Adv Syst Sci Appl 2025; 1:12–21 Published online at https://ijassa.ipu.ru. On Linear Differential Equations on the Torus and Non-Standard Analysis Vladimir P. Burskii1* 1Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia Abstract: In this paper, we consider periodic boundary value problems for differential equations whose coefficients are trigonometric polynomials. We construct the spaces of generalized functions, where such problems have solutions. In particular, the solvability space of a periodic analogue of the Mizohata equation is constructed. We build also a periodic analogue and a generalization of the construction of the nonstandard analysis, where infinitely small are not only functions, but also functional spaces. To show that not all constructions on the torus lead to a simplification in compare with the plane, we consider a periodic analogue of the hypoelliptic differential operator and show that its number-theoretic properties are significant. In particular, it turns out that if a polynomial with integer coefficients is irreducible in the rational field, then the corresponding differential operator is hypoelliptic on the torus. Keywords: differential operator on the torus, linear differential equation on the torus, Mizohata equation, nonstandard analysis, hypoellipticity 1. INTRODUCTION Periodic boundary value problems for differential equations is a famous object both in mathematical education and research (see, for example, [1], [2], [3]). As Lax noted in [4], in the periodic theory of differential equations is free of some technical difficulties that arise in non-periodic theory. This makes it possible to create a more beautiful theory. A study of periodic boundary value problems for linear differential equations brings us to the wonderful world of functions on the torus. Since the torus as the product of a finite number of circles, dealing with the torus simplifies studying the behavior of functions of several variables by each variable separately. Moreover, the basis elements are eigenfunctions of linear differential operators with constant coefficients. In addition, the topology of the torus allows us to forget about the boundary and the behavior at infinity. This allows us to focus our attention on the only infinity that we face up in this way: the infinite dimension of functional spaces. There are many works devoted to periodic boundary value problems for differential equations considered as equations on the torus. See, for example, the books [1], [2], [3] and the references therein. However, the present paper has no intersection with them. A viewpoint to the structure of the set of infinitesimals presented in this paper is essentially different from those in the non-standard analysis (see, for example, [14]). The first mention of the presented results was published in Russian in in hard-to-find publications [15],[16] and it not published in other languages. In the future it would be interesting to consider, in particular, the spectral properties of differential operators on a torus in the spirit of elliptic theory. The author expresses his gratitude to Remizov A.O. for his assistance. ∗Corresponding author: burskii.vp@phystech.edu LINEAR DIFFERENTIAL EQUATIONS ON THE TORUS... 13 2. SPACES OF PERIODIC FUNCTIONS 2.1. Spaces of Periodic Functions The number of variables in this problem is not significant, and without loss of generality we shall consider the case of two variables. It is well known that every Fourier series (trigonometric series)∑ k∈Z⊕Z ake ikx, ∑ k |ak|2 < ∞, where ak ∈ C, k = (k1, k2), x = (x1, x2) ∈ R2, and kx = k1x1 + k1x2, defines a periodic square integrable function on x. Thus, such series form the space L2(T 2), where T 2 = R2/Z2 is the torus of dimension 2. For every m ∈ Z, m ≥ 2, define by Hm the space of 2π-periodic complex-valued functions in R2 such that ∥u∥2m = ∫ T 2 u(x)(1−∆)mu(x)dx < ∞, where ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 is the Laplace operator. All Hm are Hilbert spaces, namely, the famous Sobolev spaces. It is known (see, for example, [1]) that functions exp(ikx), where kx = k1x1 + k2x2, form an orthogonal basis in Hm and, consequently, every function f ∈ Hm is expandable into the Fourier series f = ∑ k∈Z⊕Z fke ikx converging to f in the topology of Hm. Further we shall consider trigonometric series with arbitrary real coefficients, not necessarily converging (such series are usually called formal). One can consider Hm as the vector space consisting of formal Fourier series with the finite norm ∥f∥2m = ∑ k (1 + k · k)m · |fk|2. This formula defines a norm in the space Hm with any real m. Consider the vector space Hm with the topology of the space RZ, in which Hm is continuously embedded. For m < 0, the space Hm is conjugate to the space H−m in the topology of the space H0 = L2(T 2). Moreover, can prove a more general statement: Proposition 2.1: Let E be a barreled vector topological space of 2π-periodic functions continuously embedded in H0. Let the system {eikx} be a basis in E, that is, for every v ∈ E there exists a unique sequence {vk} ⊂ E such that ∑ k2≤N vke ikx → v as N → ∞. Then the dual space E∗ is naturally isomorphic to a subspace of the space F = { u = ∑ k uke ikx : ∑ vke ikx ∈ E, ⟨u, v⟩ = ∑ ukvk < ∞ ∀v ∈ E } . Here the paring ⟨·, ·⟩ gives rise to the duality. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 14 V. P. BURSKII Proof This statement follows from the known fact that the Mackey topology in barrel spaces coincides with the original topology, since it is the strongest among all topologies consistent with duality (see [5]). There are several examples, which we shall use below: Example 2.1: The space of infinitely differentiable periodic functions H∞ = ∩ m Hm, whose total element has the form u = ∑ uke ikx, k2luk k→∞−→ 0 (∀ l). Also the conjugate space (H∞)∗ = H−∞ = ∪ m Hm, which is the space of periodic distributions whose Fourier coefficients tend to infinity no faster than some power of k2l. Example 2.2: The space E0 of 2π-periodic functions u = ∑ uke ikx, ∃δ1 > 0, δ2 > 0 : ∑ e|k1|δ1+|k2|δ2|uk| < ∞, included in the space of periodic real analytic functions. Another examples is the conjugate space E∗ 0 , which consists of series ∑ uke ikx whose coefficients uk k→∞−→ ∞ slower than any exponential e|k1|δ1+|k2|δ2 . This space contains the space of hyperfunctions ([6]). Example 2.3: The space l1(|k1|!) of functions u = ∑ uke ikx, ∑ k |k1|! |uk| < ∞. Also the conjugate space l∗1(|k1|!) of series ∑ k vke ikx, vk = O(|k1|!). Let P (x1, x2) be a homogeneous polynomial of degree p with constant coefficients. Consider the differential operator P̂ : Hm → Hm−p generated by the polynomial P : P̂ u = P ( −i ∂ ∂x1 ,−i ∂ ∂x2 ) u. The obvious formula P̂ ( ∑ k∈Z⊕Z fke ikx ) = ∑ k∈Z⊕Z P (k1, k2)fke ikx allows us to consider the operator P̂ on the space F of formal trigonometric series. 2.2. Solvability of the Mizohata Equation In [7], G. Levy gave an example of a linear differential equation of the first order with infinitely differentiable coefficients that has no solutions in the space of distributions in three- dimensional space. Developing the ideas of G. Levy and P. Garabedyan [9], V. V. Grushin in [8] gave an example of a first order differential equation with infinity differentiable coefficients that has no solutions in the space of distributions on the plane: ∂u ∂x + ix ∂u ∂y = f(x, y). (2.1) Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) LINEAR DIFFERENTIAL EQUATIONS ON THE TORUS... 15 The operator in the left-hand side of equation (2.1) is one of the Mizohata operators, considered in [10]. The function f ∈ C∞ 0 (R2) is even by x, it was constructed by Grushin in a special way. Consider a periodic modification of equation (2.1): ∂u ∂x + i sinx ∂u ∂y = f̃(x, y), (2.2) where f̃ is 2π-periodic continuation of the function f mentioned above. It can be checked that the Grushin’s reasonings are also applicable to equation (2.2). We shall prove that equation (2.2) has a solution in a wider space of generalized functions than the space of Schwartz distributions. Proposition 2.2: For any even right-hand side f̃ ∈ H−∞ equation (2.2) has a unique periodic solution u(x1, x2) odd in the variable x1, which belongs to the space l∗1(|k1|!). Proof Let us write the equation (2.2) in the form ∂u ∂x1 + eix1 − e−ix1 2 ∂u ∂x2 = f̃ , which yields k1uk1,k2 + k2 2 (uk1−1,k2 − uk1+1,k2) = fk. (2.3) For a fixed k2 ̸= 0 we obtain the recurrent formula with respect to k1 uk1+1,k2 = 2 k2 (k1uk1,k2 − fk)uk2−1,k2 . (2.4) Since the function u(x1, x2) is odd in the variable x1, we have u0,k2 = 0, u−k1,k2 = −uk1,k2 . Therefore, the coefficients uk are uniquely determined by (2.4). From (2.4) we have the following estimation: |uk1+1,k2 | < ∑ j=0 (j + 1)!fk1−j < (k1 + 1)! ∑ k fk = c(k1 + 1)! Thus, the solution u = ∑ uke ikx belongs to the space l∗1(|k1|!). Note that the operations of differentiation and multiplication by a trigonometric polynomial defined formally in the space F , coincide with the analogues operations in the Banach space l∗1(|k1|!) defined as usual in spaces of generalized functions through pairing. Proposition 2.3: Every periodic solution u(x1, x2) of homogeneous equation (2.2) is even in x1 and it is uniquely determined by the functions u0(x2) := ∫ 2π 0 u(x1, x2)dx1 := ⟨u, 1⟩x1 , u1(x2) := ∫ 2π 0 u(x1, x2)e −ix1dx1 := ⟨u, eix1⟩x1 . It belongs to the space l∗1(|k1|!) if u0 and u1 have bounded sequences of coefficients. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 16 V. P. BURSKII The proof follows from formula (2.3). Here we consider the function u as a formal trigonometric series, and pairing along one coordinate is defined in the standard way: ⟨u, v⟩x1 := ∑ n 〈∑ k ukne ikx1 , ∑ m vmne imx1 〉 einx2 = = ∑ n (∑ m umnv−mn ) einx2 . It is clear that pairing on x1 does not always exist, but if t is a trigonometric polynomial, then the function ⟨t, v⟩x1(x2) exists. 2.3. Solvability of General Equations Now let us consider the general operator L : ∑ |α|≤m Tα(x)D α, where Tα(x) is a trigonometric polynomial of degree (s1α, s 2 α). It can also be written in the form L = s1∑ n1=−s1 s2∑ n2=−s2 einxPn(D), s1 := max α s1α, s2 := max α s2α. Let us assume that the operator L satisfies the following condition: Assumption 2.1: For every n the equation Pn(x) = 0 has no solutions in integers. Then the following generalization of Proposition 2.3 is true. Proposition 2.4: Under Assumption 2.1, every formal periodic solution u(x1, x2) of equation Lu = 0 is uniquely determined by the functions u02(x2) := ⟨u, 1⟩x1 , u12(x2) := ⟨u, eix1⟩x1 , . . . , us12(x2) := ⟨u, eis1x1⟩x1 , u01(x1) := ⟨u, 1⟩x2 , u11(x1) := ⟨u, eix2⟩x2 , . . . , us21(x1) := ⟨u, eis2x2⟩x2 . if they satisfy the following conditions: ⟨uq1, e ipx1⟩x1 = ⟨up2, e iqx2⟩x2 , ∀ p, q = 1, . . . ,min(s1, s2). The proof is by the direct substitution of formal series into the equation. 2.4. Linear Sections as Objects of Non-Standard Analysis Consider the vector space of formal trigonometric series F and define a relation of order u ≤ v in the following way. An element u ∈ F is more regular than v ∈ F or, equivalently, v is more singular than u if there exists an element h ∈ F with bounded positive coefficients hk such that u is the convolution of v and h: u = v ∗ h := ∑ k vkhke ikx. This is obviously equivalent to the condition ∃C > 0 : ∣∣∣un vn ∣∣∣ < C ∀n ∈ Z. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) LINEAR DIFFERENTIAL EQUATIONS ON THE TORUS... 17 Similar order relations are used in asymptotic expansions [11]. We shall the following definition: a subspace α ⊂ F is called a linear section of F if from v ∈ α it follows that u ∈ α for every u ≤ v. A trivial example: if v ∈ F , then the set of u ∈ F such that u ≤ v is a linear section of F . It is called the principal linear section of F . All the subspaces considered above are also linear sections. Note that the pairing considered in Section 1 naturally generates a Hausdorff topology on every vector subspace of F (see [5]). Therefore, every linear section α can be considered as a complete topological vector space, and the space α∗ consisting of g ∈ F such that ⟨f, g⟩ < ∞ for all f ∈ α is the dual space of α. The set M of all linear sections of F is ordered by the inclusion. For every two sections α and β there exist the linear sections sup(α, β) = α + β, inf(α, β) = α ∩ β defined as minimal (by the inclusion) linear sections that contain respectively α ∪ β or α ∩ β. It is easy to see that the distributivity relations are satisfied: α ∩ (β + γ) = α ∩ β + α ∩ γ, α + (β ∩ γ) = (α + β) ∩ (α + γ). Thus, in the set M a certain structure of the distributive lattice with additive and multiplicative identity elements is introduced. If A is an arbitrary set and {δa| a ∈ A} is a family of linear sections, then the supremum of sup δa is minimal linear section that contains all δa. By the Zorn lemma, the supremum of any family exists. 2.5. Linear Sections and Solvability of General Equations Proposition 2.5: 1. The operator L = ∑ |α|≤m Tα(x)D α sends any linear section to a linear section an, consequently, it induces a mapping (endomorphism) L̃ in the set M , which preserves the lattice structure. 2. Under Assumption 2.1, the mapping L̃ is an epimorphism of the lattice M and for every linear section G there exists the maximum β among those linear sections α for which L̃α ≤ G, and β ̸= F . Proof Since operations of differentiation, multiplication by a scalar, addition, and shifts {un} → {un+k} preserve the relation of order ≤, the mapping L sends every linear section to a linear section and the induced mapping preserves the relation of order and the lattice operations. From Assumption 2.1, it follows that every equation Lu = einx has a solution un among trigonometric polynomials. Further, if f is a formal series and the operator L satisfies condition 2.1, then the equation Lu = f = ∑ n fne inx is solvable in the class of formal series. Let such a solution be the formal series w = ∑ n,k fnunke ikx, where for each k the sum over n is finite, and the coefficients unk are uniquely defined by condition 2.1. Given linear section G, for every f ∈ G consider the set Wf of solutions to Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 18 V. P. BURSKII the equation Lu = f and the set of the corresponding principal linear sections. By the Zorn lemma, there exists the supremum s ∈ W̃f , which is the desired linear section β. Obviously, β does not coincide with F , since otherwise G = F . This completes the proof. Remark 2.1: Assumption 2.1 can be replaced with be replaced with a weaker condition: ∀m ∈ Z2 ∃n : Pn(m) ̸= 0. Definition 2.1: The linear section β constructed in Proposition 2.5 is called a solution of the equation Lu = G with linear section G in the right-hand side. For example, from what we proved above, it follows that solution of equation (2.2) with the right-hand side G = Hm is the section β ⊂ l∗1(|n1|!). Remark 2.2: The term “section” was chosen due to the obvious analogy with Dedekind sections when constructing the field of real numbers. Note also that the presented construction is consonant with some constructions of non-standard analysis related to the extension of the field of reals: the infinitesimal germs of functions from both a field and an ultrafilter. The set M of linear sections is only partially ordered, but it contains all the germs of sequences as principal sections. In the set M , there exists the operation of convolution, which is associative, commutative and distributive with respect to addition and intersection (however, the inverse exists not for all elements). Moreover, in M there exists the conjugation and all linear sections are reflexive spaces. In the set M one can also introduce an associative commutative product, which is not always defined, but covering the product of smooth functions and the product of distributions according to Mikusinsky–Hirata–Ogawa, and, in addition, the inverse element not lying in M as, so to speak, a singular section, containing not all more regular sequences, but all more singular, and so on. The most important thing here is that this set contains not only all functions, but also all functional spaces, which, along with each of their elements, also contain increasingly smooth elements. And now we can consider the question of solving the differential equation in a class of function spaces, if the given right-hand side is a space as it was in statement 5. 3. ON THE HYPOELLIPTICITY OF DIFFERENTIAL OPERATORS ON THE TORUS As we noted above, according to Lax’s statement ([4]), in the periodic case it is possible to construct a more beautiful theory. However, the periodic case is not always simpler that the non-periodic one. In this section, we characterize homogeneous differential operators with constant coefficients hypoelliptic in the space of periodic functions on the plane. This is one of the cases when the Lax’s statement is quite controversial. Let Hm, m ≥ 0, be the space of complex functions on the plane, 2π-periodic in the both arguments such that ∥u∥2m = ∫ T 2 u(x)(1−∆)mu(x)dx < ∞, ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 . The space H−m is defined as the space dual to Hm in the H0-topology. In Section 1, we noted that the functions exp(inx), nx = n1x1 + n2x2, Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) LINEAR DIFFERENTIAL EQUATIONS ON THE TORUS... 19 are an orthogonal basis in the Hilbert space Hm and, consequently, every function f ∈ Hm is presented as Fourier series f = ∑ n∈Z⊕Z fne inx converging to f in the topology of Hm. Thus, one can consider Hm, m ∈ R, as the space of formal Fourier series with the finite norm ∥f∥2m = ∑ n (1 + n · n)m · |fn|2 (see Section 1). These are the famous Sobolev spaces. Let P (x1, x2) be a homogeneous polynomial of the degree p ≥ 2 with constant coefficients. Consider the differential operator L : Hm → Hm−p acting according to the rule Lu = P ( − i ∂ ∂x1 ,−i ∂ ∂x2 ) u. According to [1], define the spaces of infinitely differentiable and generalized functions: H∞ = ∩ m Hm, H−∞ = ∪ m Hm. Definition 3.1: We shall call the operator L hypoelliptic, if for any u ∈ H−∞ the inclusion Lu ∈ H∞ implies u ∈ H∞. Lemma 3.1: The operator L is hypoelliptic if and only if the exist constants C > 0 and k1 such that |P (n)| > C(n2)k1 ∀n. (3.5) Proof Assume that there exist constants C > 0 and k1 such that (5) holds true and f = ∑ n fne inx ∈ H∞. Then fn/P (n) decreases faster than any power of n. If there exists a sequence of pairs nj such that |P (nj)| → 0 for j → ∞ faster than any power of |n|, then, for example, for the functions f = ∑ j P (nj)ein jx ∈ H∞, but solution ∑ j e injx ∈ H−2, and there is no hypoellipticity. Recall that in the case of two variables one of necessary and sufficient conditions of hypoellipticity is the following (see, e.g., [12]): There exist constants C and c such that∣∣∣P (α)(ξ) P (ξ) ∣∣∣ ≤ C |ξ|−|α|c for every multi-index α and any ξ ∈ R2 large enough. For α = p, the latter inequality gives the inequality |P (ξ)| ≥ C|ξ|pc, which coincides with inequality (3.5) on the integer lattice. This means that every operator hypoelliptic on the plane is hypoelliptic on the torus, but not vice versa. Proposition 3.1: The operator L is hypoelliptic if and only if for every real root α of the polynomial P (x, 1) Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) 20 V. P. BURSKII there exist constants C > 0 and k such that∣∣∣α− p q ∣∣∣ > C qk (3.6) for every rational p/q sufficiently close to α. Proof Let u ∈ H−∞, then u ∈ Hm with some m. Therefore, u = ∑ n une inx, Lu = ∑ n unP (n)einx. Let f = ∑ n fne inx. It is clear that f ∈ H∞ if and only if |fn| → 0 for n2 → ∞ faster than any power of n2. Note that if the operator L is hypoelliptic, then the equation P (n) = 0 has a unique integer solution n = 0. Indeed, assume that ν = (ν1, ν2) is another solution, then the function +∞∑ k=−∞ eikνx ∈ H−∞ belongs to the kernel of L, which contradicts the hypoellipticity. Therefore, the solution of the equation Lu = f can be formally written as u = ∑ n̸=0 fn P (n) einx + C0. The condition f0 = 0 is obviously a condition for the solvability of the equation Lu = f . Let us apply Lemma 3.1. Let α be a real root of the polynomial P (x, 1) of multiplicity r. The inequality |P (n)| > Cn2k1 is equivalent to∣∣P( n1 n2 , 1 )∣∣ > C(n2)(k1− p 2 ), since P ( n1 n2 , 1 ) tends to zero as n1 n2 tends to one of the roots. Thus, we get the inequality∣∣P( n1 n2 , 1 )∣∣ = ∣∣n1 n2 α ∣∣r∣∣P (r) x1 ( n1 n2 + τ ( α− n1 n2 )∣∣ > C(n2)(k1− p 2 ). For n1 n2 close enough to α, we have∣∣n1 n2 − α ∣∣ > Cn2k1 > Cn2k 2 . A direct calculation gives k = 1 r ( k1 − p 2 ) . On the contrary, if ∣∣n1 n2 − α ∣∣ > Cn2k 2 , then ∣∣n1 n2 − α ∣∣ > Cn2k(k < 0), whence we obtain |P (n)| > Cn2k1 . The proof is complete. Remark 3.1: Inequality (3.6) is not valid for some transcendental numbers α, for example, ∑∞ ν=1 1 10ν ! (see [13]). On the other hand, the Liouville theorem states that for every algebraic number α of degree ν inequality (3.6) holds true with k = ν (see [13]). In particular, we obtain the following Proposition 3.2: If a polynomial P with integer (or rational) coefficients is irreducible in the field Q, then the operator L is hypoelliptic. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) LINEAR DIFFERENTIAL EQUATIONS ON THE TORUS... 21 It is easy to see that the hypoelliptic operator L : H∞ → H∞ is reversible. Here and below, all spaces are assumed to be quotient by the subspace of constants. The inverse operator L−1 acts from Hm into Hk with some k. According to the Thue–Siegel–Roth theorem [13], for every algebraic number α of degree r ≥ 2 and any ε > 0 there exists C > 0 such that for every rational number p/q the inequality∣∣∣α− p q ∣∣∣ > c q2+ε . holds true. Then, using calculation of the exponents from the proof of Proposition 3.2, we obtain the following Proposition 3.3: Let r be the greatest multiplicity of real roots of the irreducible polynomial P . Then for every ε > 0 the operator L−1 acts from Hm into Hp/2+m−r−ε continuously. Remark 3.2: For p = 2, by the Liouville theorem, one can put ε = 0. REFERENCES 1. Bers, L., John, F., & Schechter, M. (1964) Partial differential equations, Am. Math. Soc., 196. 2. Bourbaki, N. (1987) Topological vector spaces, Springer-Verlag. 3. Burskii, V. P. (1980) On the solvability of the Garabedian-Grushin equation, In: Collection of scientific articles Boundary value problems for differential equations, Kiev: Naukova dumka, 35–39 [In Russian]. 4. Burskii, V. P. (2018) On differential operators and differential equations on torus, J. Samara State Techn. Univ., Ser. Phys. Math. Sci., 22:4, 607–619 [In Russian]. 5. Davis, M. (1977) Applied nonstandard analysis, NY: Wiley publication. 6. Dezin, A. A. (1987) Partial differential equations. An introduction to a general theory of linear boundary value problems, Springer-Verlag. 7. Fedoryuk, M. V. (1977) Metod Perevala, Moscow: Nauka [In Russian]. 8. Garabedian, P. R. (1970) An unsolvable equation, Proc. Am. Math. Soc., 25, 207–208. 9. Gelfond, A. O. (1960) Transcendental and algebraic numbers, New York: Dover Publications, Inc. VII. 10. Grushin, V. V. (1971) A differential equation without a solution, Math. Notes, 10:2, 499–501. 11. Hörmander, L. (1963) Linear Partial Differential Operators, Springer-Verlag. 12. Hörmander, L. (1983) The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients, Springer-Verlag. 13. Lax, P. D. (1955) On Cauchy’s problem for hyperbolic equations and the differentiability of elliptic equations, Comm. Pure Appl. Math., 6, 43–59. 14. Lewy, H. (1957) An example of a smooth linear partial differential equation without solution, Ann. of Math., 2:66, 155–158. 15. Mizohata, S. (1962) Solutions nulles et solution non analytiques, J. Math. Kyoto Unlv., 1, 271–302. 16. Ptashnik, B. I. (1984) Ill-posed boundary value problems for partial differential equations, Kiev: Naukova Dumka [In Russian]. Copyright © 2025 ASSA. Adv Syst Sci Appl (2025) Introduction Spaces of periodic functions Spaces of Periodic Functions Solvability of the Mizohata Equation Solvability of General Equations Linear Sections as Objects of Non-Standard Analysis Linear Sections and Solvability of General Equations On the hypoellipticity of differential operators on the torus