Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 16, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.16 L2 SOLUTIONS FOR CUBIC NLS EQUATION WITH HIGHER ORDER FRACTIONAL ELLIPTIC/HYPERBOLIC OPERATORS ON R× T AND R2 ADÁN J. CORCHO, LINDOLFO P. MALLQUI Abstract. In this work, we consider the Cauchy problem for the cubic Schrö- dinger equation posed on cylinder R × T with fractional derivatives (−∂2 y) α, α > 0, in the periodic direction. The spatial operator includes elliptic and hyperbolic regimes. We prove L2 global well-posedness results when α ≥ 1 by proving a L4-L2 Strichartz inequality for the linear equation, following the ideas in [19], where it was considered the elliptical case with α = 1. Further, these results remain valid on the Euclidean environment R2, so well-posedness in L2 are also achieved in this case. Our proof in the elliptic (hyperbolic) case does not work in the small directional dispersion case 0 < α < 1 (0 < α ≤ 1), respectively. 1. Introduction We consider the initial value problem (IVP) associated with the cubic nonlinear Schrödinger equation (3-NLS) on cylinder R× T with higher fractional derivatives in the periodic direction. More precisely, we will study the IVP i∂tu+ L± αu = ε|u|2u, (x, y) ∈ R× T, t ∈ R, u(0;x, y) = ϕ(x, y), (x, y) ∈ R× T, (1.1) where ε ∈ R∗, u = u(t;x, y) is a complex-valued function on R × R × T, with T = R/2πZ, and L± α denotes the pseudo-differential operator acting in the space variables, given by L± α := ∂2x ∓ (−∂2y)α, with α > 0, (1.2) defined in Fourier variables by L̂± α f(ξ, n) = −(ξ2 ± |n|2α)f̂x,y(ξ, n), (1.3) where f̂x,y is the Fourier transform of f on the cylinder R× T, that is, F [f ](ξ, n) = f̂x,y(ξ, n) := 1 2π ∫ ∞ −∞ ∫ 2π 0 f(x, y)e−i(xξ+ny) dx dy. (1.4) 2020 Mathematics Subject Classification. 35Q55, 35Q35, 35Q60. Key words and phrases. Elliptic/hyperbolic cubic nonlinear Schrödinger equation; Cauchy problem; well-poseddness. ©2025. This work is licensed under a CC BY 4.0 license. Submitted August 7, 2024. Published February 21, 2025. 1 2 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 So, the linear propagator U± α of (1.1) is given by U± α (t)ϕ = eitL ± α ϕ = F−1 ( e−it(ξ2±|n|2α)F [ϕ] ) . (1.5) The cases ε < 0 and ε > 0 are known as the focusing and defocusing regimes, respectively. Also, note that in the case α = 1, L± 1 = { ∂2x + ∂2y =: ∆ (Laplacian operator) in the case (+), ∂2x − ∂2y =: □ (wave operator) in the case (-). When α = 1, the elliptic case appears as a model in several physical problems (see for example the references [14, 16, 21]). On the other hand, in the hyperbolic case it describes, for instance, the gravity waves on liquid surface and ion-cyclotron waves in plasma (see [7, 13, 18]). Both cases (elliptic/hyperbolic) of the IVP (1.1) with α > 0, posed on the Euclidean domain R2, are covered in the study carried out in [8] about the existence of analytic solutions. In the case 0 < α < 1, the elliptic operators L+ α (fractional directional Laplacian) in (1.2) appear in the context of parabolic equations (see [3, 4] and references therein) as toy models to describe local diffusion occurring only in the x direction, while non-local diffusion occurs in the y-direction. Moreover, the defocusing elliptic case of the IVP (1.1) with α = 1/2 (called Half-wave-Schrödinger equation), defined on the cylinder R×T, was considered in [22] to establish a modified scattering theory between small solutions to this model and small solutions to the cubic Szegő equation. In the same work, the author infers the existence of unbounded global solutions of (1.1) in the anisotropic space L2 xH s y(R × T) for every s > 1/2. In [1], the authors considered the Half- wave-Schrödinger equation with a more general nonlinearity (|u|p−1u), posed on the Euclidean domain R2, and they showed local well-posedness (1 < p ≤ 5) in anisotropic space L2 xH s y(R2) for s > 1/2. Furthermore, they presented some results concerning the existence and orbital stability of solitary waves. Recently, in [9], the elliptic case of IVP (1.1), posed on R2, is considered when 0 < α < 1 and with nonlinearity |u|p−2u, 2 < p < 2 (1+α) 1−α . More precisely, conditions for the existence of blow-up solutions are investigated. For every α > 0, the IVP (1.1) (elliptic/hyperbolic) formally enjoys the mass conservation law: M [u](t) = ∫ R ∫ T |u(t;x, y)|2 dx dy =M [u](0), (1.6) for all t ∈ R. Therefore, it would be interesting to establish a well-posedness theory in L2(R×T) similarly to the case α = 1 (Laplacian and wave operators). The main goal of this paper is to answer this question. 1.1. Case α = 1. (some previous well-posedness results in isotropic Sobolev spaces) In the case of classical Laplacian/wave operator (α = 1), local and global well- posedness for time evolution of the flow associated to the IVP (1.1), with ini- tial data ϕ belonging to the classical Sobolev spaces Hs(D) on plane domains D = R2, T2 or R×T, has been considered for many authors. For instance, we have the following results: • D = R2: Well-posedness for in L2(R2) is a consequence of the time decay estimates coming from dispersion and the Strichartz inequalities, which have the same form for the elliptic or hyperbolic linear Schrödinger equations. For the elliptic operator EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 3 L+ 1 see [6] and for the hyperbolic case L− 1 we refer to [10], where the authors deduced Strichartz inequalities for operators eitL, where L = ∑ 1≤i,j≤n aij∂xi ∂xj , aij ∈ R, with non-degenerate quadratic form A = ( aij ) . In particular, L2(R2) is the Sobolev critical regularity (s = 0) for well-posedeness, which is suggested by the scale invariance of the model, that is, if u is a solution of (1.1) with α = 1 and initial data ϕ(x, y), then uλ(t; , x, y) = u(λ2t;λx, λy), λ > 0 is the respective solution of (1.1) with initial data λϕ(λx, λy). • D = T2 : Global well-posedness in Hs(T2), s > 0, has been proven in the elliptic case U+ 1 (see the works [2, 5]). In [20] the hyperbolic case U− 1 was considered, getting local well-posedness for s > 1 2 and ill-posedness when s < 1/2. • D = R × T: The elliptical case L+ 1 was treated in [19]. Specifically, the authors obtained global well-posedness for small data in L2(R × T) by proving a L4 − L2 Strichartz inequality for the group U+ 1 (t) = eit∆ with (x, y) ∈ R×T. More precisely, they showed that ∥U+ 1 (t)ϕ∥L4(I×R×T) ≤ CI∥ϕ∥L2(R×T), (1.7) where I ⊂ R is an interval containing t = 0 and CI is a positive constant that depends only on |I| (measure of I). By the symmetry of the laplacian operator, the results are also valid in the space L2(T×R). Similar Strichartz estimates were obtained in [11] for the energy critical nonlinear Schrödinger equation in partially periodic domains, for instance: Rm × T4−m with m = 2, 3. In view of the above comments, a natural question is to figure out what happens for the IVP (1.1) posed on domains R2, R × T or T × R, with initial data in L2, for more general α > 0 as already pointed out before the beginning of this section. The notion of criticality given below tells us that we do not expect well-posedness in L2 for IVP (1.1) for 0 < α < 1. 1.2. Notion of criticality in isotropic Sobolev spaces for α > 0. The equa- tion in (1.1) on the domain R2 has the following scaling symmetric property: if u is a solution to (1.1), uλ is also a solution to (1.1), where u(t;x, y) 7→ uλ(t;x, y) := λu(λ2t;λx, λ1/αy), λ > 0, (1.8) which establishes a notion of criticality in Sobolev spaces Hs(R2) for the IVP (1.1). More precisely, • the index s is called critical if ∥Dsuλ(0; ·, ·)∥L2(R2) ∼ ∥Dsu(0; ·, ·)∥L2(R2), • the index s is called subcritical if ∥Dsuλ(0; ·, ·)∥L2(R2) → ∞ as λ→ ∞, • the index s is called supercritical if ∥Dsuλ(0; ·, ·)∥L2(R2) → 0 as λ→ ∞. Computing ∥Dsuλ(0; ·, ·)∥L2(R2) for λ > 0 we have ∥Dsuλ(0; ·, ·)∥2L2(R2) = λ1−1/α ∫ R2 ( λ2ξ21 + λ2/αξ22 )s|û(0; ξ)|2dξ, (1.9) with ξ = (ξ1, ξ2). So one obtains ∥Dsuλ(0; ·, ·)∥2L2(R2) =  λ1−1/α+2s/αpα(λ) if 0 < α < 1, λ2s∥Dsu(0; ·, ·)∥2L2(R2) if α = 1, λ1−1/α+2sqα(λ) if α > 1, (1.10) 4 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 where pα(λ) = ∫ R2 ( λ2(1− 1 α )ξ21 + ξ22 )s|û(0; ξ)|2dξ qα(λ) = ∫ R2 ( ξ21 + λ2( 1 α−1)ξ22 )s|û(0; ξ)|2dξ. Hence, since lim λ→+∞ pα(λ) = ∥Ds yu(0; ·, ·)∥2L2(R2) and lim λ→+∞ qα(λ) = ∥Ds xu(0; ·, ·)∥2L2(R2), we conclude that sc :=  1−α 2 > 0 is the critical regularity for 0 < α < 1, 0 is the critical regularity for α = 1, 1−α 2α < 0 is the critical regularity for α > 1. (1.11) Hence, it is not expected well-posedness in L2 for 0 < α < 1. Furthermore, concerning well-posedness in Sobolev spaces to (1.1) we note that some ill-posedness results (below L2 regularity) for the one-dimensional cubic NLS on the line can be adapted to establish the same results for (1.1) on the cylinder R× T. Indeed, if we consider the following Cauchy problem i∂tu+ L± αu = ε|u|2u, (t;x, y) ∈ R× R× T, u(0;x, y) = φ(x), (1.12) with φ depending only on the x-variable and φ ∈ Hs(R), it follows that φ̃(x, y) := φ(x) ∈ Hs(R× T) with ∥φ̃∥Hs(R×T) = ∥φ∥Hs(R), and solutions of the IVP: i∂tw + ∂2xw = ε|w|2w, x ∈ R, t ∈ R, w(x, 0) = φ(x), (1.13) are also solutions of (1.12). Assuming the existence of local solutions, the IVP (1.13) is ill-posed in the following situations: (i) s ∈ (−1/2, 0) and ε < 0. In this case, for any δ > 0 the uniform continuous of the flow-map Φ : u0 ∈ Hs(R) 7→ w ∈ C([0, δ];Hs(R)) (1.14) fails (see [12]). (ii) s ≤ −1/2 and ε ̸= 0. In this case, for any δ > 0 the flow-map (1.14) is discontinuous everywhere in Hs(R) (norm inflation arguments, see [15]). Hence, the statements (i) and (ii) imply similar ill-posedness results for negative Sobolev regularity (s < 0) to the IVP (1.12) for all α > 0. In view of the previous discussion we study in this work well-posedness for IVP (1.12) with initial data in L2(R×T) and we get global well-posedness under small- ness assumption on data. This result remains valid in L2(R2), but unfortunately cannot be adapted to the case L2(T× R). EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 5 1.3. Main results. Consider the IVP (1.1), with α ≥ 1 and initial data ϕ ∈ L2(R × T). In this work we show that it is possible to get a Strichartz estimate similar to (1.7) in the case of the group U± α defined in (1.5) for α ≥ 1 in the elliptic case (+) and for α > 1 in the hyperbolic case (-). More precisely, we will prove the main result. Theorem 1.1 (Strichartz estimate on I×R×T). Let α ≥ 1 and I ⊂ Rt an interval containing t = 0. Then, there exists a positive constant Cα,I , depending only on α and the measure of I, such that∥∥U± α (t)ϕ ∥∥ L4(I×R×T) ≤ Cα,I∥ϕ∥L2(R×T), (1.15) for each ϕ ∈ L2(R × T) with α ≥ 1 for the case (+), and α > 1 for the case (-). Moreover, there exists a positive constant C̃α,I , depending only on α and the measure of I, such that∥∥∫ t 0 U± α (t− t′)f(t′; ·)dt′ ∥∥ L4(I×R×T) ≤ C̃α,I∥f∥L4/3(I×R×T), (1.16) for any f ∈ L4/3(I × R× T). Remark 1.2. We highlight that, in comparison with the case U+ 1 (t) covered in [19], the study in the case α ̸= 1 causes extra technical difficulty in the manipulation of symbol τ + ξ2 − |n|2α, since one cannot make use of the good algebraic structure of quadratic polynomial in two variables corresponding to the symbol when α = 1. Indeed, to estimate the measure of the set GK in the proof of crucial Lemma 2.1 we had to introduce two appropriate auxiliary sets, which is not necessary in the case of U+ 1 (t). As in [19], in the context of Cauchy problem for the cubic elliptic NLS, Theorem 1.1 combined with Picard iteration scheme applied to the integral equation u(t) = U± α (t)ϕ− iε ∫ t 0 U± α (t− t′)|u(t′)|2u(t′)dt′ (1.17) and the mass conservation law (1.6) imply the following result. Theorem 1.3 (Well-posedness in L2). The Cauchy problem (1.1) is globally well- posed for sufficiently small data ϕ in L2(R × T) with α ≥ 1 for the case (+) and α > 1 for the case (-). 1.4. Comments. Finally, we point out some facts. (I) From the notion of criticality given in (1.11) it is natural to expect well- posedness results for the IVP (1.1) for s > 1−α 2 > 0 when 0 < α < 1. In fact, our proof of Theorem 1.1 does not work in this setting. See remarks 2.2 and 3.2. (II) The proof of Theorem 1.1 also fails for the group U− 1 , where the critical regularity suggested by the scaling is L2. Thus, the hyperbolic case with α = 1 remains an interesting open problem on the cylinder domain. At this point it is important to remember that in the hyperbolic case with α = 1 on R × R, the critical regularity for well-posedness is L2, but on T× T it was flagged in [20] that the optimal regularity must be s = 1 2 . (III) The proof of Theorem 1.1 can also be performed in R2. Indeed, similar to the case U+ 1 treated in [19], the Bourgain’s method to obtain Strichartz inequalities on cylinder for U± α also provides a proof of the Strichartz estimate with data on R2 6 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 without using the time decay estimates coming from dispersion. Also, for subcritical nonlinearity |u|p−1u (1 < p < 3) instead the critical case |u|2u (p = 3) globall well- posedness for any data in L2(R × T) can be achieved in the same way as done in [19] in the case U+ 1 . (IV) Our approach cannot automatically adapt to the cylinder T× R since our strategy is based on the quadratic structure of the operator symbol with respect to the continuous propagation. 1.5. Notation and an elementary inequality. Throughout the paper we will use the following notation: • |J | denotes the Lebesgue measure of a set J ⊂ R, • m(·) denotes the product measure of the one-dimensional Lebesgue and counting measure, • 0 < A(v) ≲ B(v) means that there exists a positive constant c such that A(v) ≤ cB(v), for any v varying on a certain set, • ⌊x⌋ denote the integer part of x. Furthermore, Xb,s α±(R × R × T) will denote the Bourgain space associated to the group U± α , equipped with the norm ∥f∥2 Xb,s α± := ∥U± α (−t)f∥2Hb tH s(R×T) = ∫ τ ∫ ξ ∑ n∈Z ⟨|ξ|+ |n|⟩2s⟨τ + q±(ξ, n)⟩2b ∣∣f̂ t,x,y(τ ; ξ, n)∣∣2 dτ dξ, (1.18) where ⟨·⟩ = 1 + | · |, q±(ξ, n) := ξ2 ± |n|2α (1.19) and f̂ t,x,y(τ ; ξ, n) denotes the Fourier transform f̂ t,x,y(τ ; ξ, n) = ∫ ∞ −∞ ∫ ∞ −∞ ∫ 2π 0 f(t;x, y)e−i(tτ+xξ+ny)dt dx dy with (τ ; ξ, n) ∈ R × R × T. The next inequality will be useful to establish some future estimates. Lemma 1.4. Let λ be a positive number such that 0 < λ < 1. Then, it holds that (a+ b)λ ≤ aλ + bλ, (1.20) for all a, b ≥ 0. For the sake of completeness we give a proof for this inequality. Proof. The cases a = 0 or b = 0 are obvious, so we consider a, b strictly positive. Since λ ∈ (0, 1) we have a a+ b < ( a a+ b )λ and b a+ b < ( b a+ b )λ , which imply 1 < ( a a+ b )λ + ( b a+ b )λ , (1.21) for all a, b > 0. The inequality (1.20) is an immediate consequence of (1.21). □ EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 7 2. Bilinear estimate in the hyperbolic case In this section, we present the proof of the key bilinear estimate, which allows to get the L4 Strichartz estimate for U− α , with α > 1. To derive this bilinear estimate we need, as in the elliptic case treated in [19], to obtain uniform estimates of the measures for certain special sets (see [19, Lemma 2.1]). In our context the corresponding sets are unbounded and to estimate their measures is necessary to deal with series estimation. In particular, the convergence of such series occurs in the case α > 1 and fails in the case 0 < α ≤ 1. We start by showing a similar result to that present in [19, Lemma 2.1], which in our context reads as follows. Lemma 2.1. Let α > 1, ξ0 ∈ R, n0 ∈ Z and C > 0. Then for all K ≥ 1, the set GK := { (ξ, n) ∈ R× Z : C ≤ ∣∣(ξ − ξ0) 2 − 1 2 ( |n|2α + |n− n0|2α )∣∣ ≤ C +K } satisfies the estimate sup (ξ0,n0,C)∈Λ m(GK) ≲α K, (2.1) where Λ := R× Z× R+. Proof. By translation invariance it suffices to consider the case ξ0 = 0. First we write GK = G+ K ∪̇ G− K , (2.2) where G+ K = GK ∩ { (ξ, n) ∈ R× Z : ξ2 > 1 2 ( |n|2α + |n− n0|2α )} , (2.3) G− K = GK ∩ { (ξ, n) ∈ R× Z : ξ2 ≤ 1 2 ( |n|2α + |n− n0|2α )} . (2.4) Next we estimate the measures of the sets G+ K and G− K . Estimate of m(G+ K). Using (2.3) we write G+ K = { (ξ, n) ∈ R× Z : C ≤ ξ2 − 1 2 ( |n|2α + |n− n0|2α ) ≤ C +K } . (2.5) Let l such that l ≥ 1 and define h(l) := m ({ (ξ, n) ∈ R× Z : ξ2 − 1 2 ( |n|2α + |n− n0|2α ) ≤ l }) ; (2.6) then m ( G+ K ) = h(C +K)− h(C). (2.7) Note that for a fixed n ∈ Z we have∣∣{ξ ∈ R : ξ2 − 1 2 ( |n|2α + |n− n0|2α) ≤ l }∣∣ = 2 √ 1 2 ( |n|2α + |n− n0|2α ) + l; therefore, h(l) = 2 ∞∑ n=0 √ 1 2 ( |n|2α + |n− n0|2α ) + l + 2 ∞∑ n=1 √ 1 2 ( |n|2α + |n+ n0|2α ) + l. (2.8) 8 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 Figure 1. The dashed region represents the set G+ K with α = 1.3 and n0 = 0. Then, from(2.7) and (2.8) it follows that m ( G+ K ) = S+ 1 + S+ 2 , (2.9) where S+ 1 = 2 ∞∑ n=0 √ 1 2 ( |n|2α + |n− n0|2α ) + C +K − 2 ∞∑ n=0 √ 1 2 ( |n|2α + |n− n0|2α ) + C S+ 2 = 2 ∞∑ n=1 √ 1 2 ( |n|2α + |n+ n0|2α ) + C +K − 2 ∞∑ n=1 √ 1 2 ( |n|2α + |n+ n0|2α ) + C. For S+ 1 we have S+ 1 = 2 ∞∑ n=0 K√ 1 2 ( |n|2α + |n− n0|2α ) + C +K + √ 1 2 ( |n|2α + |n− n0|2α ) + C , then S+ 1 ≤ 2 √ 2 ∞∑ n=0 K√ n2α + C +K ≤ 2 √ 2 ( K√ C +K +K ∞∑ n=1 1 nα ) ≲α K, (2.10) where in the last estimate it has been used that α > 1 and C +K ≥ 1. In a similar way one obtains S+ 2 ≲α K. Therefore, from (2.9) we have m ( G+ K ) ≲α K. (2.11) EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 9 Estimate of m(G− K). This case is more delicate. We write G− K = { (ξ, n) ∈ R× Z : −(C +K) ≤ ξ2 − 1 2 ( |n|2α + |n− n0|2α ) ≤ −C } (2.12) and observe that G− K ⊂ G− 1,K ∪ G− 2,K , (2.13) where G− 1,K = { (ξ, n) ∈ R× Z : |n− n0|2α − (C +K) ≤ ξ2 ≤ |n|2α − C } , G− 2,K = { (ξ, n) ∈ R× Z : |n|2α − (C +K) ≤ ξ2 ≤ |n− n0|2α − C } . Indeed, G− 1,K contains the points of G− K with |n − n0| ≤ |n| and G− 2,K those that satisfy |n− n0| > |n|. Similar arguments to those used to estimate G+ K give us m ( G− 1,K ) = 4 ( ∞∑ n=⌊C 1 2α ⌋+1 √ n2α − C − ∞∑ n−n0=⌊(C+K) 1 2α ⌋+1 √ (n− n0)2α − C −K ) , which implies m ( G− 1,K ) = 4 ( ∞∑ n=⌊C 1 2α ⌋+1 √ n2α − C − ∞∑ n=⌊(C+K) 1 2α ⌋+1 √ n2α − C −K ) := 4(S− 1 + S− 2 ), (2.14) where S− 1 = ⌊(C+K) 1 2α ⌋∑ n=⌊C 1 2α ⌋+1 √ n2α − C, S− 2 = ∞∑ n=⌊(C+K) 1 2α ⌋+1 (√ n2α − C − √ n2α − C −K ) . To estimate S− 1 we observe that S− 1 ≤ ∫ ⌊(C+K) 1 2α ⌋ C 1 2α √ z2α − C dz + √ ⌊(C +K) 1 2α ⌋2α − C ≤ ∫ (C+K) 1 2α C 1 2α √ z2α − C dz︸ ︷︷ ︸ J1 + √ K. (2.15) To estimate the integral J1 we use that z 7→ √ z2α − C is an increasing function to obtain J1 ≤ √ K ( (C +K) 1 2α − C 1 2α ) . Then, applying Lemma 1.4 (remember that α > 1) one obtains J1 ≤ √ KK 1 2α ≤ K, where in the last estimate it has been used that K ≥ 1 and α > 1. Therefore, from (2.15) and the estimate for J1 we conclude that S− 1 ≲ K. 10 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 Now we estimate S− 2 as follows S− 2 = ∞∑ n=⌊(C+K) 1 2α ⌋+1 K√ n2α − C + √ n2α − C −K ≤ ∞∑ n=⌊(C+K) 1 2α ⌋+1 K√ n2α − C ≤ K√( ⌊(C +K) 1 2α ⌋+ 1 )2α − C + ∫ ∞ (C+K) 1 2α K√ z2α − C dz︸ ︷︷ ︸ J2 ≤ √ K + J2. (2.16) Now we proceed to estimate the integral J2. Considering the nonlinear change of variables ρ2α = z2α − C, and using that α > 1, C > 0 and K ≥ 1, we have J2 = K ∫ ∞ K 1 2α ρ2α−1 ρα ( ρ2α + C )1− 1 2α dρ ≤ K ∫ ∞ K 1 2α ρ−αdρ = 1 α− 1 KK 1−α 2α ≲α K. (2.17) Then, inserting (2.17) in (2.16) one obtains S− 2 ≲α K. Therefore, from (2.14) and the estimates obtained for S− 1 and S− 2 we have m ( G− 1,K ) ≲α K. By the same way we have m ( G− 2,K ) ≲α K. So, m ( G− K ) ≤ m ( G− 1,K ) +m ( G− 2,K ) ≲α K, which joint with (2.11) give us (2.1). This completes the proof of Lemma 2.1. □ Remark 2.2. For the case 0 < α ≤ 1 the result stated in Lemma 2.1 fails. Indeed, the series S+ 1 in (2.10) diverges for all 0 < α ≤ 1. Proposition 2.3 (Hyperbolic bilinear estimate). Let u1 and u2 be two functions in L2(R× R× T) with the following support properties: supp(ûj) ⊆ H− Kj , j = 1, 2, where H− Kj := { (τ ; ξ, n) : 1 2Kj ≤ ∣∣τ + ξ2 − |n|2α ∣∣ ≤ 2Kj } . (2.18) Then we have the inequality ∥u1u2∥L2(R×R×T) ≲ (K1K2) 1 2 ∥u1∥L2(R×R×T)∥u2∥L2(R×R×T). (2.19) EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 11 Proof. Set (τ2; ξ2, n2) := (τ−τ1; ξ−ξ1, n−n1). Using the Cauchy-Schwarz inequality and Plancherel’s theorem, we have ∥u1u2∥2L2 txy = ∫ τ ∫ ξ ∑ n∈Z ∣∣∣ ∫ τ1 ∫ ξ1 ∑ n1∈Z 1Aτξn (τ1; ξ1, n1) · û1t,x,y(τ1; ξ1, n1) × û2 t,x,y (τ2; ξ2, n2)dτ1dξ1 ∣∣∣2 dτ dξ ≲ sup τ,ξ,n m(Aτξn)∥u1∥2L2 txy ∥u2∥2L2 txy , (2.20) with Aτ,ξ,n defined as follows: (τ1; ξ1, n1) ∈ Aτ,ξ,n ⇐⇒ { (τ1; ξ1, n1) ∈ supp(û1 t,x,y ) and (τ2; ξ2, n2) ∈ supp(û2 t,x,y ). Notice that if (τ1; ξ1, n1) ∈ Aτ,ξ,n, then τ1 ∈ J1 := [ a1 − 2K1, a1 + 2K1 ] and τ1 ∈ J2 := [ a2 − 2K2, a2 + 2K2 ] , where a1 = |n1|2α − ξ21 and a2 = τ + (ξ − ξ1) 2 − |n− n1|2α. Hence τ1 ∈ J1 ∩ J2, with |J1 ∩ J2| ≤ 4min{K1,K2}. (2.21) On the other hand, for all (τ1; ξ1, n1) ∈ Aτ,ξ,n we can use the triangle inequality to eliminate τ1 and obtain∣∣∣(ξ1 − ξ 2 )2 − 1 2 ( |n1|2α + |n− n1|2α ) + τ 2 + ξ2 4 ∣∣∣ = 1 2 ∣∣τ + ξ21 − |n1|2α + (ξ − ξ1) 2 − |n− n1|2α ∣∣ ≤ 1 2 (∣∣τ1 + ξ21 − |n1|2α ∣∣+ ∣∣τ − τ1 + (ξ − ξ1) 2 − |n− n1|2α ∣∣) < K1 +K2. (2.22) So, if (τ1; ξ1, n1) ∈ Aτ,ξ,n, then (ξ1, n1) ∈ Bτ,ξ,n, (2.23) where Bτ,ξ,n := { (ξ1, n1) ∈ R×Z : ∣∣(ξ1− ξ 2 )2− 1 2 ( |n1|2α+|n1−n|2α ) + τ 2 + ξ2 4 ∣∣ ≤ K1+K2 } . This is the reason that led us to consider GK slightly different from the set that should naturally replace the used in [19, Lemma 2.1]. Now, from the triangle inequality, we note that Bτ,ξ,n ⊂ { (ξ1, n1) ∈ R×Z : C ≤ ∣∣(ξ1−ξ 2 )2−1 2 ( |n1|2α+|n1−n|2α )∣∣ ≤ C+2(K1+K2) } with C = ∣∣ τ 2 + ξ2 4 ∣∣ − (K1 +K2). Hence, if C > 0 we use Lemma 2.1 (with ξ0 = ξ 2 and n0 = n) to obtain m(Bτ,ξ,n) ≲ K1 +K2. Otherwise, if C < 0 we have m(Bτ,ξ,n) ≤ m ({ (ξ1, n1) : ∣∣(ξ1 − ξ 2 )2 − 1 2 ( |n1|2α + |n1 − n|2α )∣∣ ≤ 2(K1 +K2) }) 12 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 and consequently m(Bτ,ξ,n) ≤ lim ε↘0 m ({ (ξ1, n1) : ε < ∣∣(ξ1−ξ 2 )2−1 2 ( |n1|2α+|n1−n|2α )∣∣ ≤ 2(K1+K2) }) , so using again Lemma 2.1 it follows that m(Bτ,ξ,n) ≲ K1 +K2. (2.24) Finally, collecting the information in (2.21), (2.23) and (2.24) one obtains m(Aτ,ξ,n) ≤ |J1 ∩ J2| ·m(Bτ,ξ,n) ≲ min{K1,K2}max{K1,K2} and inserting this in (2.20) we obtain ∥u1u2∥2L2 txy ≲ K1K2∥u1∥2L2 txy ∥u2∥2L2 txy . Then, the result is proved. □ 3. Bilinear estimate in the elliptic case Now we prove the corresponding bilinear estimate for the elliptic symbol of the operator. Lemma 3.1. Let α ≥ 1, ξ0 ∈ R, n0 ∈ Z and C ≥ 1. Then for all K ≥ 1, the set G̃K := { (ξ, n) ∈ R× Z : C ≤ (ξ − ξ0) 2 + 1 2 ( |n|2α + |n− n0|2α ) ≤ C +K } satisfies the estimate sup (ξ0,n0,C)∈Λ m(G̃K) ≲α K, (3.1) where Λ := R× Z× R+. Proof. As in Lemma 2.1 it suffices to consider the case ξ0 = 0. Notice that for all K ≥ 1, G̃K ⊂ G̃1,K ∪ G̃2,K , (3.2) where G̃1,K = { (ξ, n) ∈ R× Z : C − |n|2α ≤ ξ2 ≤ C +K − |n− n0|2α } , G̃2,K = { (ξ, n) ∈ R× Z : C − |n− n0|2α ≤ ξ2 ≤ C +K − |n|2α } . Indeed, G̃1,K contains the points of G̃K with |n − n0| ≤ |n| and G̃2,K those that satisfy |n− n0| > |n|. Similar analysis as performed in the hyperbolic case shows that m(G̃1,K) = 2 ⌊(C+K) 1 2α ⌋∑ |n−n0|=0 √ C +K − |n− n0|2α − 2 ⌊C 1 2α ⌋∑ |n|=0 √ C − |n|2α = 2 ⌊(C+K) 1 2α ⌋∑ |n|=0 √ C +K − |n|2α − 2 ⌊C 1 2α ⌋∑ |n|=0 √ C − |n|2α := S̃1 + S̃2, (3.3) where S̃1 = 2 ⌊C 1 2α ⌋∑ |n|=0 (√ C +K − |n|2α − √ C − |n|2α ) , EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 13 S̃2 = 2 ⌊(C+K) 1 2α ⌋∑ |n|=⌊C 1 2α ⌋+1 √ C +K − |n|2α. To estimate S̃1 we use that S̃1 = 2 ⌊C 1 2α ⌋∑ |n|=0 K(√ C +K − |n|2α + √ C − |n|2α ) ≤ 2 ⌊C 1 2α ⌋−1∑ |n|=0 K√ C − |n|2α + 2K√ C +K − ⌊C 1 2α ⌋2α ≤ 4K ∫ C 1 2α 0 dz√ C − z2α + 2 √ K. (3.4) Making now the change of variables z = C 1 2α ρ 1 α with α ≥ 1, from (3.4) we have S̃1 ≤ 4K αC α−1 2α ∫ 1 0 dρ ρ1− 1 α √ 1− ρ2 + 2 √ K ≲ 4K αC α−1 2α (∫ 1 0 dρ ρ1− 1 α + ∫ 1 0 dρ√ 1− ρ2 ) + 2 √ K ≲α K, (3.5) where it has been used that C ≥ 1 and α ≥ 1. Now we proceed to estimate S̃2. First note that S̃2 = 4 ⌊(C+K) 1 2α ⌋∑ n=⌊C 1 2α ⌋+1 √ C +K − n2α ≤ 4 √ K + ∫ ⌊(C+K) 1 2α ⌋ ⌊C 1 2α ⌋+1 √ C +K − z2αdz ≤ 4 √ K + ∫ (C+K) 1 2α C 1 2α √ C +K − z2αdz ≤ 4 √ K + √ K ( (C +K) 1 2α − C 1 2α ) . (3.6) On the other hand, Lemma 1.20 gives us that (C +K) 1 2α − C 1 2α ≤ K 1 2α (3.7) for all C ≥ 1 and α ≥ 1. Combining (3.6) and (3.7) we have S̃2 ≲ K, then the proof is complete. □ Remark 3.2. For the case 0 < α < 1 the result stated in Lemma 3.1 fails. Indeed, it is not difficult to see that the S̃1 satisfies (3.5) in the following way S̃1 ∼ ( 1 C α−1 2α + 1 ) K, where 1 C α−1 2α → +∞ as C → +∞ whenever 0 < α < 1. 14 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 In the same way as in the hyperbolic case, Lemma 3.1 implies the following result. Proposition 3.3 (Elliptic bilinear estimate). Let u1 and u2 be two functions in L2(R× R× T) with the following support properties: supp(ûj) ⊆ H+ Kj , j = 1, 2, where H+ Kj := { (τ ; ξ, n) : 1 2Kj ≤ ∣∣τ + ξ2 + |n|2α ∣∣ ≤ 2Kj } . (3.8) Then we have the inequality ∥u1u2∥L2(R×R×T) ≲ (K1K2) 1/2 ∥u1∥L2(R×R×T) ∥u2∥L2(R×R×T). (3.9) 4. Strichartz inequality on R× T This section is devoted to the proof of Theorem 1.1, which follows the same lines as previous proofs of related results in [19] and we reproduce a sketch of it for the sake of completeness. Proof of Theorem 1.1 (1.15). The first step is the proof of the following L4 Strichartz estimate in the Bourgain space Xb,0 α±. Step 1. Let α ≥ 1 in the case (+), α > 1 in the case (-) and b > 1/2. Then ∥u∥L4(R×R×T) ≲ ∥u∥Xb,0 α±(R×R×T) (4.1) for any u ∈ Xb,0 α±(R× R× T). Proof of Step 1. Let u a smooth function and consider the dyadic decomposition u(t;x, y) = ∞∑ k=0 u2k(t;x, y), with supp(û2k t,x,y ) ∈ H± 2k defined in (2.18). Using Proposition 2.3, Proposition 3.3 and that b > 1/2 one obtains ∥u∥2L4(R×R×T) = ∥uu∥L2(R×R×T) ≤ ∞∑ k1=0 ∞∑ k2=0 ∥u2k1u2k2 ∥L2(R×R×T) ≲ ∞∑ k1=0 ∞∑ k2=0 (2k12k2)1/2∥û2k1 t,x,y∥L2(R×R×T)∥ût,x,y2k2 ∥L2(R×R×T). (4.2) Since supp(û2k t,x,y ) ∈ H± 2k we have ⟨τ + q±(ξ, n)⟩ ≥ |τ + q±(ξ, n)| ≥ 2k−1. Then, it follows that ∥û2kj t,x,y∥L2(R×R×T) = (∫ τ ∫ ξ ∑ n∈Z ∣∣û2kj t,x,y (τ ; ξ, n) ∣∣2dτdξ)1/2 ≤ (∫ τ ∫ ξ ∑ n∈Z ⟨τ + q±(ξ, n)⟩2b 2(kj−1)2b ∣∣û2kj t,x,y (τ ; ξ, n) ∣∣2dτdξ)1/2 ≲ 2−bkj∥u∥Xb,0 α± , (4.3) EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 15 for j = 1, 2. So, inserting (4.3) in (4.2) and using that b > 1/2, we have ∥u∥2L4(R×R×T) ≲ ∞∑ k1=0 (2k1)1/2−b ∞∑ k2=0 (2k2)1/2−b∥u∥2 Xb,0 α± ≲ ∥u∥2 Xb,0 α± , (4.4) as claimed in (4.1). Step 2. Let δ > 0, I = [−δ, δ] and b > 1/2. Then ∥U± α (t)ϕ∥L4(I×R×T) ≲ (δ1/2 + δ1/2−b)∥ϕ∥L2(R×T), (4.5) for each ϕ ∈ L2(R× T). Proof of Step 2. Let ψ ∈ C∞ 0 be a cut-off function such that supp(ψ) ⊂ (−2, 2) and ψ(t) ≡ 1 on [−1, 1] and define ψδ(t) := ψ( tδ ). For b > 0, one obtains ∥ψδ∥Hb t ≲ δ1/2∥ψ∥L2 t + δ1/2−b∥ψ∥Ḣb t . Now, applying (4.1) (here we need b > 1/2), we have ∥U± α (t)ϕ∥L4(I×R×T) ≲ ∥ψδ(t)U ± α (t)ϕ∥L4(R×R×T) ≲ ∥ψδ(t)U ± α (t)ϕ∥Xb,0 α± (R×R×T) = ∥ψδ∥Hb t ∥ϕ∥L2(R×T) ≲ (δ1/2 + δ1/2−b)∥ϕ∥L2(R×T), so (4.5) is proved. □ Finally, the estimate (4.5) can be extended to an arbitrary interval I in the same way as in [19]. This completes the proof of Theorem 1.1-(1.15). Proof of Theorem 1.1 (1.16). Consider the linear operator A : L2(R× T) −→ L4(I × R× T), defined by Aϕ := U± α (t)ϕ. Due to the estimate (1.15) we observe that A is bounded. Let us denote by A∗ the adjoint operator of A, then A∗(f) = ∫ I U± α (−t′)f(t′; ·)dt′, which is bounded from L4/3(I × R× T) to L2(R× T), i.e.,∥∥∫ I U± α (−t′)f(t′; ·)dt′ ∥∥ L2(R×T) ≤ C∗ I ∥f∥L4/3(I×R×T) for some constant C∗ I , depending only on the measure of I. Therefore, AA∗ = ∫ I U± α (t− t′)f(t′; ·)dt′, is bounded from L4/3(I × R× T) to L4(I × R× T), satisfying∥∥∫ I U± α (t− t′)f(t′; ·)dt′ ∥∥ L4(I×R×T) ≤ CIC ∗ I ∥f∥L4/3(I×R×T). Finally, to complete the proof of (1.16) is used the arguments in [17, Lemma 3.1]. □ 16 A. J. CORCHO, L. P. MALLQUI EJDE-2025/16 Acknowledgments. This research was partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior-Brasil (CAPES)-Finance Code 001. A. J. Corcho first author wassupported by CNPq/Brazil grant no. 307616/2020- 7 and “Beatriz Galindo” research position at the University of Córdoba/Spain. Finally, we wish to thank the referees for their comments and suggestions that actually improved this article.. 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Self-focusing and Wave Col- lapse, Appl. Math. Sci., 139, Springer (1999). [19] H. Takaoka, N. Tzvetkov; On 2D Nonlinear Schrödinger Equation with Data on R × T, Journal of Functional Analysis, 182 (2001), 427–442. [20] Y. Wang; Periodic cubic Hyperbolic Schrödinger equation on T2, Journal of Functionl Anal- ysis, 265 (2013), 424–434. [21] V. E. Zakharov, A. B. Shabat; Exact theory of two dimensional self modulation of waves in nonlinear media, Sov. Phys. J.E.T.P., 34 (1972), 62–69. [22] H. Xu; Unbounded Sobolev trajectories and modified scattering theory for a wave guide non- linear Schrödinger equation, Math. Z. 286 (2017), 443–489. EJDE-2025/16 FRACTIONAL SCHRÖDINGER EQUATION ON CYLINDERS 17 Adán J. Corcho Universidad de Córdoba - UCO, Departamento de Matemáticas, Campus de Rabanales. 14071, Córdoba, Spain Email address: a.corcho@uco.es Lindolfo P. Mallqui Universidade Federal do Rio de Janeiro - UFRJ, Instituto de Matemática, 21941-909, Rio de Janeiro - RJ, Brazil Email address: lindolfciencias@gmail.com 1. Introduction 1.1. Case bold0mu mumu =1=1=1=1=1=1 1.2. Notion of criticality in isotropic Sobolev spaces for bold0mu mumu > 0> 0> 0> 0> 0> 0 1.3. Main results 1.4. Comments 1.5. Notation and an elementary inequality 2. Bilinear estimate in the hyperbolic case 3. Bilinear estimate in the elliptic case 4. Strichartz inequality on RT Acknowledgments References