Trudinger’s inequality and continuity for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces on metric measure spaces

Nonlinear Analysis 106 (2014) 1–17

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Trudinger’s inequality and continuity for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces on metric measure spaces Takao Ohno a,∗ , Tetsu Shimomura b a

Faculty of Education and Welfare Science, Oita University, Dannoharu Oita-city 870-1192, Japan

b

Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

article

info

Article history: Received 5 February 2014 Accepted 11 April 2014 Communicated by Enzo Mitidieri

abstract In this paper we are concerned with Trudinger’s inequality and continuity for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces on metric measure spaces. © 2014 Elsevier Ltd. All rights reserved.

MSC: primary 46E35 secondary 46E30 Keywords: Musielak–Orlicz space Morrey space Trudinger’s inequality Variable exponent Continuity Metric measure space

1. Introduction A famous Trudinger inequality [1] insists that Sobolev functions in W 1,N (G) satisfy finite exponential integrability, where G is an open bounded set in RN (see also [2–5]). For 0 < α < N, we define the Riesz potential of order α for a locally integrable function f on RN by Uα f (x) =

 RN

|x − y|α−N f (y) dy.

Great progress on Trudinger type inequalities has been made for Riesz potentials of order α in the limiting case α p = N (see e.g. [6–10]). Trudinger type exponential integrability was studied on Orlicz spaces in [11–13], on generalized Morrey spaces L1,ϕ in [14,15], and on Orlicz–Morrey spaces in [16,17]. For Morrey spaces, which were introduced to estimate solutions of partial differential equations, we refer to [18,19]. Variable exponent Lebesgue spaces and Sobolev spaces were introduced to discuss nonlinear partial differential equations with non-standard growth condition. For a survey, see [20,21]. Trudinger type exponential integrability was investigated on variable exponent Lebesgue spaces Lp(·) in [22–24] and on two variable exponent spaces Lp(·) (log L)q(·) in [25]. See also [26] for two variable exponent spaces Lp(·) (log L)q(·) .

∗

Corresponding author. Tel.: +81 097 554 7566. E-mail addresses: [email protected] (T. Ohno), [email protected] (T. Shimomura).

http://dx.doi.org/10.1016/j.na.2014.04.008 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

2

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

For x ∈ RN and r > 0, we denote by B(x, r ) the open ball centered at x with radius r and dΩ = sup{d(x, y) : x, y ∈ Ω } for a set Ω ⊂ RN . For bounded measurable functions ν(·) : RN → (0, N ] and β(·) : RN → R, let Lp(·),q(·),ν(·),β(·) (G) be the set of all measurable functions f on G such that ∥f ∥Lp(·),q(·),ν(·),β(·) (G) < ∞, where r ν(x) (log(e + 1/r ))β(x)



∥f ∥Lp(·),q(·),ν(·),β(·) (G) = inf λ > 0 : 

 × B(x,r )

sup

|B(x, r )|   |f (y)| q(y) log e + dy ≤ 1 ; λ

x∈G,0
|f (y)| λ

p(y) 



we set f = 0 outside G. As an extension of Trudinger [1] and [15, Corollaries 4.6 and 4.8], Mizuta, Nakai and the authors [27] proved Trudinger type exponential integrability for two variable exponent Morrey spaces Lp(·),q(·),ν(·),β(·) (G) when p(·) and q(·) are variable exponents satisfying the log-Hölder and loglog-Hölder conditions on G, respectively. The result is an improvement of [28, Theorems 4.4 and 4.5]. In fact we proved the following: Theorem A. Suppose infx∈RN ν(x) > 0 and infx∈RN (α − ν(x)/p(x)) ≥ 0 hold. Let ε be a constant such that inf (ν(x)/p(x) − ε) > 0 and 0 < ε < α.

x∈Rn

Then there exist constants C1 , C2 > 0 such that (1) in case supx∈RN (q(x) + β(x))/p(x) < 1, r ν(z )/p(z )−ε



 exp

|B(z , r )|

|Uα f (x)|p(x)/(p(x)−q(x)−β(x))



C1

B(z ,r )

dx ≤ C2 ;

(2) in case infx∈RN (q(x) + β(x))/p(x) ≥ 1, r ν(z )/p(z )−ε

|B(z , r )|





exp exp B(z ,r )



|Uα f (x)| C1

 dx ≤ C2

for all z ∈ G, 0 < r < dG and f satisfying ∥f ∥Lp(·),q(·),ν(·),β(·) (G) ≤ 1. Recently, Theorem A was extended to Musielak–Orlicz–Morrey spaces in [29]. Our main aim in this paper is to give a general version of Trudinger type exponential integrability for Riesz potentials Iα f of functions in Musielak–Orlicz–Morrey spaces LΦ ,κ (X ) on metric measure spaces X (e.g., Corollary 4.6) as an extension of the above results (see Section 2 for the definitions of Φ and κ and Section 3 for the definition of Iα f ). Since we discuss the Morrey version, our strategy is to find an estimate of Riesz potentials by use of Riesz potentials of order ε , which plays a role of the maximal functions (see Section 3). What is new about this paper is that we can pass our results to the metric measure setting; the technique in [29] still works. Beginning with Sobolev’s embedding theorem (see e.g. [30,2]), continuity properties of Riesz potentials or Sobolev functions have been studied by many authors. Continuity of Riesz potentials of functions in Orlicz spaces was studied in [9,31–33,13] (cf. also [34]). Then such continuity was investigated on generalized Morrey spaces L1,ϕ in [14,15], on Orlicz–Morrey spaces in [35], on variable exponent Lebesgue spaces in [22,23,36] and on variable exponent Morrey spaces in [35]. In [27], Mizuta, Nakai and the authors also proved continuity for Riesz potentials of functions in two variable exponent Morrey spaces Lp(·),q(·),ν(·),β(·) (G). In [29], these results have been extended to Musielak–Orlicz–Morrey spaces. Our second aim in this paper is to give a general version of continuity for Riesz potentials Iα f of functions in Musielak–Orlicz–Morrey spaces LΦ ,κ (X ) on metric measure spaces (e.g., Corollary 5.6) as an extension of the above results. In [37], we established Trudinger type exponential integrability for Musielak–Orlicz spaces in the Euclidean setting by use of the maximal functions, which are a crucial tool as in Hedberg [38]. Our third aim in this paper is to give a general version of Trudinger type exponential integrability for Riesz potentials Iα f of functions in Musielak–Orlicz spaces LΦ (X ) on metric measure spaces (e.g., Corollary 7.2) as an extension of [23,39,37]. To obtain our results, we need the boundedness of the maximal operator on LΦ (X ) (see Lemma 6.1). In the final section, we show the continuity for Riesz potentials Iα f of functions in Musielak–Orlicz spaces LΦ (X ) on metric measure spaces (see Corollary 8.2). 2. Preliminaries Throughout this paper, let C denote various constants independent of the variables in question. We denote by (X , d, µ) a metric measure space, where X is a set, d is a metric on X and µ is a nonnegative complete Borel regular outer measure on X which is finite in every bounded set. For simplicity, we often write X instead of (X , d, µ). For x ∈ X and r > 0, we denote by B(x, r ) the open ball centered at x with radius r and dΩ = sup{d(x, y) : x, y ∈ Ω } for a set Ω ⊂ X . We say that the measure µ is a doubling measure if there exists a constant c0 > 0 such that µ(B(x, 2r )) ≤ c0 µ(B(x, r )) for every x ∈ X and 0 < r < dX . We say that X is a doubling space if µ is a doubling measure.

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

3

In this paper, we assume that X is a bounded set and a doubling space, that is dX < ∞. This implies that µ(X ) < ∞. We consider a function

Φ (x, t ) = t φ(x, t ) : X × [0, ∞) → [0, ∞) satisfying the following conditions (Φ 1)–(Φ 4): (Φ 1) φ( · , t ) is measurable on X for each t ≥ 0 and φ(x, · ) is continuous on [0, ∞) for each x ∈ X ; (Φ 2) there exists a constant A1 ≥ 1 such that 1 A− 1 ≤ φ(x, 1) ≤ A1

for all x ∈ X ;

(Φ 3) φ(x, ·) is uniformly almost increasing, namely there exists a constant A2 ≥ 1 such that φ(x, t ) ≤ A2 φ(x, s) for all x ∈ X whenever 0 ≤ t < s; (Φ 4) there exists a constant A3 ≥ 1 such that φ(x, 2t ) ≤ A3 φ(x, t ) for all x ∈ X and t > 0. Note that (Φ 2), (Φ 3) and (Φ 4) imply 0 < inf φ(x, t ) ≤ sup φ(x, t ) < ∞ x∈X

x∈X

for each t > 0. If Φ (x, ·) is convex for each x ∈ X , then (Φ 3) holds with A2 = 1; namely φ(x, ·) is non-decreasing for each x ∈ X . ¯ x, t ) = sup0≤s≤t φ(x, s) and Let φ(

Φ (x, t ) =

t



¯ x, r ) dr φ(

(2.1)

0

for x ∈ X and t ≥ 0. Then Φ (x, ·) is convex and 1 2A3

Φ (x, t ) ≤ Φ (x, t ) ≤ A2 Φ (x, t )

for all x ∈ X and t ≥ 0. We shall also consider the following condition: (Φ 5) for every γ1 , γ2 > 0, there exists a constant Bγ1 ,γ2 ≥ 1 such that

φ(x, t ) ≤ Bγ1 ,γ2 φ(y, t ) whenever d(x, y) ≤ γ1 t −1/γ2 and t ≥ 1. Example 2.1. Let p(·) and qj (·), j = 1, . . . , k, be measurable functions on X such that (P1) 1 < p− := infx∈X p(x) ≤ supx∈X p(x) =: p+ < ∞ and + (Q1) −∞ < q− j := infx∈X qj (x) ≤ supx∈X qj (x) =: qj < ∞

for all j = 1, . . . , k. (j+1) (1) Set Lc (t ) = log(c + t ) for c ≥ e and t ≥ 0, Lc (t ) = Lc (t ), Lc (t ) = Lc (L(cj) (t )) and

Φ (x, t ) = t p(x)

k 

(Lc(j) (t ))qj (x) .

j =1

Then, Φ (x, t ) satisfies (Φ 1), (Φ 2), (Φ 3) and (Φ 4). Moreover, we see that Φ (x, t ) satisfies (Φ 5) if (P2) p(·) is log-Hölder continuous, namely

|p(x) − p(y)| ≤

Cp Le (1/d(x, y))

with a constant Cp ≥ 0 and (Q2) qj (·) is j + 1-log-Hölder continuous, namely

|qj (x) − qj (y)| ≤

Cqj (j+1)

Le

(1/d(x, y))

with constants Cqj ≥ 0, j = 1, . . . , k.

(2.2)

4

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

Example 2.2. Let p(·) be a measurable function on X satisfying (P1) and (P2). Let q1 (·) be a measurable function on X satisfying (Q1) and (Q2) and let q2 (·) be a measurable function on X satisfying (Q1). Then

Φ (x, t ) = t p(x) (log(e + t ))q1 (x) (log(e + 1/t ))q2 (x) satisfies (Φ 1), (Φ 2), (Φ 3), (Φ 4) and (Φ 5). In view of (2.2), given Φ (x, t ) as above, the associated Musielak–Orlicz space Φ

L (X ) =

 f ∈

L1loc

(X );



 Φ y, |f (y)| dµ(y) < ∞ 



X

is a Banach space with respect to the norm

     ∥f ∥LΦ (X ) = inf λ > 0; Φ y, |f (y)|/λ dµ(y) ≤ 1 X

(cf. [40]). We also consider a function κ(x, r ) : X × (0, dX ] → (0, ∞) satisfying the following conditions: (κ 1) κ(x, ·) is measurable for each x ∈ X ; (κ 2) κ(x, ·) is uniformly almost increasing on (0, dX ], namely there exists a constant Q1 ≥ 1 such that

κ(x, r ) ≤ Q1 κ(x, s) for all x ∈ X whenever 0 < r < s ≤ dX ; (κ 3) there are constants Q > 0 and Q2 ≥ 1 such that Q2−1 min(1, r Q ) ≤ κ(x, r ) ≤ Q2 for all x ∈ X and 0 < r ≤ dX . Example 2.3. For Q > 0, let ν(·) and βj (·), j = 1, . . . , k be measurable functions on X such that infx∈X ν(x) > 0, supx∈X ν(x) ≤ Q and −c (Q − ν(x)) ≤ βj (x) ≤ c for all x ∈ X , j = 1, . . . , k and some constant c > 0. Then

κ(x, r ) = r ν(x)

k 

(L(ej) (1/r ))βj (x)

j =1

satisfies (κ 1), (κ 2) and (κ 3). For a locally integrable function f on X , define the LΦ ,κ norm

 ∥f ∥LΦ ,κ (X ) = inf λ > 0 :

sup

x∈X ,0
κ(x, r ) µ(B(x, r ))

 X ∩B(x,r )

 Φ (y, |f (y)|/λ) dµ(y) ≤ 1 .

See (2.1) for the definition of Φ . Let LΦ ,κ (X ) denote the set of all functions f such that ∥f ∥LΦ ,κ (X ) < ∞ (cf. [17]), which we call a Musielak–Orlicz–Morrey space. Note that LΦ ,κ (X ) = LΦ (X ) if µ(B(x, r )) ∼ κ(x, r ) for all x ∈ X and 0 < r ≤ dX . (Here h1 (x, s) ∼ h2 (x, s) means that C −1 h2 (x, s) ≤ h1 (x, s) ≤ Ch2 (x, s) for a constant C > 0.) 3. Lemmas for Musielak–Orlicz–Morrey spaces Set

Φ −1 (x, s) = sup{t > 0; Φ (x, t ) < s} for x ∈ X and s > 0. Lemma 3.1 ([41, Lemma 5.1]). Φ −1 (x, ·) is non-decreasing;

Φ −1 (x, λs) ≤ A2 λΦ −1 (x, s)

(3.1)

for all x ∈ X , s > 0 and λ ≥ 1 and



min 1,

s A1 A2



≤ Φ −1 (x, s) ≤ max{1, A1 A2 s}

for all x ∈ X and s > 0, where A1 and A2 are the constants appearing in (Φ 2) and (Φ 3).

(3.2)

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

5

Lemma 3.2. There exists a constant C > 0 such that C −1 ≤ Φ −1 (x, κ(x, r )−1 ) ≤ Cr −Q

(3.3)

for all x ∈ X and 0 < r ≤ dX . Proof. By (κ 3), Q2−1 ≤ κ(x, r )−1 ≤ Q2 max(1, r −Q ) for x ∈ X and 0 < r ≤ dX . Hence, by (3.2), we obtain (3.3).



As in [41, Lemma 5.3], we can prove the following result. Lemma 3.3 (cf. [41, Lemma 5.3]). Assume that Φ (x, t ) satisfies (Φ 5). Then there exists a constant C > 0 such that

 X ∩B(x,r )

f (y) dµ(y) ≤ C µ(B(x, r ))Φ −1 (x, κ(x, r )−1 )

for all x ∈ X , 0 < r ≤ dX and f ≥ 0 satisfying ∥f ∥LΦ ,κ (X ) ≤ 1. For α > 0, we define the Riesz potential of order α for a locally integrable function f on X by Iα f (x) =

d(x, y)α f (y)

 X

µ(B(x, d(x, y)))



dX

dµ(y)

(e.g. see [42]). Set

Γ (x, s) =

1/s

  dρ ρ α Φ −1 x, κ(x, ρ)−1 ρ

for s ≥ 2/dX and x ∈ X . For 0 ≤ s < 2/dX and x ∈ X , we set Γ (x, s) = Γ (x, 2/dX )(dX /2)s. Then note that Γ (x, ·) is strictly increasing and continuous for each x ∈ X . Lemma 3.4 (cf. [29, Lemma 3.5]). There exists a positive constant C ′ such that Γ (x, 2/dX ) ≥ C ′ > 0 for all x ∈ X . Lemma 3.5. Assume that Φ (x, t ) satisfies (Φ 5). Then there exists a constant C > 0 such that d(x, y)α f (y)

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y) ≤ C Γ



x,

1



δ

for all x ∈ X , 0 < δ ≤ dX /2 and nonnegative f ∈ LΦ ,κ (X ) with ∥f ∥LΦ ,κ (X ) ≤ 1. Proof. Let j0 be the smallest positive integer such that 2j0 δ ≥ dX . By Lemma 3.3, we have d(x, y)α f (y)

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y) =

d(x, y)α f (y)

j0  

dµ(y) µ(B(x, d(x, y)))  j0  1 ≤ (2j δ)α f (y) dµ(y) µ(B(x, 2j−1 δ)) X ∩B(x,2j δ) j =1  j0  1 ≤ c0 (2j δ)α f (y) dµ(y) µ(B(x, 2j δ)) X ∩B(x,2j δ) j=1  j −1  0  j α −1 j −1 α −1 −1 ≤C (2 δ) Φ (x, κ(x, 2 δ) ) + dX Φ (x, κ(x, dX ) ) . j =1

X ∩(B(x,2j δ)\B(x,2j−1 δ))

j=1

By (κ 2) and (3.1), we have



2j δ 2j−1 δ

t α Φ −1 (x, κ(x, t )−1 )

dt t

≥ (2j−1 δ)α Φ −1 (x, Q1−1 κ(x, 2j δ)−1 ) log 2 ≥

(2j δ)α log 2 2α A2 Q1

Φ −1 (x, κ(x, 2j δ)−1 ) = C (2j δ)α Φ −1 (x, κ(x, 2j δ)−1 )

6

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

and



dX dX /2

dt

t α Φ −1 (x, κ(x, t )−1 )

t

dαX log 2

≥

2α A2 Q1

Φ −1 (x, κ(x, dX )−1 )

= CdαX Φ −1 (x, κ(x, dX )−1 ). Hence, we obtain d(x, y)α f (y)



µ(B(x, d(x, y)))

X \B(x,δ)

dµ(y) ≤ C

 j −1  0 

2j δ

j=1

2j−1 δ

≤ CΓ as required



x,

1



δ

α

t Φ

−1

(x, κ(x, t ) ) −1

dt t

dX



α

+ dX /2

t Φ

−1

(x, κ(x, t ) ) −1

,



Lemma 3.6. Assume that Φ (x, t ) satisfies (Φ 5). Let ε > 0 and define

λε (z , r ) =

1 1+

 dX r

ρ ε Φ −1 (z , κ(z , ρ)−1 ) dρρ

for z ∈ X . Then there exists a constant CI ,ε > 0 such that

λε (z , r ) µ(B(z , r ))

 X ∩B(z ,r )

Iε f (x) dµ(x) ≤ CI ,ε

for all z ∈ X , 0 < r ≤ dX and f ≥ 0 satisfying ∥f ∥LΦ ,κ (X ) ≤ 1. Proof. Let z ∈ X . Write Iε f (x) =

d(x, y)ε f (y)



µ(B(x, d(x, y))) = I1 (x) + I2 (x) X ∩B(z ,2r )

dµ(y) +

d(x, y)ε f (y)

 X \B(z ,2r )

µ(B(x, d(x, y)))

dµ(y)

for x ∈ X . By Fubini’s theorem,

 X ∩B(z ,r )

I1 (x) dµ(x) =

d(x, y)ε



 X ∩B(z ,2r )

X ∩B(z ,r )

µ(B(x, d(x, y))) d(x, y)ε



 ≤ X ∩B(z ,2r )

X ∩B(y,3r )



 ≤

j=0



≤



µ(B(x, d(x, y)))

dµ(x) f (y) dµ(y) d(x, y)ε

∞  

X ∩B(z ,2r )





dµ(x) f (y) dµ(y)

j=0

dµ(x) f (y) dµ(y)

X ∩(B(y,2−j+2 r )\B(y,2−j+1 r ))

µ(B(x, d(x, y)))

X ∩(B(y,2−j+2 r )\B(y,2−j+1 r ))

 (2−j+2 r )ε dµ(x) f (y) dµ(y). µ(B(x, 2−j+1 r ))

∞  

X ∩B(z ,2r )



Since µ is a doubling measure, we have

 X ∩B(z ,r )

I1 (x) dµ(x) ≤

≤

≤

c02

c02

c02



 X ∩B(z ,2r )

j =0



 X ∩B(z ,2r )

X ∩B(z ,2r )

ε

X ∩(B(y,2−j+2 r )\B(y,2−j+1 r ))

∞   j =0





∞  

X ∩(B(y,2−j+2 r )\B(y,2−j+1 r ))

 ∞  −j +2 ε (2 r ) f (y) dµ(y) j =0

 ∞  



≤ C8

X ∩B(z ,2r )

 (2−j+2 r )ε dµ(x) f (y) dµ(y) µ(B(x, 2−j+3 r ))  (2−j+2 r )ε dµ(x) f (y) dµ(y) µ(B(y, 2−j+2 r ))

j =0

2−j r

t 2−j−1 r

ε

dt t

 f (y) dµ(y)

dt t



T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17 r



 ≤C X ∩B(z ,2r )

C ε r

=

t

0



ε

dt

tε

X ∩B(z ,2r )



7

f (y) dµ(y)

f (y) dµ(y).

Now, by Lemma 3.3, (κ 2) and (3.1), we have rε

 X ∩B(z ,2r )

f (y) dy ≤ Cr ε µ(B(z , 2r ))Φ −1 (z , κ(z , 2r )−1 ) 2r



≤ C µ(B(z , 2r ))

ρ ε Φ −1 (z , κ(z , ρ)−1 )

r

dρ

ρ

if 0 < r ≤ dX /2 and, by Lemma 3.3 and (3.3), we have rε

 X ∩B(z ,2r )

f (y) dy = r ε

 B(z ,dX )

f (y) dy

≤ C dX ε µ(B(z , dX ))Φ −1 (z , κ(z , dX )−1 ) ≤ C µ(B(z , r )) if dX /2 < r ≤ dX . Therefore

 X ∩B(z ,r )

C µ(B(z , r ))

I1 (x) dµ(x) ≤

ε λε (z , r )

for all 0 < r ≤ dX . For I2 , first note that I2 (x) = 0 if x ∈ X and r ≥ dX /2. Let 0 < r < dX /2. Let j0 be the smallest positive integer such that 2j0 r ≥ dX . Since I2 (x) ≤ C

d(z , y)ε f (y)

 X \B(z ,2r )

µ(B(z , d(z , y)))

dµ(y) for x ∈ X ∩ B(z , r ),

by Lemma 3.3, we have d(z , y)ε

j 0 −1

I2 (x) ≤ C

 B(z ,2j+1 r )\B(z ,2j r )

j=1 j 0 −1

≤C



(2j+1 r )ε

j=1 j 0 −1

≤C



(2j+1 r )ε

j=1

≤C

 j −2 0 

1

µ(B(z , d(z , y))) 

µ(B(z , 2j r ))

X ∩B(z ,2j+1 r )

1

f (y) dµ(y)

f (y) dµ(y)



µ(B(z , 2j+1 r ))

X ∩B(z ,2j+1 r )

f (y) dµ(y)

 (2

j +1

ε

r) Φ

−1

(x, κ(x, 2

j +1

r)

−1

ε

) + dX Φ

−1

dρ

dX

(x, κ(x, dX ) ) . −1

j =1

As in the proof of Lemma 3.5, we obtain I2 (x) ≤ C

 j −2  0 

2j r

j =1 dX

 ≤C

2j+1 r

ρ ε Φ −1 (x, κ(x, ρ)−1 )

ρ ε Φ −1 (z , κ(z , ρ)−1 )

r

≤

C

λε (z , r )

for all x ∈ X ∩ B(z , r ). Hence

 X ∩B(z ,r )

I2 (x) dµ(x) ≤ C

Thus this lemma is proved.

µ(B(z , r )) . λε (z , r )



dρ

ρ

ρ

 +

dX /2

ρ ε Φ −1 (x, κ(x, ρ)−1 )

dρ

ρ



8

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

4. Trudinger’s inequality for Musielak–Orlicz–Morrey spaces In this section, we deal with the case Γ (x, t ) satisfies the uniform log-type condition: (Γlog ) there exists a constant cΓ > 0 such that

Γ (x, t 2 ) ≤ cΓ Γ (x, t ) for all x ∈ X and t ≥ 1. Example 4.1. Let Φ and κ be as in Examples 2.1 and 2.3, respectively. Then

Γ (x, t ) ∼

dX



1/t

ρ α−ν(x)/p(x)

k  

L(ej) (1/ρ)

−(qj (x)+βj (x))/p(x) dρ ρ

j =1

(t ≥ 2/dX ),

so that it satisfies (Γlog ) if and only if

α p(x) ≥ ν(x) for all x ∈ X . By (Γlog ), together with Lemma 3.4, we see that Γ (x, t ) satisfies the uniform doubling condition in t: Lemma 4.2 (cf. [29, Lemma 4.2]). Suppose Γ (x, t ) satisfies (Γlog ). For every a > 1, there exists b > 0 such that Γ (x, at ) ≤ bΓ (x, t ) for all x ∈ X and t > 0. Theorem 4.3. Assume that Φ (x, t ) satisfies (Φ 5), Γ (x, t ) satisfies (Γlog ). For each x ∈ X , let γ (x) = sups>0 Γ (x, s). Suppose Ψ (x, t ) : X × [0, ∞) → [0, ∞] satisfies the following conditions:

(Ψ 1) Ψ (·, t ) is measurable on X for each t ∈ [0, ∞); Ψ (x, ·) is continuous on [0, ∞) for each x ∈ X ; (Ψ 2) there is a constant A′1 ≥ 1 such that Ψ (x, t ) ≤ Ψ (x, A′1 s) for all x ∈ X whenever 0 < t < s; (Ψ 3) Ψ (x, Γ (x, t )/A′2 ) ≤ A′3 t for all x ∈ X and t > 0 with constants A′2 , A′3 ≥ 1 independent of x. Then, for 0 < ε < α , there exists a constant C ∗ > 0 such that Iα f (x)/C ∗ < γ (x) for a.e. x ∈ X and    Iα f (x) λε (z , r ) Ψ x, dµ(x) ≤ 1 µ(B(z , r )) X ∩B(z ,r ) C∗ for all z ∈ X , 0 < r ≤ dX and f ≥ 0 satisfying ∥f ∥LΦ ,κ (X ) ≤ 1. Proof. Let f ≥ 0 and ∥f ∥LΦ ,κ (X ) ≤ 1. Fix x ∈ X . For 0 < δ ≤ dX /2, Lemma 3.5 implies d(x, y)α f (y)





1



dµ(y) + C Γ x, µ(B(x, d(x, y))) δ   ε d(x, y) f (y) 1 = d(x, y)α−ε dµ(y) + C Γ x, µ(B(x, d(x, y))) δ X ∩B(x,δ)    1 ≤ C δ α−ε Iε f (x) + Γ x, δ

Iα f (x) ≤

X ∩B(x,δ)



with constants C > 0 independent of x. If Iε f (x) ≤ 2/dX , then we take δ = dX /2. Then, by Lemma 3.4 Iα f (x) ≤ C Γ



x,

2



dX

.

By Lemma 4.2, there exists C1∗ > 0 independent of x such that Iα f (x) ≤ C1∗ Γ



x,

1 2A′3



if Iε f (x) ≤ 2/dX .

Next, suppose 2/dX < Iε f (x) < ∞. Let m = sups≥2/dX ,x∈X Γ (x, s)/s. By (Γlog ), m < ∞. Define δ by

δ α−ε =

(dX /2)α−ε m

Γ (x, Iε f (x))(Iε f (x))−1 .

Since Γ (x, Iε f (x))(Iε f (x))−1 ≤ m, 0 < δ ≤ dX /2. Then by Lemma 3.4 1

δ

≤ C Γ (x, Iε f (x))−1/(α−ε) (Iε f (x))1/(α−ε) ≤ C Γ (x, 2/dX )−1/(α−ε) (Iε f (x))1/(α−ε) ≤ C (Iε f (x))1/(α−ε) .

(4.1)

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

9

Hence, using (Γlog ) and Lemma 4.2, we obtain

Γ



x,

1



  ≤ Γ x, C (Iε f (x))1/(α−ε) ≤ C Γ (x, Iε f (x)).

δ

By Lemma 4.2 again, we see that there exists a constant C2∗ > 0 independent of x such that Iα f (x) ≤ C2∗ Γ



x,

1

Iε f (x) 2CI ,ε A′3



if 2/dX < Iε f (x) < ∞,

(4.2)

where CI ,ε is the constant given in Lemma 3.6. Now, let C ∗ = A′1 A′2 max(C1∗ , C2∗ ). Then, by (4.1) and (4.2), Iα f ( x ) C∗

≤

1 A′1 A′2

    1 max Γ x, , Γ x, ′ 2A3

1 2CI ,ε A′3

 Iε f (x)

whenever Iε f (x) < ∞. Since Iε f (x) < ∞ for a.e. x ∈ X by Lemma 3.6, Iα f (x)/C ∗ < γ (x) a.e. x ∈ X , and by (Ψ 2) and (Ψ 3), we have

Ψ



x,

Iα f (x) C∗



       1 ≤ max Ψ x, Γ x, ′ A′2 , Ψ x, Γ x, 2A3

≤

1 2

+

1 2CI ,ε

  

1

Iε f (x) 2CI ,ε A′3

A′2

Iε f (x)

for a.e. x ∈ X . Thus, noting that λε (z , r ) ≤ 1 and using Lemma 3.6, we have

λε (z , r ) µ(B(z , r ))

 X ∩B(z ,r )

Ψ



x,

Iα f (x)



C∗

dµ(x) ≤

≤ for all z ∈ X and 0 < r ≤ dX .

1 2 1 2

λε (z , r ) + +

1 2

1 2CI ,ε

λε (z , r ) µ(B(z , r ))

 X ∩B(z ,r )

Iε f (x) dµ(x)

=1



Remark 4.4. If Γ (x, s) is bounded, that is, dX

 sup x∈X

  ρ α Φ −1 x, κ(x, ρ)−1 dρ < ∞,

0

then by Lemma 3.5 we see that Iα |f | is bounded for every f ∈ LΦ ,κ (X ). Remark 4.5. We cannot take ε = α in Theorem 4.3. For details, see [14, Remark 2.8]. As in the proof of [29, Corollary 4.6], we obtain the following corollary applying Theorem 4.3 to special Φ and κ given in Examples 2.1 and 2.3. Corollary 4.6. Let Φ and κ be as in Examples 2.1 and 2.3. Assume that

α − ν(x)/p(x) = 0 for all x ∈ X . (1) Suppose there exists an integer 1 ≤ j0 ≤ k such that inf (p(x) − qj0 (x) − βj0 (x)) > 0

x∈X

and sup(p(x) − qj (x) − βj (x)) ≤ 0 x∈X

for all j ≤ j0 − 1 in case j0 ≥ 2. Then for 0 < ε < α there exist constants C ∗ > 0 and C ∗∗ > 0 such that r ν(z )/p(z )−ε



(j )

E+0



Iα f (x)

p(x)/(p(x)−qj

0

(x)−βj0 (x))

|B(z , r )| X ∩B(z ,r ) (qj +j (x)+βj +j (x))/(p(x)−qj (x)−βj (x))   k −j 0  0 0 0 0 Iα f (x) × L(ej) dµ(x) ≤ C ∗∗ ∗ j =1

C

C∗

10

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

for all z ∈ X , 0 < r ≤ dX and f ≥ 0 satisfying ∥f ∥LΦ ,κ (X ) ≤ 1, where E (1) (t ) = et − e, E (j+1) (t ) = exp(E j (t )) − e and (j)

E+ (t ) = max(E (j) (t ), 0). (2) If

sup(p(x) − qj (x) − βj (x)) ≤ 0 x∈X

for all j = 1, . . . , k, then for 0 < ε < α there exist constants C ∗ > 0 and C ∗∗ > 0 such that r ν(z )/p(z )−ε



|B(z , r )|

E (k+1) X ∩B(z ,r )



Iα f (x) C∗



dµ(x) ≤ C ∗∗

for all z ∈ X , 0 < r ≤ dX and f ≥ 0 satisfying ∥f ∥LΦ ,κ (X ) ≤ 1. 5. Continuity for Musielak–Orlicz–Morrey spaces In this section, we discuss the continuity of Riesz potentials Iα f of functions in Musielak–Orlicz–Morrey spaces under the condition: there are constants θ > 0 and C0 > 0 such that

   θ   d(z , y)α d(x, y)α d(x, y)α   ≤ C0 d(x, z ) −  µ(B(x, d(x, y))) µ(B(z , d(z , y)))  d(x, y) µ(B(x, d(x, y))) whenever d(x, z ) ≤ d(x, y)/2.

(5.1)

We consider the functions

ω(x, r ) =

r

 0

  dρ ρ α Φ −1 x, κ(x, ρ)−1 ρ

and

ωθ (x, r ) = r θ

dX

 r

  dρ ρ α−θ Φ −1 x, κ(x, ρ)−1 ρ

for θ > 0 and 0 < r ≤ dX . Lemma 5.1 (cf. [29, Lemma 5.1]). Let E ⊂ X . If ω(x, r ) → 0 as r → 0+ uniformly in x ∈ E, then ωθ (x, r ) → 0 as r → 0+ uniformly in x ∈ E. Lemma 5.2 (cf. [29, Lemma 5.2]). There exists a constant C > 0 such that

ω(x, 2r ) ≤ C ω(x, r ) for all x ∈ X and 0 < r ≤ dX /2. Theorem 5.3. Assume that Φ (x, t ) satisfies (Φ 5). Then there exists a constant C > 0 such that

|Iα f (x) − Iα f (z )| ≤ C {ω(x, d(x, z )) + ω(z , d(x, z )) + ωθ (x, d(x, z ))} for all x, z ∈ X with d(x, z ) ≤ dX /4 and nonnegative f ∈ LΦ ,κ (X ) with ∥f ∥LΦ ,κ (X ) ≤ 1. Before giving a proof of Theorem 5.3, we prepare two more lemmas. Lemma 5.4. Assume that Φ (x, t ) satisfies (Φ 5). Let f be a nonnegative function on X such that ∥f ∥LΦ ,κ (X ) ≤ 1. Then there exists a constant C > 0 such that d(x, y)α f (y)

 X ∩B(x,δ)

µ(B(x, d(x, y)))

dµ(y) ≤ C ω(x, δ)

for all x ∈ X and 0 < δ ≤ dX . Proof. Let f be a nonnegative µ-measurable function on X with ∥f ∥LΦ ,κ (X ) ≤ 1. As usual we start by decomposing B(x, δ) dyadically: d(x, y)α f (y)

 X ∩B(x,δ)

µ(B(x, d(x, y)))

dµ(y) =

∞  

d(x, y)α f (y)

dµ(y) µ(B(x, d(x, y)))  ∞  1 ≤ (2−j+1 δ)α f (y) dµ(y) µ(B(x, 2−j δ)) B(x,2−j+1 δ) j =1  ∞  1 (2−j+1 δ)α f (y) dµ(y). ≤ c0 µ(B(x, 2−j+1 δ)) B(x,2−j+1 δ) j =1 j =1

X ∩(B(x,2−j+1 δ)\B(x,2−j δ))

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

11

By Lemma 3.3, we have d(x, y)α f (y)

 X ∩B(x,δ)

µ(B(x, d(x, y)))

dµ(y) ≤ C

∞  (2−j+1 δ)α Φ −1 (x, κ(x, 2−j+1 δ)−1 ) j =1

δ

  dρ ρ α Φ −1 x, κ(x, ρ)−1 ρ 0 = C ω(x, δ).  

≤C

The following lemma can be proved in the same manner as Lemma 3.5. Lemma 5.5. Assume that Φ (x, t ) satisfies (Φ 5). Let θ ∈ R. Let f be a nonnegative function on X such that ∥f ∥LΦ ,κ (X ) ≤ 1. Then there exists a constant C > 0 such that d(x, y)α−θ f (y)

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y) ≤ C δ −θ ωθ (x, δ)

for all x ∈ X and 0 < δ ≤ dX /2. Proof of Theorem 5.3. Let f be a nonnegative µ-measurable function on X with ∥f ∥LΦ ,κ (X ) ≤ 1 and x, z ∈ X with d(x, z ) ≤ dX /4. Write Iα f (x) − Iα f (z ) =

d(x, y)α f (y)



dµ(y) −



d(z , y)α f (y)

µ(B(x, d(x, y))) X ∩B(x,2d(x,z )) µ(B(z , d(z , y)))    α d(z , y)α d(x, y) − f (y) dµ(y). + µ(B(x, d(x, y))) µ(B(z , d(z , y))) X \B(x,2d(x,z )) X ∩B(x,2d(x,z ))

dµ(y)

Using Lemmas 5.2 and 5.4, we have d(x, y)α f (y)

 X ∩B(x,2d(x,z ))

µ(B(x, d(x, y)))

dµ(y) ≤ C ω(x, 2d(x, z )) ≤ C ω(x, d(x, z ))

and d(z , y)α f (y)

 X ∩B(x,2d(x,z ))

µ(B(z , d(z , y)))

dµ(y) ≤

d(z , y)α f (y)



dµ(y) µ(B(z , d(z , y))) ≤ C ω(z , 3d(x, z )) ≤ C ω(z , d(x, z )). X ∩B(z ,3d(x,z ))

On the other hand, by (5.1) and Lemma 5.5, we have



    d(x, y)α d(z , y)α   −  µ(B(x, d(x, y))) µ(B(z , d(z , y)))  f (y) dµ(y) X \B(x,2d(x,z ))  d(x, y)α−θ f (y) ≤ Cd(x, z )θ dµ(y) X \B(x,2d(x,z )) µ(B(x, d(x, y))) ≤ C ωθ (x, 2d(x, z )) ≤ C ωθ (x, d(x, z )).

Then we have the conclusion.



In view of Lemma 5.1, we obtain the following corollary. Corollary 5.6. Assume that Φ (x, t ) satisfies (Φ 5). (a) Let x0 ∈ X and suppose ω(x, r ) → 0 as r → 0+ uniformly in x ∈ X ∩ B(x0 , δ) for some δ > 0. Then Iα f is continuous at x0 for every f ∈ LΦ ,κ (X ). (b) Suppose ω(x, r ) → 0 as r → 0+ uniformly in x ∈ X . Then Iα f is uniformly continuous on X for every f ∈ LΦ ,κ (X ). 6. Lemmas for Musielak–Orlicz spaces For a measurable function Q (·) satisfying 0 < Q − := inf Q (x) ≤ sup Q (x) =: Q + < ∞, x∈X

x∈X

(6.1)

12

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

we say that a measure µ is lower Ahlfors Q (x)-regular if there exists a constant c1 > 0 such that

µ(B(x, r )) ≥ c1 r Q (x) for all x ∈ X and 0 < r < dX . Recall that we say that the measure µ is a doubling measure if there exists a constant c0 > 0 such that µ(B(x, 2r )) ≤ c0 µ(B(x, r )) for every x ∈ X and 0 < r < dX . Here note that if µ is a doubling measure and dX < ∞, then µ is lower Ahlfors log2 c0 -regular since

µ(B(x, r )) ≥ c0−2 µ(B(x, dX ))



r

log2 c0

dX

for all x ∈ X and 0 < r < dX (see e.g. [43, Lemma 3.3]). For a locally integrable function f on X , the Hardy–Littlewood maximal function Mf is defined by Mf (x) = sup r >0



1

µ(B(x, r ))

B(x,r )∩X

|f (y)| dµ(y).

As in the proof of [41, Theorem 4.1], we can show the following boundedness of the maximal operator on LΦ (X ). Lemma 6.1 (c.f. [41, Theorem 4.1]). Suppose that Φ (x, t ) satisfies (Φ 5) and further assume: (Φ 3∗ ) t → t −ε0 φ(x, t ) is uniformly almost increasing on (0, ∞) for some ε0 > 0. Then the maximal operator M is bounded from LΦ (X ) into itself, namely, there is a constant C > 0 such that

∥Mf ∥LΦ (X ) ≤ C ∥f ∥LΦ (X ) for all f ∈ LΦ (X ). We consider the function

γ (x, t ) : X × (0, dX ) → (0, ∞) satisfying the following conditions: (γ 1) γ ( ·, t ) is measurable on X for each 0 < t < dX and γ ( x, · ) is continuous on (0, dX ) for each x ∈ X ; (γ 2) there exist constants γ0 > 0 and B0 ≥ 1 such that 1 −γ0 B− 0 ≤ γ (x, t ) ≤ B0 t

for all x ∈ X whenever 0 < t < dX .

(γ 3) there exists a constant B1 ≥ 1 such that 1 B− 1 γ (x, s) ≤ γ (x, t ) ≤ B1 γ (x, s) for all x ∈ X and 1 ≤ t /s ≤ 2.

Further we consider the function

(x, t ) : X × [0, ∞) → [0, ∞) Γ satisfying the following conditions (Γ 1), (Γ 2) and (Γ 3):

( · , t ) is measurable on X for each t ≥ 0 and Γ (x, · ) is continuous on [0, ∞) for each x ∈ X ; (Γ 1) Γ (x, ·) is uniformly almost increasing, namely there exists a constant B2 ≥ 1 such that (Γ 2) Γ (x, t ) ≤ B2 Γ (x, s) for all x ∈ X whenever 0 ≤ t < s; Γ (Γ 3) For a measurable function Q (·) satisfying (6.1), there exist constants α0 > 0, B3 ≥ 1 and B4 ≥ 1 such that

(x, 1/t ) t α−Q (x) φ(x, γ (x, t ))−1 ≤ B3 Γ for all x ∈ X and α ≥ α0 whenever 0 < t < dX and dX

 t

ρ α γ (x, ρ)

dρ

ρ

(x, 1/t ) ≤ B4 Γ

for all x ∈ X , 0 < t ≤ dX /2 and α ≥ α0 . Example 6.2. Let Φ be as in Example 2.1. (1) Suppose there exists an integer 1 ≤ j0 ≤ k such that inf (p(x) − qj0 (x) − 1) > 0

x∈X

and sup(p(x) − qj (x) − 1) ≤ 0 x∈X

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

13

for all j ≤ j0 − 1 in case j0 ≥ 2. Set

   j −1  k 0   ( x )+ 1 )/ p ( x ) −( q [L(ej) (1/t )]−qj (x)/p(x) γ (x, t ) = t −Q (x)/p(x) [L(ej) (1/t )]−1 [L(ej0 ) (1/t )] j0 j =j 0 +1

j =1

and (j0 )

(x, t ) = [Le (t )] Γ



(p(x)−qj0 (x)−1)/p(x)

k 

 (j)

[Le (t )]

−qj (x)/p(x)

.

j=j0 +1

(x, t ) satisfies (Γ 1), (Γ 2) and (Γ 3) for all α ≥ Q + /p− . Then γ (x, t ) satisfies (γ 1), (γ 2) and (γ 3) and Γ (2) Suppose that sup(p(x) − qj (x) − 1) ≤ 0 x∈X

for all j = 1, . . . , k. Set

 γ (x, t ) = t

−Q (x)/p(x)

k 

 (j)

[Le (1/t )]

[L(ek+1) (1/t )]−1/p(x)

−1

j=1

and

(x, t ) = [Le(k+1) (1/t )]1−1/p(x) . Γ (x, t ) satisfies (Γ 1), (Γ 2) and (Γ 3) for all α ≥ Q + /p− . Then γ (x, t ) satisfies (γ 1), (γ 2) and (γ 3) and Γ In fact, see the proof of [37, Corollary 4.2]. Lemma 6.3. Assume that µ is lower Ahlfors Q (x)-regular. Suppose that Φ (x, t ) satisfies (85). Let α ≥ α0 . Then there exists a constant C > 0 such that d(x, y)α f (y)

 X \B(x,δ)

  1  dµ(y) ≤ C Γ x, µ(B(x, d(x, y))) δ

for all x ∈ X , 0 < δ ≤ dX /2 and nonnegative f ∈ LΦ (X ) with ∥f ∥LΦ (X ) ≤ 1. Proof. Let f be a nonnegative µ-measurable function on X with ∥f ∥LΦ (X ) ≤ 1. Let j0 be the smallest integer j0 such that 2j0 δ ≥ dX . Since 1 −γ0 B− 0 ≤ γ (x, d(x, y)) ≤ B0 d(x, y)

in view of (γ 2), we have 2/γ0

d(x, y) ≤ B0

(B0 γ (x, d(x, y)))−1/γ0 .

Hence, by (Φ 3), (Φ 4) and (Φ 5), we obtain

φ(y, γ (x, d(x, y)))−1 ≤ B′ φ(x, γ (x, d(x, y)))−1 with some constant B′ > 0. By (γ 3), (Φ 3), (Γ 2) and (Γ 3), we have d(x, y)α f (y)

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y) ≤

d(x, y)α γ (x, d(x, y))



µ(B(x, d(x, y)))

X \B(x,δ)

d(x, y)α f (y)

 + A2

≤

X \B(x,δ)

φ(y, f (y)) dµ(y) µ(B(x, d(x, y))) φ(y, γ (x, d(x, y))) d(x, y)α γ (x, d(x, y))

j0   B(x,2j δ)\B(x,2j−1 δ)

j =1

−1

′

+ c0 A2 B ≤ 2α B1

dµ(y)

j0  j =1

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y)

d(x, y)α−Q (x) φ(x, γ (x, d(x, y)))−1 Φ (y, f (y)) dµ(y)

(2j−1 δ)α γ (x, 2j−1 δ)



1 B(x,2j δ)\B(x,2j−1 δ)

µ(B(x, 2j−1 δ))

dµ(y)

14

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

(x, 1/δ) + c0−1 A2 B2 B3 B′ Γ ≤ 2α c2 B1

 X \B(x,δ)

Φ (y, f (y)) dµ(y)

j0  (x, 1/δ). (2j−1 δ)α γ (x, 2j−1 δ) + c0−1 A2 B2 B3 B′ Γ j =1

Since dX

 δ

α

ρ γ (x, ρ)

dρ

ρ

j 0 −1

≥



2j δ 2j−1 δ

j =1

ρ α γ (x, ρ)

dρ

ρ

≥

j 0 −1 log 2 

B1

(2j−1 δ)α γ (x, 2j−1 δ)

j=1

and dX

 δ

ρ α γ (x, ρ)

dρ

ρ

dX

 ≥

dX /2

ρ α γ (x, ρ)

dρ

ρ

≥

log 2 2α B1

(2j0 −1 δ)α γ (x, 2j0 −1 δ),

we have

 j0  B1 α j−1 α j −1 (2 + 1) (2 δ) γ (x, 2 δ) ≤ log 2

j =1

dX

δ

ρ α γ (x, ρ)

dρ

ρ

.

Hence, we obtain d(x, y)α f (y)

 X \B(x,δ)

µ(B(x, d(x, y)))

dµ(y)

≤ (log 2)−1 2α (2α + 1)c2 B21

dX

 δ

ρ α γ (x, ρ)

dρ

ρ

(x, 1/δ) + c0−1 A2 B2 B3 B′ Γ

(x, 1/δ) + c0−1 A2 B2 B3 B′ Γ (x, 1/δ) ≤ (log 2)−1 2α (2α + 1)c2 B21 B4 Γ (x, 1/δ), = ((log 2)−1 2α (2α + 1)c2 B21 B4 + c0−1 A2 B2 B3 B′ )Γ as required.



(x, 2/dX ) ≥ C ′ for all x ∈ X . Lemma 6.4 (cf. [37, Lemma 3.3]). Let α ≥ α0 . Then there exists a constant C ′ > 0 such that Γ (x, t ) satisfies the uniform log-type condition: Lemma 6.5 (cf. [37, Lemma 3.4]). Suppose Γ log ) there exists a constant cΓ > 0 such that (Γ (x, t ) ≤ Γ (x, t 2 ) ≤ cΓ Γ (x, t ) cΓ−1 Γ for all x ∈ X and t > 0.

(x, at ) ≤ bΓ (x, t ) for all x ∈ X and t > 0. Then, for every a > 1, there exists b > 0 such that Γ 7. Trudinger’s inequality for Musielak–Orlicz spaces Theorem 7.1. Suppose that µ is lower Ahlfors Q (x)-regular. Assume that Φ (x, t ) satisfies (Φ 5) and (Φ 3∗ ). Further, assume (x, t ) satisfies (Γ log ). For each x ∈ X , let  (x, s). Suppose Ψ  (x, t ) : X × [0, ∞) → [0, ∞] satisfies the that Γ γ (x) = sups>0 Γ following conditions:

 1) (Ψ  2) (Ψ  (Ψ 3)

 (·, t ) is measurable on X for each t ∈ [0, ∞) and Ψ  (x, ·) is continuous on [0, ∞) for each x ∈ X ; Ψ   there is a constant B5 ≥ 1 such that Ψ (x, t ) ≤ Ψ (x, B5 s) for all x ∈ X whenever 0 < t < s;  (x, Γ (x, t )/B6 ) ≤ B7 t for all x ∈ X and t ≥ t0 . there are constants B6 , B7 ≥ 1 and t0 > 0 such that Ψ

Then there exist constants c1 , c2 > 0 such that Iα f (x)/c1 ≤  γ (x) for µ-a.e. x ∈ X and

  I f (x)  x, α dµ(x) ≤ c2 Ψ

 X

c1

for all α ≥ α0 and nonnegative functions f ∈ LΦ (X ) satisfying ∥f ∥LΦ (X ) ≤ 1.

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

15

Proof. Let f be a nonnegative µ-measurable function on X with ∥f ∥LΦ (X ) ≤ 1. Note from Lemma 6.1 that



Mf (x) dµ(x) ≤ µ(X ) + A1 A2



X

Φ (x, Mf (x)) dµ(x) ≤ CM .

(7.1)

X

Fix x ∈ X . For 0 < δ ≤ dX /2, Lemma 6.3 implies d(x, y)α f (y)



Iα f (x) =

dµ(y) +

µ(B(x, d(x, y)))    1  x, ≤ C δ α Mf (x) + Γ δ B(x,δ)

d(x, y)α f (y)



µ(B(x, d(x, y)))

X \B(x,δ)

dµ(y)

with a constant C > 0 independent of x. If Mf (x) ≤ 2/dX , then we take δ = dX /2. Then, by Lemma 6.4



 x, Iα f (x) ≤ C Γ



2 dX

.

By Lemma 6.5 and (Γ 2), there exists C1∗ > 0 independent of x such that

 (x, t0 ) Iα f (x) ≤ C1∗ Γ

if Mf (x) ≤ 2/dX .

(7.2)

(x, s)/s. By (Γ log ), m < ∞. Define δ by Next, suppose 2/dX < Mf (x) < ∞. Let m = sups≥2/dX ,x∈X Γ δα =

(dX /2)α m

(x, Mf (x))(Mf (x))−1 . Γ

(x, Mf (x))(Mf (x))−1 ≤ m, 0 < δ ≤ dX /2. Then by Lemma 6.4 and (Γ 2) Since Γ 1

δ

= ≤

m1/α dX /2 m1/α

(x, Mf (x))−1/α (Mf (x))1/α Γ 1/α

dX /2

(x, 2/dX )−1/α (Mf (x))1/α ≤ C (Mf (x))1/α . B2 Γ

log ) and Lemma 6.5, we obtain Hence, using (Γ 2), (Γ     1   x, C (Mf (x))1/α ≤ C Γ (x, Mf (x)). Γ x, ≤ B2 Γ δ By Lemma 6.5 again, we see from (Γ 2) that there exists a constant C2∗ > 0 independent of x such that Iα f (x) ≤ C2 Γ

∗



x,

t 0 dX 2

Mf (x)



if 2/dX < Mf (x) < ∞.

(7.3)

Now, let c1 = B5 B6 max(C1∗ , C2∗ ). Then, by (7.2) and (7.3), Iα f ( x ) c1

≤

1 B5 B6





 x,  (x, t0 ) , Γ max Γ

t 0 dX 2

Mf (x)



 2) and whenever Mf (x) < ∞. Since Mf (x) < ∞ for µ-a.e. x ∈ X by Lemma 6.1, Iα f (x)/c1 ≤  γ (x) µ-a.e. x ∈ X , and by (Ψ  3), we have (Ψ           I f (x) t d  x, α  x, Γ  (x, t0 ) /B6 , Ψ  x, Γ  x, 0 X Mf (x) Ψ ≤ max Ψ B6 c1

2

≤ B7 t0 +

B7 t0 dX 2

Mf (x)

for µ-a.e. x ∈ X . Thus, we have by (7.1)

   Iα f (x) B7 t0 dX  Ψ x, dµ(x) ≤ B7 t0 µ(X ) + Mf (x) dµ(x)

 X

c1

2

≤ B7 t0 µ(X ) +

X

B7 t0 dX CM 2

= c2 . 

We obtain the following corollary applying Theorem 7.1 to special Φ given in Example 2.1.

16

T. Ohno, T. Shimomura / Nonlinear Analysis 106 (2014) 1–17

Corollary 7.2. Let Φ be as in Example 2.1. Assume that µ is lower Ahlfors Q (x)-regular. (1) Suppose there exists an integer 1 ≤ j0 ≤ k such that inf (p(x) − qj0 (x) − 1) > 0

(7.4)

sup(p(x) − qj (x) − 1) ≤ 0

(7.5)

x∈X

and x∈X

for all j ≤ j0 − 1 in case j0 ≥ 2. Then there exist constants c1 , c2 > 0 such that



(j ) E+0



Iα f (x)

p(x)/(p(x)−qj

0

−j 0 (x)−1) k

c1

X



(j)



Le

j =1

Iα f (x)

qj

0 +j

(x)/(p(x)−qj0 (x)−1)

c1



dµ(x) ≤ c2

for all α ≥ Q + /p− and nonnegative functions f ∈ LΦ (X ) satisfying ∥f ∥LΦ (X ) ≤ 1. (2) If sup(p(x) − qj (x) − 1) ≤ 0 x∈X

for all j = 1, . . . , k, then there exist constants c1 , c2 > 0 such that

 E

(k+1)



Iα f (x) c1

X

p(x)/(p(x)−1) 

dµ(x) ≤ c2

for all α ≥ Q + /p− and nonnegative functions f ∈ LΦ (X ) satisfying ∥f ∥LΦ (X ) ≤ 1. 8. Continuity for Musielak–Orlicz spaces For a measurable function Q (·) satisfying (6.1), we consider the functions

 ω(x, r ) =

r

 0

  dρ ρ α Φ −1 x, ρ −Q (x) ρ

and

 ωθ (x, r ) = r θ

dX

 r

  dρ ρ α−θ Φ −1 x, ρ −Q (x) ρ

for θ > 0 and 0 < r ≤ dX . As in the proof of Theorem 5.3, we can obtain the continuity of Riesz potentials Iα f of functions in Musielak–Orlicz spaces under the condition (5.1). Theorem 8.1. Assume that µ is lower Ahlfors Q (x)-regular. Suppose that Φ (x, t ) satisfies (Φ 5). Suppose that (5.1) holds. Then there exists a constant C > 0 such that

|Iα f (x) − Iα f (z )| ≤ C { ω(x, d(x, z )) +  ω(z , d(x, z )) +  ωθ (x, d(x, z ))} for all x, z ∈ X with 0 < d(x, z ) ≤ dX /2 whenever f ∈ LΦ (X ) is a nonnegative function on X satisfying ∥f ∥LΦ (X ) ≤ 1. Corollary 8.2. Assume that µ is lower Ahlfors Q (x)-regular. Suppose that Φ (x, t ) satisfies (Φ 5). Suppose that (5.1) holds. (a) Let x0 ∈ X and suppose  ω(x, r ) → 0 as r → 0+ uniformly in x ∈ B(x0 , δ) ∩ X for some δ > 0. Then Iα f is continuous at x0 for every f ∈ LΦ (X ). (b) Suppose  ω(x, r ) → 0 as r → 0+ uniformly in x ∈ X . Then Iα f is uniformly continuous on X for every f ∈ LΦ (X ). Acknowledgments We would like to express our thanks to the referees for their kind comments and suggestions. References [1] N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483. [2] D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, Heidelberg, 1996. [3] H. Brézis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980) 773–789.

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