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Differentiability properties of functions from Orlicz-Sobolev spaces

Angela Alberico Istituto per le Applicazioni del Calcolo "M. Picone" Sez. Napoli - C.N.R. Via P. Castellino 111, 80131 Napoli (Italy) e-mail: [email protected] Andrea Cianchi Dipartimento di Matematica e Applicazioni per l'Architettura Universit` di Firenze a Piazza Ghiberti 27, 50122 Firenze (Italy) e-mail: [email protected]

1

Introduction and main results

A classical theorem by Rademacher ([R]) states that any locally Lipschitz continuous function 1, in an open subset of IRn (and hence any function from the Sobolev space Wloc ()) is differentiable at a.e. point in , and that its classical gradient agrees with its weak gradient a.e. in . An extension of this result ensures that the same conclusion remains true even for functions 1,p from the Sobolev space Wloc (), as long as p > n (see e.g. [EG, Par.6.2, Thm. 1], [MZ, Thm. 1.72]). Counterexamples show that exponents p n cannot be allowed; in fact, for these values 1,p of p, functions from Wloc () need not even be continuous at any point of , nor bounded in a neighborhood of any point of ([S1, Chap. V, 6.3]). 1,p When functions from Wloc (), with 1 p < n, are taken into account, a substitute of the above results holds, provided that the notion of classical differentiability is replaced by that np of differentiability in Lp sense, where p = n-p , the Sobolev conjugate of p. Precisely, if u

1,p Wloc () for some p [1, n), then p

(1.1)

r0+ Br (x)

lim -

u(y) - u(x)- < u(x), y - x > r

dy = 0,

for a.e. x ([EG, Sec. 6.1.2, Thm. 2], [Z, Thm. 3.4.2]). Here, Br (x) is the ball centered at x 1 and having radius r, -E u(y)dy stands for |E| E u(y)dy, when E is a measurable set with finite Lebesgue measure |E|, < ·, · > denotes scalar product, and u is the gradient of u. In fact, any such function u enjoys a slightly stronger property: it is a.e. approximately differentiable in Lp , in the sense that (1.2) for a.e. x . lim -

Br (x)

r0+

u(y) - u(x)- < u(x), y - x > |y - x|

p

dy = 0,

1

2 The present paper contains optimal versions of the results mentioned above in the more 1,A general setting of the Orlicz-Sobolev spaces W 1,A () (or Wloc ()) associated with any Young function A, namely a (non constant) convex function from [0, +) into [0, +] vanishing at 0. In 1,n particular, we are able to deal with the limiting space Wloc (), which does not seem to have been considered in the literature, and to obtain, in the spirit of Trudinger's embedding theorem ([Tr]), an a.e. integral differentiability of exponential type for functions from this space. Moreover, in its strongest form, our result slightly sharpens also the classical result (1.1), corresponding to the case where A(s) = sp with 1 p < n. To begin with, a sharp assumption on A ensuring the a.e. differentiability of functions from 1,A 1,p Wloc (), extending the condition p > n for the ordinary Sobolev spaces Wloc (), is that A grows so fast at infinity that

+

(1.3)

t A(t)

1 n-1

dt < +.

1,A Assumption (1.3) is already known to be necessary and sufficient for Wloc () to be continuously embedded into the space of locally bounded ([M, Par. 5.4], [T1], [C1]) and also continuous ([C1]) functions. Here, we prove

Theorem 1.1 Let be an open subset of IRn and let A be a Young function satisfying (1.3). 1,A If u Wloc (), then u is equivalent to a function which is differentiable a.e. in and whose gradient agrees with its weak gradient a.e. in . Remark 1.2 A theorem by Stein ([S2]) ensures that any weakly differentiable function whose gradient is in the Lorentz space Ln,1 () (a space strictly contained in Ln ()) is a.e. differentiable loc loc in . Since LA (), Ln,1 () = loc loc

A satisfies (1.3)

(see [KKM]), then Theorem 1.1 turns out to be equivalent to the result of [S2]. We present here a direct proof of Theorem 1.1, which, in particular, provides an alternate approach to Stein's theorem. Assume now that

+

(1.4)

t A(t)

1 n-1

dt = +.

1,A With condition (1.4) in force, the a.e. differentiability of functions from Wloc () is not guaranteed anymore, as demonstrated by the following proposition.

Proposition 1.3 Let be an open subset of IRn and let A be a Young function satisfying (1.4). 1,A Then there exists u Wloc () such that every function equivalent to u is nowhere differentiable in . On the other hand, under (1.4) results in the spirit of (1.1)-(1.2) can be shown to hold, where the role of the function tp is played by the Sobolev conjugate An of A defined by (1.5) An (t) = A Hn -1 (t) for t 0,

3 where Hn -1 is the (generalized) right-continuous inverse of the function Hn : [0, ) [0, ) given by

r

(1.6)

Hn (r) =

0

t A(t)

1 n-1

1 n

dt

for r 0,

n and n = n-1 , the H¨lder conjugate of n. Obviously, for Hn , and hence An , to be well defined, o A has to fulfill

(1.7)

0

t A(t)

1 n-1

dt < +.

1,A However, this is by no means a restriction, since Wloc () is unchanged if A is modified near 0; thus, we may always assume that (1.7) is satisfied on replacing, if necessary, A by an equivalent Young function near infinity. Under customary regularity conditions on , LAn () turns out to be the smallest Orlicz space into which W 1,A () is continuously embedded ([C3]; see also [C2] for an equivalent formulation). A corresponding Poincar´ type inequality suitable for our e applications is recalled in Theorem 2.1, Section 2. The relevant results about LAn differentiability are contained in the next two theorems. In the statements, - An (Br (x)) denotes an averaged L Luxemburg norm - see Subsection 2.1.

Theorem 1.4 Let be an open subset of IRn and let A be a Young function satisfying (1.4) 1,A and (1.7). Assume that u Wloc (). Then, for every > 0, (1.8)

r0+

lim -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | dy = 0 r

for a.e. x , and (1.9) for a.e. x . Theorem 1.5 Under the same assumptions as in Theorem 1.4, for every > 0, (1.10)

r0+ r0+

lim

u( · ) - u(x)- < u(x), · - x > r

=0

- An (Br (x)) L

lim -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | dy = 0 |y - x|

for a.e. x , and (1.11) for a.e. x . Let us emphasize that the conclusions of Theorems 1.4-1.5 are stated both in terms of integral and of norm quantities since, unlike the case of Lebesgue spaces, they are not equivalent in general Orlicz spaces. Since An (t) is equivalent to tp when A(t) = tp for some p [1, n), then Theorems 1.4-1.5 recover (1.1)-(1.2). In the following corollary, Theorems 1.1 and 1.5 are applied to the borderline 1,A spaces Wloc (), with A(t) = tn log (e + t), which have attracted much attention in recent years (see e.g. [FLS], [EGO], [CP], [AH]). Obviously, the important special case to which we alluded 1,A 1,n above, where Wloc () agrees with Wloc (), is included here.

r0+

lim

u( · ) - u(x)- < u(x), · - x > | · -x|

=0

- An (Br (x)) L

4 Corollary 1.6 Let be an open subset of IRn and let A(t) = tn log (e + t), with 0. Let 1,A u Wloc (). If < n - 1, then, for every > 0, (1.12)

r0+

lim -

Br (x)

exp

|u(y) - u(x)- < u(x), y - x > | |y - x|

n n-1-

- 1 dy = 0

for a.e. x , and (1.13)

r0+

lim

u( · ) - u(x)- < u(x), · - x > | · -x|

q

- exp n-1- (Br (x)) L

n

=0

for a.e. x . Here, Lexp (Br (x)) stands for the Orlicz space associated with the Young function q es - 1 with q 1. If = n - 1, then, for every > 0, (1.14) lim -

Br (x)

r0+

exp exp

|u(y) - u(x)- < u(x), y - x > | |y - x|

n

- e dy = 0

for a.e. x , and (1.15)

r0+

lim

u( · ) - u(x)- < u(x), · - x > | · -x|

n

- exp exp L

n

=0

(Br (x))

for a.e. x . Here, Lexp exp (Br (x)) stands for the Orlicz space associated with the Young function exp(exp(sn )) - e. If > n - 1, then u is equivalent to a function which is differentiable a.e. in . We now address ourselves to the problem of whether the conclusion (1.1), and, more gener ally, (1.8), can be somewhat sharpened. Consider first (1.1), involving the norm in Lp (), the Lebesgue space into which W 1,p () is continuously embedded when is a sufficiently smooth open subset of IRn . This embedding is optimal, as long as Lebesgue (and also Orlicz) range spaces are allowed, but it can be improved if Lorentz spaces are employed. Actually, W 1,p () is continuously embedded into the Lorentz space Lp , p (), a space strictly contained in Lp (), , p whenever 1 p < n. Moreover, Lp () is known to be the smallest rearrangement invariant range space for Sobolev embeddings of W 1,p () ([CP], [EKP]). Thus, one may expect that a result in the spirit of (1.1) holds with the Lp -norm replaced by the Lp , p -norm. This is indeed the case, and it follows as a special instance of Theorem 1.7 below. This theorem relies on a version for Orlicz-Sobolev spaces W 1,A (), recently proved in [C4], of the Sobolev embedding into Lorentz spaces, which involves certain spaces of Orlicz-Lorentz type and yields the best possible rearrangement invariant range space - see Theorem 2.3, Subsection 2.3. Here, a key role is played by the Young function DA,n associated with any Young function A satisfying (1.7) as follows. Let a : [0, +) [0, +] be the non-decreasing left-continuous function such that

t

A(t) =

0

a(r) dr for t 0 ,

and let d : [0, +) [0, +) be the left-continuous function whose (generalized) left-continuous inverse obeys 1 -n 1-n 1 t n-1 dt 1 -1 d (s) = dr for s 0. a(r) a(t)n a-1 (s) 0

5 Then

t

(1.16)

DA,n (t) =

d(r) dr

0

for t 0.

Let us notice that A always dominates DA,n , and is in fact equivalent to DA,n if and only if A(t) is strictly below tn (see [C4, Prop. 5.2] for a precise statement of this fact). For instance, if A(t) = tp with 1 p < n, then DA,n (t) is equivalent to tp , but if A(t) is equivalent to tn near infinity (and satisfies (1.7)), then DA,n (t) is equivalent to tn log-n (e + t) near infinity. Theorem 1.7 Let be an open subset of IRn and let A be a Young function satisfying (1.4) and (1.7). Let u W 1,A (). Then there exists a constant > 0 such that

|Br (x)|

(1.17)

r0+

lim -

0

DA,n s-1/n (u( · ) - u(x)- <

u(x), · - x >) (s) ds = 0

for a.e. x . Here " " stands for decreasing rearrangement. Remark 1.8 Theorem 1.7 strengthens Theorem 1.4, since a constant k, depending only on n, exists such that, for any measurable subset of IRn and any Young function A satisfying (1.4) and (1.7), (1.18)

An

k |f (x)| 4

-1/n f (r)) dr 1/n 0 DA,n (r

dx

0

DA,n (r-1/n f (r)) dr

for every measurable function f in . Thus, in particular, Theorem 1.7 could be deduced from Theorem 1.4, via (1.18). We are not going to prove inequality (1.18) for a general A; let us just mention that it can be established by the methods of [C4, Thm. 4.1]. Instead, we illustrate the situation in the classical setting where A(s) = sp with p [1, n). Let u W 1,p (). Then (1.17) yields (1.19)

r0+

lim

u( · ) - u(x)- <

u(x), · - x >

- p,p (Br (x)) L

=0

for a.e. x . Since a constant c, depending only on p and n, exists such that f - p (Br (x)) L c r f - p ,p (Br (x)) for every f Lp , p (Br (x)), then (1.19) implies (1.10). Equation (1.19) contains L however a more accurate information, due to the strict inclusion of Lp , p (Br (x)) into Lp (Br (x)). Let us also notice that, when A(t) is equivalent to tn near infinity, and hence DA,n (t) is equivalent to tn log-n (e + t) near infinity, then (1.17) is related to the embedding of [BW] and [H]. The remaining part of the paper is organized as follows. Section 2 contains the necessary background from the theory of Orlicz spaces and, more generally, rearrangement invariant spaces, as well as some preliminary results about Poincar´ type inequalities in Orlicz-Sobolev spaces. e Proofs of the results stated above are presented in Section 3.

2

2.1

Background and preliminary results

Rearrangements and rearrangement invariant spaces

Let be a measurable subset of IRn . Given any real-valued measurable function u in , we denote by u : [0, +) [0, +] its decreasing rearrangement, defined as u (s) = sup{t > 0 : |{x : |u(x)| > t}| > s} for s 0.

6 It is easily checked that u is non-increasing and right-continuous in [0, +), and that u and u are equidistributed. Note that the support of u is contained in [0, ||]. The function u is s defined by u = 1 0 u (r) dr for s > 0. s The signed decreasing rearrangement u of u is the function from [0, ||] into IR given by u (s) = sup{t IR : |{x : u(x) > t}| > s} for s [0, ||].

A rearrangement invariant space - briefly, an r.i. space - X() is a Banach function space whose norm · X() satisfies (2.1) v

X()

= u

X() ,

if

u = v .

The associate space X () of X() is the r.i. space defined as X () = v : v is a real-valued measurable function in and

|uv|dx < for all u X()

and is endowed with the norm (2.2) v

X ()

= sup

u=0

|uv|dx

u

.

X()

As a consequence, the H¨lder type inequality o (2.3)

|uv|dx u

X()

v

X () ,

holds for every u X() and v X (). The fundamental function X of X() is defined in [0, ||] as X (t) = E

X()

for t [0, ||],

where E is any measurable subset of such that |E| = t. The fundamental functions of X() and of X () are related by the equality (2.4) X (t) X (t) = t for t [0, ||].

We refer to [BS] for more details on these topics.

2.2

Orlicz, Lorentz, and Orlicz-Lorentz spaces

Let be a measurable subset of IRn , and let A be a Young function, as defined in Section 1. Then the Orlicz space LA () is the set of all real-valued measurable functions u in such that (2.5) u

LA ()

= inf > 0 :

A

|u(x)|

dx 1

is finite. The expression · LA () is called the Luxemburg norm; clearly, LA () is an r.i. space equipped with this norm. The space LA () is defined as the set of those functions which loc belong to LA ( ) for every compact subset of . When || < , we define the averaged norm · - A () as in (2.5) with replaced by - . Notice that the Lebesgue spaces Lp () are L recovered as special instances of Orlicz spaces with A(s) = sp , if 1 p < , and with A(s) = 0

7 for 0 s 1 and A(s) = + for s > 1, if p = +. In both cases, usual norm in Lp (). Notice also that (2.6) E

LA ()

·

LA ()

agrees with the

=

1 , A-1 ( |E| )

1

for every subset E of having finite measure. Hereafter, A-1 denotes the (generalized) rightcontinuous inverse of A. The associate space of LA () is, up to equivalent norms, LA (), where A is the Young conjugate of A defined as A(s) = sup{rs - A(r) : r 0} for s 0. In fact, one has (2.7) v

LA ()

v

(LA ) ()

2 v

LA ()

for every v LA (). A function A is said to dominate another function D near infinity if positive constants k and s exist such that D(s) A(ks) for s s . If this inequality holds for every s 0, then A is said to dominate D globally. The functions A and D are called equivalent near infinity [resp. globally equivalent] if they dominate each other near infinity [globally]. If A and D are Young functions, then the inclusion LA () LD () holds if and only if either || = and A dominates D globally, or || < and A dominates D near infinity. Hence, (2.8) LA () LD () if and only if A dominates D near infinity. loc loc

Lorentz spaces represent another example of r.i. spaces. Recall that, if either 1 < p < and 1 q , or p = q = , the Lorentz space Lp,q () is the space of real-valued measurable functions u in G such that the quantity (2.9) u

Lp,q ()

= sp

1

-1 q

u (s)

Lq (0,||)

is finite. Such a quantity is a norm in Lp,q () if q p. In general, it can be turned into an equivalent norm on replacing u by u on the right-hand side of (2.9). The averaged norm · - p,q () is defined accordingly, with · Lq (0,||) replaced by · - q (0,||) in (2.9). L L Various notions of Orlicz-Lorentz spaces have been introduced in the literature, in the attempt of providing a unified framework for Orlicz and Lorentz spaces. Here, we need to work with spaces from a family of Orlicz-Lorentz spaces considered in [C4] and defined as follows. D(t) Given any q (1, ] and any Young function D satisfying dt < (if q < ), we call t1+q L(q, D)() the r.i. space of those real-valued measurable functions u on such that the norm -1 u L(q,D)() = s q u (s) LD (0,||) is finite. Plainly, the Orlicz spaces LA () and the Lorentz spaces Lp,q (), with q p, are recovered as special cases of the spaces L(q, D)().

2.3

Orlicz-Sobolev spaces

Let be an open subset of IRn and let A be a Young function. The Orlicz-Sobolev space W 1,A () is defined as W 1,A () = {u : u LA (), u is weakly differentiable in and | u| LA ()}.

1,A 1,A The space Wloc () is defined accordingly. Furthermore, we denote by W0 () the subspace of W 1,A () of those functions u which vanish on , in the sense that the continuation of u outside by 0 is a weakly differentiable function in IRn . A Poincar´ type inequality with sharp e Orlicz range norm is given by the following result.

8 Theorem 2.1 Let B be any ball in IRn . Let A be a Young function satisfying (1.7) and let An be the Sobolev conjugate of A defined by (1.5). Then a constant k1 (n), depending only on n, exists such that (2.10) u - uB

LAn (B)

k1 (n) | u|

LA (B)

for every weakly differentiable function u in B such that | u| LA (B). Here, uB = -B u(x) dx, the mean value of u over B. A proof of inequality (2.10) is given in [C2, Thm 2] (for a much larger class of ground domains ), with An replaced by the Young function An given by

s

(2.11) An (s) =

0

rn -1 -1 (rn ) n

n

s

dr

for s 0, where n (s) = n

A(t) t1+n

dt for s 0.

0

Thus, Theorem 2.1 follows from this result and the next lemma, a combination of [C4, Lemma 2.4] and [C3, Lemma 2]. Lemma 2.2 Let A be a Young function. Then

+

(2.12) and (2.13)

0

t A(t)

1 n-1

+

dt < +

if and only if

A(t) t1+n

dt < +

t A(t)

1 n-1

dt < +

if and only if

A(t)

0 t1+n

dt < +.

If (1.7) holds, then the Young functions An and An defined by (1.5) and (2.11), respectively, are globally equivalent with constants depending only on n. Moreover, the function given by ¯ An (s) = s -1 (sn ) n satisfies (2.14) ¯ A-1 (1/s) = ( · )-1/n (s,) ( · ) n

LA (0,) n

for

s0

for s > 0,

and is globally equivalent to An and to An , with constants depending only on n. Theorem 2.1 is a key tool in our proof of Theorem 1.4. The proof of Theorem 1.7 requires the stronger Poincar´ inequality, involving norms of Orlicz-Lorentz type defined in Subsection 2.1, e contained in the next result. Theorem 2.3 Let B be any ball in IRn . Let A be a Young function satisfying (1.7) and let DA,n be the Young function associated with A and n as in (1.16). Then there exists a constant k2 (n), depending only on n, such that (2.15) u - uB

L(n,DA,n )(B)

k2 (n) | u|

LA (B)

for every weakly differentiable function u in B such that | u| LA (B).

9 A version of inequality (2.15), with u - uB

1,A W0 (B),

L(n,DA,n )(B)

replaced by u

L(n,DA,n )(B)

and for

functions u is established in [C4] via symmetrization and interpolation techniques. The proof of Theorem 2.3 follows an analogous scheme. However, the first part of the proof, whose task is to reduce (2.15) to a one-dimensional inequality, is more delicate, due to the fact that functions are taken into account which do not necessarily vanish on B. The symmetrization argument in this case rests upon a form of the P´lya-Szeg¨ principle (see e.g. [C1]), which tells o o 1,A (B), then u is locally absolutely continuous in us that if A is a Young function and u W (0, |B|), and a constant k3 (n), depending only on n, exists such that (2.16) k3 (n) min1/n {( · ), |B| - ( · )} - du (·) ds | u|

LA (0, |B|) LA (B) .

Proof of Theorem 2.3. We have (2.17) u - uB

L(n,DA,n )(B)

= u - uB

L(n,DA,n )(0, |B|) .

By the triangle inequality and the very definition of the norm in L(n, DA,n )(0, |B|), (2.18) u - uB

L(n,DA,n )(0, |B|)

(u - uB )(0, |B|/2) = +

+ (u - uB )(|B|/2, |B|) L(n,D )(0, |B|) L(n,DA,n )(0, |B|) A,n -1/n (·) (u - uB )(0, |B|/2) ( · ) LDA,n (0, |B|) ( · )-1/n (u - uB )(|B|/2, |B|) ( · ) LDA,n (0, |B|) .

Since u is locally absolutely continuous in (0, |B|), it is easily verified that (2.19) u (s) - uB =

|B| 0

(s, |B|) (r) -

r |B|

-

du dr

dr

for s (0, |B|).

Let us call (s) the right-hand side of (2.19). The function is non-increasing in (0, |B|/2) and non-decreasing in (|B|/2, |B|). Hence, (2.20) (0, |B|/2) (s) = (s)(0, |B|/2) (s) and

(|B|/2, |B|) (s) = (|B| - s)(0, |B|/2) (s)

for s 0. From (2.18)-(2.20) we infer, after a change of variable, that (2.21) u - uB

L(n,DA,n )(0, |B|) |B| 0 |B| 0

( · )-1/n (0, |B|/2) ( · ) + ( · )-1/n (0, |B|/2) ( · )

( ·, |B|) (r) - ( ·, |B|) (r) -

r |B| r |B|

-

du (r) dr

dr

L

DA,n (0, |B|)

-

du (|B| - r) dr

dr

L

DA,n

.

(0, |B|)

Now, define the linear operator T , at a locally integrable function on (0, |B|), as T (s) = s-1/n (0, |B|/2) (s)

|B| 0

r-1/n (s, |B|) (r) -

r (r) dr |B|

for s (0, |B|).

If we prove that a constant c, depending only on n, exists such that (2.22) T

L

DA,n

(0, |B|)

c

LA (0, |B|)

10 for every LA (0, |B|), then we deduce from (2.21) and from the P´lya-Szeg¨ principle (2.16) o o that du u - uB L(n,DA,n )(0, |B|) c ( · )1/n - (·) (2.23) ds LA (0, |B|) + c ( · )1/n - du (·) ds 2 c k3 (n) | u|

LA (0, |B|) LA (B) .

Hence, (2.15) follows with k2 (n) = 2 c k3 (n). As for (2.22), it is not difficult to show that constants c1 and c2 , depending only on n, exist such that (2.24) for L1 (0, |B|), and (2.25) T

Ln, (0, |B|)

T

L1 (0, |B|)

c1

L1 (0, |B|)

c2

Ln, 1 (0, |B|)

for Ln, 1 (0, |B|). Thus, by the interpolation theorem [C4, Thm. 3.1], inequality (2.22) holds with c = max{c1 , c2 }.

3

Proof of the main results

Our approach exploits some recent developments in the theory of Orlicz-Sobolev spaces, as well as techniques employed in [S1] and [EG] for ordinary Sobolev spaces. An underling idea is to make use of the Lebesgue differentiation theorem applied to the gradient of a weakly differentiable function. In this connection, a basic result in the present setting is contained in the following lemma. Lemma 3.1 Let be a measurable subset of IRn and let A be a finite-valued Young function. Let f LA (). Then there exists > 0 such that (3.1) for a.e. x . The proof of Lemma 3.1 can be accomplished, a part minor modifications, via an analogous argument as in the proof of [EG, Cor. 1, Sec. 1.7.1]. We omit the details for brevity. Proof of Theorem 1.1. Thanks to (2.8), on replacing, if necessary, A with another Young function 1,A still satisfying (1.3), we may assume that A is finite-valued. Let v Wloc (). By (1.3) and by [C1, Thm. 1b], v is (equivalent to) a continuous function. Fixed any x and any r > 0 such that Br (x) , denote by w the restriction of v to Br (x). Then, (3.2) |v(y) - v(x)| sup w - inf w 1 = |Br (x)| |Br (x)| |Br (x)| 2 k3 (n)-1 min-1/n {( · ), |Br (x)| - ( · )} × k3 (n)min1/n {( · ), |Br (x)| - ( · )} 4 s-1/n k3 (n)

- A (0, |B L

r (x)|)

r0+

lim -

Br (x)

A

|f (y) - f (x)|

dy = 0

|Br (x)|

-

0

dw ds

ds

-

- A (0, |Br (x)|) L dw

ds

(·)

- A (0, |Br (x)|) L

k3 (n)min1/n {( · ), |Br (x)| - ( · )}

-

dw (·) ds

- A (0, |Br (x)|) L

11 for y Br (x). The P´lya-Szeg¨ principle (2.16) entails that o o (3.3) dw k3 (n)min1/n {( · ), |Br (x)| - ( · )} - (·) | w| ds - A (0, |Br (x)|) L On the other hand, on setting (3.4) we have (3.5) 1 |Br (x)|

|Br (x)| t

- A (Br (x)) L

= | v|

- A (Br (x)) . L

F (t) = n tn

A( ) 1+n

d

for t > 0,

A

0

s-1/n

ds = F

1 |Br (x)|1/n

for > 0.

Notice that, by (2.12), F (t) < + for every t > 0; moreover, F strictly increases from 0 to + as t goes from 0 to +. Hence, (3.6) s-1/n

- A (0, |Br (x)|) L

=

1 |Br (x)|1/n F -1 (1)

.

Combining (3.2), (3.3) and (3.6) yields (3.7) |v(y) - v(x)| 4 n r | v| k3 (n) F -1 (1)

1/n - A (Br (x)) L

for y Br (x), where n denotes the measure of the unit ball in IRn . Now, let be any positive number such that Br (x) A | v| dy < +. Let us set M = -Br (x) A | v| dy and A(s) for s 0. M By the very definition of the averaged Luxemburg norm, we have that | v|/ - AM (Br (x)) 1. L Moreover, if we call FM the function defined as in (3.4) with A replaced by AM , then FM (t) = -1 1 1 -1 (M s) for s > 0. Consequently, on setting M F (tM ) for t > 0, whence FM (s) = M F s L(s) = -1 for s > 0 F (s) (3.8) AM (s) = and applying (3.7) with v replaced by v/ and with A replaced by AM , we get (3.9) |v(y) - v(x)| 4 n r | v(y)| L - A k3 (n) Br (x)

1/n 1/n

dy

for y Br (x). Choosing v(y) = u(y) - u(x)- < (3.10)

u(x), y - x > in (3.9) yields u(x)| dy .

|u(y) - u(x)- < u(x), y - x > | 4 n | u(y) - L - A r k3 (n) Br (x)

t0 s0

An application of De L'Hospital rule shows that lim F (t)/t = 0, whence lim L(s) = 0. By Lemma 3.1, fixed any open set , the averaged integral on the right-hand side of (3.10) converges to 0 as r goes to 0 for a.e. x if is sufficiently large. Thus, (3.10) applied with r = |y - x| ensures that |u(y) - u(x)- < u(x), y - x > | (3.11) lim =0 y0 | y - x| for a.e. x . Hence, the conclusion follows.

12 Proof of Proposition 1.3. When A satisfies (1.4), then, as a consequence of [C1, Thms. 1a1b] and of Lemma 2.2, W 1,A (B1 (0)) is not continuously embedded into L (B1 (0)). In particular, an inspection of the proofs of those theorems reveals that a sequence of nonnegative 1,A spherically symmetric functions {uk } can be chosen in such a way that uk W0 (B1 (0)), k B1 (0) A(| uk |) dx 1 and ess sup uk = uk (0) 4 . Let us still denote by uk the continuation by 0 of uk outside B1 (0). Thus, uk is weakly differentiable in the whole of IRn . Let {xk } be the sequence of points in IRn with rational coordinates, and let u : IRn [0, +) be the function defined as u(x) = 21 uk (x - xk ) for x IRn . k=1 k Since ess sup uk (xk ) 21 uk (0) 2k , then u is not essentially bounded in any neighborhood of k any point of B1 (0). Hence, u is not equivalent to any function which is differentiable at some point of B1 (0). On the other hand, inasmuch as B1 (0) A(|uk |) dx B1 (0) A(| uk |) dx for every k (see [T2, Lemma 3]), then

A(|u(x)|) dx

B1 (0) B1 (0)

A

k=1

1 |uk (x - xk )| dx 2k

k=1

1 2k

B1 (0)

A(|uk (x - xk )|) dx

k=1

1 = 1, 2k

where the second inequality is due to the convexity of A. Thus, u LA (B1 (0)), and since

A

B1 (0) k=1

1 | uk (x - xk )| dx 2k

k=1

1 2k

B1 (0)

A(| uk (x - xk )|) dx

k=1 B1 (0) A(|

1 = 1, 2k

then u is easily seen to be a weakly differentiable function in B1 (0) with Hence, u W 1,A (B1 (0)).

u(x)|) dx 1.

The proofs of Theorems 1.4 and 1.7 require the next lemma, containing a weak version of (1.8). Lemma 3.2 Let be an open subset of IRn and let A be a Young function. Assume that 1,1 u Wloc (). Then (3.12) -

Br (x)

A

|u(y) - u(x)- < u(x), y - x > | r

dy sup -

0s1 Bsr (x)

A | u(y) - u(x)| dy

for a.e. x and for every r > 0 such that Br (x) .

1,1 Proof. Let u Wloc (). On replacing, if necessary, u by an equivalent function, we may assume that, for a.e. x , u is absolutely continuous on a.e. ray issued from x (see e.g. [Z, Chap.3, Exercise 3.15]). Fix any such x and let r be a positive number such that Br (x) . Then, for a.e. y Br (x), the function s u(x + s(y - x)) is absolutely continuous in [0, 1]. Moreover,

(3.13)

d u(x + s(y - x)) =< ds

u(x + s(y - x)), y - x >

for a.e. s [0, 1] (see e.g. [AFP, Thm. 3.108]). Consequently,

1

(3.14)

u(y) - u(x)- <

u(x), y - x >=

0

<

u(x + s(y - x)) -

u(x), y - x > ds

13 for a.e. y Br (x). Hence, (3.15) -

Br (x)

A

|u(y) - u(x)- < u(x), y - x > | r

1

dy dy

-

Br (x)

A

0 1

1 r 1 r 1 sr

u(x + s(y - x)) - u(x + s(y - x)) - u(z) - u(z) -

u(x) y - x ds u(x) y - x dz ds

-

Br (x) 1 0

A - A A

ds dy

=

0 1

u(x) z - x

Bsr (x)

0

-

Bsr (x)

u(x)

dz ds sup -

0s1 Bsr (x)

A

u(z) -

u(x)

dz,

where the second inequality holds in (3.15) owing to Jensen's inequality. Proof of Theorem 1.4. Fix any open set . Let x and r > 0 be such that Br (x) 1,A . Let v Wloc (). Then (3.16) v

LAn (Br (x))

v - vBr (x)

LA (Br (x)) LA (Br (x))

LAn (Br (x))

+ vBr (x)

LAn (Br (x))

k1 (n) | v| k1 (n) | v|

+ vBr (x) +

LAn (Br (x))

1 |v| dx 1 LAn (Br (x)) |Br (x)| Br (x) 2 v LA (Br (x)) 1 LA (Br (x)) 1 LAn (Br (x)) k1 (n) | v| LA (Br (x)) + |Br (x)| 2 1 1 = k1 (n) | v| LA (Br (x)) + v LA (Br (x)) . -1 1 |Br (x)| An |B 1 A-1 (x)|

|Br (x)|

r

Notice that the second inequality holds thanks to the Poincar´ inequality (2.10) and that the e last equality follows from (2.6). [C2, Inequality (3.26)] and Lemma 2.2 ensure that a constant c1 (n), depending only on n, exists such that (3.17) 1 t1/n A-1 1 t A-1 n

1 t

c1 (n)

for t > 0.

Combining (3.16)-(3.17) yields (3.18) v

LAn (Br (x))

k1 (n) | v|

LA (Br (x))

-1/n + 2c1 (n) n v/r

LA (Br (x)) .

Assume for a moment that the quantity (3.19)

Br (x)

A(| v|) dy +

Br (x)

A

|v| r

dy

is finite, and call it M . For such a choice of M , define AM as in (3.8), and observe that, if (AM )n is the function associated with AM as in (1.5), then (3.20) (AM )n (t) = 1 t An M M 1/n for t 0.

14 Since | v|

LAM (Br (x))

1

and

v/r

LAM (Br (x))

1,

then, on replacing A by AM in (3.18), one gets (3.21) where c2 (n) = k1 (n) + 2c(n)n (3.22) -

Br (x) -1/n

v

L(AM )n (Br (x))

c2 (n),

. From (3.20)-(3.21) we deduce that |v(y)| dy

An

c3 (n)r -Br (x) A(| v|) dz + -Br (x) A A

Br (x)

|v| r

dz

1/n

-

Br (x)

A(| v|) dy + -

1/n

|v| r

dy,

where c3 (n) = c2 (n)n . Obviously, inequality (3.22) continues to hold even if the expression (3.19) is infinite. Applying (3.22) with v( · ) = for some > 0, yields (3.23) -

Br (x)

1 u( · ) - u(x)- <

u(x), · - x > ,

An A

u(y) - u(x)- < c3 (n)r -Br (x) A | u(y) - u(x)|

| u(z)- u(x)|

u(x), y - x >

|u(z)-u(x)-< u(x),z-x>| r

dz + -Br (x) A A

dz dy.

1/n

dy

-

Br (x)

dy + -

Br (x)

|u(y) - u(x)- < u(x), y - x > | r

On setting (3.24) (r, x) = 2 sup -

0s1 Bs r (x)

A

| u(y) -

u(x)|

dy,

we get from (3.23) and from Lemma 3.2 (3.25) -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | c3 (n) r (r, x)1/n

dy (r, x)

for a.e. x . By Lemma 3.1, a number exists such that limr0 (r, x) = 0 for a.e. x . Fix any such x. Given any > 0, there exists r such that (r, x) < if 0 < r < r . Thus, by (3.25), (3.26) -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | c3 (n) r 1/n

n

dy <

if 0 < r < r . Choosing < c3 (n) in (3.26) yields (1.8). Finally, if < 1, then (3.26) entails that u( · ) - u(x)- < u(x), · - x > c3 (n) 1/n r An (B (x)) - L r

if 0 < r < r . Hence (1.9) follows, owing to the arbitrariness of .

15 Proof of Theorem 1.5. The conclusion will be derived from Theorem 1.4, via a discretization argument (see e.g. [AFP, Ex. 3.16]). Fix any open set . Let x and r > 0 be such 1,A that Br (x) . Let u Wloc () and let be the function, non-decreasing in r, defined by (3.24). Let c3 (n) be the constant appearing in (3.25). Given any > 0, we have (3.27) -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | 2 c3 (n) |y - x|((r, x))1/n

Br 2-i (x)\Br 2-i-1 (x)

dy dy

=

i=0

1 n rn 1 n rn 2-in -

An

|u(y) - u(x)- < u(x), y - x > | 2 c3 (n) |y - x|((r, x))1/n dy dy.

i=0

Br 2-i (x)

An

|u(y) - u(x)- < u(x), y - x > | r 2-i c3 (n) ((r 2-i , x))1/n |u(y) - u(x)- < u(x), y - x > | r 2-i c3 (n) ((r 2-i , x))1/n

=

i=0

Br 2-i (x)

An

By inequality (3.25) applied with r replaced by r 2-i for i = 0, 1, ..., the last sum does not exceed 2-in (r 2-i , x) for a.e. x , and this expression is smaller than or equal to i=0 -in (r, x) = 2n (r, x). Thus, i=0 2 2n -1 (3.28) -

Br (x)

An

|u(y) - u(x)- < u(x), y - x > | 2 c3 (n) |y - x|((r, x))1/n

dy

2n (r, x) 2n - 1

for a.e. x . Starting from (3.28) instead of (3.25) and arguing as in the proof of Theorem 1.4 yield (1.10)-(1.11). Proof of Corollary 1.6. Assume first that 0 < n - 1. Let A(t) be any Young function which n is equivalent to t n- near 0 and is equivalent to A(t) near infinity. Then A satisfies (1.4) and 1,A 1,A 1,A (1.7). Moreover, Wloc () = Wloc (). Thus, u Wloc (). It is easily verified that the function n exp t n-1- - 1 and the function An (t) associated with A as in (1.5) are globally equivalent. Hence, (1.12) and (1.13) follow from (1.10) and (1.11), respectively. Consider now the case where = n - 1, and let A(t) be any Young function which is equivalent to t near 0 and to A(t) near infinity. Since An (t) is globally equivalent to exp exp tn - e, then the conclusion follows as above. Finally, if > n - 1, then A fulfills (1.3). Hence, by Theorem 1.1, u is equivalent to a function which is differentiable a.e. in . Our last task is the proof of Theorem 1.7. We shall need the following lemma. Lemma 3.3 Let A be a Young function satisfying (1.7). Then a positive constant k4 (n), depending only on n, exists such that (3.29) (0,t)

L(n, DA,n )(0, )

k4 (n)

1 A-1 1 n t

for t > 0.

Proof. By [C4, Inequality (3.1)] a constant c, depending only on n, exists such that

(3.30)

(·)

r-1/n (r) dr

L(n, DA,n )(0, )

c

LA (0, )

16 for every LA (0, ). Thus, (3.31) c sup

LA (0, ) -1/n (·) r

|(r)| dr

L(n, DA,n )(0, )

LA (0, ) 0 (s) -1/n s r

=

sup

sup

LA (0, ) L(n, DA,n ) (0, )

L(n, DA,n ) (0, ) r 0

|(r)| dr ds LA (0, ) (s) ds dr LA (0, ) .

=

sup

sup ( · )-1/n

-1/n 0 |(r)| r

L(n, DA,n ) (0, ) LA (0, )

L(n, DA,n ) (0, )

sup

L(n, DA,n ) (0, )

(·) (s) 0

ds

LA (0,)

L(n, DA,n ) (0, )

Notice that the first equality relies on (2.2), and the last inequality is due to (2.7). We have from (3.31) that (3.32) ( · )-1/n

0 (·)

(s) ds

LA (0,)

c

L(n, DA,n ) (0, ) r 0

for L(n, DA,n ) (0, ). Given t > 0, choose (s) = (0,t) (s) and observe that r-1/n t r-1/n if r t. Thus, (3.32) yields (3.33) t ( · )-1/n (t,) ( · )

LA (0,)

(0,t) (s) ds =

c (0,t)

L(n, DA,n ) (0, )

for t > 0.

Hence, by Lemma 2.2, a constant c , depending only on n, exists such that (3.34) t (0,t)

L(n, DA,n ) (0, )

c A-1 1 n t

for t > 0. and the second inequality in

By (2.4), the left-hand side of (3.34) equals (0,t) (3.29) follows.

L(n, DA,n )(0, ) ,

Proof of Theorem 1.7. Fixed any x and any r > 0 such that Br () , and given any v W 1,A (), we start as in the proof of Theorem 1.4. On making use of inequality (2.15) instead of (2.10) we get (3.35) v - vBr (x) L(n, DA,n )(Br (x)) + vBr (x)) L(n, DA,n )(Br (x)) 1 k2 (n) | v| LA (Br (x)) + |v(y)| dy 1 L(n, DA,n )(Br (x)) |Br (x)| Br (x) 2 k2 (n) | v| LA (Br (x)) + v LA (Br (x)) 1 LA (Br (x)) 1 L(n, DA,n )(Br (x)) |Br (x)| 2 1 = k2 (n) | v| LA (Br (x)) + v LA (Br (x)) (0, |Br (x)|) L(n, DA,n )(0, ) . 1 |Br (x)| A-1

L(n, DA,n )(Br (x)) |Br (x)|

v

Inequalities (3.29) and (3.17) ensure that (3.36) (0,t)

L(n, DA,n )(0, ) A-1 1 t

t1/n

c1 (n) k4 (n)

for t > 0.

17 Combining (3.35)-(3.36) yields (3.37) v

L(n, DA,n )(Br (x))

k4 (n) | v|

LA (Br (x))

+

2

1/n n

c1 (n) k4 (n) v/r

LA (Br (x)) .

Now, let M denote the quantity (3.19), let AM be given by (3.8), and let DM be an abridged notation for DAM ,n , the function defined as in (1.16) with A replaced by AM . Then an analogous argument as in the proof of Theorem 1.4 enables us to deduce from (3.37) that (3.38)

-1/n

v

L(n, DM )(Br (x))

c4 (n),

where c4 (n) = k4 (n)+2n c1 (n) k4 (n). It is easily verified that DM (t) = DA,n (t)/M for t 0. Thus, inequality (3.38) implies that

|Br (x)|

(3.39)

-

0

DA,n

s-1/n v (s) c4 (n)

1

ds -

Br (x)

A(| v|) dy + -

Br (x)

A

|v| r

dy.

On applying (3.39) with v( · ) = 3.2, we get

|Br (x)|

u( · ) - u(x)- <

u(x), · - x >

and making use of Lemma

(3.40)

s-1/n u( · ) - u(x)- < u(x), · - x > (s) ds - DA,n c4 (n) 0 | u(y) - u(x)| |u(y) - u(x)- < u(x), y - x > | - dy + - A A r Br (x) Br (x) 2 sup -

0s1 Bsr (x)

dy

A

| u(y) -

u(x)|

dy .

By Lemma 3.1, a positive number exists such that the last expression converges to 0 as r goes to 0 for a.e. x . Hence (1.17) follows.

References

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