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Our aim in this paper is to provide an optimal description for the existence and nonexistence of positive solutions to the integral inequality (1). For any α> 0, the fractional Laplacian ( ∆)α/2 is defined by means of the Fourier trans- form −
\ ( ∆)α/2u(ξ) := ξ αuˆ(ξ) − | |
for all u ′ such that ξ αu ′, here ′ stands for the space of tempered distributions on RN which∈ is S the dual of| the| Schwartz∈ S spaceS . S α/2 Since for α (0,N) the Riesz potential Iα can be interpreted as the inverse of ( ∆) (cf. [13, Sect.5.1]∈ or [2, Section 2.1]), under some extra integrability conditions on−u 0, inequality (1) is equivalent to the elliptic inequality ≥
(4) ( ∆)α/2u (I up)uq a.e. in RN , − ≥ β ∗
provided that both (1) and (4) are well-defined. This is the case, for instance, if (3) holds and u belongs to the homogeneous Sobolev space H˙ α/2(RN ), so that (4) is understood in the weak sense. Pointwise interpretations of the inequality (4) for non-integer α/2 are also possible, cf. [2, Theorem 2.13]. For a comparison of different definitions of the higher order fractional Laplacian ( ∆)α/2 see [1]. − Inequality (4) is a Choquard type inequality. Equations and inequalities of such structure originate from mathematical physics and have attracted considerable interest of mathemati- cians in the past decade, see [12] for a survey. In the 2nd order elliptic case α = 2 optimal regimes for the existence and nonexistence of positive solutions to inequality (4) were fully investigated in [11]. The higher–order polyharmonic case α/2 = m N was recently stud- ied in [5], where (amongst other results) optimal existence and nonexistence∈ regimes for the equation (4) were obtained for the exponents p 1 and q > 1, see [5, Theorem 1.4]. ≥ In this work we extend earlier results in [11] and [5] to the full admissible range α (0,N) and exponents p,q > 0. Our approach is different from the techniques in [5], which were∈ based on the poly–superharmonic properties of ( ∆)m in the elliptic framework of equation (4). Instead, we work entirely with the double–nonlocal− inequality (1). Our analysis of (1) employs only direct Riesz kernel estimates, and a new version of the nonlocal positivity principle in Lemma 3.1, inspired by [11, Proposition 3.2]. This has the advantage of incorporating the fractional case of noninteger α/2 in a seemingly effortless way, and does not rely on comparison principles or Harnack type inequalities, which are commonly used for similar Liouville type results in the elliptic framework, but which are generally speaking not available in the case of the higher–order fractional Laplacians ( ∆)α/2 with α> 2. − The main result of this work related to the existence of positive solutions to (1) reads as follows. NONLINEAR INEQUALITIES WITH DOUBLE RIESZ POTENTIALS 3
Theorem 1.1. Let p,q > 0. Then, inequality (1) has a nontrivial nonnegative solution u L1 (RN ) which satisfies (2) if and only if ∈ loc β p> N α −N + β p + q > N α − β q > if β > N α, N α − − q 1 if β = N α, ≥ − N α β q > 1 − − p if β < N α. − N − The necessary part of the proof follows directly from Propositions 2.3, 4.1-4.5 below. The sufficiency follows from Propositions 5.4-5.7, where we construct explicitly smooth positive radial solutions to (1). In the case α = 2 our results are fully consistent with the results established in [11, Theorem 1] for the 2nd order elliptic inequality (4). The nonexistence of positive solutions to double–nonlocal inequality (1) with p> 1, q > 0 and p + q N+β was ≤ N−α established by different methods in [8, Theorem 1]. Remark 1.2. In Section 5 we also discuss the optimal decay of solutions to (1) in terms of the parameters α, β, p and q. Clearly (see (9)), if u 0 is a nontrivial solution of (1) then lim inf u(x) x N−α > 0. In particular, for R 1≥ we have |x|→∞ | | ≫ (5) u cRα. ≥ ZB2R\BR In Proposition 5.1 we establish an integral lower bound
α+β−Nq (6) u cR 1−q , ≥ ZB2R\BR β which is stronger than (5) when α + β < N and q < N−α < 1. In Propositions 5.4 and 5.5 we β construct positive radial solutions u to (1) that confirm the optimality of (5) when q > N−α β β and of (6) when q < N−α . When q = N−α the bounds in (5) and (6) coincide. In that case in Propositions 5.6 and 5.7 we construct positive radial solutions to (1) that satisfy (5) up to a log (q < 1) or arbitrary small polynomial (q = 1) corrections. In the case α = 2 such corrections are necessary, see [11, Proposition 4.12, 4.13].
2. Preliminaries In this section we collect some useful facts for our approach. Lemma 2.1. Let f : RN R be a nonnegative measurable function. Then, the Riesz potential I f of order α →(0,N) is well defined, in the sense that α ∗ ∈ (7) I f < + a.e. in RN , α ∗ ∞ if and only if (8) f L1 (1 + x )−(N−α)dx, RN . ∈ | | 4 MARIUSGHERGU, ZENG LIU, YASUHITO MIYAMOTO,ANDVITALY MOROZ
N Moreover, if (7) fails then Iα f =+ everywhere in R , see [7, p.61-62]. We present the proof of the lemma for completeness.∗ ∞
Proof. Assume first that (7) holds. Then, for any x,y RN , x = 0 we have ∈ 6 x y N−α c( x N−α+ y N−α) c max 1, x N−α (1+ y N−α) c max 1, x N−α (1+ y )N−α. | − | ≤ | | | | ≤ { | | } | | ≤ { | | } | | Thus, for any x RN 0 such that (7) holds, we have ∈ \ { } 1 f(y)dy > (Iα f)(x) N−α N−α , ∞ ∗ ≥ c max 1, x RN (1 + y ) { | | } Z | | which yields (8). RN 1 RN Conversely, assume now that (8) holds and let x 0 . From (8) we have f Lloc( ). Then ∈ \{ } ∈ f(y)dy f(y)dy (I f)(x)= + α ∗ x y N−α x y N−α Z|y|≤2|x| | − | Z|y|>2|x| | − | f(y)dy f(y)dy + 2N−α ≤ x y N−α y N−α Z|y|≤2|x| | − | Z|y|>2|x| | | < . ∞
In the same spirit to the above proof, if f 0 and (7) (or, equivalently (8)) holds, then ≥ c (9) I f(x) f(y) dy. α ∗ ≥ x N−α | | ZB|x|(0) One of the elementary yet important for our approach consequences of (9) is the following estimate, which we will be using frequently, and which to some extent is the counterpart of the Harnack inequalities on the annuli in the study of (4). Lemma 2.2. Let α (0,N), θ > 0 and 0 f L1 (1 + x )−(N−α)dx, RN . Then for all R> 0 we have ∈ ≤ ∈ | | θ (10) I f θ CRN−(N−α)θ f . α ∗ ≥ ZB2R\BR ZBR Proof. Follows from (9) by integration.
An obvious implication of (9) is that Iα f can not decay faster than Iα at infinity, even if the function f is compactly supported. Recall∗ also that if f 0 then an elementary estimate shows that for every x RN , ≥ ∈ A (11) I f(x) α f(y)dy. α ∗ ≥ RN−α ZBR(x) As a consequence, if u 0 is a solution of (1) that is positive on a set of positive measure, then u is everywhere strictly≥ positive on RN and the following lower bounds must hold: (12) u(x) c(1 + x )−(N−α), ≥ | | (13) I up(x) c(1 + x )−(N−β). β ∗ ≥ | | NONLINEAR INEQUALITIES WITH DOUBLE RIESZ POTENTIALS 5
On the other hand, (2) requires (14) up L1 (1 + x )−(N−β)dx, RN , ∈ | | p q 1 −(N−α) N (15) (Iβ u )u L (1 + x ) dx, R . ∗ ∈ | | Combining the competing upper and lower bounds immediately leads to the following nonex- istence result. Proposition 2.3. Let p,q > 0 and assume that either p β , or α + β > N and q ≤ N−α ≤ β 1. If u 0 is a solution of (1) then u 0. N−α − ≥ ≡ Proof. First we note that (12) and (14) are incompatible when p β . Then we observe ≤ N−α that (12), (13) and (15) are incompatible when 0 3. Nonlocal positivity principle The nonexistence result in Proposition 2.3 ”decouples” the values of p and q. In order to deduce an estimate which involves the quantity p + q which appears in Theorem 1.1, we need the following lemma, inspired by [10, Proposition 2.1] and [11, Section 3]. Lemma 3.1 (Nonlocal positivity principle). Let α (0,N) and V : RN [0, ) be a measurable function. Assume that there exists u L∈1 (RN ) such that u >→0 a.e.∞ in RN , ∈ loc V u L1 (1 + x )−(N−α)dx, RN and ∈ | | N (16) u Iα (V u) a.e. in R . ≥ ∗ Then for every R> 0 and 0 ϕ C∞(B ), ≤ ∈ c R 2 (17) ϕ2 CRα−N √V ϕ . ≥ ZBR ZBR ϕ2 Proof. Take ψ := u as a test function in (16). Then ϕ2(x) ϕ2 I ( x y )V (y)u(y) dy dx ≥ α | − | u(x) ZBR ZZBR×BR 1 ϕ2(y) ϕ2(x) = I ( x y ) V (x)u(x) + V (y)u(y) dxdy 2 α | − | u(y) u(x) ZZBR×BR = Iα( x y ) V (x)V (y)ϕ(y)ϕ(x)dxdy BR×BR | − | ZZ p 1 ϕ(y) ϕ(x) 2 + I ( x y )u(x)u(y) V (x) V (y) dxdy 2 α | − | u(y) − u(x) BR×BR ZZ p p A 2 α V (x)ϕ(x)dx , ≥ 2N−αRN−α ZBR since p A (18) I (x y) α (x,y B ), α − ≥ 2N−αRN−α ∈ R 6 MARIUSGHERGU, ZENG LIU, YASUHITO MIYAMOTO,ANDVITALY MOROZ which completes the proof. Remark 3.2. Nonlocal inequality (16) can be interpreted as the “inversion” of the fractional Schr¨odinger inequality ( ∆)α/2u V u 0 in RN . − − ≥ In this context Lemma 3.1 can be seen as a higher–order version of the fractional Agmon– Allegretto–Piepenbrink’s positivity principle: if (16) has a positive supersolution then a cer- tain variational inequality which involves the potential V must hold. We will see that Lemma 3.1 alongside with the standard integral estimate (10) of the Riesz potentials are sufficient for the complete analysis of the existence and nonexistence of positive solutions of the nonlinear inequality (1). Using Lemma 3.1 we establish the following estimate. Proposition 3.3. Let p,q > 0 and u > 0 be a solution of (1). Then, for every R > 0 and every ϕ C∞(B ), ∈ c R 2 2 α+β−2N p q−1 (19) ϕ CR u u 2 ϕ . ≥ ZBR ZBR ZBR ∞ p q−1 Proof. For every ϕ Cc (BR), by Lemma 3.1 with V = (Iβ u )u , and using a similar inequality to (18) for∈ I up, we obtain ∗ β ∗ 2 ϕ2 CRα−N (I up)uq−1 1/2ϕ ≥ β ∗ ZBR ZBR 2 ′ α+β−2N p q−1 C R u u 2 ϕ . ≥ ZBR ZBR One of the principal tools in the subsequent analysis is the following decay estimate on the solutions of (1), which is an adaptation of (19). Note that for q < 1 our estimate contains a lower bound on the solution, since the 2nd integral involves a negative power of u. Corollary 3.4. Let p,q > 0 and u> 0 be a solution of (1). Then for every R> 0, 2 p q−1 3N−α−β (20) u u 2 CR . ≤ ZB2R ZB2R\BR ! Proof. Take ϕ (x) = ϕ(x/R), where ϕ C∞(RN ) is such that supp(ϕ) B B , ϕ 1 R ∈ c ⊂ 4 \ 1/2 ≡ on B B and 0 ϕ 1. Then,by (19) we find 2 \ 1 ≤ ≤ 2 N 2 ′ α+β−2N p q−1 cR = ϕ C R u u 2 ϕ R ≥ R ZB4R ZB4R ZB4R 2 ′ α+β−2N p q−1 C R u u 2 . ≥ ZB2R ZB2R\BR ! NONLINEAR INEQUALITIES WITH DOUBLE RIESZ POTENTIALS 7 4. Nonexistence In this section we derive several nonexistence result for (1). Our approach is inspired by [11] which studied the inequality (4) in the semilinear the case α = 2, yet with substantial modifications. In particular, in this work we completely avoid the use of the comparison principle and Harnack’s inequalities, which are not applicable in the framework of the double– nonlocal inequality (1). It turns out that Harnack inequality estimates in the context of (1) can be replaced by the estimate (10). Proposition 4.1. Let p,q > 0 and assume that p + q < 1. If u 0 is a solution of (1) then u 0. ≥ ≡ Proof. Since up L1((1 + x )−(N−β)dx) and β < N, by Lebesgue’s dominated convergence theorem ∈ | | up = o RN−β = o RN as R . →∞ ZB2R\BR Since p + q < 1, we may apply H¨older and then the estimate (20) to obtain p(1−q) (q−1)p cRN = 1= u 2p+1−q u 2p+1−q ZB2R\BR ZB2R\BR 1−q 2p p 2p+1−q q−1 2p+1−q u u 2 ≤ B2R\BR B2R\BR Z Z p 1−q−p 2 2p+1−q p 2p+1−q p q−1 u u u 2 ≤ ZB2R\BR ZB2R\BR ZB2R\BR 1−q−p p up 2p+1−q CR3N−α−β 2p+1−q ≤ B \B Z 2R R N 1−q−p p o(1)R 2p+1−q R3N 2p+1−q ≤ N = o(1)R , which raises a contradiction. Proposition 4.2. Let p,q > 0 and assume that 1 p + q N+β . If u 0 is a solution of ≤ ≤ N−α ≥ (1) then u 0. ≡ N+β Proof. Assume first p + q < N−α . By the Cauchy–Schwarz inequality, 2 p −N p u cR u 2 , ≥ ZB2R ZB2R and so (20) implies 2 2 4N−α−β p q−1 (21) CR u 2 u 2 . ≥ ZB2R ZB2R\BR ! 8 MARIUSGHERGU, ZENG LIU, YASUHITO MIYAMOTO,ANDVITALY MOROZ −(N−α) N Using (21) and the Cauchy–Schwarz again together with u c x in R B1 (that follows from (12)), we obtain ≥ | | \ 2 2 4N−α−β p q−1 CR u 2 u 2 ≥ B B \B Z 2R Z 2R R (22) p+q−1 4 u 4 ≥ B \B Z 2R R cR4N−(N−α)(p+q−1), ≥ α+β which is a contradiction since 0 0 we deduce p q p+q p+q (Iβ u )u u(x) 2 Iβ(x y)u(y) 2 dx dy RN ∗ ≥ RN RN − Z Z \B1 Z \B1 1 1 c N+β Iβ(x y) N+β dx dy = . N N ≥ R \B1 R \B1 x 2 − y 2 ∞ Z Z | | | | Hence, p q (23) lim (Iβ u )u = . R→∞ ∗ ∞ ZBR Since u satisfies (1), from Lemma 2.2 with θ = p+q−1 = α+β and f = (I up)uq we find 4 4(N−α) β ∗ p+q−1 p+q−1 p+q−1 N− α+β p q 4 u 4 (Iα f) 4 CR 4 (Iβ u )u . B \B ≥ B \B ∗ ≥ B ∗ Z 2R R Z 2R R Z R From the above estimate and (23) we deduce 1 p+q−1 lim u 4 = , R→∞ N− α+β ∞ R 4 ZB2R\BR which contradicts the upper bound in (22). If α+β N we give precise lower bounds on uq−1 to obtain a further nonexistence ≥ B2R\BR result. R Proposition 4.3. Let p,q > 0 and assume that α + β > N and 1 Proof. Assume that u> 0 on a set of positive measure. From (12) we obtain 2 q−1 2N−(N−α)(q−1) (24) u 2 cR . ≥ ZB2R\BR ! On the other hand, by Corollary 3.4, 2 3N−β−α q−1 CR ′ 3N−β−α (25) u 2 C R . ≤ ≤ ZB2R\BR ! up ZB2R β This yields a contradiction if q < N−α . β In the critical case q = N−α using (9), (13) we obtain u(x)q (I up)uq c up dx β ∗ ≥ x N−β ZBR ZB1 ZBR\B1 | | 1 c′ dx = c′′ log(R), ≥ x N−β+(N−α)q ZBR\B1 | | since (N α)q = β. By Lemma 2.2 with θ = q−1 > 0, − 2 q−1 q−1 p q 2 u 2 I (I u )u ≥ α ∗ β ∗ ZB2R\BR ZB2R\BR q−1 2 N−(N−α) q−1 p q cR 2 (I u )u (26) ≥ β ∗ ZBR ′ 2N−(N−α)(q−1) q−1 c R 2 log 2 (R) ≥ ′ 3N−α−β q−1 = c R 2 log 2 (R), which contradicts (25). The transitional locally linear case q = 1 requires a separate consideration. Proposition 4.4. Let p > 0 and assume that α + β > N and q = 1. If u 0 is a solution of (1) then u 0. ≥ ≡ Proof. Using Corollary 3.4, for any R> 0 we have 2 R3N−α−β c up 1 = cR2N up . ≥ B B \B B Z R Z 2R R Z R Since α + β > N, it follows that u 0. ≡ In the sublinear case q < 1 we deduce an additional restriction on the admissible range of the exponent q. Proposition 4.5. Let p,q > 0 and assume that p + q 1, q < 1 and ≥ N α β q 1 − − p. ≤ − N If u 0 is a solution of (1) then u 0. ≥ ≡ 10 MARIUS GHERGU, ZENG LIU, YASUHITO MIYAMOTO,AND VITALY MOROZ Proof. Since q < 1, by H¨older’s inequality we deduce p 1−q q−1 2p cRN = 1= u 2p+1−q u 2 2p+1−q ZB2R\BR ZB2R\BR 1−q 2p p 2p+1−q q−1 2p+1−q u u 2 (27) ≤ ZB2R\BR ZB2R\BR 1−q p+q−1 2 2p+1−q 2 2p+1−q p q−1 q−1 = u u 2 u 2 . B \B B \B B \B Z 2R R Z 2R R Z 2R R By Corollary 3.4 we have 2 p q−1 3N−α−β u u 2 CR , ≤ ZB2R ZB2R\BR ! which yields 2 p q−1 3N−α−β (28) u u 2 CR , B \B B \B ≤ Z 2R R Z 2R R and on the other hand 2 3N−β−α q−1 R ′ 3N−α−β (29) u 2 C C R . B \B ≤ p ≤ Z 2R R u ZB2R If q < 1 N−β−α p, we use (28)-(29) in (27) to raise a contradiction since p + q 1. − N ≥ If q = 1 N−β−α p, we use (29) and H¨older’s inequality to further estimate − N 1+ 2p 1 1−q B \B p p 2R R N− 1−q (N−α−β) u Z 2p cR = c. 1−q B2R\BR ≥ 1−q ≥ Z u 2 B \B Z 2R R This shows that up L1(RN ) and 6∈ lim up = . R→∞ ∞ ZB2R Using this fact in (29) we deduce 2 q−1 3N−α−β u 2 = o(R ) as R . B \B →∞ Z 2R R We now use this last estimate and (28) in (27) to conclude. 5. Optimal decay and existence If u 0 is a solution of (1), then either u 0 or u must obey the ”natural” lower bound (12), which≥ implies in particular, the integral≡ lower bound (30) u cRα. ≥ ZB2R\BR NONLINEAR INEQUALITIES WITH DOUBLE RIESZ POTENTIALS 11 In the region q < 1 the estimate (20) of Corollary 3.4 leads to an integral lower bound which β improves upon (30) when α + β < N and q < N−α . Proposition 5.1. Let p,q > 0 and assume that α + β < N and q < β < 1. If u 0 is a N−α ≥ solution of (1), then either u 0 or ≡ α+β−Nq (31) u cR 1−q . ≥ ZB2R\BR β As pointed out in Remark 1.2, since α + β < N and q < N−α < 1, the exponent of R in (31) is greater than α. Proof. From Corollary 3.4 we have 2 3N−α−β − 1−q R ′ 3N−α−β u 2 c c R . B \B ≤ p ≤ Z 2R R u ZB2R Further, by H¨older’s inequality (since 0 2 −(1−q) 3−q − 1−q u 2 u 1 . B \B ≥ B \B B \B Z 2R R Z 2R R Z 2R R Now, the above estimates yield −(1−q) u RNq−α−β, B \B ≤ Z 2R R which leads to (31). Our next step is to construct explicit solutions with the decay which match or near-match the lower bounds in (30) and (31). Before we do this, we recall the following simple estimates, cf. [11, Lemma A.1 and A.2] which are frequently used in the proofs below. Lemma 5.2. Let v L1 (RN ), γ (0,N) and s > γ. If ∈ loc ∈ lim sup v(x) x s < , |x|→∞ | | ∞ then s−γ lim sup(Iγ v)(x) x < if γ Lemma 5.3. Let v L1 (RN ), γ (0,N) and σ R. If ∈ loc ∈ ∈ x N lim sup v(x) | | < , (log x )σ ∞ |x|→∞ | | 12 MARIUS GHERGU, ZENG LIU, YASUHITO MIYAMOTO,AND VITALY MOROZ then N−γ lim sup(Iγ v)(x) x < if σ < 1, |x|→∞ ∗ | | ∞ − x N−γ lim sup(I v)(x) | | < if σ = 1, γ ∗ (log(log x )) ∞ − |x|→∞ | | x N−γ lim sup(I v)(x) | | < if σ > 1. γ ∗ (log x )σ+1 ∞ − |x|→∞ | | Proposition 5.4. Assume that β N + β β (32) p> , p + q > and q > . N α N α N α − − − Then, (1) admits a positive radial solution u C(RN ) which satisfies ∈ (33) lim sup u(x) x N−α < . |x|→∞ | | ∞ Proof. Let 0 <ε Proof. Let u(x)=(1+ x )−k where k = (N α β)/(1 q). Since 1 (N α β)p/N < q, we have pk > N, and hence| | by the third estimate− − of Lemma− 5.2 we have− − − I up c (1 + x )β−N in RN , β ∗ ≤ 1 | | for some constant c > 0. Since β N kq = N−β−αq , we have 1 − − − 1−q − N−β−αq (I up)uq c (1 + x ) 1−q in RN . β ∗ ≤ 2 | | β N−β−αq Since q < N−α < 1, we have α < 1−q < N. Hence, by the first estimate of Lemma 5.2 we have I (I up)uq c (1 + x )−k in RN , α ∗ β ∗ ≤ 2 | | h i −1/(p+q−1) −k where c2 > 0 is a constant. Thus, U(x) = c2 (1 + x ) is a continuous solution of (1). Moreover, | | lim sup U(x) x k < . |x|→∞ | | ∞ Proposition 5.6. Assume that N α + β = N, p> and q = 1. N α − Then, for every m > 0 inequality (1) admits a positive radial solution u C(RN ) which satisfies ∈ lim sup u(x) x N−α−m < . |x|→∞ | | ∞ Proof. Let m> 0. Since p> N = N , we see that β N > 0. Set N−α β − p m if 0 Proposition 5.7. Assume that N β p> and q = < 1. N α N α − − Then, for m N−α inequality (1) admits a positive radial solution u C(RN ), which ≥ N−α−β ∈ satisfies lim sup u(x) x N−α log x −m < . |x|→∞ | | | | ∞ Proof. Take u(x)=(1+ x )−(N−α)(log(e + x ))m. | | | | Since (N α)p< N, by the third estimate of Lemma 5.2 we see that I up c (1+ x )β−N − − − β ∗ ≤ 1 | | for some constant c1 > 0, and hence (I up)uq c (1 + x )β−N−(N−α)q(log(e + x ))mq = c (1 + x )−N (log(e + x ))mq. β ∗ ≤ 1 | | | | 1 | | | | By the third estimate of Lemma 5.3 we see that I (I up)uq c (1 + x )−(N−α)(log(e + x ))mq+1 in RN , α ∗ β ∗ ≤ 2 | | | | for some constant hc > 0. Sincei m N−α , we have 2 ≥ N−α−β I (I up)uq c u. α ∗ β ∗ ≤ 2 −1/(p+q−1) h i Thus, U(x)= c2 u is a continuous solution of (1). NONLINEAR INEQUALITIES WITH DOUBLE RIESZ POTENTIALS 15 References [1] N. Abatangelo, S. Jarohs and A. Salda˜na, Positive powers of the Laplacian: from hypersingular integrals to boundary value problems, Commun. Pure Appl. Anal. 17 (2018), 899–922. [2] G. Caristi, L. D’Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math. 76 (2008), 27–67. 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School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland Institute of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei St., 010702 Bucharest, Romania Email address: [email protected] Department of Mathematics, Suzhou University of Science and Technology, Suzhou, 215009, P.R. China Email address: [email protected] Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro- ku, Tokyo 153-8914, Japan Email address: [email protected] Mathematics Department, Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, Wales, United Kingdom Email address: [email protected]