SOBOLEV AND TRUDINGER TYPE INEQUALITIES ON GRAND MUSIELAK ORLICZ MORREY SPACES

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 205, SOBOLEV AND TRUDINGER TYPE INEQUALITIES ON GRAND MUSIELAK ORLICZ MORREY SPACES Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura 4-24 Furue-higashi-machi, Nishi-ku, Hiroshima , Japan; Hiroshima Institute of Technology, Department of Mechanical Systems Engineering 2-- Miyake Saeki-ku Hiroshima , Japan; Oita University, Faculty of Education and Welfare Science Dannoharu Oita-city , Japan; Hiroshima University, Graduate School of Education, Department of Mathematics Higashi-Hiroshima , Japan; Abstract. Our aim in this paper is to establish generalizations of Sobolev s inequality and Trudinger s inequality for general potentials of functions in grand Musielak Orlicz Morrey spaces.. Introduction Grand Lebesgue spaces were introduced in [5] for the study of Jacobian. They play important roles also in the theory of partial differential equations see [0], [6] and [29], etc.. The generalized grand Lebesgue spaces appeared in [2], where the existence and uniqueness of the non-homogeneous N-harmonic equations were studied. The boundedness of the maximal operator on the grand Lebesgue spaces was studied in [9]. For variable exponent Lebesgue spaces, see [6] and [7]. In [2] and [7], grand Morrey spaces and generalized grand Morrey spaces were introduced. For Morrey spaces, we refer to [24] and [27]. Further, grand Morrey spaces of variable exponent were considered in []. On the other hand, the classical Sobolev s inequality for Riesz potentials of L p - functions see, e.g. [2, Theorem 3..4 b] has been extended to various function spaces. For Morrey spaces, Sobolev s inequality was studied in [], [27], [5], [25], etc., for Morrey spaces of variable exponent in [3], [3], [4], [22], [23], etc., for grand Morrey spaces in [2] and [7], and also for grand Morrey spaces of variable exponent in []. Recently, Sobolev s inequality has been extended by the authors [9] to an inequality for general potentials of functions in Musielak Orlicz Morrey spaces. The classical Trudinger s inequality for Riesz potentials of L p -functions see, e.g. [2, Theorem 3..4 c] has been also extended to function spaces as above; see [22], [23] for Morrey spaces of variable exponent, [] for grand Morrey spaces of variable exponent and [20] for Musielak Orlicz Morrey spaces. In this paper, we define generalized grand Musielak Orlicz Morrey space on a bounded open set in R N and give a Sobolev type inequality as well as a Trudinger type inequality for general potentials of functions in such spaces. doi:0.586/aasfm Mathematics Subject Classification: Primary 3B5, 46E35. Key words: Riesz potentials, maximal functions, Sobolev s inequality, Trudinger s inequality, grand Musielak Orlicz Morrey spaces.

2 404 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura Throughout this paper, let C denote various constants independent of the variables in question. The symbols g h and g h means that g Ch and C h g Ch for some constant C > 0 respectively. 2. Preliminaries Let G be a bounded open set in R N and let d G denote the diameter of G. We consider a function Φx,t = tφx,t: G [0, [0, satisfying the following conditions Φ Φ4: Φ φ,t is measurable on G for each t 0 and φx, is continuous on [0, for each x G; Φ2 there exists a constant A such that A φx, A for all x G; Φ3 there exists a constant ε 0 > 0 such that t t ε 0 φx,t is uniformly almost increasing, namely there exists a constant A 2 such that t ε 0 φx,t A 2 s ε 0 φx,s for all x G whenever 0 < t < s; Φ4 there exists a constant A 3 such that Note that Φ3 implies that φx,2t A 3 φx,t for all x G and t > 0. t ε φx,t A 2 s ε φx,s for all x G and 0 < ε ε 0 whenever 0 < t < s. Also note that Φ2, Φ3 and Φ4 imply 0 < inf φx,t supφx,t < x G x G for each t > 0 and there exists ω > such that 2. A A 2 t +ε 0 Φx,t A A 2 A 3 t ω for t ; in fact we can take ω +loga 3 /log2. We shall also consider the following condition: Φ5 for every γ > 0, there exists a constant B γ such that φx,t B γ φy,t whenever x y γt /N and t. Let φx,t = sup 0 s t φx,s and Then Φx, is convex and for all x G and t 0. Φx,t = t 0 φx,rdr. 2A 3 Φx,t Φx,t A 2 Φx,t

3 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 405 Example 2.. Let p and q j, j =,...,k, be measurable functions on G such that < p := inf px suppx =: p + < x G x G and < q j := inf q jx supq j x =: q + j <, j =,... k. x G x G Set Lt := loge+t, L t = Lt and L j t = LL j t, j = 2,... Then, k Φ p,{qj }x,t = t px L j t q j x satisfies Φ, Φ2, Φ3 with 0 < ε 0 < p and Φ4. 2. holds for any ω > p +. Φ p,{qj }x,t satisfies Φ5 if p is log-hölder continuous, namely j= px py C p L/ x y and q j is j +-log-hölder continuous, namely x y q j x q j y for j =,...,k cf. [9, Example 2.]. C qj L j+ / x y x y We also consider a function κx,r: G 0,d G 0, satisfying the following conditions: κ κx, is continuous on 0,d G for each x G and satisfies the uniform doubling condition: there is a constant Q such that Q κx,r κx,r Q κx,r for all x G whenever 0 < r r 2r < d G ; κ2 r r δ κx,r is uniformly almost increasing for some δ > 0, namely there is a constant Q 2 > 0 such that r δ κx,r Q 2 s δ κx,s for all x G whenever 0 < r < s < d G ; κ3 there is a constant Q 3 such that for all x G and 0 < r < d G. Q 3 min, r N κx,r Q 3 Example 2.2. Let ν andβ be functions ongsuch thatν := inf x G νx > 0, ν + := sup x G νx N and cn νx βx c for all x G and some constant c > 0. Then κx,r = r νx loge+/r βx satisfies κ, κ2 and κ3; we can take any 0 < δ < ν for κ2. by Given Φx,t and κx,r, we define the Musielak Orlicz Morrey space L Φ,κ G L Φ,κ G = { f L loc G; sup κx,r Φ y, fy } dy <. x G,0<r<d G Bx, r

4 406 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura It is a Banach space with respect to the norm { κx,r f Φ,κ;G = inf λ > 0; sup x G,0<r<d G Bx, r Φ y, fy /λ } dy cf. [26]. In case κx,r = r N, L Φ,κ G is the Musielak Orlicz space } L Φ loc G = {f L G; Φy, fy dy < G with the norm { f Φ;G = inf λ > 0; Φ y, fy /λ } dy. G Remark 2.3. The Musielak Orlicz spaces L Φ G include Orlicz spaces defined by Young functions satisfying the doubling condition; variable exponent Lebesgue spaces. The Musielak Orlicz Morrey spaces L Φ,κ G include Morrey spaces as well as variable exponent Morrey spaces. 3. Grand Musielak Orlicz Morrey space For ε 0, set Φ ε x,t := t ε Φx,t = t ε φx,t. Then, Φ ε x,t satisfies Φ, Φ2 with the same A and Φ4 with the same A 3. If Φx,t satisfies Φ5, then so does Φ ε x,t with the same {B γ } γ>0. If 0 ε < ε 0, then Φ ε x,t satisfies Φ3 with ε 0 replaced by ε 0 ε and the same A 2. It follows that 3. Φ ε x,t Φ ε x,t A 2 Φ ε x,t 2A 3 for all x G, t 0 and 0 ε ε 0. By Φ3, we see that for 0 ε ε 0 { A 2 aφ ε x,t if 0 a, 3.2 Φ ε x,at A 2 aφ εx,t if a. Let σ = sup{σ 0: r N σ κx,r is bounded on G 0,min,d G }. By κ2, 0 σ N. If σ = 0, then let σ 0 = 0; otherwise fix any σ 0 0, σ. We also take δ 0 such that 0 < δ 0 < δ for δ in κ2. For δ 0 σ σ 0, set κ σ x,r = r σ κx,r for x G and 0 < r < d G. Then κ σ x,r satisfies κ, κ2 and κ3 with constants independent of σ. Lemma 3.. For 0 ε ε 0, let Φ ε x,s = sup{t > 0 : Φ ε x,t < s} x G, s > 0. Then there exists r 0 0,min,d G such that κ σ x,r and Φ ε x,κσ x,r for all x G, 0 < r r 0, δ 0 σ σ 0 and 0 < ε ε 0.

5 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 407 Proof. By κ2 and κ3, κ σ x,r Q 2 Q 3 min,d G δ r δ+σ Q 2 Q 3 min,d G δ r δ δ 0 for x G, 0 < r < min,d G and δ 0 σ σ 0. Hence, there is r 0,min,d G such that κ σ x,r for x G, 0 < r r and δ 0 σ σ 0. By 2., we see that Φ ε x,κσ x,r C κ σ x,r /ω C r δ δ 0/ω whenever x G, 0 < r r, δ 0 σ σ 0 and 0 < ε ε 0 with constants C, C > 0 independent ofx,r, σ,ε. Hence the assertion of the lemma holds if we taker 0 0,r ] satisfying r δ δ 0/ω 0 C. Proposition 3.2. Assume that Φx,t satisfies Φ5. If 0 ε ε 2 ε 0, δ 0 σ j σ 0, j =,2 and then L Φε,κσ G L Φ ε2,κ σ2g and σ + δ δ 0 ω ε σ 2 + δ δ 0 ω ε 2, f Φε2,κ σ2 ;G C f Φε,κ σ ;G for all f L Φε,κσ G with C > 0 independent of ε, ε 2, σ, σ 2. In particular, L Φ,κ G L Φε,κσ G if 0 ε ε 0, δ 0 σ σ 0 and σ +δ δ 0 /ωε 0. Proof. Let f Φε,κ σ ;G. Then κ σ x,r Bx, r for x G and 0 < r < d G. For x G and 0 < r < d G, let and Bx,r Φ ε y, fy dy kx,r = Φ ε x,κσ x,r Ix,r = Bx,r We write Ix,r = I x,r+i 2 x,r, where I x,r = and I 2 x,r = If fy kx,r, then Bx,r {y: fy kx,r} Bx,r {y: fy >kx,r} Φ ε2 y, fy dy. Φ ε2 y, fy dy Φ ε2 y, fy dy. Φ ε2 y, fy A 2 Φ ε2 y,kx,r = A 2 kx,r ε ε 2 Φ ε y,kx,r. Let r 0 0,min,d G be the number given in Lemma 3.. Then, 3.2 implies kx,r Cκ σ x,r Cr N

6 408 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura for 0 < r r 0 with constants independent of x, σ, ε. Hence, by Φ5, there is a constant B > 0 independent of x, σ, ε, such that whenever x y < r r 0. Therefore, Φ ε y,kx,r BΦ ε x,kx,r I x,r C Bx,r kx,r ε ε 2 Φ ε x,kx,r = C Bx,r kx,r ε ε 2 κ σ x,r for 0 < r r 0. On the other hand, if fy > kx,r, then so that Φ ε2 y, fy = fy ε ε 2 Φ ε y, fy kx,r ε ε 2 Φ ε y, fy, I 2 x,r kx,r ε ε 2 Bx,r Φ ε y, fy dy Bx,r kx,r ε ε 2 κ σ x,r for 0 < r r 0. Therefore, Ix,r C Bx,r kx,r ε ε 2 κ σ x,r, which implies κ σ2 x,r Φ ε2 y, fy dy Cr σ 2 σ kx,r ε ε 2 Bx, r Bx,r for 0 < r r 0. Since kx,r Cr δ δ 0/ω and σ 2 σ +δ δ 0 /ωε 2 ε 0 by assumption, κ σ2 x,r Φ ε2 y, fy dy Cr σ 2 σ +δ δ 0 /ωε 2 ε C Bx, r Bx,r for 0 < r r 0 with positive constants C s independent of x, σ j, ε j j =,2. In case r 0 < r < d G, we see Ix,r A 2 Φ ε2 y,dy+ Φ ε y, fy dy Bx,r Bx,r A A 2 Bx,r + Bx,r κ σ x,r, so that κ σ2 x,r Φ ε2 y, fy dy A A 2 κ σ2 x,r+r σ 2 σ C Bx, r Bx,r with C independent of r, x, σ σ 2. Therefore, f Φε2,κ σ2 ;G C with C > 0 independent of ε, ε 2, σ, σ 2. Let ηε be an increasing positive function on 0, such that η0+ = 0. Let ξε be a function on 0,ε ] with some ε 0,ε 0 /2] such that δ 0 ξε σ 0 for 0 < ε ε, ξ0+ = 0 and ε ξε+δ δ 0 /ωε is non-decreasing; in particular, ξε+δ δ 0 /ωε 0 for 0 < ε ε. Given Φx, t, κx, r, ηε and ξε, the associated generalized grand Musielak Orlicz Morrey space is defined by cf. [7] for generalized grand Morrey space { L Φ,κ η,ξ G = f } L Φε,κξε G; f Φ,κ;η,ξ;G <, 0<ε ε

7 where Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 409 f Φ,κ;η,ξ;G = sup 0<ε ε ηε f Φε,κ ξε ;G. L Φ,κ η,ξ G is a Banach space with the norm f Φ,κ;η,ξ;G. Note that, in view of Proposition 3.2, this space is determined independent of the choice of ε. In case ξε 0, the symbol ξ may be omitted. If κx,r = r N and ξε 0, then the symbol κ will be also omitted; namely { } L Φ ηg = f 0<ε ε 0 L Φε G; f Φ;η;G := sup 0<ε ε 0 ηε f Φε;G < This space may be called a grand Musielak Orlicz space. Remark 3.3. The grand Musielak Orlicz space L Φ η G include the following spaces: generalized grand Lebesgue spaces introduced in [4]; grand Orlicz spaces introduced in [8] where Φx,t = Φt satisfying see also [8]. sup ηε 0<ε ε 0 t N ε Φt dt t < The generalized grand Musielak Orlicz Morrey space following spaces: Φ,κ L η,ξ G include also the grand Morrey spaces introduced in [2] where ξε 0; grand grand Morrey spaces introduced in [28] and generalized grand Morrey spaces introduced in [7] where ξε is an increasing positive function on 0,. 4. Boundedness of the maximal operator Hereafter, we shall always assume that Φx, t satisfies Φ5. For a nonnegative f L loc G, x G, 0 < r < d G and ε > 0, set If;x,r := fydy Bx, r and J ε f;x,r := Φ ε y,fy dy. Bx, r We show a Jensen type inequality for functions in L Φε,κσ G. Lemma 4.. There exists a constant C > 0 independent of ε and σ such that Φ ε x,if;x,r CJ ε f;x,r for all x G, 0 < r < d G, 0 < ε ε 0 and for all nonnegative f L loc G such that fy or fy = 0 for each y G and f Φε,κ σ;g with δ 0 σ σ 0. Proof. Let f be as in the statement of the lemma and let I = If;x,r and J ε = J ε f;x,r for x G, 0 < r < d G and 0 < ε ε 0. Note that f Φε,κ σ;g implies J ε 2A 3 κ σ x,r by 3...

8 40 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura By Φ2 and 3.2, Φ ε y,fy A A 2 fy, since fy or fy = 0. Hence I A A 2 J ε. Thus, if J ε, then Φ ε x,i A A 2 J ε A 2 φx,a A 2 CJ ε. Next, suppose J ε >. Since Φ ε x,t as t, there exists K ε such that Φ ε x,k ε = Φ ε x,j ε. Then K ε A 2 J ε by 3.2. With this K ε, we have fydy K ε Bx,r +A 2 fy fy ε φ y,fy dy. Kε ε φy,k ε Since κ σ x,rj ε 2A 3, K ε A 2 J ε 2A 2 A 3 κ σ x,r Cr N with a constant C > 0 independent of ε and σ. Hence, by Φ5 there is β, independent of f, x, r, ε and σ such that for all y Bx,r. Thus, we have fydy K ε Bx,r + φx,k ε βφy,k ε A 2 β Kε ε φx,k ε = K ε Bx,r +A 2 β Bx,r J ε K ε ε φx,k ε. Φ ε y,fy dy Since Kε ε φx,k ε = Kε Φ ε x,k ε = Kε J ε Φ ε x, A Kε J ε, it follows that I +A A 2 βk ε, so that by Φ2, Φ3 and Φ4 Φ ε x,i CΦ ε x,k ε CJ ε with constants C > 0 independent of f, x, r, ε and σ as required. For a locally integrable function f on G, the Hardy Littlewood maximal function Mf is defined by Mfx = sup fy dy. r>0 Bx, r The following lemma can be proved in a way similar to the proof of [25, Theorem ]: Lemma 4.2. Let p 0 > and δ 0 σ σ 0. Then there exists a constant C > 0 independent of σ for which the following holds: If f is a measurable function such that fy p 0 dy Bx,r κ σ x,r for all x G and 0 < r < d G, then [Mfy] p 0 dy C Bx,r κ σ x,r

9 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 4 for all x G and 0 < r < d G. Lemma 4.3. There is a constant C > 0 independent of ε and σ such that Mf Φε,κ σ;g C f Φε,κ σ;g for all f L Φε,κσ G whenever 0 < ε ε 0 /2 and δ 0 σ σ 0. Proof. Set p 0 = +ε 0 /2 and consider the function Φx,t = Φx,t /p 0. Then Φx,t also satisfies all the conditions Φj, j =,2,..., 5 with ε 0 replaced by ε 0 = ε 0 /2+ε 0. In fact, it trivially satisfies Φj for j =,2,4,5. Since t ε 0 t Φx,t = [t ε 0 φx,t] /p 0, condition Φ3 implies that Φx,t satisfies Φ3 with ε 0 replaced by ε 0. Let 0 < ε ε 0 /2, δ 0 σ σ 0 and f 0 and f Φε,κ σ;g. Let f = fχ {z:fz } and f 2 = f f, where χ E is the characteristic function of E. Since Φ ε x,t /A A 2 for t, we see that if t, so that Φ ε/p0 x,t = Φ ε x,t /p 0 A A 2 /p 0 Φ ε x,t Φ ε/p0 y,f y dy 2A A 2 /p 0 A 3 Bx,r κ σ x,r for every x G and 0 < r < d G. Hence f Φε/p0,κ σ;g c 0 with c 0 > 0 independent of ε and σ. Let F ε x = Φ ε x,fx. Then Φ ε/p0 x,fx = F ε x /p 0. Applying Lemma 4. to Φ ε/p0 and f /c 0, we have Φ ε x,mf x = [ Φε/p0 x,mf x ] p 0 C[MF /p 0 ε x] p 0. On the other hand, since Mf 2, we have by Φ2 and Φ3 Thus, we obtain Φ ε x,mf 2 x A A 2. ] Φ ε x,mfx C {[MF /p p0 } 0 ε x + for x G with a constant C > 0 independent of f and ε. Hence { [ ] } p0dy Φ ε y,mfy dy C MF /p 0 ε y + Bx,r for x G and 0 < r < d G. Since f Φε,κ σ;g and Φ ε y,fy = F ε y = F /p 0 ε y p 0, Lemma 4.2 implies [ ] p0dy MF /p 0 ε y C Bx,r κσ x,r with a constant C > 0 independent of x, r, ε, σ. Hence, Φ ε y,mfy dy C Bx,r κσ x,r,

10 42 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura which shows Mf Φε,κ σ;g C f Φε,κ σ;g with a constant C > 0 independent of ε and σ. From this lemma we obtain the boundedness of the maximal operator on Theorem 4.4. The maximal operator M is bounded from namely there exists a constant C > 0 such that Mf Φ,κ;η,ξ;G C f Φ,κ;η,ξ;G L Φ,κ η,ξ G. Φ,κ L η,ξ G into itself; Φ,κ for all f L η,ξ G. Corollary 4.5. If Φ p,{qj }x,t satisfies the conditions in Example 2., then the maximal operator M is bounded from L Φ p,{q j } η G into itself. 5. Sobolev type inequality Lemma 5.. [9, Lemma 5.] Let Fx,t be a positive function on G 0, satisfying the following conditions: F Fx, is continuous on 0, for each x G; F2 there exists a constant K such that Set K Fx, K for all x G; F3 t t ε Fx,t is uniformly almost increasing for ε > 0; namely there exists a constant K 2 such that t ε Fx,t K 2 s ε Fx,s for all x G whenever 0 < t < s. for x G and s > 0. Then: F x,s = sup{t > 0; Fx,t < s} F x, is non-decreasing; 2 F x,λs K 2 λ /ε F x,s for all x G, s > 0 and λ ; 3 Fx,F x,t = t for all x G and t > 0; 4 K /ε 2 { t F x,fx,t K 2/ε 2 t for all x G and t > 0; } /ε s 5 min, F x,s max{,k K 2 s /ε } for all x G and K K 2 s > 0. Remark 5.2. Fx,t = Φ ε x,t 0 < ε ε 0 satisfies F, F2 and F3 with ε =, K = A and K 2 = A 2 and Fx,t = κ σ x,t δ 0 σ σ 0 satisfies F, F2 and F3 with ε = δ δ 0, K = Q and K 2 = Q 2. Lemma 5.3. There exists a constant C > 0 such that ηε fydy CΦ ε Bx, r x,κ ξεx,r for all x G, 0 < r < d G, 0 < ε ε and nonnegative functions f on G such that f Φ,κ;η,ξ;G.

11 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 43 Proof. Let f be a nonnegative function on G such that f Φ,κ;η,ξ;G. Then we have by 3. κ ξε x,r Φ ε y,ηεfydy 2A 3 Bx, r for all x G, 0 < r < d G and 0 < ε ε. Fix ε and let f = fχ {x:ηεfx } and f 2 = f f. By Lemma 4., Φ ε x, ηεf ydy Cκ ξε x,r Bx, r for all x G, 0 < r < d G and 0 < ε ε 0 /2 with a constant C > 0 independent of x, r, ε. Since Φ ε x, ηεf 2 ydy A 2 Φ ε x, A A 2, Bx, r we have Φ ε x, ηεfydy C κ ξε x,r Bx, r with a constant C independent of x, r, ε. Hence, we find by Lemma 5. with F = Φ ε and ε = ηεfydy A 2 Φ ε x,c κ ξε x,r Bx, r as required. As a potential kernel, we consider a function Jx,r: G 0,d G [0, C A 2 2Φ ε x,κξε x,r, satisfying the following conditions: J J,r is measurable on G for each r 0,d G ; J2 Jx, is non-increasing on 0,d G for each x G; J3 d G 0 Jx,rr N dr J 0 < for every x G. Example 5.4. Let α be a measurable function on G such that 0 < α := inf x G αx supαx =: α + < N. x G Then, Jx,r = r αx N satisfies J, J2 and J3. For a nonnegative measurable function f on G, its J-potential Jf is defined by Jfx = Jx, x y fydy x G. Set G Jx,r = N N r Jx,ρρ N dρ r 0 for x G and 0 < r < d G. Then Jx,r Jx,r. Further, Jx, is non-increasing and continuous on 0,d G for each x G. Also, set Y J x,r = r N Jx,r

12 44 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura for x G and 0 < r < d G. We consider the following condition: ΦκJ there exist constants δ > 0 and A 4 such that s δ Y J x,sφ ε x,κ σ x,s A 4 t δ Y J x,tφ ε x,κ σ x,t for all x G whenever 0 < t < s < d G, 0 < ε ε 0 /2, δ 0 σ σ 0 and σ +δ δ 0 /ωε 0. Lemma 5.5. Assume ΦκJ. Then there exists a constant C > 0 such that dg r ρ N Φ ε x,κ σ x,ρ d Jx, ρ CY J x,rφ ε x,κ σ x,r for all x G, 0 < r d G /2, 0 < ε ε 0 /2 and minδ 0,δ δ 0 /ωε σ σ 0. Proof. We follow the proof of [9, Lemma 6.2], noting that the constants are independent of ε and σ. Lemma 5.6. Assume ΦκJ. Then there exists a constant C > 0 such that ηε Jx, x y fydy CY J x,rφ ε x,κ ξε x,r G\Bx,r for all x G, 0 < r d G /2, 0 < ε ε and f 0 satisfying f Φ,κ;η,ξ;G. Proof. By the integration by parts, we have Jx, x y fydy G\Bx,r Jx,d G 0 G fydy+ dg r Bx,ρ G fydy d Jx, ρ, where Jx,d G 0 = lim ρ dg 0Jx,ρ. Hence, by Lemma 5.3, we have { ηε Jx, x y fydy C Y J x,d G Φ ε x,κ ξεx,d G G\Bx,r + dg r } Bx,ρ Φ ε x,κ ξεx,ρ d Jx, ρ. Hence by ΦκJ and the previous lemma we obtain the required result. Lemma 5.7. Assume ΦκJ. Then there exists a constant C > 0 such that ηεjfx C {ηεmfxy J x,κξε x,φε x,ηεmfx } + for all x G, 0 < ε ε and f 0 satisfying f Φ,κ;η,ξ;G. Proof. Let f be a nonnegative function on G such that f Φ,κ;η,ξ;G. For 0 < r d G /2, we write Jfx = Jx, x y fydy+ Jx, x y fydy First note that = J x+j 2 x. G\Bx,r J x CY J x,rmfx

13 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 45 see, e.g., [30, p. 63, 6]. By Lemma 5.6, we have Hence ηεj 2 x CY J x,rφ ε x,κ ξεx,r. 5. ηεjfx CY J x,r { ηεmfx+φ ε x,κ ξε x,r } for x G, 0 < r d G /2 and 0 < ε ε. We consider two cases. Case : d G /2 < κ ξε x,φε x,ηεmfx. In this case, let r = d G /2. Since Φ ε x,ηεmfx Q 2 κ ξε x,d G /2 Q 2 Q 3 max,d G /2 N, it follows that ηεmfx C with a constant C > 0 independent of x and ε. Also, Φ ε x,κ ξεx,r = Φ ε x,κ ξεx,d G /2 C 2 with a constant C 2 > 0 independent of x and ε. Hence, by 5. and J3, ηεjfx C with a constant C > 0 independent of x and ε. Case 2: d G /2 κ ξε x,φε x,ηεmfx. In this case, take r = κ ξε x,φε x,ηεmfx. Then κ ξε x,r = Φ ε x,ηεmfx, so that by Lemma 5.4 Φ ε x,κ ξε x,r CηεMfx with a constant C > 0 independent of x and ε. Hence, by 5. ηεjfx CY J x,rηεmfx = CηεMfxY J x,κ ξε x,φε x,ηεmfx with a constant C > 0 independent of x and ε. The following theorem gives a Sobolev type inequality for potentials Jf of f L Φ,κ η,ξ G. Example 5.9 below shows that this theorem includes known Sobolev type inequalities as special cases. Theorem 5.8. Assume ΦκJ. Suppose a function Ψx,t: G [0, [0, satisfies Φ Φ4 with ε 0 replaced by some ε 0 in Φ3 and ΨΦ there exist a constanta and a strictly increasing continuous functionζε on [0,ε ] such that ζ0 = 0, ε ξε+δ δ 0 /ω ζε is non-decreasing with ω > such that Ψx,t Ct ω for t, and Ψ ζε x,ty J x,κξε x,φε x,t A Φ ε x,t for all x G, t and 0 < ε ε. Then there exists a constant C > 0 such that for all f L Φ,κ η,ξ G. Jf Ψ,κ;η ζ,ξ ζ ;G C f Φ,κ;η,ξ;G

14 46 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura Proof. Let f be a nonnegative function on G such that f Φ,κ;η,ξ;G. Choose ε 0,ε ] such that ζε ε 0. Let x G, 0 < r < d G and 0 < ε ε. By Lemma 5.7 and ΨΦ we have Ψ ζε x,ηεjfx C {Ψ ζε x,ηεmfxy J x,κξε x,φε x,ηεmfx } + C { Φ ε x,ηεmfx + }. Note that f Φ,κ;η,ξ;G implies Mf Φ,κ;η,ξ;G C by Theorem 4.4. Hence there is a constant C > 0 such that κ ξε x,r Φ ε y,ηεmfy dy C Bx, r for all x G, 0 < r < d G and 0 < ε ε. Therefore, there is another constant C 2 > 0 such that κ ξεx,r Bx, r Ψ ζε y,ηεjfy dy C 2 for all x G, 0 < r < d G and 0 < ε ε, so that κ ξ ζ ε x,r Ψ ε y,η ζ ε Jfy dy C 2 Bx, r for all x G, 0 < r < d G and 0 < ε ζε, which implies the required result. Example 5.9. Let Φx,t = Φ p,{qj }x,t be as in Example 2., κx,r = r νx loge+/r βx be as in Example 2.2 andjx,r = r αx N be as in Example 5.4. Note that σ 0 = 0 if ν + := sup x G νx = N and 0 < σ 0 < N ν + if ν + < N. We may take 0 < δ 0 < δ < ν and ω > p +. Then, Hence, if σ +δ δ 0 /ωε 0, then 5.2 Since σ + δ δ 0 ω ε < σ + ν νx p +ε σ + px ε. νx+σ px ε νx px. Y J x,rφ ε x,κ σx,r r αx νx+σ/px ε[ Qx,/rloge+/r βx] /px ε, where Qx,t = k j= L j t q j x, we see that condition ΦκJ holds if νx inf x G px αx > 0. Set Ψx,t = [ Φ p,{qj }x,t ] p x/px loge+t p xαxβx/νx, where /p x = /px αx/νx. We see ty J x,κ σ x,φ εx,t [ ] αx/νx+σ, t px/p σx+εαx/νx+σ Qx,tloge+t βx

15 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 47 where /p σ x = /px αx/νx+σ. Hence Ψ x, ty J x,κ σ x,φ εx,t t pxp x/p σ x+εp xαx/νx+σ [ Qx,tloge+t βx] p xαx/νx+σ Qx,t p x/px loge+t p xαxβx/νx = Φ ε x,tt σp x px/νx+σ+ε[p xαx/νx+σ+] Qx,t σp x px/[pxνx+σ] loge+t σp xαxβx/[νxνx+σ]. Here, note that ξε + ν /p + ε 0 implies νx + ξε > νx/2 if 0 < ε /2. Let 0 < ε min/2,ε. Let θ = δ δ 0 /ω. Since ξε νx+ξε ξε+θε and νx Ψ x, ty J x,κ ξε x,φ εx,t p xαx νx+ξε + x 2p px, Φ ε x,tt ξε+θεp x px/νx+2εp x/px [loge+t] m ξε+θε for t with a constant m 0. In view of 5.2, we also see that ty J x,κ ξε x,φ εx,t t px/p x [loge+t] m 2 with a constant m 2 0, which implies Ψ ζε x, ty J x,κ ξε x,φ εx,t Φ ε x,t { t px/p x [loge+t] m 2 } ζε t 2p x/pxε { t p x px/νx [loge+t] m } ξε+θε for t. Now, let ζε = aε+bξε+θε a,b > 0. If a > 2sup x G p x/px 2, then { t px/p x [loge+t] } m a 2 t 2p x/px < sup x G, t and if b > sup x G p xp x px/pxνx, then { t px/p x [loge+t] } b{ m 2 t p x px/νx [loge+t] } m <, sup x G, t so that Ψx,t satisfies condition ΨΦ with ζε = aε + bξε + θε 0 < ε min/2,ε. 6. Trudinger type inequality In this section, we consider Trudinger type inequality on Lemma 6.. Let t, t 2 > 0. If Φx,t KΦx,t 2 for some x G with K A 2, then t A 2 Kt 2. L Φ,κ η,ξ G.

16 48 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura Proof. Assume t > A 2 Kt 2. Note that t > t 2. Using Φ3, we have which contradicts the assumption. Φx,t = t φx,t > Kt 2 φx,t 2 = KΦx,t 2, In this section, we assume: Ξ ξε aε for 0 < ε ε with some a 0. Recall that ξε δ δ 0 /ωε by assumption. Let εr = loge+/r for r > 0 and let r 0,min,d G be such that εr ε for 0 < r r. Lemma 6.2. There exists a constant C such that C Φ x,κx,r Φ εr x,κ ξεrx,r CΦ x,κx,r for all x G and 0 < r r. Proof. Fix x G and set for 0 < r r. Then 6. t 0 r = Φ x,κx,r and Φx,t 0 r = κx,r = r ξεr κ ξεr x,r tr = Φ εr x,κ ξεrx,r = r ξεr Φ εr x,tr = r ξεr tr εr Φx,tr. Thus, in view of Lemma 6., it is enough to show that there exists a constant K independent of x such that 6.2 K r ξεr tr εr K for all 0 < r r. Note that 6.3 e a r aεr r ξεr r δ δ 0/ωεr e δ δ 0/ω for 0 < r r and that by κ3. If tr, then by 6. and 6.3 Q 3 κx,r Q 3 + N r Q 3 κx,r = r ξεr tr εr Φx,tr e δ δ 0/ω tr εr φx,tr e δ δ 0/ω A A 2 tr εd G, so that tr C with a constant C independent of x. Thus C εd G tr εr if tr. If tr, then by 6. and 6.3 again Q 3 + r N κx,r e a tr εr φx,tr e a A A 2 tr εd G,

17 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 49 so that tr C 2 [ + /r N ] / εdg with C 2 independent of x. Since + /r εr is bounded for r > 0, it follows that [ tr εr C εd G 2 + ] N εr/ εdg C 3 r if tr, with a constant C 3 independent of x. Therefore, 6.2 holds with K = max{e δ δ 0/ω C εd G, e a C 3 }. Lemma 6.3. There exists a constant C > 0 such that 6.4 fydy CΦ x,κx,r η loge+/r Bx, r Φ,κ for all x G, 0 < r < d G and nonnegative f L η,ξ G with f Φ,κ;η,ξ;G. Proof. Let f be a nonnegative measurable function on G such that f Φ,κ;η,ξ;G. If 0 < r r, then by Lemma 5.3 fydy CΦ εr Bx, r x,κ ξεrx,r ηεr for all x G. Hence, using the above lemma we obtain 6.4. In case r < r < d G, note that Φ εr x,κξεr x,r CΦ x,κx,r by κ3 and Lemma 5.5. Hence, by Lemma 5.3 with ε = εr, we obtain 6.4 in this case, too. In this section, we also assume that J3 Jx,r C J r ς for x G and 0 < r d G with constants 0 ς < N and C J > 0; J4 there is r 0 0,d G such that inf Jx,r 0 > 0 and x G Here note that J3 implies J3. inf x G Jx,r 0 Jx,d G >. Example 6.4. Let α and Jx,r be as in Example 5.4. Then, Jx,r satisfies J3 and J4 with ς = N α. In particular, it satisfies J4 with any r 0 0,d G. We consider the function dg ρ N Φ x,κx,ρ η loge+/ρ d Jx, ρ if s, Γx,s = /s r 0 Γx,/r 0 r 0 s if 0 s /r 0 for every x G, where r 0 is the number given in J4. Γx, is strictly increasing and continuous for each x G. Lemma 6.5. There exist positive constants C and C such that a Γx,s C s ς ηloge+s for all x G and s /r 0 with ς in condition J3 ;

18 420 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura b Γx,/r 0 C > 0 for all x G. Proof. First note from κ3 and Lemma 5.5 that 6.5 C Φ x,κx,r Cr N. By 6.5 and J3, Γx,s Cη loge+s dg /s d Jx, ρ Cη loge+s Jx,/s C s ς η loge+s for all x G and s /r 0 ; and Γx,/r 0 C dg dg ρ N d Jx, ρ C r0 N r 0 = C r0 N Jx,r0 Jx,d G C > 0, where we used J4 to obtain the inequalities in the last line. r 0 d Jx, ρ Lemma 6.6. There exists a constant C > 0 such that Jx, x y fydy CΓ x, δ G\Bx,δ for all x G, 0 < δ r 0 and nonnegative f L Φ,κ η,ξ G with f Φ,κ;η,ξ;G. Proof. By integration by parts, Lemma 6.3, 6.5, J3 and Lemma 6.5b, we havḙ Jx, x y fydy Jx, x y fydy G\Bx,δ G\Bx,δ { C d N GJx,d G Φ x,κx,d G η loge+/d G + dg δ ρ N Φ x,κx,ρ η loge+/ρ } d Jx, ρ C { Γx,/r 0 +Γx,/δ } CΓx,/δ. Lemma 6.7. Let 0 < λ < N and define I λ fx = x y λ N fydy for a nonnegative measurable function f on G and ω λ z,r = + dg r G ρ λ Φ z,κz,ρ η loge+/ρ dρ ρ for z G. Then there exists a constant C I,λ > 0 such that ω λ z,r I λ fxdx C I,λ Bz, r Bz,r G for all z G, 0 < r < d G and nonnegative f L Φ,κ η,ξ G with f Φ,κ;η,ξ;G.

19 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 42 Proof. Let z G. Let fx = 0 for x R N \G and write I λ fx = x y λ N fydy+ x y λ N fydy Bz,2r = I x+i 2 x for x G. By Fubini s theorem, I xdx = Bz,r G C Bz,2r Bz,2r Bz,2r Bz,r G By,3r 3r 0 G\Bz,2r x y λ N dx fydy x y λ N dx fydy t λ dt t fydy Cλ rλ Bz,2r fydy. Now, λ by Lemma 6.3, κ2 and Lemma 5.2, we have r fydy Cr λ Bz,2r Φ z,κz,2r η loge+/2r Bz,2r 2r C Bz,r ρ λ Φ z,κz,ρ η loge+/ρ dρ r ρ if 0 < r λ < d G /2 and, by Lemma λ 6.3 and 6.5, we have r fydy = r fydy Bz,2r Bz,d G Cd G λ Bz,d G Φ z,κz,d G η loge+/d G C Bz,r if d G /2 r < d G. Thereforḙ Bz,r GI xdx C λ Bz, r ω λ z,r for all 0 < r < d G. For I 2, first note that I 2 x = 0 if x G and r d G /2. Let 0 < r < d G /2. Since I 2 x C z y λ N fydy for x Bz,r G, G\Bz,2r by integration by parts and Lemma 6.3, we have { I 2 x C d λ G Φ z,κz,d G η loge+/d G for all x Bz,r G. Hence Thus this lemma is proved. dg + ρ λ Φ z,κz,ρ η loge+/ρ } dρ 2r ρ C ω λ z,r Bz,r G I 2 xdx C Bz,r ω λ z,r.

20 422 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura From now on, we deal with the case Γx,r satisfies the uniform log-type condition: Γ log there exists a constant c Γ > 0 such that for all x G and s. Γx,s 2 c Γ Γx,s ByΓ log, together with Lemma 6.5, we see that Γx,s satisfies the uniform doubling condition in s: Lemma 6.8. [20, Lemma 4.2] For every a >, there exists b > 0 such that Γx,as bγx,s for all x G and s > 0. Now we consider the following condition J5: J5 there exists 0 < λ < N ς such that r r N λ Jx,r is uniformly almost increasing on 0,d G for ς in condition J3. Example 6.9. Let J be as in Example 5.4. It satisfies J5 with 0 < λ < α. Theorem 6.0. Assume that Γ satisfies Γ log and J satisfies J5. For each x G, let γx = sup s>0 Γx,s. Suppose Λx,t: G [0, [0, ] satisfies the following conditions: Λ Λ, t is measurable on G for each t [0, ; Λx, is continuous on [0, for each x G; Λ2 there is a constant A such that Λx,t Λx,A s for all x G whenever 0 < t < s; Λ3 Λx,Γx,s/A 2 A 3 s for all x G and s > 0 with constants A 2, A 3 independent of x. Then, for λ given in J5, there exists a constant C > 0 such that Jfx/C γx for a.e. x G and ω λ z,r Λ x, Jfx dx Bz, r Bz,r G C Φ,κ for all z G, 0 < r < d G and nonnegative f L η,ξ G with f Φ,κ;η,ξ;G. ByΓ log andλ3, the assertion of this theorem can be considered as exponential integrability of Jf; cf. Corollary 6.2 below. Proof. Let f be a nonnegative measurable function on G such that f Φ,κ;η,ξ;G. Fix x G. For 0 < δ r 0, Lemma 6.6, J5 and J3 imply Jfx Jx, x y fydy+cγ x, Bx,δ δ = x y N λ Jx, x y x y λ N fydy+cγ x, Bx,δ δ { C δ N λ Jx,δI λ fx+γ x, } δ { C δ N ς λ I λ fx+γ x, } δ with constants C > 0 independent of x.

21 Sobolev and Trudinger type inequalities on grand Musielak Orlicz Morrey spaces 423 If I λ fx /r 0, then we take δ = r 0. Then, by Lemma 6.5b Jfx CΓ x, r0. By Lemma 6.8, there exists C > 0 independent of x such that 6.6 Jfx C x, Γ if I λ fx /r 0. 2A 3 Next, suppose /r 0 < I λ fx <. Let m = sup s /r0,x GΓx,s/s. By Γ log, m <. Define δ by δ N ς λ = rn ς λ 0 m Γx,I λfxi λ fx. Since Γx,I λ fxi λ fx m, 0 < δ r 0. Then by Lemma 6.5b δ CΓx,I λfx /N ς λ I λ fx /N ς λ CΓx,/r 0 /N ς λ I λ fx /N ς λ CI λ fx /N ς λ. Hence, using Γ log and Lemma 6.8, we obtain Γ x, Γ x,ci λ fx /N ς λ CΓx,I λ fx. δ By Lemma 6.8 again, we see that there exists a constant C2 > 0 independent of x such that 6.7 Jfx C2 x, Γ I 2C I,λ A λ fx if /r 0 < I λ fx <, 3 where C I,λ is the constant given in Lemma 6.7. Now, let C = A A 2 maxc,c 2. Then, by 6.6 and 6.7, Jfx { } max Γ x,, Γ x, I C A A 2 2A 3 2C I,λ A λ fx 3 whenever I λ fx <. Since I λ fx < for a.e. x G by Lemma 6.7, Jfx/C γx a.e. x G, and by Λ2 and Λ3, we have Λ x, Jfx { } max Λ x,γ x, /A C 2A 2, Λ x,γ x, I 3 2C I,λ A λ fx /A I λ fx 2C I,λ for a.e. x G. Thus, noting that ω λ z,r and using Lemma 6.7, we have ω λ z,r Λ x, Jfx dx Bz, r Bz,r G C 2 ω λz,r+ ω λ z,r I λ fxdx 2C I,λ Bz, r = for all z G and 0 < r < d G. Bz,r G

22 424 Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura Remark 6.. If Γx,s is bounded, that is, sup x G dg 0 ρ N Φ x,κx,ρ η loge+/ρ d Jx, ρ <, then by Lemma 6.6 we see that J f is bounded for every f if ω N ς x,r is bounded, that is, sup x G dg 0 Φ,κ L η,ξ G. In particular, ρ N ς Φ x,κx,ρ η loge+/ρ dρ ρ <, then Γx,s is bounded by J3 Φ,κ, and hence J f is bounded for every f L η,ξ G. If we further assume a continuity of the potential kernel J like condition J5 Φ,κ in our paper [20], then we can show a continuity of Jf for f L η,ξ G, as in [20, Theorem 5.3]. Applying Theorem 6.0 to special Φ, κ and J, we obtain the following corollary: Corollary 6.2. Let κx,r and αx be as in Examples 2.2 and 5.4 and let px and qx be as in Examples 2.. Set ηt = t θ for θ > 0, Φx,t = t px loge+t qx and I α fx = x y αx N fydy for a nonnegative locally integrable function f on G. Assume that Suppose that G αx νx/px = 0 for all x G. inf qx/px βx/px+θ+ > 0. x G Then for 0 < λ < α there exist constants C > 0 and C > 0 such that r νz/pz λ Iα fx px/px+θpx βx qx exp dx C Bz, r C Bz,r G for all z G, 0 < r < d G and nonnegative f 2 If L Φ,κ η,ξ G with f Φ,κ;η,ξ;G. sup qx/px βx/px+θ+ 0, x G then for 0 < λ < α there exist constants C > 0 and C > 0 such that r νz/pz λ Bz, r Bz,r G exp exp Iα fx dx C Φ,κ for all z G, 0 < r < d G and nonnegative f L η,ξ G with f Φ,κ;η,ξ;G. Proof. In the present situation, we see that { loge+s qx/px βx/px+θ+ in case, Γx,s logloge + s in case 2 for all x G and s /r 0 = 2/d G. Hence, we may take { exp t px/px+θpx qx βx in case, Λx,t = expexpt in case 2. C

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