Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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A measure µ ∈ M

such that µ is concentrated on X and ||k ∗ µ||

∗

≤ 1

is called a test measure for D

k,A

(X). If in addition D

k,A

(X) = ||µ||, we

say that µ is a D

k,A

−capacitary distribution for X and k ∗ µ is called a

k,A

−capacitary potential for X.

For m > 0, the Bessel kernel G

is deﬁned through its Fourier trans-

form IF (G

) as [IF (G

)] (x) = (2π)

−



1 + |x|



−

where [IF (f )] (x) =

(2π)

−

f(y)e

−ixy

dy for f ∈ L

. G

is positive, in L

and veriﬁes the

equality: G

r+s

= G

∗ G

For more details on Bessel kernels, see [44], [45] and [92].

We note I

(x) = |x|

m−N

the Riesz kernel.

We put B

m,A

= C

, B

m,A

= C

, R

m,A

= C

and R

m,A

3. Capacity and non linear potential in Orlicz spaces

The essential results concerning the notion of capacity in Orlicz spaces are

resumed in the following theorem; see [28] and [42].

Theorem 3.1. Let A be an N-function. Then

a) C

k,A

is an outer capacity deﬁned on all subsets of IR

b) C

k,A

(X) = 0 if and only if there exists f ∈ L

such that k ∗ f = +∞.

c) If f

→ f strongly in L

, then

(1) k ∗ f

→ k ∗ f in C

k,A

−capacity,

(2) there exists a subsequence (f”

)

of the sequence (f

)

such that

k ∗ f”

→ k ∗ f C

k,A

-quasi uniformly,

(3) k ∗ f”

→ k ∗ f C

k,A

-quasi everywhere.

d) If (K

)

is a decreasing sequence of compact set and K = ∩

, then

lim

n→∞

k,A

) = C

k,A

e) If (X

)

is a increasing sequence of sets in IR

and if L

is a reﬂexive

Orlicz space, then lim

n→∞

k,A

) = C

k,A

(∪

f) If L

is a reﬂexive Orlicz space, then analytic sets are C

k,A

-capacitable.

g) If A is an N-function and D

∗

k,A

is the outer capacity associated to D

k,A

then for any set X, D

∗

k,A

(X) = C

k,A

(X).

h) If L

is a reﬂexive Orlicz space, then for all analytic sets X

k,A

(X) = C

k,A

(X).

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Let A be an N-function and k a positive and measurable function deﬁned

on IR

. Let X be any set such that C

k,A

(X) < ∞. We consider the following

variational problem: Find f

∈ L

such that k ∗f

≥ 1 C

k,A

-q.e. on X and

|||f

|||

= inf{|||f|||

: f ∈ L

, k ∗ f ≥ 1 C

k,A

-q.e. on X}.

An answer is given in the following theorem; see [43].

Theorem 3.2. 1) Let L

be a reﬂexive Orlicz space and X be any set such

that C

k,A

(X) < ∞. Then X has a unique C

k,A

-capacitary distribution f ;

f ∈ L

, k ∗ f ≥ 1 on X and C

k,A

(X) = |||f|||

2) Let L

be a reﬂexive Orlicz space and X be an analytic set. If γ

is the D

k,A

−capacitary distribution for X and f is the C

k,A

-capacitary

distribution for X, then

k ∗ γ =





a ◦ (f .|||f |||

−1

)



∗



−1

a ◦(f .|||f|||

−1

) a.e.

Moreover k ∗ f ≤ 1 on suppγ, where a is the derivative of A.

4. On the Bessel potentials in Orlicz spaces

4.1. On the continuity of Bessel potentials in Orlicz spaces

The results in this subsection show that Bessel capacities in reﬂexive Orlicz

spaces are non increasing under orthogonal projection of sets; see [29]. This

is used to get a continuity of potentials on some subspaces. The obtained

results generalize those of Meyers and Reshetnyak in the case of Lebesgue

classes.

Theorem 4.1. 1) Let A be an N-function such that A and A

∗

satisfy the

∆

condition. Let k be a kernel on IR

which is spherically symmetric and

non-increasing as |x| increases. If S is an aﬃne subspace of IR

and X a

subspace of IR

, then

k,A

X) ≤ C

k,A

(X) .

2) Let A be an N-function such that A and A

∗

satisfy the ∆

condi-

tion. Let k be a kernel on IR

which is spherically symmetric and non-

increasing as |x| increases. Further, suppose that k is locally integrable with

lim

|x|→∞

k(x) = 0. Let S be an aﬃne subspace of IR

. Then

a) For f ∈ L

and ε > 0, there exists a closed set F ⊂ S such that

k,A

(S − F ) < ε and k ∗ f ∈ C

(F + S

⊥

Hence

k ∗ f ∈ C

(x + S

⊥

) C

k,A

− q.e. in S.

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b) Let (f

)

be a sequence convergent to f in L

. Then there is a subse-

quence (f

)

, such that given ε > 0, there exists a closed set F ⊂ S

with the property

k,A

(S − F ) < ε and k ∗ f

→ k ∗ f in C

(F + S

⊥

Hence

k ∗ f

→ k ∗ f in C

(x + S

⊥

) C

k,A

− q.e. in S.

Theorem 4.2. Let A be an N-function such that A and A

∗

satisfy the ∆

condition. Let S be an aﬃne subspace of IR

and 0 ≤ s < m. Then

(1) For f ∈ L

and ε > 0, there exists a closed set F ⊂ S such that

m−s,A

(S − F ) < ε

and

∗ f) ∈ C

(F + S

⊥

) for all j, |j| ≤ s.

Hence for such j,

∗ f) ∈ C

(x + S

⊥

) B

m−s,A

− q.e. in S.

(2) Let (f

)

be a sequence convergent to f in L

. Then there is a subse-

quence (f

)

, such that given ε > 0, there exists a closed set F ⊂ S

with the property

m−s,A

(S − F ) < ε

and

∗ f

) → D

∗ f ) in C

(F + S

⊥

) for all j, |j| ≤ s.

Hence for such j,

∗ f

) → D

∗ f ) in C

(x + S

⊥

) B

m−s,A

− q.e. in S.

Theorem 4.3. Let A be an N-function and T be a one to one map of IR

onto itself. Suppose that T and its inverse T

−1

satisfy a Lipschitz condition.

Let ρ, 0 < ρ < ∞, and X ⊂ IR

be such that diam X ≤ ρ. Then there

exists a constant C, independent of X such that

m,A

[T (X)] ≤ CB

m,A

(X).

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If IR

is the aﬃne direct sum of G and H, we deﬁne P

as the pro-

jection of IR

onto G, parallel to H.

Theorem 4.4. Let A be an N-function such that A and A

∗

satisfy the ∆

condition. Let ρ, 0 < ρ < ∞ and X ⊂ IR

be such that diam P

X ≤ ρ.

Then there exists a constant C, independent of X such that

m,A

X) ≤ CB

m,A

(X).

4.2. Instability of capacity in Orlicz spaces

In this subsection we describe the instability of certain capacities in Orlicz

spaces. Our results generalize some of those obtained by C. Fernstr¨om in

[48] and the Theorem 9 of Hedberg in [7] in the case of Lebesgue spaces.

See [30].

For m > 0 we deﬁne the Riesz kernel, I

, by I

(x) = |x|

m−N

for

x ∈ IR

We suppose that m < N. If A is an N-function and X any subset of

, we deﬁne R

m,A

(X) as

m,A

(X) = inf{|||f|||

: f ∈ L

and I

∗ f ≥ 1 on X}.

We know that R

m,A

is an outer capacity; see [40]. We let R

m,A

A ◦R

m,A

Deﬁnition 4.1. Let X be a Borel set and m be the Lebesgue measure on

. Then x is a density point for X if

lim

r→0

m(B(x, r))

−1

m(X ∩ B(x, r)) = 1.

Theorem 4.5. Let A be an N -function verifying the ∆

condition. Suppose

that m <

. Let X be a Borel set and x a density point for X. Then

lim

r→0



m,A

(B(x, r))



−1

m,A

(X ∩ B(x, r)) = 1.

Corollary 4.1. Let A be an N-function verifying the ∆

condition. Sup-

pose that m <

. Let X be a Borel set. Then

lim

r→0



m,A

(B(x, r))



−1

m,A

(X ∩ B(x, r)) = 1, a.e. on X.

Theorem 4.6. Let A be an N-function verifying the ∆

condition and E

be a Borel set. Suppose that m <

. Then a.e. on IR

one of the following

relations holds

lim

r→0



m,A

(B(x, r))



−1

m,A

(E ∩ B(x, r)) = 1

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lim

r→0

limr

−N

m,A

(E ∩ B(x, r)) = 0.

Theorem 4.7. Let A be an N-function verifying the ∆

condition and E

be an everywhere dense Borel set. Suppose that m <

. Then the following

are equivalent.

(i) R

m,A

(E ∩ O) = R

m,A

(O) for every open O.

(ii) R

m,A

(E ∩ B(x, r)) = R

m,A

(B(x, r)) for all x and r.

(iii) For almost all x (with respect to Lebesgue measure)

lim

r→0

sup r

−N

m,A

(E ∩ B(x, r)) > 0.

The next Theorem gives a connection between Riesz and Bessel capac-

ities in the case of Orlicz spaces.

Theorem 4.8. (a) Let m be a positive number such that m < N. Then

there exists a constant δ, such that for all N-functions A and all sets X ⊂

m,A

(X) ≤ δB

m,A

(X) .

(b) Let A be an N-function verifying the ∆

condition such that m <

Let B

be the ball centered at 0 with radius r. Then for all sets X ⊂ B

there exists a constant δ, independent of X, but dependent on r, such that

m,A

(X) ≤ δR

m,A

(X) .

Remark 4.1. Since A veriﬁes the ∆

condition, there exists a constant γ

independent of X, but dependent of r, such that

m,A

(X) ≤ γ

m,A

(X) .

4.3. Bessel potentials in Orlicz spaces

In this subsection it is shown that Bessel potentials have a representation in

term of measure when the underlying space is Orlicz. A comparison between

capacities and Lebesgue measure is given and geometric properties of Bessel

capacities in this space are studied. Moreover it is shown that if the capacity

of a set is null, then the variation of all signed measures of this set is null

when these measures are in the dual of an Orlicz-Sobolev space; see [31].

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4.3.1. Representation of potentials and comparison with Lebesgue

measure

Theorem 4.9. Let A be an N-function such that A and A

∗

satisfy the

∆

condition. Let m be a positive integer and X a set in IR

such that

0 < B

m,A

(X) < ∞.

Let f be the B

m,A

−capacitary distribution of X. Then there exists a

positive measure µ

such that:

1) G

∗ f = B

m,A

(X).G

∗



−1

◦ (G

∗ µ

)



, where a is the derivative

of A.

2) supp µ

⊂ X.

If in addition we suppose that X is compact, then

3) G

∗ f ≤ 1 on supp µ

Theorem 4.10. Let A be an N-function such that A and A

∗

satisfy the

∆

condition. Let m be a positive integer. Then

(1) If m ≤ J(A, N ) there exists a constant C = C(A, N, m) > 0 such that

m,A

(X) ≥ C



−1



∗

(X)



−1

for all set X such that m

∗

(X) 6= 0. (Here m is the Lebesgue measure

on IR

and m

∗

is the outer measure associated to m).

(2) If m > J(A, N), there exists a constant C = C(A, N, m) > 0 such that

for all set X 6= ∅,

m,A

(X) ≥ C.

Theorem 4.11.

(1) Let A be an N-function and m be such that 0 < m < N. Let S

B(x, ρ) be the open ball centered at x and with radius ρ. Then there

exists a constant C independent of ρ such that

m,A

) ≤ Cρ

−m

for 0 < ρ ≤ 1.

(2) Let A be an N -function satisfying the ∆

condition and let m be such

that 0 < m < N. Let C(A) be the smallest constant C

such that:

A(2t) ≤ C

A(t), ∀t.

Then there exists a constant C independent of ρ such that: B

m,A

) ≤

−q

−m

for 0 < ρ ≤ 1, where q is the greatest positive integer such

that q ≤

Logρ

−N

LogC(A)

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4.3.2. Relation between capacity and Hausdorﬀ measure

Theorem 4.12. Let A be an N-function such that A and A

∗

satisfy the

∆

condition. Let m be a positive real such that αm < N, where α = α(A).

Let bu be a positive, decreasing function deﬁned on IR, continuous from the

right and such that

(r) ≤ bu(r) and lim

r→0

bu(r)G

(r)

−1

= +∞.

If B

= B

bu,A

, then lim

ρ→0

m,A

)

−1

= 0.

In particular, if αm < N , then lim

ρ→0

m,A

) = 0.

Deﬁnition 4.2. Let ϕ(r) be a positive, increasing function in some interval

[0, r

[ and such that lim

r→0

ϕ(r) = 0. If X is an arbitrary set, the Hausdorﬀ

ϕ-measure of X is given by

ϕ(r)

(X) = lim

s→0

(inf

i≥1

ϕ(r

)),

where the above inﬁmum is taken over all countable coverings of X by

spheres S(x

, r

) such that r

≤ s.

Note that H

ϕ(r)

is a capacity which has the property

ϕ(r)

(X) = H

ϕ(r)

(Y ),

where Y is a G-set containing X.

Theorem 4.13. Let A be an N-function such that A and A

∗

satisfy the

∆

condition and let X be a subset of IR

. Let m be a positive real such

that αm < N, where α = α(A) and let ϕ(r) = B

m,A

). Then

m,A

(X) = 0 if H

ϕ(r)

(X) < ∞.

4.3.3. Capacities and measures in Orlicz-Sobolev spaces

Theorem 4.14. 1) Let A be an N-function such that A and A

∗

satisfy

the ∆

condition and let m be a positive integer such that m ≤ J(A, N).

Let T ∈ W

−m

∗





∩ M





and let K be a compact set such that

m,A

(K) = 0 and T

−

(K) = 0.

Then kT k(K) = 0.

2) Let A be an N-function such that A and A

∗

satisfy the ∆

condi-

tion and let m be a positive integer such that m ≤ J(A, N ). Let T ∈

−m

∗





∩ M





and let X be a kT k-measurable set such that

m,A

(X) = 0 and T

−

(X) = 0.

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Then kT k(X) = 0.

3) Let A be an N-function such that A and A

∗

satisfy the ∆

condi-

tion and let m be a positive integer such that m ≤ J(A, N). Let T ∈

−m

∗





∩M





and K be a compact set such that B

m,A

(K) =

0. Then kT k(K) = 0.

4) Let A be an N-function such that A and A

∗

satisfy the ∆

condi-

tion and let m be a positive integer such that m ≤ J(A, N). Let T ∈

−m

∗





∩ M





and let X be a set such that B

m,A

(X) = 0.

Then kT k(X) = 0.

5. Another developments of strongly nonlinear potential

theory

We establish some relations between potentials and maximal functions in

Orlicz spaces. We give a deﬁnition of quasicontinuity and obtain a descrip-

tion of quasicontinuous representative in some potential spaces. We also

give a result on smooth truncation of potentials in Orlicz-Sobolev spaces

and compare some capacities. As a consequence: a compact is removable

in Orlicz space for an elliptic linear operator of order m with constant

coeﬃcients if and only if its Bessel capacity of order m is null. See [32].

5.1. Maximal operators and potentials

Let f be a locally integrable function. The Hardy-Littlewood max-

imal function associated to f is deﬁned by Mf(x) = M

f(x) =

sup

r>0

|B(x, r)|

−1

B(x,r)

|f(y)|dy.

Here |B(x, r)| is the Lebesgue measure of B(x, r) on IR

The fractional maximal function associated to f is deﬁned for 0 < α <

N, by M

f(x) = sup

r>0

|B(x, r)|

α−N

B(x,r)

|f(y)|dy.

And for 0 ≤ α < N, and δ > 0, the inhomogeneous version of these

functions is deﬁned by M

α,δ

f(x) = sup

δ≥r>0

|B(x, r)|

α−N

B(x,r)

|f(y)|dy.

For 0 ≤ α < N, and δ > 0, we deﬁne the modiﬁed Riesz kernel, I

α,δ

(x) = I

(x), if |x| < δ,

α,δ

(x) = 0, if |x| ≥ δ.

The Riesz potential I

∗ µ, 0 < α < N , where µ is a positive measure,

can be estimated below by the fractional maximal function associated to µ.

In fact, for every r > 0,

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|x − y|

−Ndµ(y) ≥

|x−y|≤r

|x − y|

α−N

dµ(y) ≥

|x−y|≤r

dµ(y).

The reverse inequality is false in general.

In the ﬁrst part of the following theorem, we give a generalization to

Orlicz spaces, of the classical theorem of B. Muckenhoupt and R.I. Wheeden

[85], which establishes the opposite inequality in term of L

norms.

Theorem 5.1. Let A be an N-function satisfying the ∆

condition, and

let 0 < α < N. Then

(1) There is a constant C > 0, such that for any positive measure µ,

i) C

−1

A(M

µ)dx ≤

A(I

∗ µ)dx ≤ C

A(M

µ)dx

ii) C

−1

|||M

µ|||

≤ |||I

∗ µ|||

≤ C|||M

µ|||

(2) If δ is a positive number, there are positive constants C

, C

and C

such that for any positive measure µ,

|||M

α,δ

µ|||

≤ C

|||I

α,δ

∗ µ|||

≤ C

|||G

∗ µ|||

≤ C

|||M

α,δ

µ|||

5.2. Quasicontinuity

We recall the general deﬁnition of quasicontinuity.

Deﬁnition 5.1. Let C be a capacity on IR

and let f be a function de-

ﬁned C−q.e. on IR

or on some open subset of IR

. Then f is said to be

C−quasicontinuous if for every  > 0, there is an open set O such that

C(O) <  and f |

∈ C(O

In other words, the restriction of f to the complement of O is continuous

in the induced topology.

For Bessel capacity B

m,A

, we write (m, A)-quasicontinuous in place of

m,A

-quasicontinuous.

Theorem 5.2. 1) Let A be an N -function satisfying the ∆

condition. If

f ∈ L

, then the potential G

∗ f, m > 0, is (m, A)−quasicontinuous.

Hence every element in L

m,A

has an (m, A)−quasicontinuous represen-

tative.

2) Let A be any N-function. Let f = G

∗ g ∈ L

m,A

, m > 0. Then

lim

r→0

|B(x,r)|

B(x,r)

f(y)dy = G

∗ g(x), wherever G

∗ |g|(x) < ∞,

i.e., (m, A) − q.e.

Theorem 5.3. 1) Let A be any N-function. Let f

and f

be two

(m, A)−quasicontinuous functions, m > 0. Suppose that f

(x) = f

(x)

almost everywhere. Then

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(x) = f

(x) (m, A)−quasieverywhere.

2) Let A be an N-function satisfying the ∆

condition. Let f, g ∈ L

m,A

m > 1. Then D(fg) = f(x)Dg(x) + Df(x)g(x) (m − 1, A) − q.e.

We extend the last theorem for any kernel and with a less restrictive

property that the quasicontinuity; but for reﬂexive Orlicz space. This is the

tribute to pay!

By B(IR

) we note the family of Borelian sets in IR

Theorem 5.4. Let k be any kernel and suppose that the N-function A is

such that A and A

∗

satisfy the ∆

condition. Let g

and g

be two functions

verifying the following: ∀ > 0, ∃X ∈ B(IR

) : C

k,A

(X) <  and the

restrictions of g

and g

X are continuous.

Suppose that {x : g

(x) 6= g

(x)} ∈ B(IR

) and that g

(x) = g

(x) a.e.

Then

(x) = g

(x)(k, A) − q.e.

5.3. Operations on potentials, Other deﬁnition of capacity

and removable singularities

For reﬂexive Orlicz spaces, we establish that composition of Bessel potential

with a smooth operator, is a potential. This is an extension of a well known

Theorem of V.G. Maz’ya which is a substitute of the fact that the Sobolev

spaces W

m,p

(m 6= 1) are not closed under contractions. An immediate

consequence is the equivalence of capacities N

m,A

and B

m,A

. Note that in

the case of L

Lebesgue spaces, this two capacities are equivalent even if m

is not integer. See [21]. The correspondent case for Orlicz spaces remains

open.

On the other hand, we show that in reﬂexive Orlicz spaces, a compact

set K is removable for an elliptic linear operator of order m, with constant

coeﬃcients, if and only if its Bessel capacity is null (i.e. B

m,A

(K) = 0).

This is the ﬁrst relation between the Strongly Nonlinear Potential Theory,

and Partial Diﬀerential Equations.

Theorem 5.5. Let m be an integer such that 0 < m < N and A be an

N-function such that A and A

∗

satisfy the ∆

condition. Let k be an integer

such that k ≥ m and T ∈ C

(IR

) veriﬁes the following condition

sup



i−1

(i)

(x)



≤ L < ∞, i = 1, 2, ..., k.

Then T ◦ (G

∗ f) ∈ L

m,A

, for all f ∈ L

, and there is a constant C,

which depends only on A, m and N, such that

|||T ◦ (G

∗ f)|||

m,A

≤ CL|||G

∗ f|||

m,A

= CL|||f |||