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# Associated spaces – a new tool of real and complex analysis

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... The concept of associated spaces comes from complex analysis: The space of holomorphic functions is associated to the complex differentiation d dz because the complex derivative of a holomorphic function is again holomorphic. Generalizing this idea, we say that a function space X is called an associated space to a given differential operator F if F transform X into itself [13,15]. Associated spaces are used to solve initial value problems of the type ...
... In order to apply a fixed-point theorem such as the contraction mapping principle, the operator (5) has to be estimated in a suitable function space. As F (τ, x, u, ∂ j u) in (5) also depends on the derivatives ∂ j u, the operator (5) maps a certain Banach function space into itself in case the derivatives ∂ j (Tu(t, x)) do exist and can be estimated in a suitable way, see [14,15]. There are two basic problems in the theory of associated spaces. ...
... There are two basic problems in the theory of associated spaces. The first one is the direct problem, which consists on the construction of an associated space X to a given operator F , whereas the second one is the inverse problem, dealing on finding an operator F defined on a given space X such that X is associated to F [15]. ...
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We are giving a characterization of all linear first order partial differential operators with Clifford-algebra-valued coefficients that are associated to the meta-q-monogenic operator. As an application, the solvability of initial value problems involving these operators is shown.
... ( where ' satisfies a partial differential equation G.!/ D 0. Then this problem is solvable provided that the initial function ' belongs to the associated space X of F containing all the solutions for G.!/ D 0, and that the elements of X satisfy an interior estimate, i.e., an estimate for the first order derivatives of the solutions (see [5]). Generalized Complex Numbers are defined as complex numbers of the form z D x C iy where the product of two complex numbers is induced by the relation i 2 D ˇi ˛; for˛andˇreal numbers subject to the ellipticity condition 4˛ ˇ2 >0 (see [6]). ...
... First order interior estimates can be obtained via a generalized version of the Cauchy Pompeiu operator for elliptic numbers [1] and thus the method of associated operators [5] is applied to solve initial value problems with initial functions that are generalized analytic in elliptic complex numbers. ...
... b.z/G.z/ D 0 and so G is generalized analytic and the term G 4 can be omitted from (5). We now look for a function satisfying @ z ! ...
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We find all linear first order partial differential operators with elliptic complex numbers-valued coefficients that are associated to an elliptic generalized-analytic operator. As an application, the solvability of initial value problems involving these operators is shown.
... The pair of operators (l, L) is also called an associated pair. See [17]. ...
... The concept of associated spaces generalizes the connection between holomorphic right-hand sides and holomorphic initial data: if the initial data belong to an associated space of the right-hand side, whose elements satisfy an interior estimate, then the initial value problem is uniquely solvable via the Banach fixed point theorem. See [17]. The integro-differential equation ...
... • The operator maps the space into itself. This means we have to use the concept associated pair (see [17,19]). • The elements of the associated space satisfy a first-order interior estimate (see [17,20,22]), that is, the norms of the derivatives with respect to space-like variables of the elements u of the associated space can be estimated by the norms of the elements. ...
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Consider the initial value problem $$\label{ivp} \begin{array}{lll} \partial_{t}u & = & \cL(t,x,u,\partial_{x_{i}}u),\\ u(0,x) & = & \varphi(x), \end{array}$$ where $t$ is the time, $\cL$ is a linear first order differential operator and $\varphi$ is a generalized $q-$metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by W. Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work we formulate sufficient conditions on the coefficients of the operator $\cL$ under which this operator is associated to the space of generalized $q-$metamonogenic functions satisfying a differential equation with anti$-q-$me\-tamonogenic right-hand side, when $q$ and $\lambda$ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases $\cA^{*}_{2,2}$ and $\cA^{*}_{3,2}$. In conical domains, the initial value problem (\ref{ivp}) is uniquely solvable for an operator $\cL$ and for any generalized $q-$metamonogenic initial function $\varphi$, provided an interior estimate hold for generalized $q-$metamonogenic functions satisfying a differential equation with anti$-q-$metamonogenic right hand side. The solution is also a generalized $q-$metamonogenic function for each fixed $t$. This work generalizes the results given in (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).
... In order to apply a fixed-point theorem, the operator (1.3) has to be estimated in a suitable function space whose elements depend only on the spacelike variable x. As F in (1.3) also depends on the derivatives ∂ xi u, the operator (1.3) maps a certain Banach function space into itself in case that the derivatives ∂ xi (T u(x, t)) exist and can be estimated in a suitable way [9,10]. This can be done by the so-called interior estimate for the elements of the associated space. ...
... In this work we are interested in dealing with the problem of finding first order differential operators F associated to the given space of q-monogenic functions, where q is an (n+1)-dimensional constant vector or a vector-valued function of x. This problem is one of the so-called inverse problems in the theory of associated spaces [10]. We show, in the context of a Clifford-type algebra and in the quaternions, the necessary and sufficient conditions on the coefficients of the operator F that make this possible. ...
... , q n real constants. Therefore we can apply the following theorem [10]: ...
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This paper is aimed at finding all linear first order partial differential operators $${\mathcal{F}}$$ with parameter-depending Clifford-algebra-valued coefficients, that are associated to the generalized Cauchy-Riemann operator. In order to obtain the conditions on the coefficients of $${\mathcal{F}}$$, a Leibniz rule for functions with values in a more general Clifford-type algebra is proved. Using the theory of associated spaces, we show the construction of solutions of initial value problems involving these operators.
... Applying the method of associated spaces [6,7], N. Q. Hung [3] has recently solved the initial value problem ...
... In order to investigate for which operators the initial value problem (1.1), (1.2) is soluble with arbitrary generalized regular initial functions, one has to determine operators L which are associated with the operator D λ . Definition 1.1 ( [6,7]). Let F be a first order differential operator depending on t, x, u and on the first order partial derivatives ∂ xi u, while G is a differential operator with respect to the space variables x i with coefficients not depending on time t. ...
... The function space containing all solutions to the differential equation Gu = 0 is called an associated space of F. Initial value problem (1.1), (1.2) is soluble in the space of generalized regular functions satisfying the differential equation (1.3) by the method of associated spaces (see [6,7]) provided the following conditions hold: ...
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This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$where $${t \in [0, T]}$$ is the time variable, x runs in a bounded and simply connected domain in $${\mathbb{R}^{4}, \lambda}$$ is a real number, and $${\mathcal{D}}$$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator $${\mathcal{L}}$$ under which $${\mathcal{L}}$$ is associated with the operator $${\mathcal{D}_{\lambda}}$$ , i.e. $${\mathcal{L}}$$ transforms the set of all solutions of the differential equation $${\mathcal{D}_{\lambda}u = 0}$$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators $${\mathcal{L}}$$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u 0 by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.
... This proof generalizes a construction which was originally given by W. Walter in [11] for a proof of the classical Cauchy-Kovalevskaya Theorem. In order to solve this initial value problem not only in the holomorphic case, one can apply "the method of associated spaces" which is constructed by W.Tutschke in [2], [9] and [10]. The basic idea of this method is the following: ...
... • The operator maps the space into itself. Which means we have to use the concept "associated pair" (See [2], [9], [10]) • For element of the associated space one has an "interior estimate" (See [10], [6]), that is, the norms of the derivatives with respect to spacelike variables of the elements of the associated space can be estimated by the norms of the elements. ...
... • The operator maps the space into itself. Which means we have to use the concept "associated pair" (See [2], [9], [10]) • For element of the associated space one has an "interior estimate" (See [10], [6]), that is, the norms of the derivatives with respect to spacelike variables of the elements of the associated space can be estimated by the norms of the elements. ...
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Initial value problems (IVPs) of type $$\frac{\partial u}{\partial t} = L ( t, x, u, \frac{\partial u}{\partial x_j} ), \quad u(0, x) = \varphi (x)$$ can be solved by applying the method of associated spaces which is constructed by W.Tutschke (Teubner Leipzig and Springer-Verlag 1989). The present paper considers above IVPs in the space of Helmholtz-type generalized regular functions in the sense of quaternionic analysis. Using the Poisson integral formula, we shall prove an interior estimate for Helmholtz-type generalized regular functions and then give out conditions under which these IVPs are uniquely solvable.
... This shows that the equation (3) does not always have a solution even if F (t, x, u, ∂ j u) and ϕ(x) are infinitely many differen- tiables. The concept of associated spaces [2, 7, 9] leads to conditions under which the equation (3) has solution. This concept comes from complex analysis: In the holomorphic case (5), (6) the associated space is the space of holomorphic functions and the right hand side F (t, z, u, ∂ z u) transforms this space into itself. ...
... The solutions of the differential equation Gu = 0 form a function space called associated space to G. In [7, 8, 9] we can see that, if the initial function (2) satisfies an associated equation Gu = 0 and the elements of the associated space satisfy an interior estimate, then there exists a (uniquely determined) solution of the initial problem (1), (2) also satisfying the associated equation for each t. In this paper, we will use this technique in order to solve the initial problem (1), (2) when the operator G is given by ...
... Using the theory of associated spaces [2, 7, 9] the initial value problem (1), (2) can be solved. In this work, conditions for the associated pair (F , D) have been obtained in two ways: Given D, if the equations (18) or (19)-(21) are satisfied, we can get the conditions on the functions A (i) which allow to find F , reciprocally given F , through (26), we can determine the operator D. Thus one obtains immediately that the operator F sends monogenic functions into monogenic functions. ...
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We consider an initial value problem of type $$\frac{\partial u}{\partial t}={\cal F}(t,x,u,\partial_j u), \quad u(0,x)=\phi(x),$$ where $t$ is the time, $x \in \mathbb{R}^n$ and $u_0$ is a Clifford type algebra-valued function satisfying ${\bf D}u=\displaystyle\sum_{j=0}^{n}\lambda_j(x)e_j\partial_ju = 0$, $\lambda_j(x)\in \mathbb{R}$ for all $j$. We will solve this problem using the technique of associated spaces. In order to do that, we give sufficient conditions on the coefficients of the operators ${\cal F}$ and ${\bf D}$, where ${\cal F}(u)= \displaystyle\sum_{i=0}^{n}A^{(i)}(x)\displaystyle\partial_iu$ for $A^{(i)}(x) \in \mathbb{R}$ or $A^{(i)}(x)$ belonging to a Clifford-type algebra, such that these operators are an associated pair.
... This implies, especially, that no initial value problem is solvable for Lewy's equation. However the concept of associated spaces [13,14] leads to conditions under which initial value problems of type (1.1), (1.2) are uniquely solvable. This concept is originated from complex analysis: In the holomorphic case (1.3), (1.4) the associated space is the space of holomorphic functions and the right-hand side F(t, z, w, ∂ z w) transforms the space of holomorphic functions into itself. ...
... (see [13,14]). Then the function space containing all solutions to the differential equation Gu = 0 is called an associated space to F. The conditions for associated pairs are, generally speaking, only sufficient. If one has necessary and sufficient conditions, then one can determine all evolution equations for which all solutions of a given associated elliptic equation are possible initial data. ...
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Recently the initial value problem (Formula presented.) has been solved uniquely by N. Q. Hung (Adv. appl. Clifford alg., Vol. 22, Issue 4 (2012), pp. 1061-1068) using the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) in the space of generalized regular functions in the sense of quaternionic analysis satisfying the equation (Formula presented.), where (Formula presented.) and (Formula presented.) is the Dirac operator, x = (x1, x2, x3) is the space like variable running in a bounded domain in (Formula presented.), and t is the time. The author has proven only sufficient conditions on the coefficients of the operator (Formula presented.) under which (Formula presented.) is associated with the operator (Formula presented.) transforms the set of all solutions of the differential equation (Formula presented.) into solutions of the same equation for fixedly chosen t. In the present paper we prove necessary and sufficient conditions for the underlined operators to be associated. This criterion makes it possible to construct all linear operators (Formula presented.) for which the initial value problem with an arbitrary initial generalized regular function is always solvable.
... • Applications to differential equations, boundary and initial valued problems. • Construction of the associated spaces [41] or the necessary conditions in terms of structural sets [1]. • Study of the Vekua equation in the context of fractional calculus. ...
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In this paper, we present the development of fractional bicomplex calculus in the Riemann–Liouville sense, based on the modification of the Cauchy–Riemann operator using the one-dimensional Riemann–Liouville derivative in each direction of the bicomplex basis. We introduce elementary functions such as analytic polynomials, exponential, trigonometric, and some properties of these functions. Furthermore, we present the fractional bicomplex Laplace operator connected with the fractional Cauchy–Riemann operator.
... This implies, especially, that no initial value problem is solvable for Lewy's equation. However, the concept of associated spaces [41,42] leads to conditions under which initial value problems of type (1.1) and (1.2) are uniquely solvable. Null solutions of the right-hand side of Eq. (1.3) are called generalized analytic functions. ...
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In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy–Riemann operator in the Riemann–Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.
... En los últimos años, el tema de los problemas de valores iniciales en las álgebras de Clifford ha sido abordado a través del método de los espacios asociados [7,11,12]. En el 2015, con la ayuda de la ecuación (1.1) editada en [6], los autores de [4] lograron resolver los problemas de valores iniciales planteados en el álgebra A n . Posteriormente en el artículo [2], estos problemas se abordan en las álgebras de Clifford dependientes de parámetros A n (2, α j , γ ij ). ...
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Las álgebras de Clifford son álgebras asociativas y no conmutativas definidas a través de ciertas estructuras multiplicativas. En estas álgebras no siempre existe una fórmula explícita que permita expresar el producto entre los vectores de la base del espacio vectorial, tal como está propuesto en el álgebra An (ver [6]). En esta investigación se ofrece una expresión explícita para el producto de determinados elementos de la base del álgebra An(2, αj , γij ), lo cual representa la apertura para deducir cálculos que arrojen nuevos aportes en el análisis de Clifford con parámetros.
... Definition 1 (see [4,5]) Let F be a first order differential operator depending on t, x, u and on the first order partial derivatives ∂ xi u, while G is a differential operator with respect to the space variables x i with coefficients not depending on time t. Then F is said to be 'associated' with G if F transforms the set of all solutions to the differential equation Gu = 0 into solutions of the same equation for fixedly chosen t, i.e. ...
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Consider the initial value problem $$\begin{array}{rcl} \dfrac{\partial u(t,x)}{\partial t} & \hspace{-0.1cm}=\hspace{-0.1cm} & \mathcal{L}u(t,x):=\sum\limits_{A,B,i}C_{B,i}^{(A)}(t,x)\dfrac{\partial u_{B}(t,x)}{\partial x_{i}}e_{A}, \\ u(0,x) & \hspace{-0.1cm}=\hspace{-0.1cm} & u_{0}(x)% \end{array} \label{ivp}$$% where the desired function $u(t,x)=\sum\limits_{B}u_{B}(t,x)e_{B}$ defined in $[0,T]\times \Omega \subset \mathbb{R}_{0}^{+}\times \mathbb{R}^{n+1}$ is a \textsc{Clifford}-Algebra-valued function with real-valued components $% u_{B}(t,x).$ This paper is aimed at solving the initial value problem (\ref{ivp}) by the contraction mapping principle in Banach Spaces of monogenic functions equipped with a weighted sup-norm.
... ABSTRACT In this paper we make a comprehensive literature review on the method of associated spaces proposed by Tutschke 7,9,11 in order to guarantee the existence and uniqueness of the solution of initial value problems in Clifford analysis. We also present an interior estimate for monogenic functions using the Cauchy integral formula. ...
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RESUMEN En el presente artículo se hace una revisión bibliográfica exhaustiva sobre el método de espacios asociados propuesto por Tutschke 7, 9, 11 para garantizar la existencia y unicidad de la solución de problemas de valores iniciales en análisis de Clifford. Además presentamos un estimado interior para funciones monogénicas haciendo uso de la fórmula integral de Cauchy. Palabras clave: Problemas de valor inicial, Operadores asociados, Espacios asociados, Estimados interiores, Principio de contracción, ´ Algebras de Clifford dependiendo de parámetros, funciones Monogénicas. ABSTRACT In this paper we make a comprehensive literature review on the method of associated spaces proposed by Tutschke 7, 9, 11 in order to guarantee the existence and uniqueness of the solution of initial value problems in Clifford analysis. We also present an interior estimate for monogenic functions using the Cauchy integral formula.
... u Ω (see [13]). ...
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This paper deals with the initial value problem of the type $$\partial _{t} u(t,x) = {\mathcal{L}}u \left( {t,x} \right), \quad u(0,x) = u_0(x)$$ (0.1) where $$t \in \mathbb{R}^+_0$$ is the time, $$x \in \mathbb{R}^{n+1}, \, u_{0}(x)$$ is a generalized monogenic function and the operator $$\mathcal{L}$$, acting on a Clifford-algebra-valued function $$u\left({t,x} \right) = \sum\limits_{B} u _{B} \left( {t,x} \right)e_{B}$$ with real-valued components u B (t, x), is defined by $${\mathcal{L}}u(t, x) := \sum\limits_{A,B,i} {c_{B,i}^{(A)} \left(t,x \right) \partial _{x_{i}} u_{B} \,\left( t,x \right)e_A }+ \sum\limits_{A,B} {d_{B}^{(A)}} \left( t,x \right)\,u_{B}\left( {t,x} \right)e_{A} + \sum\limits_{A} g_{A} \left( t,x \right)e_{A}$$ We formulate sufficient conditions on the coefficients of the operator $${\mathcal{L}}$$ under which $${\mathcal{L}}$$ transforms generalized monogenic functions again into generalized monogenic functions. For such an operator the initial value problem (0.1) is solvable for an arbitrary generalized monogenic initial function u 0 and the solution is also generalized monogenic for each t.
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In this paper, we introduce dualistic contractive mappings and use such map- pings to prove some fixed point theorems. The results extend various comparable results existing in the literature. Moreover, we give examples that show the superiority and effectiveness of our results among corresponding fixed point theorems in partial metric spaces.
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In this paper we characterize all the first order differential operators, with paravector coefficients, associated with the q-monogenic operator. Then initial value problems involving these operators are always solvable provided that the initial function is a q-monogenic function. The underlying algebra is of Clifford type.
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Using the fixed point method and the weakly Picard operator technique, we obtain some abstract Ulam–Hyers stability results of the initial value problem of fractional differential equations in quaternionic analysis. Sufficient conditions for the existence of solutions of the initial value problem are given by the application of the method of associated spaces. An example is provided to illustrate these results.
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Initial value problems of type $$\partial_t{u} = L(t, x, u, \partial_{{x}_i}u),$$ (0.1) $$u(0, x) = \varphi(x),$$ (0.2)where t is the time, L is a linear first order operator in a Clifford Analysis and φ is a generalized metamonogenic function, can be solved by applying the method of associated spaces which is constructed by W.Tutschke (Teubner Leipzig and Springer-Verlag 1989). The present paper formulates sufficient conditions on the coefficients of operator L under which L is associated to differential equations with anti-monogenic right-hand sides. We shall exhibit an interior estimate for the generalized metamonogenic functions using the Cauchy integral formula and then give out conditions under which this initial value problem (0.1), (0.2) is uniquely solvable in the context of Clifford type algebras.
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In this paper we give a characterization of all linear first order partial differential operators with Clifford-algebra-valued coefficients that are associated to the generalized Cauchy-Riemann operator. In order to achieve our goal, we make use of a rule for D(u·v), where u and v are Clifford-algebra-valued functions. As an application, the solvability of initial value problems involving these operators is shown.
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In this paper we give estimates for the first order derivatives of Clifford-type algebras valued functions, in the L p -norm. As the functions are solutions of elliptic partial differential equations, the estimates follow from integral representations obtained using fundamental solutions.
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In this paper, we establish sufficient conditions for the existence of solutions for the initial value problem of fractional differential equations. We consider the initial value problem in the space of Helmholtz-type generalized regular functions in the sense of quaternionic analysis. The results are obtained by the application of the method of associated spaces and the fixed point theorem. An example is provide to illustrate results.
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Initial value problems of type can be solved by the contraction-mapping principle in case the initial function  belongs to an associated space whose elements satisfy an interior estimate. The present article proves such an interior estimate in the sup-norm for generalized monogenic functions. The proof is based on a representation of generalized monogenic functions by harmonic functions. That way the article is another example for the technique of transforming solutions of simpler partial differential equations into solutions of more general ones (an overview on such methods can be found in Begehr's and Gilbert's two volumes [Begehr, H.G.W. and Gilbert, R.P., 1992/1993, Transformations, Transmutations and Kernel Functions, Vol. I and II (Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc.)].
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