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volume 7, issue 1, article 33, 2006.

Received 24 November, 2005;

accepted 28 November, 2005.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

A MULTIPLICATIVE EMBEDDING INEQUALITY IN ORLICZ-SOBOLEV SPACES

MARIA ROSARIA FORMICA

Dipartimento di Statistica e Matematica per la Ricerca Economica Università degli studi di Napoli “Parthenope", via Medina 40 80133 Napoli (NA) - ITALY.

EMail:mara.formica@uniparthenope.it

c

2000Victoria University ISSN (electronic): 1443-5756 347-05

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A Multiplicative Embedding Inequality in Orlicz-Sobolev

Spaces Maria Rosaria Formica

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J. Ineq. Pure and Appl. Math. 7(1) Art. 33, 2006

http://jipam.vu.edu.au

Abstract

We prove an Orlicz type version of the multiplicative embedding inequality for Sobolev spaces.

2000 Mathematics Subject Classification:46E35, 26D15, 46E30.

Key words: Orlicz spaces, Sobolev embedding theorem, Orlicz-Sobolev spaces.

1. Introduction and Preliminary Results

Let Ω be a non-empty bounded open set in R, n > 1 and let 1 ≤ p < n.

The most important result of Sobolev space theory is the well-known Sobolev imbedding theorem (see e.g. [1]), which - in the case of functions vanishing on the boundary - gives an estimate of the norm in the Lebesgue space L^q(Ω), q = np/(n−p)of a functionuin the Sobolev space W₀^1,p(Ω), in terms of its W₀^1,p(Ω)-norm. Such an estimate, due to Gagliardo and Nirenberg ([6], [12]) can be stated in the following multiplicative form (see e.g. [4], [10]).

Theorem 1.1. Let Ω be a non-empty bounded open set in R, n > 1 and let 1 ≤ p < n. Letu ∈ W₀^1,p(Ω)T

L^r(Ω) for somer ≥ 1. Ifq lies in the closed interval bounded by the numbersrandnp/(n−p), then the following inequality holds

(1.1) kuk_q≤ck|Du|k^θ_pkuk^1−θ_r , where

θ =

1 r −¹_q

1

n− ¹_p + ¹_r ∈[0,1]

and

c=c(n, p, θ) =

p(n−1) n−p

θ

.

The constantc=c(n, p, θ)is not optimal (see [16], [7] for details).

The goal of this paper is to provide an Orlicz version of inequality (1.1), in which the role of the parameterθ is played by a certain concave function. Our

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approach uses a generalized Hölder inequality proved in [8] (see Lemma 1.2 below).

We summarize some basic facts of Orlicz space theory; we refer the reader to Krasnosel’ski˘ı and Ruticki˘ı [9], Maligranda [11], or Rao and Ren [14] for further details.

A functionA: [0,∞)→[0,∞)is anN-function if it is continuous, convex and strictly increasing, and ifA(0) = 0,A(t)/t →0ast →0,A(t)/t→ +∞

ast→+∞.

IfA, B areN-functions (in the following we will adopt the next symbol for the inverse function of N-functions, too), we writeA(t) ≈ B(t) if there are constantsc₁, c₂ >0such thatc₁A(t)≤ B(t) ≤c₂A(t)for allt >0. Also, we say thatB dominatesA, and denote this byA B, if there exists c > 0such that for all t > 0, A(t) ≤ B(ct). If this is true for allt ≥ t₀ > 0, we say that A B near infinity.

AnN-functionAis said to be doubling if there exists a positive constant c such thatA(2t)≤ cA(t)for allt >0;Ais called submultiplicative ifA(st)≤ cA(s)A(t) for alls, t > 0. ClearlyA(t) = t^r, r ≥ 1, is submultiplicative. A straightforward computation shows thatA(t) = t^a[log(e+t)]^b,a≥1,b >0, is also submultiplicative.

Given an N−function A, the Orlicz space L_A(Ω) is the Banach space of Lebesgue measurable functions f such that A(|f|/λ)is (Lebesgue) integrable on A for some λ > 0. It is equipped with the Luxemburg norm kfk_A = infn

λ >0 :R

ΩA_|f|

λ

dx≤1o .

IfAB near infinity then there exists a constantc, depending onAandB,

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such that for all functionsf,

(1.2) kfkA≤ckfkB.

This follows from the standard embedding theorem which shows thatL_B(Ω)⊂ L_A(Ω).

Given anN-functionA, the complementaryN-functionAeis defined by A(t) = supe

s>0

{st−A(s)}, t ≥0.

The N-functions A andAesatisfy the following inequality (see e.g. [1, (7) p.

230]):

(1.3) t≤A⁻¹(t)Ae⁻¹(t)≤2t.

The Hölder’s inequality in Orlicz spaces reads as Z

Ω

|f g|dx≤2kfk_Akgk

Ae.

We will need the following generalization of Hölder’s inequality to Orlicz spaces due to Hogan, Li, McIntosh, Zhang [8] (see also [3] and references therein).

Lemma 1.2. IfA,BandCareN−functions such that for allt >0, B⁻¹(t)C⁻¹(t)≤A⁻¹(t),

then

kf gk_A≤2kfk_Bkgk_C.

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IfAis anN−function, let us denote byW^1,A(Ω)the space of all functions inL^A(Ω)such that the distributional partial derivatives belong toL^A(Ω), and by W₀^1,A(Ω)the closure of theC₀^∞(Ω)functions in this space. Such spaces are well- known in the literature as Orlicz-Sobolev spaces (see e.g. [1]) and share various properties of the classical Sobolev spaces. References for main properties and applications are for instance [5] and [15].

Ifu∈W₀^1,A(Ω)and Z ∞

1

A(s)˜

sⁿ⁰⁺¹ds= +∞, n⁰ =n/(n−1) then the continuous embedding inequality

(1.4) kuk_A^∗ ≤ck|Du|k_A

holds, whereA^∗ is the so-called Sobolev conjugate ofA, defined in [1], andc is a positive constant depending only on A andn. In the following it will be not essential, for our purposes, to know the exact expression of A^∗. However, we stress here that one could consider the best functionA^∗ such that inequality (1.4) holds (see [2], [13] for details).

In the sequel we will need the following definition.

Definition 1.1. Given anN−functionA, define the functionh_Aby h_A(s) = sup

t>0

A(st)

A(t) , 0≤s <∞.

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Remark 1. The functionh_Acould be infinite ifs >1, but ifAis doubling then it is finite for all 0 < s < ∞ (see Maligranda [11, Theorem 11.7]). If A is submultiplicative thenh_A ≈ A. More generally, given anyA, for alls, t ≥ 0, A(st)≤h_A(s)A(t).

The property of the function h_A which will play a role in the following is that it can be inverted, in fact the following lemma holds.

Lemma 1.3. If Ais a doublingN−function thenh_A is nonnegative, submulti- plicative, strictly increasing in[0,∞)andh_A(1) = 1.

For the (easy) proof see [3, Lemma 3.1] or [11, p. 84].

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2. The Main Result

We will begin by proving two auxiliary results. The first one concerns two functions that we callK = K(t)andH = H(t): they are a way to “measure”, in the final multiplicative inequality, how far the right hand side is with respect to the norms ofuand of|Du|. In the standard case it isK(t) = t^θ, 0≤ θ ≤ 1 andH(t) =t^1−θ.

Lemma 2.1. LetK ∈ C([0,+∞[)T

C²(]0,+∞[)be:

- a positive, constant function, or

-K(t) =αtfor someα >0, or

- the inverse function of anN−function which is doubling together with its complementaryN−function.

Then the functionH : [0,+∞[→[0,+∞[defined by

H(t) =







t

K(t) ift >0 limt→0

t

K(t) ift= 0 belongs toC([0,+∞[)T

C²(]0,+∞[), and is:

- a positive, constant function, or

-H(t) = βtfor someβ >0, or

- is equivalent to the inverse function of an N−function which is doubling together with its complementaryN−function.

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Proof. In the first two possibilities for K the statement is easy to prove. IfK is the inverse of a doubling N-functionA, it is sufficient to observe that from inequality (1.3) it isH ≈Ae⁻¹.

Lemma 2.2. Let Φ be an N−function, and let F be a doubling N−function such thatΦ◦F⁻¹ is anN−function. The following inequality holds for every u∈L^Φ(Ω):

(2.1) kuk_Φ ≤ξ_F⁻¹(kF ◦ |u|kΦ◦F⁻¹), whereξ_F⁻¹ is the increasing function defined by

(2.2) ξ_F⁻¹(µ) = 1

h⁻¹_F

1 µ

∀µ >0.

Proof. By definition ofh_F (see Definition1.1; note that by the assumption that F is doubling,h_F is everywhere finite, see Remark1) we have

F(s)h_F(t)≥F(st) ∀s, t >0 and therefore

sh_F(t)≥F(F⁻¹(s)t) ∀s, t >0,

(2.3) F⁻¹(sh_F(t))≥F⁻¹(s)t ∀s, t >0.

Setting

µ=µ(λ) = 1 h_F _λ¹

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it is

λ = 1

h⁻¹_F

1 µ

:=ξ_F⁻¹(µ),

therefore from inequality (2.3), fort = ¹_λ ands = F(|u|), taking into account thatξ_F⁻¹ is increasing, we have

kuk_Φ = inf

λ >0 : Z

Ω

Φ |u|

λ

dx≤1

= inf

λ >0 : Z

Ω

Φ

F⁻¹(F(|u|)) λ

dx≤1

≤inf

λ >0 : Z

Ω

Φ

F⁻¹

F(|u|)h_F 1

λ

dx≤1

= inf

ξ_F⁻¹(µ)>0 : Z

Ω

Φ

F⁻¹

F(|u|) µ

dx≤1

=ξ_F⁻¹

inf

µ >0 : Z

Ω

Φ

F⁻¹

F(|u|) µ

dx≤1

=ξ_F⁻¹(kF ◦ |u|k_Φ◦F⁻¹)

We can prove now the main theorem of the paper. The symbol ξ_K which appears in the statement is the function considered in Lemma 2.2, defined in equation (2.2). However, since this symbol is used for any functionK considered in Lemma2.1, we agree to denote

ξK(µ) := 1 ∀µ≥0 if K is constant

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and

ξK(µ) := µ ∀µ≥0 if K(t) = αt for someα >0.

The same conventions will be adopted for the symbol ξ_H. Note that from Lemma2.1we know thatHis equivalent to the inverse of a doublingN−function, let us call itB⁻¹. We will agree to denote still byξ_H the function that we should denote byξ_B⁻¹. This convention does not create ambiguities because ifB ≈C thenhB ≈hCandξ_B⁻¹ ≈ξ_C⁻¹, thereforeξH is well defined up to a multiplicative positive constant.

Theorem 2.3. Let Ωbe a non-empty bounded open set in R,n > 1and let P be anN−function satisfying

Z ∞ 1

P˜(s)

sⁿ⁰⁺¹ ds= +∞, n⁰ =n/(n−1).

Let u ∈ W₀^1,P(Ω)T

L^R(Ω) for some N−functionR. If Q is an N−function such that

(2.4) K((P^∗)⁻¹(s))·H(R⁻¹(s))≤Q⁻¹(s) ∀s >0 then the following inequality holds

(2.5) kuk_Q ≤ξ_K(ck|Du|k_P)ξ_H(kuk_R),

where K andH are functions as in Lemma 2.1 andcis a constant depending only onn, P, K.

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Proof. LetK andHbe functions as in Lemma 2.1. IfKis a positive, constant function orK(t) =αtfor someα >0, then the statement reduces respectively to a direct consequence of inequality (1.2) (withAandBreplaced respectively by QandR) or to inequality (1.4) (withAreplaced by P). We may therefore assume in the following thatKis the inverse function of anN−function which is doubling together with its complementaryN−function. Let

Φ₁ =P^∗◦K⁻¹ Φ₂ =R◦H⁻¹.

It is easy to verify that Φ₁ andΦ₂ areN−functions. By assumption (2.4) and Lemma1.2we have

(2.6) kukQ =kK(u)H(u)kQ≤ kK(u)kΦ1 kH(u)kΦ2 . By inequality (2.1),

(2.7) kK(u)k_Φ₁ ≤ξ_K(kuk_Φ₁◦K) =ξ_K(kuk_P^∗)≤ξ_K(ck|Du|k_P),

wherecis a positive constant depending onnandP only. On the other hand, (2.8) kH(u)k_Φ₂ ≤ξ_H(kuk_Φ₂◦H) = ξ_H(kuk_R).

From inequalities (2.6), (2.7), (2.8), we get the inequality (2.5) and the theorem is therefore proved.

We remark that the natural choice of powers forP, Q, R, K, H reduce The- orem 2.3to Theorem 1.1(in Theorem 2.3also the case p = n is allowed); on the other hand, if inequality (2.5) allows growths ofξ_K different power types, in general it is not true thatξ_K(t)ξ_H(t) =t, and this is the “price” to pay for the major “freedom” given to the growthK.

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References

[1] R.A. ADAMS, Sobolev Spaces, Academic Press, New York 1975.

[2] A. CIANCHI, Some results in the theory of Orlicz spaces and applications to variational problems, Nonlinear Analysis, Function Spaces and Appli- cations, Vol. 6, (M. Krbec and A. Kufner eds.), Proceedings of the Spring School held in Prague (1998) (Prague), Acad. Sci. Czech Rep., (1999), 50–92.

[3] D. CRUZ-URIBE, SFOANDA. FIORENZA, TheA∞property for Young functions and weighted norm inequalities, Houston J. Math., 28 (2002), 169–182.

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[9] M.A. KRASNOSEL’SK˘I AND YA.B. RUTICKI˘I, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen 1961.

[10] O.A. LADYZHENSKAYA AND N.N. URAL’CEVA, Linear and Quasi- linear Elliptic Equations, Academic Press, New York 1968.

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