. y1_{Ž < Ž .<.}

Remark 3. Observe that the expression A_␣ 1r B r , appearing inx

Ž3.13 , is nothing but 1. r 15 5_LA_␣ .

Ž B Ž r ..x

Remark 4. An inspection of the proof of Theorem 3, below, and

Remark 2, Section 1, show that, for functions f supported in a set of finite measure, similar conclusions as in Theorem 3 hold even without

assump-Ž . Ž .

tion 3.10 . If such assumption is dropped, A has to be replaced in 1.10 by any Young function which is equivalent to A near infinity and makes

Ž .

the integral in 3.10 converge.

Ž . Ž .

As a consequence of Eqs. 3.8᎐ 3.9 , we have the following corollary of Theorem 3.

COROLLARY 3. Under the same assumptions as Theorem 3, we ha¨e for

e¨ery ⑀ ) 0 ŽirI y⑀. 1 y1 5 5 A lim_q

ž

A␣

ž

<_{B r} <

/

/

g

Ž .

⭈ y g x

Ž .

L ␣Ž B Ž r ..x s 0

Ž .

rª0 x Ž . n for ␣, A q.e. x g ⺢ . Ž . nr␣ Ž ny␣ .r ␣Ž .

EXAMPLE. Assume that A s is equivalent to s log 1q s

near infinity. Since is I s␣rn, from Corollary 3 and Remark 4 we have

< < Ž .

that, if sprt f - ⬁, then for any␴ - n y ␣ rn ␴

1

nrŽ ny␣ .

5 5

lim_q

ž

log log 1

ž

ž

q < <

/

/

/

g

Ž .

⭈ y g x

Ž .

ExpŽExpŽ L ..Ž B Ž r ..x s 0

B r

Ž .

rª0 x

Ž . n Ž Ž nrŽ ny␣ ...

for ␣, A q.e. x g ⺢ . Here, Exp Exp L stands for the Orlicz Ž Ž nrŽ ny␣ ...

space associated with the Young function exp exp s y e.

w x Our Proof of Theorem 3 is patterned on that of Theorem 6.2.1 of 1 . Preliminary steps are certain capacitary estimates for the level sets of the maximal function of I_␣) f and of a suitable fractional maximal function of

I_␣) f which will be established in Lemma 3 and Lemma 4, respectively,

LEMMA 3. Let 0-␣ - n and let A be a Young function such that

˜

A_Ž n_.

Ag ⌬ . Let f be any nonnegati₂ ¨e function from L ⺢ and set gs I ) f._␣ Then there exists a constant C, independent of f, such that

C A n 5 5 C␣ , A

Ž



x : Mg x

Ž .

)␭ F

4

.

_␭ f L Ž⺢ .

Ž

3.14

.

for ␭ ) 0. n _Ž < <.

Proof. Given any subset E of⺢ , we define ␹ s 1r E ␹ . Then, forE E

every xg ⺢n _{and r) 0, we have}

␹B Ž r .x ) g x s

Ž .

␹B Ž r .x ) I ) f x s I )␣

Ž .

␣ ␹B Ž r .x ) f x F I ) Mf x .

Ž .

␣

Ž .

Ž . Ž . n

Hence, Mg x F I ) Mf x for x g ⺢ . Thus, by the very definition of_␣

A

Ž n.

C_{␣, A}and by the maximal theorem in L ⺢ , a constant C exists such that

1 C

A n A n

5 5 5 5

C␣ , A

Ž



x : Mg x

Ž .

)␭ F

4

.

_␭ Mf L Ž⺢ .F _␭ f L Ž⺢ .

for␭ ) 0.

LEMMA 4. Under the same assumptions and with the same notation as

Theorem 3, define prq 1 y1 _{5 5 A} n M␣ , Ag x

Ž .

s sup A

ž

␣

ž

<_{B r} <

/

/

g L ␣Ž B Ž r ..x for xg ⺢ .

Ž .

r)0 x 3.15

Ž

.

Then there exist constants C and C, independent of f, such that

qrp 1 A n 5 5 C␣ , A

Ž



x : M␣ , Ag x

Ž .

)␭ F C

4

.

ž

_␭ f L Ž⺢ .

/

Ž

3.16

.

A n 5 5 for ␭ ) C f L Ž⺢ ..

The proof of Lemma 4 requires the following propositions.

PROPOSITION1. Let 0-␣ - n and let A be a Young function such that

Ž3.10 holds. Let B r. Ž . be any ball of radius r in ⺢ . Then a constant C,n

depending only on ␣ and n, exists such that

C

C␣ , A

Ž

B r

Ž .

.

F _Ay1 _{1r B r< <} for r) 0.

Ž

3.17

.

Ž .

Ž

.

Proof. An application of the minimax theorem yields ˜ q A n 5 5 C␣ , A

Ž

K

.

s sup 1r I )



␣ ␮ L Ž⺢ .:␮ g MM K , ␮ K s 1 , 3.18

Ž

.

Ž

.

4

Ž

.

n q_{Ž .}

for every compact subset K of ⺢ , where MM K is the set of positive

Ž w x.

measures supported in K see, e.g., 2, proof of Theorem 11 . Now, let

q_{Ž Ž .. Ž Ž .. < < < <}

␮ g MM B r be such that ␮ B r s 1. Since x y y F 2 x whenever

Ž . Ž . Ž . Ž .

yg B r and x f B r , then, owing to 2.17 and 2.18 , one has

1 _␣yn ˜ A n 5I␣)␮5L Ž⺢ .G ₂ny␣ < <⭈ ␹ x :< x < G r4

Ž .

⭈ LA˜Ž⺢n. Cn1y␣ r n _y1 1 _y1 1

ˆ

s ₂ny␣ A_␣

ž

_{<B r <}

/

G CA_␣

ž

_{<B r <}

/

Ž

3.19

.

Ž .

Ž .

Ž . Ž .

for some positive constant C. The conclusion follows from 3.18᎐ 3.19 .

PROPOSITION 2. Let A be a Young function. Assume that p is a

num-A

Ž n.

ber G1 such that L ⺢ satisfies an upper p-estimate. Then

C_{␣ , A}p

ž

D

Ei

/

F Np

_Ý

C␣ , Ap

Ž

Ei

.

Ž

3.20

.

i i

 4

for e¨ery countable family Ei of disjoint sets. Here, N is the constantp

Ž .

appearing in 3.5 .

Proof. Let ⑀ ) 0 and let f be nonnegative functions such thati

Ž . 5 5pA n p Ž . yi Ž .

I␣) f x G 1 for x g E and fi i i L Ž⺢ .F C␣ , A Ei q⑀ 2 . Set f x s

Ž . Ž . Ž .

sup f x and fi i m x s supiF m if x . Thus,

m fm

Ž .

x s

_Ý

f xi

Ž .

␹ x ,F_i

Ž .

where is1 iy1 Fis x: f x s f x _



m

Ž .

i

Ž .

4

D

½

x : fm

Ž .

x s f x .j

Ž .

5

js1 Ž . Inasmuch as F are disjoint sets, then, by 3.5 ,i

p m m p p A n A n 5 5fm L Ž⺢ .F

_Ý

fi␹F_i F Np

_Ý

5 5fi L Ž⺢ . A_Ž n_. is1 L ⺢ is1 ⬁ p F Np

_Ý

C␣ , A

Ž

Ei

.

q Np⑀ .

Ž

3.21

.

is1

Ž . 5 5pA n

On passing to the limit in 3.21 as m goes to infinity we get f L Ž⺢ .F

⬁ p _{Ž .} 5 5 _{A n} 5 5 _{A n}

N Ýp is1C␣ , A Ei q Np⑀, since limmª⬁ fm L Ž⺢ .s f L Ž⺢ ., fm being an

⬁ p _{Ž .}

increasing sequence converging to f. On the other hand, Ýis1C␣ , A Ei F

5 5f L ŽpA⺢ .n, inasmuch as I␣) f x G 1 for x g DŽ . ⬁is1 E . The conclusioni

follows, thanks to the arbitrariness of ⑀.

Proof of Lemma 4. Without loss of generality, we may assume that f is

 n Ž . 4

nonnegative. Set E_␭s x g ⺢ : M_{␣ , A}g x )␭ and let x g E . Then0 ␭ there exists r) 0 such that

␭ A 5I␣) f5L␣Ž Bx0Ž r ..) _y1 prq.

Ž

3.22

.

< < A

Ž

1r B r

Ž .

.

ž

␣ x₀

/

Ž . Ž .

Combining 3.22 with inequality 1.11 tells us that

prq

y1 A n

<Bx0

Ž .

r <- 1 provided that ␭ ) C A

Ž

␣

Ž .

1

.

5 5f L Ž⺢ .,

Ž

3.23

.

Ž .

where C is the constant appearing in 1.11 . Now, let us split f as

Ž . Ž . Ž .

fs f q f , where f x equals f x in B 2r and vanishes elsewhere._{1 2 1 x}

0

Ž .

From 3.22 , via the triangle inequality, we deduce that one of the following alternatives holds:

␭ A 5I␣) f15L␣Ž Bx0Ž r ..) _{2 Ay1 1rC r}n prq

Ž

3.24

.

Ž

.

Ž

␣ n

.

or ␭ A 5I␣) f25L␣Ž Bx0Ž r ..) _y1 n prq.

Ž

3.25

.

2 A

Ž

␣

Ž

1rC rn

.

.

Ž . Ž .

In the case where 3.24 is in force, on exploiting inequality 1.11 again we get that a constant C exists such that

␭ A 5 5 - C f L Ž B Ž2 r ..x .

Ž

3.26

.

prq 0 y1 n A

Ž

1rC r

.

Ž

␣ n

.

Ž .

Assume now that 3.25 holds. It is not difficult to verify that a positive

Ž . Ž .

constant C exists such that inf_{xg B Žr .}I_␣) f x G CI ) f y for every_{␣ 2}

x 0

Ž . Ž . Ž .

yg B r . Hence, owing to 3.25 and 3.22 and to the fact that p F q, we_x

0

have

qr p_{5 5}Ž p_Ayq.r p_n

I_␣) f x G C␭

Ž

0

.

f L Ž⺢ .

Ž

3.27

.

for some constant C) 0.

Ž .

Denote by U the set of those xg E for which 3.26 holds for some_␭

Ž .

rs r . If_x ␭ is as in 3.23 , then by Vitali’s covering lemma there exists a

 Ž .4 Ž .

sequence B 2 rx_i x_i of disjoint balls such that xig U and U ; D B 10r .i x_i x_i

Therefore, there exists a constant C such that

1 p p C_{␣ , A}

Ž

U

.

F Np

Ý

C␣ , A

Ž

B 10 rx_i

Ž

x_i

.

.

F N Cp

Ý

_n p y1 i i

ž

A␣

_ž

1rC 10rn

Ž

xi

.

_/

/

1 10n p_{N C} p q n p 5 5 _A F 10 N Cp

_Ý

_Ay1 n p F ␭q

_Ý

f L Ž B Ž2 r ..xi xi 1rC r i

_Ž

_␣

_Ž

_{n x}

_.

_.

i i 10n p_{N C} p q A n 5 5 F _{␭ N}q f L Ž⺢ ..

Ž

3.28

.

q Ž .

Notice that the first inequality in 3.28 is due to Proposition 2, the second to Proposition 1, the third to the fact that A_␣ is a Young function, the

Ž . Ž .

fourth to 3.26 , and the last one to 3.6 .

Ž .

On the other hand, inequality 3.27 must be true for every xg E _ U,_␭ whence 1 _qrp A n 5 5 C␣ , A

Ž

E␭_ U F

.

_C

Ž

f L Ž⺢ .r␭

.

.

Ž

3.29

.

Ž . Ž .

The conclusion follows from 3.28 and 3.29 .

Ž . n

Proof of Theorem 3. Consider Eq. 3.12 . Define for␦ ) 0 and x g ⺢

⌳ g x s sup_␦

Ž .

␹B Ž r ._x ) g x y inf

Ž .

␹B Ž r ._x ) g x .

Ž .

0-r-␦ 0-r-␦

⬁_Ž n_.

Since Ag ⌬ , the set C ⺢ of smooth compactly supported functions in2 0

n A_Ž n_. ⬁_Ž n_.

⺢ is dense in L ⺢ . Thus, for every ⑀ ) 0 there exists f g C ⺢0 0

5 5 A n

such that fy f0 L Ž⺢ .-⑀. Set g s I ) f . Then g is smooth and0 ␣ 0 0 Ž .

decays to zero at infinity. Consequently, limrª 0 ␹B Ž r .x ) g x s g uni-0 0

n

Ž . Ž .

formly for xg ⺢ and there exists ␦ ⑀ ) 0 such that ⌳ g x - ⑀ if_{␦ 0}

Ž . Ž .Ž . Ž .Ž . n

␦ - ␦ ⑀ . Moreover, ⌳ g y g_{␦ 0} x F M g y g₀ x for xg ⺢ . Hence,

Ž . Ž .Ž . Ž . Ž .Ž . n

⌳ g x F ⌳ g y g_{␦ ␦ 0} x q ⌳ g x F M g y g_{␦ 0 0} x q⑀ for x g ⺢ if

Ž .  Ž . 4  Ž .Ž . 4

␦ - ␦ ⑀ . Thus, for ⑀ - ␭r2, x: ⌳ g x ) ␭ : x: M g y g_{␦ 0} x )␭r2 ,

and, by Lemma 3, there exists a constant C such that

C C⑀

A n

5 5

C␣ , A

Ž



x :⌳ g x )␦

Ž .

␭ F

4

.

fy f0 L Ž⺢ .F .

Ž

3.30

.

On choosing ␭ s 2ym and ⑀ s 4ym for mg ⺞, and setting ⬁ ym ym ␦ s ␦ 4m

Ž

.

, Ems x: ⌳ g x ) 2



␦m

Ž .

4

, Fjs

D

E ,m msj Ž . Ž .

one easily deduces from 3.30 that limjª ⬁ C␣ , A Fj s 0 and

⬁

Ž . Ž .

C_{␣, A}Fjs1 Fj s 0. The last two equations ensure that limrª 0 ␹B Ž r ._x ) g x

⬁ _{Ž .}

exists for xf Fjs1F , uniformly outside every F . The proof of 3.14 isj j

Ž . n

complete. As far as 3.15 is concerned, we set for␦ ) 0 and x g ⺢

prq 1 y1 5 5 A ⌳␣ , A , ␦

Ž

g

. Ž .

x s sup

ž

A␣

ž

< <

/

/

g

Ž .

⭈ y g x

Ž .

L ␣Ž B Ž r ..x . B r

Ž .

0-r-␦ x 3.31

Ž

.

If f and g are as above, then g0 0 0' g ; moreover, given0 ⑀ ) 0, ␦ can be

Ž . n

chosen so small that ⌳_{␣, A, ␦} g₀ -⑀ for x g ⺢ . On adding and

subtract-Ž .

ing g0y g x in the argument of the norm on the right-hand side of0 Ž3.31 , it is not difficult to verify that.

< <

⌳␣ , A , ␦

Ž

g

. Ž .

x F M

Ž

␣ , A

Ž

gy g0

. Ž .

x q g x y g x q0

Ž .

Ž .

⑀

.

for xg ⺢n _{and for sufficiently small ␦. On choosing ⑀ - ␭r3, one gets}

x :⌳ g x )␭ : x: M gy g x )␭r3



␣ , A , ␦

Ž

. Ž .

4



␣ , A

Ž

0

. Ž .

4

< < j x: g x y g x )



0

Ž .

Ž .

␭r3 .

4

n < Ž . Ž .< < <Ž . Ž .

Therefore, since g₀ x y g x F I ) f y f x for_{␣ 0} ␣, A a.e. x g ⺢ ,

Ž .

we infer from Lemma 4 and the definition of ␣, A capacity that, if ⑀r␭ - 1, then a constant C exists such that

C␣ , A

Ž



x :⌳␣ , A

Ž

g

. Ž .

x )␭

4

.

qrp 1 1 A n A n 5 5 5 5 F C

ž

ž

fy f0 L Ž⺢ .

/

q fy f0 L Ž⺢ .

/

␭ ␭ qrp ⑀ ⑀ ⑀ F C

ž

ž /

q

/

F 2C .

Ž

3.32

.

␭ ␭ ␭ Ž . Ž . Ž .

On starting from 3.32 instead of 3.30 , Eq. 3.15 can be established via the same argument as before.

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