CHARACTERIZATION OF GENERALIZED ORLICZ SPACES

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1 CHARACTERIZATION OF GENERALIZED ORLICZ SPACES RITA FERREIRA, PETER HÄSTÖ, AND ANA MARGARIDA RIEIRO Abstract. The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. ecause the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.. Introduction ourgain, rézis, and Mironescu [5, 6] studied the limit behavior of the Gagliardo semi-norms f p W = fx fy p dx dy, < s <, s,p Ω Ω x y n+sp as s, and established the appropriate scaling factor for comparing the limit with the L p -norm of the gradient of f. They characterized the Sobolev space W,p and proved the convergence of certain imaging models of Aubert and Kornprobst [2] to the well-known total variation model of Rudin, Osher, and Fatemi [3]. Our aim in this paper is to extend the characterization to generalized Orlicz spaces defined on open subsets of R n. Generalized Orlicz spaces L ϕ have been studied since the 94 s. A major synthesis of functional analysis in these spaces is given in the monograph of Musielak [28] from 983, for which reason they have also been called Musielak Orlicz spaces. These spaces are similar to Orlicz spaces, but defined by a more general function ϕx, t that may vary with the location in space: the norm is defined by means of the integral ϕx, fx dx, R n whereas in an Orlicz space, ϕ would be independent of x, ϕ fx. When ϕt = t p, we obtain the Lebesgue spaces, L p. Generalized Orlicz spaces are motivated by applications to image processing [7, 2], fluid dynamics [32], and differential equations [4, 7]. Recently, harmonic analysis in this setting has been studied e.g. in [2, 22, 26]. We have in mind two principal classes of examples of generalized Orlicz spaces: variable exponent spaces L p, where ϕx, t := t px [, 4], and dual phase spaces, where ϕx, t := t p + axt q [3, 4, 8, 9, ]. It is interesting to note that our general methods give optimal results in these two disparate cases, cf. [22]. Also covered are variants of the variable exponent case such as t px loge + t [7, 27, 29, 3]. It is not difficult to see that a direct generalization of the difference quotient to the nontranslation invariant generalized Orlicz case is not possible [24, Section ]. For instance esov and Triebel Lizorkin spaces in this context have been defined using Fourier theoretic approach [, 5]. Hästö and Ribeiro [24], following [3], adopted a more direct approach with a smoothed difference quotient expressed by means of the sharp averaging operator Date: December 4, Mathematics Subject Classification. 46E35, 46E3. Key words and phrases. Musielak Orlicz spaces, Orlicz space, Sobolev space, variable exponent, Poincaré inequality.

2 M #. This is the general approach adopted also in this paper. This paper improves [24] in three major ways:. Instead of variable exponent spaces, we consider more general generalized Orlicz spaces; 2. Instead of R n, we allow arbitrary open sets Ω R n ; and 3. In our main result, we relax the technical assumption f L Ω to its natural form; i.e., f L loc Ω. The latter two generalizations have been previously established in the case L p by Leoni and Spector [25], see also [6, Section ]. In order to achieve these goals, the methods of the main results Section 4 are completely different from those in [24] and involve a new bootstrapping scheme. We introduce some notation to state our main result. We refer to the next section for the precise definition of L ϕ and the assumptions in the theorem. Let Ω R n be an open set, ϕ Φ w Ω, and ε >. Let ψ ε be a set of functions such that. ψ ε L,, ψ ε, and, for every γ >,.2 lim γ ψ ε r dr =. ψ ε r dr =, We define a weak quasi-semimodular, ϱ ε #,Ω f, on L loc Ω by setting ϱ ε #,Ωf := x, r M # f dx ψ ε r dr Ω r ϕ for f L loc Ω, where Ω r := {x Ω : distx, Ω > r} and M # f := fy f dy with f := The associated quasi-norm, f ε #,Ω, is defined by fy dy. f ε #,Ω := inf { λ > ϱ ε #,Ωf/λ }. The following is our main result stated for a Φ-function it is also possible to state it for weak Φ-functions, see Theorem 4.6. The proof follows from Propositions 4. and 4.5. Theorem.. Let Ω R n be an open set, let ϕ ΦΩ satisfy Assumptions A, ainc, and adec, and let ψ ε ε be a family of functions satisfying. and.2. Assume that f L loc Ω. Then, f L ϕ Ω; R n lim sup ϱ ε #,Ωf <. In this case, lim ϱε #,Ωf = ϱ ϕ,ω c n f and lim f ε #,Ω = c n f ϕ,ω, where c n := ffl, x e dx. Remark.2. Note that the previous result is new even in the case of classical Orlicz spaces. In this case, Assumption A automatically holds. 2

3 Remark.3. In the case ϕx, t = t px, Assumptions A and loc always hold, while Assumptions A and A2 are equivalent to the local log-hölder continuity and Nekvinda s decay condition, respectively. Moreover, if p := inf x Ω px >, then ainc holds; and if p + := sup x Ω px <, then adec holds. 2. Preliminaries This section is organized as follows. In Subsection 2., we collect some notation used throughout this paper. Then, in Subsection 2.2, we recall the definition of Φ-functions and of some of its generalizations; we recall also the associated Orlicz spaces, norms, and semimodulars. Finally, in Subsection 2.3, we introduce and discuss our main assumptions on the generalized weak Φ-functions and relate them with other assumptions in the literature. We conclude by proving two auxiliary results, Lemmas 2. and 2.2. The first one can be interpreted as a counterpart in our setting of the weighted power-mean inequality for the function ϕx, t = t p for some p ; the second one is a Jensen-type inequality in the spirit of [22, Lemma 4.4] and [24, Lemma 2.2]. 2.. Notation. We denote by R n the n-dimensional real Euclidean space. We write x, r for the open ball in R n centered at x R n and with radius r >. We use c as a generic positive constant; i.e., a constant whose value may change from appearance to appearance. If E R n is a measurable set, then E stands for its Lebesgue measure and χ E denotes its characteristic function; we denote by L E the space of all Lebesgue measurable functions on E. Let g, h : D R m [, ] be two functions. We write g h to mean that there exists a positive constant, C, such that gz Chz for all z D. If g h g, we write g h. Also, given a sequence g ε ε of non-negative functions in D, the notation lim ε g ε h means that there is a positive constant, C, such that h lim inf C ε g ε lim sup ε g ε Ch in D; i.e., for the equivalence we do not require the limit to exist, only the upper and lower limits to be within a constant of each other. Moreover, if D = [,, we say that g and h are equivalent, written g h, if there exists L such that for all t, we have h t gt hlt. In this case, L is said L to be the equivalence constant. Let ϕ : [, [, ] be an increasing function. We denote by ϕ : [, ] [, ] the left-continuous generalized inverse of ϕ; that is, for all s [, ], 2. ϕ s := inf{t ϕt s}. Let Ω R n be open, let ϕ : Ω [, [, ], and let R n. Then, ϕ + [, ] and ϕ : [, [, ] are the functions defined for all t [, by : [, ϕ + t := sup ϕx, t and ϕ t := inf ϕx, t. x Ω x Ω Note that if ϕx, is increasing for every x Ω, then so are ϕ + and ϕ ; thus, these functions admit a left-continuous generalized inverse in the sense of Φ-functions and generalized Orlicz spaces. We start this subsection by introducing the notion of almost increasing and almost decreasing functions, after which we recall the definition of Φ-functions and of some of its generalizations. Definition 2.. We say that a function g : D R [, ] is almost increasing if gt c gt 2 for every t t 2 in D and some c. We say that c is the monotonicity constant of g. Almost decreasing is defined analogously. 3

4 Definition 2.2. Let ϕ : [, [, ] be an increasing function satisfying ϕ = lim t + ϕt = and lim t ϕt =. We say that ϕ is a i weak Φ-function if t ϕt is almost increasing. t ii Φ-function if it is left-continuous and convex. We denote by Φ w the set of all weak Φ-functions and by Φ the set of all Φ-functions. Remark 2.3. If ϕ Φ, then, by convexity and because ϕ =, for < t < t 2, ϕt = ϕ t t2 t 2 + t t2 t t2 ϕt 2 ; thus, t ϕt is increasing. Hence, Φ Φ t w. Conversely, if ϕ Φ w, then there exists ψ Φ such that ϕ ψ [9, Proposition 2.3]. Definition 2.4. Let Ω R n be an open set, and let ϕ : Ω [, [, ] be a function such that ϕ, t L Ω for every t [,. We say that ϕ is a i generalized weak Φ-function on Ω if ϕx, Φ w uniformly in x Ω; i.e., the monotonicity constant is independent of x. ii generalized Φ-function on Ω if ϕx, Φ for every x Ω. We denote by Φ w Ω and ΦΩ the sets of generalized weak Φ-functions and generalized Φ-functions, respectively. y this definition, it is clear that properties of weak Φ-functions carry over to generalized weak Φ-functions point-wise uniformly. In particular, this holds for Remark 2.3. Similarly, ϕ ψ means that ϕx, ψx, with constant uniform in x, etc. Next, we recall the definition of the generalized Orlicz space, quasi-norm, and quasisemimodular associated with a generalized weak Φ-function. Definition 2.5. Let Ω R n be an open set, let ϕ Φ w Ω, and consider the weak quasi-semimodular, ϱ ϕ,ω, on L Ω defined by ϱ ϕ,ω f := ϕx, fx dx for all f L Ω. The generalized Orlicz space, L ϕ Ω, is given by Ω L ϕ Ω := { f L Ω ϱ ϕ,ω λf < for some λ > }. We endow L ϕ Ω with the quasi-norm f ϕ,ω := inf { λ > ϱ ϕ,ω f/λ }. If, in the Definition 2.5, ϕ ΦΩ, then ϱ ϕ,ω defines a semimodular on L Ω and ϕ,ω a norm on L ϕ Ω see [4]. If ϕ, ψ Φ w Ω and ϕ ψ, then L ϕ = L ψ with equivalent quasi- Remark 2.6. norms Main assumptions. We begin by introducing our main assumptions on the generalized weak Φ-functions on Ω. The first three assumptions, Δ 2, ainc, and adec, extend three known properties for Φ-functions to generalized Φ-functions point-wise uniformly. The fourth assumption, A, relates the behavior of generalized Φ-functions at different values of the variable in Ω. The last assumption, loc, implies that simple functions belong to the generalized Orlicz space. These last two assumptions hold trivially for Orlicz spaces. Let Ω R n be open and ϕ Φ w Ω. We denote by ϕ the generalized inverse of ϕ with respect to the second variable see 2.. 4

5 Δ 2 ϕ is doubling; i.e., there exists A > such that ϕx, 2t Aϕx, t for a.e. x Ω and for all t. ainc There exists a constant ϕ > such that for a.e. x Ω, the map s s ϕ ϕx, s is almost increasing with monotonicity constant c independent of x. adec There exists a constant ϕ > such that for a.e. x Ω, the map s s ϕ ϕx, s is almost decreasing with monotonicity constant c independent of x. A There exist β, σ > for which: A ϕx, βσ ϕx, σ for all x Ω; A ϕx, βt ϕy, t for every ball Ω, x, y, and t [ σ, ϕ y, ] ; A2 there exists h L Ω L Ω such that for a.e. x, y Ω and for all t [, σ], we have ϕx, βt ϕy, t + hx + hy. loc There exists t > such that ϕ, t L loc Ω. The notation ainc is used for a version of ainc with ϕ ; i.e., equality included. Note that, for any weak Φ-function, ainc holds for ϕ =. Remark 2.7. Let us collect several observations regarding the assumptions above.. Each of the previous conditions is invariant under equivalence of weak Φ-functions; i.e., if ϕ ψ, then ϕ satisfies a condition if and only if ψ satisfies it. 2. For doubling weak Φ-functions, and are equivalent. 3. adec and Δ 2 are equivalent [2, Lemma 2.6]. 4. A implies loc choose t := βσ. 5. If ainc and adec hold, then ϕ ϕ. 6. Finally, note that if ϕ satisfies adec and loc, then ϕ, t L loc Ω for every t. It follows directly from the definition of the left-inverse that ϕ ψ implies ϕ ψ. Furthermore, if ϕ ψ, then ϕ ψ and so ϕ ψ. The next proposition shows that Assumption A is equivalent to its counterpart in [22]. Lemma 2.8. Let ϕ Φ w Ω, σ >, and β,. Then, ϕ satisfies A for σ, β if and only if for every ball Ω, and for all finite t [σ, ϕ / ], we have 2.2 ϕ + βt ϕ t. Proof. ecause ϕ ϕ y,, it follows that if ϕ satisfies 2.2, then it also satisfies A. Conversely, assume that ϕ satisfies A. We first consider the case when t [ σ, ϕ. In this case, we can find y i i N such that t [ σ, ϕ y i, ] for all i N and ϕ t = lim i ϕy i, t. Then, by A, we have ϕx, βt ϕy i, t for a.e. x and for all i N. Taking the supremum over x and then letting i in the previous estimate, we obtain ϕ + βt ϕ t. Finally, assume that t = ϕ <, and let t [σ, t. y the previous case, 2.3 ϕx, βt ϕ + βt ϕ t ϕ t for a.e. x, where in the last inequality we used the fact that ϕ is increasing. Taking the limit t t in 2.3, the left-continuity of ϕx, yields ϕx, βt ϕ t for a.e. x. Hence, taking the supremum over x, we conclude that ϕ satisfies

6 The next lemma shows that left-inverse commutes with infimum, even when the function is not continuous. Lemma 2.9. Let Ω R n be an open set, let Ω, and let ϕ Φ w Ω. Then, ϕ = ϕ +. Proof. Fix s [, ]. For every x, we have ϕ s = inf { t ϕ t s} inf { t ϕx, t s } = ϕ x, s. Thus, taking the supremum over x, ϕ s ϕ + s. To prove the converse inequality, we may assume that ˉt := ϕ + s < without loss of generality. Fix ε >. y definition of ˉt and because ϕx, is increasing for every x, we have ϕx, ˉt + ε s for all x. Hence, ϕ s ˉt + ε. Letting ε, we conclude the desired inequality. The following lemma allows us to relate Assumption A with its counterpart in [8]. Lemma 2.. Let Ω R n be open and assume that ϕ ΦΩ is doubling and satisfies A. Then, A is equivalent to the following condition: A ϕ x, s ϕ y, s for every ball Ω, x, y, and s [, ]. Proof. ecause ϕ belongs to ΦΩ and is doubling, it is a bijection with respect to the second variable from [, to [,. Applying ϕ to A and A, we find that 2.4 A βσ ϕ x, σ for a.e. x Ω; A βϕ x, s ϕ y, s for a.e. x, y and for all s [ ] ϕx, σ,. If s [, ϕx, σ], then βϕ x, βϕ x, s βσ because ϕ x, is increasing. Thus, using 2.4, βϕ x, s ϕ y, s holds for all such s. As mentioned at the beginning of Section 2, the following lemma can be interpreted as a counterpart in our setting of the weighted power-mean inequality for the function ϕx, t = t p for some p. Lemma 2.. Let Ω R n be open and assume that ϕ Φ w Ω satisfies adec. Then, for all δ > and a, b and for a.e. x Ω, we have 2.5 ϕx, a + b ϕx, + δa + + ϕ ϕx, b c δ and 2.6 ϕx, a + b c [ + δ ϕ ϕx, a + Proof. If b δa, then the monotonicity of ϕx, yields ϕx, a + b ϕx, + δa. If a < δ b, then the monotonicity of ϕx, and adec yield + ϕ ] ϕx, b. δ ϕx, a + b ϕx, + δ b c + δ ϕ ϕx, b. Thus, 2.5 holds. Further, 2.6 follows from 2.5 by adec. 6

7 The following lemma is a variant of [22, Lemma 4.4] without the assumption ρ ϕ fχ { f >σ} < and correspondingly weaker conclusion see also [24, Lemma 2.2]. Lemma 2.2. Let Ω R n be open and assume that ϕ Φ w Ω satisfies Assumption A. Then, there exists β > such that, for every ball Ω, f L ϕ Ω, and a.e. x, we have ϕ x, β min { ϕ, } fy dy where h is the function provided by A2. ϕy, fy dy + hx + hy dy, Proof. Without loss of generality, we may assume that f. y Remarks 2.3 and 2.7, we may also assume that ϕ ΦΩ. Then, t ϕx,t is increasing for every x Ω. t Fix a ball Ω, and denote α := ϕ. Let σ, β be given by Assumption A, and set f := fχ {f>σ}, f 2 := f f, and F i := ffl f i dy for i {, 2}. ecause ϕx, is convex and increasing, 2.7 ϕ x, β4 { } min α, f dy ϕ x, β min{α, F 2 } + ϕ x, βf 2. We start by estimating the first term on the right-hand side of 2.7. Suppose first that min{α, F 2 } > σ. Then, by definition of α, min{α, F 2 } [σ, ϕ ]. Thus, Lemma 2.8 and the monotonicity of ϕx, and ϕ yield ϕ x, β min{α, F 2 } ϕ F 2. Using now [22, Lemma 4.3], we obtain ϕ x, β min{α, F 2 } ϕ F 2 ϕ f y dy ϕy, f y dy. Next, suppose that 2 min{α, F } σ. y convexity and monotonicity of ϕx,, by A, and by convexity again, we conclude that ϕ x, β 2 min{α, F } ϕ x, βσ F 2σ 2σ f y dy 2 ϕy, f y dy, where, in the last inequality, we used also A together with the fact that f y > σ in {y Ω: f y }. To estimate the second term on the right-hand side of 2.7, we invoke the convexity and monotonicity of ϕx, and Assumption A2 to obtain ϕ x, βf 2 ϕx, βf 2 y dy ϕy, f 2 y dy + hx + hy dy. Recalling that α = ϕ and ϕx, =, the claim follows. 3. Auxiliary results Lemma 3.. Let Ω and U be two open subsets of R n such that U Ω. Let ϕ Φ w Ω satisfy Assumptions adec and loc, and let ψ ε ε be a family of functions satisfying. and.2. Then, 3. lim ϱε #,Uf ϱ ϕ,u c n f for all f C Ω, where c n := ffl, x e dx. If, in addition, ϕx, is continuous for all x U, then 3. holds with equality. Remark 3.2. Note that ϕx, is continuous for all x Ω if ϕ ΦΩ satisfies adec. 7

8 Proof. Let f C Ω. We start by treating the case in which ϕx, is continuous for all x U. We claim that 3.2 lim ϱε #,Uf = ϱ ϕ,u c n f. y the Taylor expansion formula, for any x Ω and y Ω such that y x < distx, Ω, we have fy = fx + fx y x + Rx, y, ffl where Rx, y = o x y as y x. Denote hx, r := 2 Rx, y dy, for r > and r x Ω r, and c n = ffl x e, dx. As proved in [24, Lemma 3.], we have the point-wise estimate c n r fx rhx, r M # f c n r fx + rhx, r. Consequently, we can define a function α : Ω R + [, ] such that M # r f = c n fx + αx, rhx, r when x Ω r. Then, 3.3 ϱ ε #,Uf = ϕx, c n fx + αx, rhx, r dx ψ ε r dr. U r Next, we prove that lim sup ϱ ε #,U f ϱ ϕ,uc n f. Fix δ >. ecause c n fx + αx, rhx, r c n fx + hx, r, the monotonicity of ϕ and 2.5 yield ϕx, c n fx + αx, rhx, r ϕx, + δc n fx + + ϕ ϕx, hx, r. c δ Hence, invoking 3.3 and., we obtain 3.4 We claim that lim sup ϱ ε #,Uf ϱ ϕ,u + δc n f + + ϕ lim sup c δ 3.5 lim U r ϕx, hx, r dx ψ ε r dr =, U r ϕx, hx, r dx ψ ε r dr. from which the estimate on the upper limit follows by dominated convergence as δ + in 3.4 taking also into account the continuity of ϕx,. To prove 3.5, we start by observing that because U is bounded, the set U r is empty for all r > sufficiently large. Thus, there exists r > for which we have r ϕx, hx, r dx ψ ε r dr = ϕx, hx, r dx ψ ε r dr. U r U r Moreover, since f C 2, Rx, y f W 2, U x y 2 for all r, r, x U r, and y x, r. Set C := f W 2, U. Then, for all r, r and x U r, hx, r 2Cr. Fix < γ < min{, r }. We have that ϕx, hx, r ϕx, 2Cr whenever r γ, r ; moreover, denoting by c the monotonicity constant of t ϕx,t, we have also t ϕx, hx, r cγϕx, 2C whenever r, γ]. Using, in addition, the condition ϕ and., it follows that r ϕx, hx, r dx ψ ε r dr cγ U r U ϕx, 2C dx + U ϕx, 2Cr dx ψ ε r dr. γ In view of.2 and Assumption loc, together with Remark 2.7, letting ε + in this estimate first, and then γ +, we obtain

9 Finally, we prove that lim inf ϱ ε #,U f ϱ ϕ,uc n f. Fix δ >, and denote a := c n fx and b := αx, rhx, r. If r > and x U r are such that αx, r <, then applying 2.5 with a := a +b and b := b gives +δ +δ 3.6 ϕx, c n fx + αx, rhx, r = ϕx, + δa ϕx, a + b ϕ x, cn fx +δ c + δ ϕ ϕx, b c + δ ϕ ϕx, hx, r, where we used the monotonicity of ϕ and the estimate b hx, r. If r > and +δ x U r are such that αx, r, then the monotonicity of ϕ yields 3.7 ϕx, c n fx + αx, rhx, r ϕ x, c n fx +δ Thus, by 3.6 and 3.7, for every r > and x U r, we have ϕx, c n fx + αx, rhx, r ϕ x, cn +δ fx c + δ ϕ ϕx, hx, r. Consequently, by 3.3, it follows that [ 3.8 ϱ ε #,Uf ϕ ] x, cn fx +δ c + δ ϕ ϕx, hx, r dx ψ ε r dr. U r Fix < γ < and r > such that U r. y. and.2, there exists ε = ε γ, r such that, for all ε, ε, we have Thus, we have also, for < ε < ε, r ψ ε r dr γ. r ϕ x, c n fx dx ψ +δ ε r dr ϕ x, c n fx dx ψ +δ ε r dr U r U r γ ϕ x, c n fx dx. +δ U r This estimate, 3.8, and 3.5 yield lim inf ϱε #,Uf γϱ ϕ,ur cn +δ f. The conclusion then follows by using the continuity of ϕx, and by letting δ +, γ +, and r + in this order. This completes the proof of 3.2 under the continuity assumption. Suppose now that ϕ is a general weak Φ-function. y Remarks 2.3 and 2.7, there exists ψ ΦΩ satisfying the same assumptions as ϕ and such that ψ ϕ. Then, there is C > such that ϕ ψ Cϕ. Hence, by the first part of the proof, C C ϱ ϕ,uc 2 n f C ϱ ψ,uc n f = lim inf C ϱε #,U,ψf lim inf ϱε #,U,ϕf lim sup ϱ ε #,U,ϕf C lim sup ϱ ε #,U,ψf = Cϱ ψ,u c n f C 2 ϱ ϕ,u c n f. Next, we derive an auxiliary upper bound which holds for all functions f L loc Ω with f L ϕ Ω. Lemma 3.3. Let Ω be an open set of R n, let ϕ Φ w Ω satisfy Assumptions A and adec, and let ψ ε ε be a family of functions satisfying. and.2. If f L loc Ω and f L ϕ Ω, then ϱ ε #,Ωf c max { f ϕ ϕ,ω, f ϕ ϕ,ω}. 9.

10 Proof. y Remarks 2.3, 2.6, and 2.7 we may assume without loss of generality that ϕ ΦΩ. Then, in view of Assumptions A and ainc, we have L ϕ L for every bounded set Ω by [9, Lemma 4.4]. Let f L loc Ω with f Lϕ Ω be such that f ϕ,ω. Fix r > and x Ω r. y the Poincaré inequality in L, we have 3.9 r M # f = r fy f dy c fy dy. On the other hand, in view of Lemma 2.8, we may invoke [23, Lemma 4.4] with γ = that gives the existence of a constant, β,, depending only on the constants in A, such that 3. ϕ x, β fy dy ϕy, fy dy + hx + hy dy. Using the monotonicity of ϕ, adec, 3.9, and 3., we obtain 3. ϱ ε #,Ωf ϕ x, β fy dy dx ψ ε r dr Ω r ϕy, fy dy + hx + hy dy dx ψ ε r dr. Ω r Next, by changing the order of integration, we observe that ϕy, fy dy dx = ϕy, fy χ y,rx dy dx Ω r Ω r Ω x, r ϕy, fy dy = ϱ ϕ,ω f. Similarly, Ω r Ω hy dy dx Ω hy dy. Consequently, because h L Ω and ψ ε r dr =, from 3. we conclude that 3.2 ϱ ε #,Ωf c for all f L locω with f L ϕ Ω and f ϕ,ω. y considering the cases λ and λ > and use Assumptions ainc and adec, respectively, we find that ϱ ε #,Ω f cϱε #,Ω f λ max{λϕ, λ ϕ } for all λ > and f L loc Ω with f L ϕ Ω. Using this estimate with λ := f ϕ,ω + δ for δ > and then invoking 3.2, it follows that ϱ ε #,Ωf cϱ ε #,Ω f f ϕ,ω + δ max { f ϕ,ω + δ ϕ, f ϕ,ω + δ ϕ } c max { f ϕ,ω + δ ϕ, f ϕ,ω + δ ϕ }. Letting δ +, we conclude the proof of Lemma Main results In this section we prove our main result, which provides a characterization of generalized Orlicz spaces. Theorem. is an immediate consequence of Propositions 4., 4.3 and 4.5 below.

11 Proposition 4.. Let Ω R n be open, let ϕ Φ w Ω satisfy Assumptions A, ainc, and adec, and let ψ ε ε be a family of functions satisfying. and.2. Assume that f L loc Ω with f Lϕ Ω. Then, 4. lim ϱε #,Ωf ϱ ϕ,ω c n f < where c n = ffl, x e dx. If, in addition, ϕx, is continuous for all x Ω, then 4. holds with equality. Proof. We start with the upper bound. Let U Ω. Using the same arguments as in [9, Theorems 6.5 and 6.6], we find g ν C Ω, ν N, such that g ν n N converges to f in L ϕ U; R n. For r > and x U r, we have M # f M # f g ν + M # g ν by the triangle inequality. Combining this estimate with 2.5 applied to a = r M # g ν and b = r M # f g ν, we obtain ϱ ε #,Uf ϱ ε #,U + δg ν + c δ ϱ ε #,Uf g ν for all δ >, where we also used the monotonicity of ϕx,. Invoking now Lemmas 3. and 3.3, we conclude that 4.2 lim sup ϱ ε #,Uf ϱ ϕ,u c n + δ g ν + c δ max { f g ν ϕ ϕ,u, f g ν ϕ ϕ,u}. ecause f g ν ϕ,u, it follows from adec that lim ν ϱ ϕ,u c n +δ f g ν =. Thus, letting ν in 4.2 and using 2.5 again, we obtain 4.3 lim sup ϱ ε #,Uf ϱ ϕ,u c n + δ 2 f ϱ ϕ,ω c n + δ 2 f. y adec, ϕx, c n +δ 2 fx cϕx, c n fx. This proves that lim sup ϱ ε #,Uf cϱ ϕ,u c n f for a general ϕ. If, in addition, ϕx, is continuous, we use cϕx, c n fx as a majorant and let δ in 4.3. Then, Lebesgue s dominated convergence theorem yields We claim that 4.4 lim sup U Ω lim sup ϱ ε #,Uf ϱ ϕ,ω c n f. lim sup ϕ x, M # r f dx ψ ε r dr =, Ω r\u r which, together with the bound on lim sup ϱ ε #,Uf established above, concludes the proof of the upper bound. To prove 4.4, we observe that replacing f by f := f ϕ,ωf if necessary, we may assume that f ϕ,ω. Indeed, by adec, if 4.4 holds for f, then it also holds for f. Arguing as in Lemma 3.3, we find that 4.5 ϕ x, M # r f dx ψ ε r dr Ω r\u r c ϕy, fy dy + hx + Ω r\u r hy dy dx ψ ε r dr.

12 Let γ >. If r γ, then y Ω\U 2 γ whenever x Ω r \U r. Therefore, we may estimate the right-hand side of 4.5 as follows: ϕy, fy dy + hx + hy dy dx ψ ε r dr 4.6 Ω r\u r + Ω r\u r γ 2 Ω r ϕy, fy + hy χy,rx,r Ω\U γ ϕy, fy dy + hx + dy + hx dx ψ ε r dr hy dy dx ψ ε r dr. Changing the order of integration as in Lemma 3.3, from 4.5 and 4.6, we obtain ϕ x, M # r f dx ψ ε r dr Ω r \U r ϱ ϕ,ω\uγ f + 2 h L Ω\U γ + ϱ ϕ,ω f + 2 h L Ω ψ ε r dr. Then, 4.4 follows by.2 as ε, γ, and U Ω, in this order. It remains to prove the lower bound. The proof of this estimate is similar to the proof of the upper bound once we show that for all U Ω, we have 4.7 ϱ ε #,Uf ϱ ε #,U +δ g ν c δ ϱ ε #,Uf g ν, where c δ = c + ϕ. Fix r > and x U δ r, and set a := M # r g ν and b := M # r f g ν. Note that M # r f a b. We consider two cases: If a b, then using 2.5 with a = a b and b = b, we obtain +δ +δ ϕx, a b ϕ a x, +δ cδ ϕ b x, +δ ϕ x, a +δ cδ ϕx, b, where in the last inequality we used the fact that ϕx, is increasing and +δ If a b, then ϕ a x, +δ ϕ x, b +δ cδ ϕx, b because ϕx, is increasing, +δ, and c δ. Thus, ϕ a x, +δ cδ ϕx, b. Splitting the inner integral defining ϱ ε #,Uf according to these two cases, we conclude that 4.7 holds. Remark 4.2. In the proof of the previous proposition the assumption ainc was used only for the density of smooth functions in the Sobolev space. Presumably, ainc is not really needed for this, but it was used in the cited reference. Proposition 4.3. Under the assumptions of Proposition 4., we have 4.8 lim f ε #,Ω c n f ϕ,ω. If, in addition, ϕ ΦΩ, then 4.8 holds with equality. Proof. Suppose first that ϕ ΦΩ. Since ϕ is also doubling, ϕx, is continuous; thus, the modular inequality 4. holds with equality. Fixing δ > and applying this equality to the function f g δ := c n + δ f ϕ,ω + δ, 2 γ 2.

13 we obtain lim ϱε #,Ωg δ = ϱ ϕ,ω c n g δ +δ Ω f ϕ x, dx <, f ϕ,ω + δ where we used convexity in the last estimate. Thus, ϱ ε #,Ω g δ for all sufficiently small ε >. Consequently, also g δ ε #,Ω for all sufficiently small ε > ; that is, f ε #,Ω c n + δ f ϕ,ω + δ. Letting ε first and then δ, we obtain lim sup f ε #,Ω c n f ϕ,ω. Next, we prove the opposite inequality. Fix δ >, let ε j as j be such that lim inf f ε #,Ω = lim j f ε j #,Ω, and set g δ := f lim j f ε j #,Ω + δ. ecause lim j g δ ε j #,Ω <, we conclude that g δ ε j #,Ω for all sufficiently large j N. For all such j N, ϱ ε j #,Ω g δ by the definition of the norm. Letting j, the previous proposition yields ϱ ϕ,ω c n g δ ; so, c n g δ ϕ,ω. Therefore, c n f ϕ,ω lim j f ε j #,Ω + δ = lim inf f ε #,Ω + δ. Letting δ, we obtain the desired inequality. This completes the proof in the case ϕ ΦΩ. If ϕ Φ w Ω, we find ψ ΦΩ with ϕ ψ Remarks 2.3 and 2.7. Then, f ϕ f ψ and similarly for the #-norm; so, the claim follows from the first part. The next lemma shows that the condition lim sup ϱ ε #,Ωf < implies that f is locally in a Sobolev space. The estimate for the norm obtained in this way is not uniform, however. Nevertheless, this information is used later to prove a uniform estimate. Lemma 4.4. Let Ω R n be open, let ϕ Φ w Ω satisfy Assumptions A, ainc, and adec, and let ψ ε ε be a family of functions satisfying. and.2. Assume that f L loc Ω and lim sup ϱ ε #,Ωf <. Then, there is a constant c > such that, for every U Ω, U ϕ U fx dx c. Moreover, f L ϕ loc Ω; Rn. Proof. Let U Ω and note that ϕ U Φ w the only nontrivial condition is the limit at infinity, which follows from A. y [9, Lemma 2.2], there exists ξ Φ with ξ ϕ U for which the equivalent constant depends only on the monotonicity constant of t ϕx,t. t Then, by adec, we can find c >, independent of U, such that c ϕ U ξ c ϕ U. Fix δ >, and let G δ be a standard mollifier; that is, G δ x = δ n Gx/δ, where G C, is a non-negative function, radially symmetric, and Gx dx =., Let τ > be such that U Ω τ. Then, for < δ < τ, G δ f C Ω τ and, by Lemma 3. with ϕ = ξ, U := U δ, and Ω := Ω τ, 4.9 lim ξ r G δ f U δ+r dx ψ ε r dr = ϱ ξ,uδ c n G δ f ϱ ξ,uτ c n G δ f. 3

14 On the other hand, by the triangle inequality and a change on the order of integration, we obtain, for all r > and x U r, 4. M # G δ f = = Hence, the monotonicity of ξ yields G δ fy G δ f dy G δ fy G δ f,r x dy G δ M #,r fx. ξ r M # G δ f ξ G δ r M #,r fx. For x Ω r, define g r x := M # r f. Since G δ dx is a probability measure and ξ is convex, ξ M # r G δ f ξg δ g r x G δ ξg r x. Integrating G δ ξg r over U r+δ and changing variables and the order of integration, we obtain G δ x zξg r z dz dx = G δ w ξg r x w dxdw U δ+r x,δ,δ U δ+r ξg r y dy. U r Combining the previous two estimates, we conclude that ξ M # r G δ f dx ξg r x dx c U δ+r U r U r ϕ x, r M # f dx, where in the last inequality we used the definition of g r and the estimate ξt c ϕx, t for x U. Consequently, 4. lim sup U δ+r ξ M # r G δ f dx ψ ε r dr c lim sup ϱ ε #,Uf c lim sup ϱ ε #,Ωf <, which, together with 4.9, shows that G δ f δ is bounded in L ξ U τ ; R n. y A, ainc, and adec, L ξ U τ ; R n is reflexive see [9, Lemma 2.4 and Proposition 4.6]. Since G δ f δ is a bounded sequence, it has a subsequence which converges weakly in L ξ U τ ; R n to a function l L ξ U τ ; R n. Using the fact that G δ f converges to f in L U, it follows from the definition of weak derivative that l = f. y weak semi-continuity [4, Theorem 2.2.8], 4.9, and 4., we obtain that ϕ U c n fx dx cρ ξ,uτ c n f c lim inf ρ ξ,u τ c n G δ f U τ δ + c lim sup ϱ ε #,Ωf <. Letting τ, we obtain U ϕ U fx dx c by monotone convergence and adec. Finally, we observe that Assumptions ainc and A imply that t ϕ cϕ U t + with c > independent of U. Hence, fx ϕ dx c ϕ U fx + dx c + c U, which yields f L ϕ loc Ω; Rn. U We now remove the local-condition from the previous lemma: 4 U

15 Proposition 4.5. Let Ω R n be open, let ϕ Φ w Ω satisfy Assumptions A, ainc, and adec, and let ψ ε ε be a family of functions satisfying. and.2. Assume that f L loc Ω and lim Then, f L ϕ Ω; R n. sup ϱ ε #,Ωf <. Proof. Let U V Ω and δ, be such that U δ V Ω 6 δ. Let G δ and g r be as in the previous lemma. y Remarks 2.3 and 2.7, there exists ξ ΦΩ equivalent to ϕ which satisfies the same assumptions. In Lemma 4.4, we proved that U ξ U f dx c for every U Ω. In view of adec, by scaling the function f, if necessary, we may assume that 4.2 ξ fx dx 3 n for every ball Ω with. Let r, δ] and x U r. y the Poincaré and triangle inequalities, r M # G δ f c G δ fy dy c G δ y z fz dz dy. y,δ Note that G δ c χ,δ ; hence, by [22, Lemma 4.3] and 4.2, together with the previous,δ estimate, we obtain M # r G δ f c c c y,δ ξ y,δ ξ y,δ fz dz dy y,δ 3 n y,δ ξ y,δ fz dz dy dy. Observe that y, δ x, 3δ for every y x, r, and y, δ = x, δ and x, 3δ = 3 n x, δ. y Lemmas 2.9 and 2., ξ y,δ 3 n y,δ ξ x,3δ 3 n x,δ ξ x,δ 3 n x,δ when y x, r. Then, because ξ x,δ is increasing, we obtain M # r G δ f cξ x,δ 3 n x, δ cξ x,δ x, δ. On the other hand, by 4., we have also M # r G δ f G δ g r x c ffl g x,δ r dy. Thus, { r M # G δ f c min g r dy, ξ x,δ x,δ }. x,δ Then, by the monotonicity of ξx, and Lemma 2.2, invoking adec if necessary, it follows that ξ x, M # r G δ f c ξy, gr y + hy dy + hx. x,δ 5

16 Integrating this estimate over x U r, changing the order of integration as in Lemma 3.3, and using the inclusion U r δ V r, we obtain ξ x, M # r G δ f dx c ξy, g r y + hy dy + hx dx U r U r δ U r c ξ y, M # r y,r f dy + 2 h L Ω V r for r δ. Next, we consider r δ,. Fix x U r, and recall that f L ϕ V ; R n by Lemma 4.4. y the Poincaré and Hölder inequalities, we have M # r G δ f c G δ fy dy c c δ f ϕ,u δ c δ f ϕ,v. G δ ϕ,,δ f ϕ,y,δ dy Hence, ξ x, M # r G δ f dx ξ x, c δ f ϕ,v dx U r U r for r > δ. Combining the two cases, r δ and r > δ, we obtain ξ x, M # r G δ f dx ψ ε r dr U r δ c ξ x, M # r f dx + 2 h L Ω ψ ε r dr + ξ x, c δ f ϕ,v dx ψε r dr. V r δ U r The second term on the right-hand side of the previous inequality tends to zero as ε by loc and.2. Consequently, lim sup x, M # r G δ f dx ψ ε r dr c lim sup ρ ε #,V f + c < ε ε U r ξ by hypothesis, where we also used. and the fact that ξ ϕ. Thus, arguing as in Lemma 4.4, we conclude that f L ξ U; R n. f L ϕ U; R n see Remark 2.6. Finally, we appeal to Proposition 4. to conclude that Hence, also ϱ ϕ,u c n f c lim sup ϱ ε #,Uf c lim sup ϱ ε #,Ωf <. ecause the upper bound in the last estimate is independent of U, we conclude the proof of Proposition 4.5 by monotone convergence as U Ω. Combining the propositions of this section, we arrive at the following theorem. Theorem 4.6. Let Ω R n be an open set, let ϕ Φ w Ω satisfy Assumptions A, ainc, and adec, and let ψ ε ε be a family of functions satisfying. and.2. Assume that f L loc Ω. Then, In this case, f L ϕ Ω; R n lim sup ϱ ε #,Ωf <. lim ϱε #,Ωf ϱ ϕ,ω c n f and lim f ε #,Ω c n f ϕ,ω, where c n := ffl, x e dx. 6

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18 [24] P. Hästö and A. Ribeiro: Characterization of the variable exponent Sobolev norm without derivatives, Commun. Contemp. Math., to appear. DOI:.42/S X [25] G. Leoni and D. Spector: Characterization of Sobolev and V spaces, J. Funct. Anal. 26 2, [26] F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura: oundedness of maximal operators and Sobolev s inequality on Musielak-Orlicz-Morrey spaces, ull. Sci. Math , [27] Y. Mizuta, T. Ohno and T. Shimomura: Sobolev s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space L p log L q, J. Math. Anal. Appl , [28] J. Musielak: Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 34. Springer, erlin, 983. [29] J. Ok: Gradient estimates for elliptic equations with L p log L growth, Calc. Var. Partial Differential Equations 55 26, no. 2, 3. [3] J. Ok: Regularity results for a class of obstacle problems with nonstandard growth, J. Math. Anal. Appl , no. 2, [3] L. Rudin, S. Osher, and E. Fatemi: Nonlinear total variation based noise removal algorithms, Physica D 6, 992, no.-4, [32] A. Świerczewska-Gwiazda: Nonlinear parabolic problems in Musielak Orlicz spaces, Nonlinear Anal , R. Ferreira King Abdullah University of Science and Technology KAUST, CEMSE Division, Thuwal , Saudi Arabia. address: rita.ferreira@kaust.edu.sa URL: P. Hästö Department of Mathematical Sciences, P.O. ox 3, FI-94 University of Oulu, Finland, and, Department of Mathematics and Statistics, University of Turku, Finland address: peter.hasto@oulu.fi URL: phasto/ A.M. Ribeiro Centro de Matemática e Aplicações CMA and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, Caparica, Portugal address: amfr@fct.unl.pt URL: 8