SOBOLEV EMBEDDINGS FOR VARIABLE EXPONENT RIESZ POTENTIALS ON METRIC SPACES

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 3, 2006, 495 522 SOBOLEV EMBEDDINS FOR VARIABLE EXPONENT RIESZ POTENTIALS ON METRIC SPACES Toshihide Futamura, Yoshihiro Mizuta and Tetsu Shimomura Daido Institute of Technology, Department of Mathematics Nagoya 457-8530, Japan; futamura@daido-itacjp Hiroshima University, The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences, Higashi-Hiroshima 739-852, Japan; mizuta@mishiroshima-uacjp Hiroshima University, raduate School of Education Department of Mathematics, Higashi-Hiroshima 739-8524, Japan; tshimo@hiroshima-uacjp Abstract In the metric space setting, our aim in this paper is to deal with the boundedness of Hardy Littlewood maximal functions in generalized Lebesgue spaces L p ) when p ) satisfies a log-hölder condition As an application of the boundedness of maximal functions, we study Sobolev s embedding theorem for variable exponent Riesz potentials on metric space Introduction Let X be a metric space with a metric d Write dx, y) = x y for simplicity We denote by Bx, r) the open ball centered at x X of radius r > 0 Let µ be a Borel measure on X Assume that 0 < µb) < and there exist constants C > 0 and s such that µb ) ) r s ) µb) C r for all balls B = Bx, r) and B = Bx, r ) with x B and 0 < r r Note that µ is a doubling measure on X, that is, there exists a constant C > 0 such that 2) µ Bx, 2r) ) µ Bx, r) ) for all x X and r > 0 We define the Riesz potential of order α for a locally integrable function f on X defined by x y α fy) U α fx) = µ Bx, x y ) ) dµy) 2000 Mathematics Subject Classification: Primary 3B5, 46E35 X

496 T Futamura, Y Mizuta and T Shimomura Here α > 0 Following Orlicz [26] and Kováčik and Rákosník [9], we consider a positive continuous function p ) on X and a function f satisfying X fy) py) dµy) < In this paper we treat p ) such that p > on X and p satisfies a log-hölder condition: px) py) a log log/ x y ) ) log/ x y ) + a 2 log/ x y ) whenever x y < 4, where a and a 2 are nonnegative constants If a > 0, then we can not expect the usual boundedness of maximal functions in L p ), according to the recent works by Diening [3], [4], Pick and R užička [27] and Cruz- Uribe, Fiorenza and Neugebauer [2] Our typical example is a variable exponent p ) on X such that px) = p 0 + a log log /ϱ K x) )) log /ϱ K x) ) + a 2 log /ϱ K x) ) when ϱ K x) is small, where p 0 >, a 0, a 2 0 and ϱ K x) denotes the distance of x from a compact subset K of X Our first task is then to establish the boundedness of Hardy Littlewood maximal functions from L p ) to some Orlicz classes, as an extension of Harjulehto Hästö Pere [3] with a = 0 in metric setting and the authors [, Theorem 24] in Euclidean setting As an application of the boundedness of maximal functions, we establish Sobolev s embedding theorem for variable exponent Riesz potentials on metric space; in the case a = 0, see also Diening [4] and Kokilashvili Samko [7], [8], In the borderline case of Sobolev s theorem, we are concerned with exponential integrabilities of Trudinger type, which extend the results by Edmunds urka Opic [5], [6], Edmunds Krbec [7] and the authors [9], [22] We also discuss the pointwise continuity of Riesz potentials defined in the n-dimensional Euclidean space, as an extension of the authors [9], [23], [24] For related results, see Adams Hedberg [], Heinonen [6], Musielak [25] and R užička [28] 2 Variable exponents Throughout this paper, let C denote various constants independent of the variables in question

Sobolev embeddings for variable exponent Riesz potentials 497 Let be a bounded open set in X In this section let us assume that p ) is a positive continuous function on satisfying: p) < p ) = inf px) sup px) = p + ) < ; p2) px) py) a log log/ x y ) ) a 2 + log/ x y ) log/ x y ), whenever x y < 4, x and y, for some constants a 0 and a 2 0 Example 2 Let F be a closed subset of For a 0 and b 0, consider px) = p 0 + ω a,b ϱf x) ), where < p 0 <, ϱ F x) denotes the distance of x from F and ω a,b t) = a log log/t) ) b + log/t) log/t) for 0 < t r 0 < 4 ); set ω a,bt) = ω a,b r 0 ) when t > r 0 and ω a,b 0) = 0 Then we can find r 0 > 0 sufficiently small that p satisfies p) and p2) For a proof, we prepare the following result Lemma 22 Let ω be a nonnegative continuous function on the interval [0, r 0 ] such that i) ω0) = 0; ii) ω t) 0 for 0 < t r 0 ; iii) ω t) 0 for 0 < t r 0 Then 2) ωs + t) ωs) + ωt) for s, t 0 and s + t r 0 It is easy to find r 0 0, 4) such that ωa,b satisfies i) iii) on [0, r 0 ] Let /p x) = /px) Then, noting that 22) p y) p x) = we have the following result = px) py) px) ) py) ) px) py) px) ) 2 + {px) py) 2 px) ) 2 py) ), Lemma 23 There exists a positive constant c such that p x) p y) ω a,c x y ) whenever x and y, where a = ax) = a px) ) 2

498 T Futamura, Y Mizuta and T Shimomura 3 Boundedness of maximal functions Define the L p ) ) norm by f p ) = f p ), = inf { λ > 0 : fy) λ py) dµy) and denote by L p ) ) the space of all measurable functions f on with f p ) < By the decay condition ), we have 3) µ Bx, r) ) Cr s for all x and 0 < r < d, where d denotes the diameter of For f L p ) ), define the maximal function Mfx) = sup r>0 µ Bx, r) ) fy) dµy) Bx,r) = sup 0<r<d µ Bx, r) ) fy) dµy) Bx,r) Our first aim is to discuss the boundedness of the maximal functions Theorem 3 Let a > a when a > 0 and a = 0 when a = 0 Set Ax) = as/px) 2 If f p ), then { )) Ax) px) Mfx) log e + Mfx) dµx) When a = 0, Theorem 3 was proved by Harjulehto Hästö Pere [3], which is an extension of Diening [3] For the boundedness of maximal functions in general domains, see Cruz-Uribe, Fiorenza and Neugebauer [2] Remark 32 Set Φr, x) = { r loge + r) ) Ax) px) for r > 0 and x Then Theorem 3 assures the existence of C > 0 such that Φ Mfx)/C, x ) dµx) whenever f p ) As in Edmunds and Rákosník [8], we define { f Φ = f Φ, = inf λ > 0 : then it follows that Mf Φ f p ) Φ fx) /λ, x ) dµx) ; for f L p ) ) Theorem 3 is proved along the same lines as in the authors [, Theorem 24], but we give a proof of Theorem 3 for the readers convenience To complete the proof, we prepare the following lemma

Sobolev embeddings for variable exponent Riesz potentials 499 Lemma 33 Let f be a nonnegative measurable function on with f p ) Then Mfx) { Mgx) /px) log e + Mgx) )) A x) +, where gy) = fy) py) and A x) = a s/px) 2 Proof Let f be a nonnegative measurable function on with f p ) First note that 32) fy) py) dµy) Then, if r r 0, then 33) µ Bx, r) ) fy) dµy) Bx,r) by our assumption For 0 < k and r > 0, we have by Lemma 23 µ Bx, r) ) fy) dµy) Bx,r) { k µ Bx, r) ) Bx,r) k{/k) p x)+ωr) + F, /k) p y) dµy) + µ Bx, r) ) {+fy) py) dµy) Bx,r) µ Bx, r) ) fy) py) dµy) Bx,r) where F = µ Bx, r) )) Bx,r) fy)py) dµy) and ωr) = ω a,c r) as in Example 2 Here, considering k = F /{p x)+ωr) = F /p x)+βx) with βx) = ωr)/ { p x) p x) + ωr) ) when F, we have µ Bx, r) ) fy) dµy) 2F /px) F ωr)/p x) 2 ; Bx,r) if F <, then we can take k = to obtain µ Bx, r) ) fy) dµy) 2 Bx,r)

500 T Futamura, Y Mizuta and T Shimomura Hence it follows that 34) µ Bx, r) ) Bx,r) If r F, then we see from 34) that fy) dµy) 2 { F /px) F ωr)/p x) 2 + µ Bx, r) ) fy) dµy) { F /px) loge + F ) ) A x) + Bx,r) If r 0 > r > F, then we have by the lower bound 3) F /px)+ωr)/p x) 2 µ Bx, r) ) /px) r sωr)/p x) 2 In view of 32), we find { Bx,r) fy) py) dµy) /px)+ωr)/p x)2 F /px)+ωr)/p x) 2 µ Bx, r) ) { /px) ) A x) log/r) µ Bx, r) ) { /px) ) A x) log/r) µ Bx, r) ) { /px) log F ) A x) = CF /px) log F ) A x) Bx,r) Bx,r) Bx,r) /px)+ωr)/p x) 2 fy) py) dµy) /px) fy) py) dµy) fy) py) dµy) /px) Now we have established 35) µ Bx, r) ) fy) dµy) { F /px) loge + F ) ) A x) + Bx,r) for all r > 0 and x, which completes the proof Proof of Theorem 3 Let p x) = px)/p for < p < p ) Then Lemma 33 yields {Mfx) p x) { Mgx) log e + Mgx) )) ã s/p x) +

Sobolev embeddings for variable exponent Riesz potentials 50 for x, where gy) = fy) p y) and ã = a /p Letting a > a when a > 0 and a = 0 when a = 0, we set Ax) = as/px) 2 Then we can choose p so that a p a and which yields {Mfx) px) { Mgx) log e + Mgx) )) Ax)px)/p + p, { Mfx) log e + Mfx) )) Ax) px) {Mgx) + p Now Theorem 3 follows from the boundedness of maximal functions in L p the case of constant exponent) Remark 34 Let p ) be a positive continuous function on such that px) p + ) < Then, as in Harjulehto Hästö Pere [3], we can prove the following weak type result for maximal functions: E f t) fy) t py) dµy) whenever t > 0 and f L p ) ), where E f t) = {x : Mfx) t; see also Cruz-Uribe, Fiorenza and Neugebauer [2, Theorem 8] To prove this, we may assume that t = We have for k > µ Bx, r) ) fy) dµy) Bx,r) { k µ Bx, r) ) /k) p y) dµy) + Bx,r) k { /k) p +) + F, in µ Bx, r) ) fy) py) dµy) Bx,r) where F = µ Bx, r) )) Bx,r) fy) py) dµy) Here, considering k = F /p +) when F <, we find 2F /p +, so that 2) p+ M f p ) ) x) for x E f ), which proves the required assertion Remark 35 For 0 < r < 2, let = {x = x, x 2 ) : 0 < x <, < x 2 <

502 T Futamura, Y Mizuta and T Shimomura Define px, x 2 ) = { p0 + a log log/x2 ) )) / log/x 2 ) when 0 < x 2 r 0, p 0 when x 2 0; and px, x 2 ) = px, r 0 ) when x 2 > r 0 Setting we consider r) = {x = x, x 2 ) : 0 < x < r, r < x 2 < 0, f r y) = χ r) y) and set g r = f r / f r p ),, where χ E denotes the characteristic function of a measurable set E Then we claim for 0 < r < 2 r 0 : i) f r p ), = r 2/p 0 ; ii) Mg r )x) C r 2/p 0 for 0 < x < r and r < x 2 < 2r ; { iii) Mgr )x) log e + Mg r )x) )) Ax) px) ) 2a a)/p 0 dx C2 log/r) for Ax) = 2a/px) 2, so that the conclusion of Theorem 3 does not hold for 0 < a < a by 4 Sobolev s inequality For 0 < α < s, we consider the Riesz potential U α f of f L p ) ) defined U α fx) = x y α fy) µ Bx, x y ) ) dµy); recall that s is the decay constant in ) In this section, suppose p ) satisfies p), p2) and p3) p + ) < s/α Let /p x) = /px) α/s In what follows we establish Sobolev s inequality for α-potentials on, as an extension of the case of constant exponent which was discussed by Haj lasz and Koskela [2] and Heinonen [6]; for the Euclidean case, see the books by Adams and Hedberg [] and the second author [2] In the next two sections, we are concerned with the exponential integrability, which extends the results by Edmunds, urka and Opic [5], [6], and the authors [22] Now we show our result, which gives an extension of Diening [4]; for further investigations we also refer the reader to the results by Kokilashvili Samko [7], [8], where the index α is a variable exponent

Sobolev embeddings for variable exponent Riesz potentials 503 Theorem 4 Letting a > a when a > 0 and a = 0 when a = 0, we set Ax) = as/px) 2 Suppose p + ) < s/α Let f be a nonnegative measurable function on with f p ) Then { Uα fx) log e + U α fx) )) Ax) p x) dµx) In spite of the fact that the proof of Theorem 4 is quite similar to that of Theorem 34 in [], we give a proof of Theorem 4 for the readers convenience For this purpose, we prepare the following two lemmas Lemma 42 Let f be a nonnegative measurable function on with f p ) Then \Bx,δ) x y α fy) µ ) dµy) Cδ s/p x) log/δ) A x) Bx, x y ) for x and 0 < δ < 4, where A x) = a s/px) 2 as before Proof Let f be a nonnegative measurable function on with f p ) Then, for k >, we have x y α fy) µ Bx, x y ) ) dµy) { x y α k \Bx,δ) kµ Bx, x y ) ) { x y α k kµ Bx, x y ) ) \Bx,δ) \Bx,δ) Consider the set ) p y) ) p y) dµy) + dµy) + \Bx,δ) E = { y : x y α / kµ Bx, x y ) )) > Note here that by 2), 3) and Lemma 23 fy) py) dµy)

504 T Futamura, Y Mizuta and T Shimomura E\Bx,δ) j x y α kµ Bx, x y ) ) E\Bx,δ) ) p y) x y α kµ Bx, x y ) ) Bx,2 j δ)\bx,2 j δ) dµy) ) p x)+ω x y ) 2 j δ) α kµ Bx, 2 j δ) ) dµy) ) p x)+ω2 j δ) dµy) k p x) ωδ) j k p x) ωδ) j 2 j δ) αp x)+ω2 j δ)) µbx, 2 j δ) ) p x)+ω2 j δ))+ 2 j δ) α s)p x)+ω2 j δ))+s k p x) ωδ) δ t α s)p x)+ωt))+s t dt k p x) ωδ) δ α s)p x)+ωδ))+s k p x) ωδ) δ p x)α s/px)) log/δ) ) s α)a /px) ) 2 = Ck p x) ωδ) δ p x)s/p x) log/δ) ) s α)a /px) ) 2, where ωr) = ω a,c r) Now, letting k = δ s/p x) log/δ) ) A x), we see that Further we find E\Bx,δ) \E Consequently it follows that \Bx,δ) x y α kµ Bx, x y ) ) x y α kµ Bx, x y ) ) for x and 0 < δ <, as required 4 ) p y) ) p y) dµy) dµy) x y α fy) µ ) dµy) x) Cδ s/p log/δ) ) A x) Bx, x y )

Sobolev embeddings for variable exponent Riesz potentials 505 Lemma 43 Let f be a nonnegative measurable function on with f p ) Then U α fx) { Mfx) px)/p x) log e + Mfx) )) a α/px) + Proof To give the required estimate, we borrow the idea of Hedberg [5] In fact, for x and 0 < δ < we have by Lemma 4 4 U α fx) = Bx,δ) x y α fy) µ Bx, x y ) ) dµy) + \Bx,δ) δ α Mfx) + Cδ s/p x) log/δ) ) A x) x y α fy) µ Bx, x y ) ) dµy) Considering δ = Mfx) px)/s log e + Mfx) )) a /px) enough, we obtain the required inequality when M fx) is large Proof of Theorem 4 Let a > a > 0 or a = a = 0, and set Ax) = as/px) 2 Then Lemma 43 yields {U α fx) log e+u α fx) )) Ax) p x) [ {Mfx) log e+mfx) )) Ax) px)+ ], which together with Theorem 3 completes the proof Remark 44 In Remark 35, we see that U α g r x) C 3 r 2/p x) log/r) ) A x) for 0 < x < r and r < x 2 < 2r, where A x) = 2a /px) 2 Hence we have { Uα g r x) log e + U α g r x) )) Ax) p x) ) 2a a)/p 0, dx C4 log/r) where Ax) = 2a/px) 2 This implies that the conclusion of Theorem 4 does not hold when a < a Remark 45 By Theorem 4 we see that U α f L p ) ) whenever f L p ) ) Then, as was pointed out by Lerner [20], the inequality U α fx) px) dµx) fy) py) dµy) holds whenever f L p ) ) if and only if p is constant, under the additional assumption that µe) = sup { µk) K E, K : compact

506 T Futamura, Y Mizuta and T Shimomura for every measurable set E X In fact, the if part is clear We here assume that p is not constant Then we can find numbers p and p 2 such that p < p 2 <, and both E = {x : px) p and E 2 = {x : px) p 2 have positive µ measure Further by our assumption, there exist compact sets K i, i =, 2, such that K i E i If f = kχ K with k >, then so that On the other hand, U α fx) CkµK ) for x K 2, U α fx) px) dµx) Ck p 2 µk 2 ) fx) px) dµx) k p µk ) If the inequality holds, then we should have k p 2 k p, which gives a contradiction by letting k In the same manner, we see that the inequality {Mfx) px) dµx) fy) py) dµy) holds whenever f L p ) ) if and only if p is constant Remark 46 Let ωr) be a continuous function on 0, ) such that ωr) = a log log/r) ) log/r) + a 2 log/r) for 0 < r r 0 < 4, with a > 0 and a 2 > 0; set ωr) = ωr 0 ) for r > r 0 Consider a variable exponent p ) on the unit ball B in R n defined by px) = p 0 + ω ϱx) ), where < p 0 < n/α and ϱx) = x Take r 0 so small that px) < n/α for all x B In view of Theorem 4, we see that if a > a and Ax) = an/px) 2, then B { U α fx) log e + U α fx) )) Ax) p x) dx whenever f is a nonnegative measurable function on B with f p )

Sobolev embeddings for variable exponent Riesz potentials 507 5 Exponential integrability For fixed x 0, let us assume that an exponent px) is a continuous function on satisfying 5) px) > p 0 when x x 0 and 52) { px) p 0 + a log log/ x 0 x ) ) log/ x 0 x ) b log/ x 0 x ) for x Bx 0, r 0 ), where 0 < r 0 < 4, p 0 = s/α, 0 < a s α)/α 2 and b > 0 Our aim in this section is to give an exponential integrability of Trudinger type Before doing so, we prepare several lemmas In view of 22) and 52), we have the following result Lemma 5 There exist C > 0 and 0 < r 0 < 4 such that 53) p y) p 0 ω x 0 y ) for all y B 0 = Bx 0, r 0 ), where p 0 = p 0/p 0 ) = s/s α) and ω is a nonnegative nondecreasing function on 0, ) such that ωr) = ) aα2 log log/r) s α) 2 log/r) C log/r) when 0 < r r 0 ; set ωr) = ωr 0 ) when r > r 0 as before Lemma 52 If 0 < a < s α)/α 2, then B 0 \Bx,δ) x y α µ Bx, x y ) ) and if a = s α)/α 2, then B 0 \Bx,δ) ) p y) x y α µ Bx, x y ) ) dµy) log/δ) ) aα 2 /s α) ) p y) for x B 0 and 0 < δ < δ 0, where 0 < δ 0 < 4 dµy) log log/δ) )

508 T Futamura, Y Mizuta and T Shimomura Proof First consider the case 0 < a < s α)/α 2 Let E = { y B 0 : x y α /µ Bx, x y ) ) > for fixed x B 0 Let j 0 be the smallest integer such that 2 j 0 δ > 2r 0 Since x y 3 x 0 y for y B 0 \ Bx 0, x 0 x /2), we have by 2), 3) and 53) I = E\{Bx 0, x 0 x /2) Bx,δ) j 0 j= j 0 j= j 0 j= j 0 j= 3r0 δ Bx,2 j δ)\bx,2 j δ) x y α ) p y) µ Bx, x y ) ) dµy) 2 j δ) α ) p µ Bx, 2 j δ) ) 0 ω2j δ/3) dµy) 2 j δ) αp 0 ω2j δ/3)) µ Bx, 2 j δ) )) p 0 ω2j δ/3))+ 2 j δ) α s)p 0 ω2j δ/3))+s log /2 j δ) ) aα 2 /s α) log/t) ) aα 2 /s α) t dt log/δ) ) aα 2 /s α) for 0 < δ < δ 0, since aα 2 /s α) > 0 Next we give an estimate for I 2 = Bx 0, x x 0 /2)\Bx,δ) x y α µ Bx, x y ) ) ) p y) dµy) We may assume that 2 x x 0 > δ Then we see from Lemma 5 that if y Bx 0, x x 0 /2), then p y) p 0 + η, where η = C/ log/ x 0 x ) Hence we obtain by ) and 3) I 2 Bx 0, x x 0 /2) Bx 0, x x 0 /2) x x 0 α ) p y) µ Bx, x 0 x /2) ) dµy) { x x 0 α ) p µ Bx, x 0 x /2) ) 0 +η + dµy) µ Bx 0, x 0 x /2) ){ x x 0 α µ Bx, x 0 x /2) ) ) p 0 +η + { x x 0 αp 0 +η) µ Bx 0, x 0 x /2) ) p 0 +η)+ +

Sobolev embeddings for variable exponent Riesz potentials 509 Thus it follows that x y α µ Bx, x y ) ) B 0 \Bx,δ) ) p y) for 0 < δ <, which proves the first case 4 The second case a = s α)/α 2 is similarly proved dµy) log/δ) ) aα 2 /s α) Lemma 53 Let f be a nonnegative measurable function on B 0 with f p ) If β > β = aα 2 /s α) ) /p 0 = s α aα2 )/s > 0, then 54) B 0 \Bx,δ) x y α fy) µ Bx, x y ) ) dµy) log/δ) ) β for x B 0 and 0 < δ < δ 0, where 0 < δ 0 < 4 Proof Take p such that < p < p 0 and β > γ = aα 2 /s α) ) /p > β We may assume that p y) > p for y B 0 Let f be a nonnegative measurable function on B 0 with f p ) For k > and 0 < δ <, we have by Lemma 52 4 x y α fy) B 0 \Bx,δ) µ Bx, x y ) ) dµy) { x y α ) p y) k kµ Bx, x y ) ) dµy) + fy) py) dµy) B 0 \Bx,δ) k { Ck p log/δ) ) aα 2 /s α) + B 0 \Bx,δ) ) Now, considering k such that k p aα log/δ) 2 /s α) =, we have x y α fy) µ Bx, x y ) ) dµy) log/δ) ) γ ) β, log/δ) as required B 0 \Bx,δ) In what follows we show that 54) remains true with β replaced by β = s α aα 2 )/s Lemma 54 Let f be a nonnegative measurable function on B 0 with f p ) If β = s α aα 2 )/s > 0, then B 0 \Bx,δ) x y α fy) µ Bx, x y ) ) dµy) log/δ) ) β for x B 0 and 0 < δ < δ 0, where 0 < δ 0 < 4

50 T Futamura, Y Mizuta and T Shimomura Proof Let f be a nonnegative measurable function on B 0 with f p ) Let η = log/δ) ) log log/δ) for small δ, say 0 < δ < δ0 < Then note from 4 Lemma 53 that x y α fy) 55) µ Bx, x y ) ) dµy) log/η) ) β ) β log/δ) B 0 \Bx,η) Letting k = log/δ) ) β and Bx) = Bx0, x 0 x /2), we find k ωη/3), so that we obtain from Lemmas 5 and 52 that x y α ) p y) kµ Bx, x y ) ) dµy) Bx,η)\{Bx,δ) Bx) Bx,η)\{Bx,δ) Bx) k p 0 B 0 \Bx,δ) { x y α ) p kµ Bx, x y ) ) 0 ω x y /3) + dµy) x y α ) p µ Bx, x y ) ) 0 ω x y /3) dµy) + C k p 0 log/δ) ) aα 2 /s α) + C Hence it follows from the proof of Lemma 53 that x y α fy) 56) µ Bx, x y ) ) dµy) log/δ))β Bx,η)\{Bx,δ) Bx) Next we show that 57) Bx)\Bx,δ) x y α fy) µ Bx, x y ) ) dµy) log/δ) ) β Since a > 0, we have by the latter half of the proof of Lemma 52 x y α fy) Bx)\Bx,δ) µ Bx, x y ) ) dµy) x y α ) p y) µ Bx, x y ) ) dµy) + fy) py) dµy) Bx)\Bx,δ) Bx)\Bx,δ) Now we claim from 55), 56) and 57) that x y α fy) µ Bx, x y ) ) dµy) log/δ) ) β B 0 \Bx,δ) Thus the proof is completed

Sobolev embeddings for variable exponent Riesz potentials 5 Lemma 55 Let f be a nonnegative measurable function on B 0 with f p ) If β = s α aα 2 )/s > 0, then U α fx) log e + Mfx) )) β for x B 0 Proof We see from Lemma 54 that x y α fy) U α fx) = µ Bx, x y ) ) dµy) + Here, letting Bx,δ) δ α Mfx) + C log/δ) ) β when Mfx) is large enough, we have as required It follows from Lemma 55 that B 0 \Bx,δ) δ = Mfx) ) /α log e + Mfx) )) β/α U α fx) log e + Mfx) )) β, exp C U α fx) ) /β) e + Mfx) x y α fy) µ Bx, x y ) ) dµy) whenever f is a nonnegative measurable function on B 0 with f p ) By the classical fact that Mf L p B 0 ), we establish the following exponential inequality of Trudinger type Theorem 56 Let 0 < a < s α)/α 2 If β = s α aα 2 )/s, then there exist positive constants c and c 2 such that B 0 exp c Uα fx) ) /β) dµx) c2 for all nonnegative measurable functions f on B 0 with f p ) Remark 57 Let B 0 be a ball in the n-dimensional space R n If f is a nonnegative measurable function on B 0 such that fy) py) dy <, B 0 then we claim by applying an idea by Hästö [4] that 58) fy) n/α log e + fy) )) aα dy < B 0

52 T Futamura, Y Mizuta and T Shimomura In fact, if y E = { x B 0 : fx) x 0 x α log e + x 0 x ) ), then so that fy) py) Cfy) n/α log e + fy) )) aα E fy) n/α log e + fy) )) aα dy <, which proves 58), since 0 < a < n α)/α 2 With the aid of Edmunds Krbec [7] and the authors [22] we also obtain Theorem 56 in the Euclidean case Finally we are concerned with the case a = s α)/α 2 Lemma 58 Let f be a nonnegative measurable function on B 0 with f p ) If a = s α)/α 2, then U α fx) log e + log e + Mfx) ))) p 0 for x B 0 Proof Let f be a nonnegative measurable function on B 0 with f p ) For k > and 0 < δ < δ 0 < 4, we have by applications of the arguments in the proof of Lemma 54 x y α fy) µ Bx, x y ) ) dµy) log log/δ) )) /p 0 B 0 \Bx,δ) Consequently it follows that x y α fy) U α fx) = µ Bx, x y ) ) dµy) + Here let Bx,δ) δ α Mfx) + C log log/δ) )) /p 0 B 0 \Bx,δ) δ = Mfx) /α log e + log e + Mfx) ))) /{αp 0 when Mfx) is large enough Then we have as required U α fx) log e + log e + Mfx) ))) /p 0, x y α fy) µ Bx, x y ) ) dµy) By Lemma 58 and the fact that Mf L p 0 B 0 ), we establish the following double exponential inequality for f L p ) B 0 ) Theorem 59 If a = s α)/α 2, then there exist positive constants c and c 2 such that B 0 exp exp c Uα fx) ) s/s α))) dµx) c2 for all nonnegative measurable functions f on B 0 with f p ) Remark 50 In case f belongs to more general variable exponent Lebesgue spaces, we will be expected to discuss the corresponding exponential integrability as in Edmunds, urka and Opic [5], [6] But we do not go into details any more

Sobolev embeddings for variable exponent Riesz potentials 53 6 Exponential integrability, II In this section, let B = B0, ) be the unit ball in R n We consider a variable exponent p ) on B which is a continuous function on B satisfying 6) px) > p 0 on B and 62) { px) p 0 + a log log /ϱx) )) log /ϱx) ) b log /ϱx) ) when ϱx) < r 0, where 0 < r 0 < 4, a > 0, b > 0, p 0 = n/α and ϱx) = x denotes the distance of x from the boundary B For f L p ) B), the Riesz potential of order α, 0 < α < n, is defined by U α fx) = B x y α n fy) dy If a > n α)/α 2, then we see from Theorem 77 below that U α fx) U α fz) log/ x z ) ) n α aα 2 )/n whenever x, z B and x z < ; for this fact, see also [0, Theorem 43] 2 In what follows, when 0 < a n α)/α 2, we discuss exponential inequalities of U α f as in Section 5 As in Lemma 5, we have the following result Lemma 6 There exist positive constants t 0 < 4 p x) p 0 ω ϱx) ) and C such that for x B, where ωt) = aα 2 /n α) 2) log log/t) ) / log/t) C/ log/t) for 0 < t t 0 and ωt) = ωt 0 ) for t > t 0 Lemma 62 If 0 < a < n α)/α 2, then I B\Bx,r) x y α n)p y) dy log/r) ) γ for all x B and 0 < r < 2, where γ = aα2 /n α)

54 T Futamura, Y Mizuta and T Shimomura Proof First consider the case 2 ϱx) r < 2 Letting E = { y B\Bx, r) : ϱx) ϱy) > 2r, we find by polar coordinates, I x y α n)p y) dy E t ϱx) α n)p 0 ωt))+n dt {t: t ϱx) >2r {t:t>2r /2 2r t n α)ωt) dt log/t) ) aα 2 /n α) t dt + C log/r) ) γ Letting E 2 = { y B \ Bx, r) : ϱx) ϱy) 2r, we find by polar coordinates, I 2 x y α n)p y) dy E 2 4r r α n)p 0 +n dt r dt Hence it follows that {t: t ϱx) 2r B\Bx,r) x y α n)p y) dy log/r) ) γ when 2 ϱx) r < In particular, we obtain 2 63) x y α n)p y) dy log /ϱx) )) γ B\Bx,ϱx)/2) Next consider the case 0 < r < 2 ϱx) Let E 3 = B x, 2 ϱx)) \ Bx, r) In view of Lemma 6, we find p y) p 0 ω ϱx) ) + C/ log /ϱx) ) p 0 ω x y ) for y E 3, where ω t) = ωt) C/ log/t) for small t > 0 Hence, we see that I 3 x y α n)p y) dy E 3 x y α n){p 0 ω x y ) dy E 3 ϱx)/2 r log/t) ) aα 2 /n α) t dt log/r) ) γ In view of 63), we establish x y α n)p y) dy log/r) ) γ B\Bx,r) when 0 < r < ϱx) Thus the required result is proved 2 0

Sobolev embeddings for variable exponent Riesz potentials 55 As in the proof of Lemma 54, we can prove the following result Lemma 63 Let f be a nonnegative function on B such that f p ) If 0 < a < n α)/α 2 and β = γ/p 0 = n α aα2 )/n, then x y α n fy) dy log/r) ) β whenever 0 < r < 2 B\Bx,r) We see from Lemma 63 that U α fx) = x y α n fy) dy + Here, letting Bx,δ) δ α Mfx) + C log/δ) ) β when Mfx) is large enough, we have so that B\Bx,δ) δ = Mfx) ) /α log e + Mfx) )) β/α U α fx) log e + Mfx) )) β, exp C U α fx) ) /β) e + Mfx) x y α n fy) dy whenever f is a nonnegative measurable function on B 0 with f p ) By the classical fact that Mf L p 0 B), we establish the following exponential inequality of Trudinger type Theorem 64 Let 0 < a < n α)/α 2 If β = n α aα 2 )/n, then there exist positive constants c and c 2 such that exp c Uα fx) ) /β) dx c2 B for all nonnegative measurable functions f on B with f p ) Finally we are concerned with the case a = n α)/α 2 The following can be proved in the same way as Lemma 54 Lemma 65 Let f be a nonnegative function on B such that f p ) If a = n α)/α 2, then x y α n)p y) fy) dy log log/r) )) n α)/n B\Bx,r) for small r > 0

56 T Futamura, Y Mizuta and T Shimomura As in the proof of Theorem 64, we establish the following double exponential inequality for f L p ) B) Theorem 66 If a = n α)/α 2, then there exist positive constants c and c 2 such that exp exp c Uα fx) ) n/n α))) dx c2 B for all nonnegative measurable functions f on B with f p ) 7 Continuity Let be a bounded open set in the n-dimensional space R n, and fix x 0 In this section, we deduce the continuity at x 0 of Riesz potentials U α f when f L p ) ) with p ) satisfying { n px) α + a log log/ x 0 x ) ) b log/ x 0 x ) log/ x 0 x ), where a > n α)/α 2, b > 0 and x runs over the small ball B 0 = Bx 0, r 0 ) Consider a positive continuous nonincreasing function ϕ on the interval 0, ) such that ϕ) log/t) ) ε 0ϕt) is nondecreasing on 0, r0 ] for some ε 0 > 0 and r 0 > 0; set ϕr) = ϕr 0 ) for r > r 0 We see from condition ϕ) that ϕ satisfies the doubling condition Set r n α)/n Φr) = ϕt) α2 /n α) t dt) 0 Our final goal is to establish the following result, which deals with the continuity of α-potentials in R n Theorem 7 Let p ) satisfy px) = n α + log ϕ x 0 x ) log/ x 0 x ) for x B 0 = Bx 0, r 0 ) and f L p ) B 0 ) If Φ) <, then U α f is continuous at x 0 ; in this case, whenever x, z B x 0, 2 r 0) U α fx) U α fz) Φ x z ) Remark 72 Let ϕr) = loge + /r) ) a Then Φ) < if and only if a > n α)/α 2, so that Theorem 7 gives an extension of the authors [9] For a proof of Theorem 7, we may assume that x 0 = 0 without loss of generality Before the proof we prepare the following two results

Sobolev embeddings for variable exponent Riesz potentials 57 Lemma 73 For x B 0, 2 r 0) and small δ > 0, Bx,δ) x y p y)α n) dy δ 0 ϕr) α2 /n α) r dr Proof First note from 22) and ϕ) that p y) p 0 ω y ) for y B 0, where p 0 = n/n α) and ωr) = α 2 /n α) 2) log ϕr) ) / log/r) C/ log/r) for 0 < r r 0 ; set ωr) = ωr 0 ) for r > r 0 If 0 < δ 2 x, then we have x y p y)α n) dy x y p y)α n) dy Bx,δ) j Bx,2 j+ δ)\bx,2 j δ) j 2 j δ) α n)p 0 ω2 j δ)) σ n 2 j+ δ) n j ϕ2 j δ) α2 /n α) δ 0 ϕr) α2 /n α) r dr, where σ n denotes the volume of the unit ball Similarly, if 2 x < δ < 3 r 0, we have x y p y)α n) dy y p y)α n) dy Bx,δ)\Bx, x /2) B0,3δ) 3δ Therefore it follows from the doubling property that Bx,δ) x y p y)α n) dy δ when 0 < δ < 3 r 0 Now the proof is completed 0 0 ϕr) α2 /n α) r dr ϕr) α2 /n α) r dr Lemma 74 Let f be a nonnegative measurable function on B 0 with f p ) Then x y α n fy) dy δ ϕδ) α2 /n B 0 \{B0,δ) Bx,δ) for x B 0, 2 r 0) and small δ > 0

58 T Futamura, Y Mizuta and T Shimomura Proof Let f be a nonnegative measurable function on B 0 with f p ) For k > we have x y α n fy) dy B 0 \{B0,δ) Bx,δ) { k x y α n /k) p y) dy B 0 \{Bx,δ) B0,δ) + B 0 \{Bx,δ) B0,δ) fy) py) dy { k x y α n /k) p y) dy + B 0 \{Bx,δ) B0,δ) In view of the assumption of ϕ, we obtain B 0 \{Bx,δ) B0,δ) x y α n /k) p y) dy { x y α n /k) p 0 ωδ) dy + B 0 \{Bx,δ) B0,δ) {k p 0 +ωδ) t α n )p 0 ωδ))+n t dt + k p 0 +ωδ) δ α n )p 0 ωδ))+n δ Considering k such that k p 0 +ωδ) δ α n )p 0 ωδ))+n =, we see that as required B 0 \{B0,δ) Bx,δ) x y α n fy) dy δ ϕδ) α2 /n, Proof of Theorem 7 Let f be a nonnegative measurable function on B 0 with f p ) For 0 < k <, we have by Lemma 73 Bx,δ) x y α n fy)dy k Bx,δ) k {k n/n α) { x y α n /k) p y) + fy) py) dy Bx,δ) k { Ck n/n α) Φδ) n/n α) + x y α n)p y) dy +

Sobolev embeddings for variable exponent Riesz potentials 59 whenever x B 0, 2 r 0) and 0 < δ < 2 r 0 Now, considering k = Φδ), we find 7) x y α n fy) dy Φδ) B 0 Bx,δ) Hence, if x, z B 0, 2 r 0) and x z < 4 r 0, then we have 72) x y α n fy) dy Φ x z ) Bx,2 x z ) On the other hand we have x y α n z y α n fy) dy B 0 \Bx,2 x z ) x z { = C x z + B 0 \Bx,2 x z ) x y α n fy) dy B 0 \{Bx,2 x z ) B0,2 x z ) {B 0 B0,2 x z )\Bx,2 x z ) x y α n fy) dy x y α n fy) dy It follows from Lemma 74 that x y α n fy) dy x z ϕ x z ) α2 /n B 0 \{Bx,2 x z ) B0,2 x z ) Moreover we see from 7) that {B 0 B0,2 x z )\Bx,2 x z ) x z B 0 B0,2 x z ) x y α n fy) dy y α n fy) dy x z Φ x z ) Since ϕr) α2 /n Φr) by the doubling property of ϕ, we obtain x y α n z y α n fy) dy Φ x z ) B 0 \Bx,2 x z ) Further we obtain by 72) Now, we establish as required Bx,2 x z ) z y α n fy) dy Φ x z ) U α fx) U α fz) Φ x z ),

520 T Futamura, Y Mizuta and T Shimomura Remark 75 If Φ) =, then we can find f L p ) B 0 ) such that U α f0) =, which means that U α f is not continuous at 0 For this purpose set and ψr) = r ϕt) α2 /n α) t dt fy) = y n α)/py) ) ψ y ) Take r 0 so small that ψr) > e when 0 < r < r 0 Note that r n α)p/p )+n = r n αp)/p ) = ϕr) α/p ) for r = y and p = py) By ϕ) we have so that ϕr) α/p ) Cϕr) α2 /n α), U α f0) = y α n n α)/py) ) ψ y ) dy B 0 C r0 0 ϕt) α2 /n α) ψt) dt/t = since ψ0) = by our assumption On the other hand, taking a number δ such that < δ < n/α and noting by ϕ) that ϕr) α/p ) ϕr) α2 /n α), we have fy) py) dy = y n α)py)/py) ) ψ y ) py) dy B 0 B 0 r0 0 ϕt) α2 /n α) ψt) δ dt/t < since < δ < n/α py) and ψ0) =, as required Finally, we consider a variable exponent p ) on the unit ball B such that 73) px) = p 0 + log ϕ ϱx) ) log e/ϱx) ) for x B, where p 0 = n/α; assume as above that px) > p 0 on B Theorem 76 If Φ) < and f L p ) B), then whenever x, z B U α fx) U α fz) Φ x z )

Sobolev embeddings for variable exponent Riesz potentials 52 For a proof of Theorem 76, it suffices to show that Bx,δ) x y p y)α n) dy δ 0 ϕr) α2 /n α) r dr for x B and small δ > 0, as in Lemma 73 We obtain, in fact, this inequality in the same way as in Lemmas 62 and 73 Remark 77 We do not know the best condition which assures the continuity of Riesz potentials in the metric space setting References [] Adams, D R, and L I Hedberg: Function Spaces and Potential Theory - Springer- Verlag, Berlin, 996 [2] Cruz-Uribe, D, A Fiorenza, and C J Neugebauer: The maximal function on variable L p spaces - Ann Acad Sci Fenn Ser Math 28, 2003, 223 238; 29, 2004, 247 249 [3] Diening, L: Maximal functions on generalized L p ) spaces - Math Inequal Appl 72), 2004, 245 253 [4] Diening, L: Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p ) and W k,p ) - Math Nachr 263), 2004, 3 43 [5] Edmunds, D E, P urka, and B Opic: Double exponential integrability, Bessel potentials and embedding theorems - Studia Math 5, 995, 5 8 [6] Edmunds, D E, P urka, and B Opic: Sharpness of embeddings in logarithmic Bessel-potential spaces - Proc Roy Soc Edinburgh 26, 996, 995 009 [7] Edmunds, D E, and M Krbec: Two limiting cases of Sobolev imbeddings - Houston J Math 2, 995, 9 28 [8] Edmunds, D E, and J Rákosník: Sobolev embedding with variable exponent, II - Math Nachr 246 247, 2002, 53 67 [9] Futamura, T, and Y Mizuta: Continuity properties of Riesz potentials for functions in L p ) of variable exponent - Math Inequal Appl 8, 2005, 69 63 [0] Futamura, T, and Y Mizuta: Continuity of weakly monotone Sobolev functions of variable exponent - Proceedings of the IWPT to appear) [] Futamura, T, Y Mizuta, and T Shimomura: Sobolev embeddings for Riesz potential space of variable exponent - Math Nachr to appear) [2] Haj lasz, P, and P Koskela: Sobolev met Poincaré - Mem Amer Math Soc 45, 2000, no 688 [3] Harjulehto, P, P Hästö, and M Pere: Variable exponent Lebesgue spaces on metric spaces: the Hardy Littlezood maximal operator - Real Anal Exchange 30, 2004/2005, 87 04 [4] Hästö, P: The maximal operator in Lebesgue spaces with variable exponent near - Math Nachr to appear) [5] Hedberg, L I: On certain convolution inequalities - Proc Amer Math Soc 36, 972, 505 50 [6] Heinonen, J: Lectures on Analysis on Metric Spaces - Springer-Verlag, New York, 200

522 T Futamura, Y Mizuta and T Shimomura [7] Kokilashvili, V, and S Samko: On Sobolev theorem for Riesz type potentials in the Lebesgue spaces with variable exponent - Z Anal Anwendungen 22, 2003, 899 90 [8] Kokilashvili, V, and S Samko: Maximal and fractional operators in weighted L px) spaces - Rev Mat Iberoamericana 20, 2004, 493 55 [9] Kováčik, O, and J Rákosník: On spaces L px) and W k,px) - Czechoslovak Math J 4, 99, 592 68 [20] Lerner, A K: On modular inequalities in variable L p spaces - Arch Math 85, 2005, 538 543 [2] Mizuta, Y: Potential Theory in Euclidean Spaces - akkōtosho, Tokyo, 996 [22] Mizuta, Y, and T Shimomura: Exponential integrability for Riesz potentials of functions in Orlicz classes - Hiroshima Math J 28, 998, 355 37 [23] Mizuta, Y, and T Shimomura: Continuity and differentiability for weighted Sobolev spaces - Proc Amer Math Soc 30, 2002, 2985 2994 [24] Mizuta, Y, and T Shimomura: Continuity of Sobolev functions of variable exponent on metric spaces - Proc Japan Acad Ser A Math Sci 80, 2004, 96 99 [25] Musielak, J: Orlicz Spaces and Modular Spaces - Lecture Notes in Math 034, Springer- Verlag, Berlin, 983 [26] Orlicz, W: Über konjugierte Exponentenfolgen - Studia Math 3, 93, 200 2 [27] Pick, L, and M R užička: An example of a space L px) on which the Hardy Littlewood maximal operator is not bounded - Expo Math 9, 200, 369 37 [28] R užička, M: Electrorheological Fluids: Modeling and Mathematical Theory - Lecture Notes in Math 748, Springer, 2000 Received 20 June 2005