Characterizations similar to (3.0.3) **and** (3.0.4) can be obtained for these **spaces**. One important thing when we research in complex analysis is to have a good variety **of** examples **of** **functions** which belong to the **spaces** we are working with. Because **of** that, the next section is devoted to explore the structure **of** Morrey **spaces**, characterizing for some typical classes **of** **analytic** **functions** C those **functions** in C which lie in the Morrey **spaces**, **and** paying attention to the differences **and** similarities with Hardy **spaces** **and** BM OA.

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Among **them**, maps that preserve the disjointness **of** cozeroes defined **between** **spaces** **of** scalar- valued continuous **functions** on locally compact **and** compact **spaces**, as a generalization **of** the concept **of** homomorphism, have a long history in functional analysis in the context **of** rings, algebras, or vector lattices under several names such as Lamperti **operators**, separating maps, disjointness preserving **operators**, etc. (see, for example, [1, 2, 3, 4, 5, 7, 8, 13, 16]). In recent years, certain attention has been given to such maps when defined on **spaces** **of** vector-valued continuous **functions** (see, e.g., [10, 14]). However, we do not know much about disjointness preserving maps on vector- valued settings in comparison with scalar-valued contexts **and** something similar can be said with regard to (algebra) homomorphisms **between** vector-valued group algebras.

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Besides, Chan **and** Shapiro studied in [5] the hypercyclicity **of** translations in the setting **of** Hilbert **spaces** **of** entire **functions** **of** “slow growth”; in [5] the posed question was whether translations in a “reasonable” space **of** entire **functions** are always hypercyclic, **and** it was shown that this is not true. For example, differentiation **and** translation **operators** in the Paley-Wiener space P W a are bounded but not hypercyclic; in particular, T w is an isometry **of**

Atomic decompositions, **and** even partial results **of** the same fashion, for func- tions in **spaces** **of** **analytic** **functions** are very useful in operator theory. In particular, they can be used to describe dual **spaces** [63] or to study basic questions such as the boundedness, the compactness or the Schatten class membership **of** concrete opera- tors [5, 7, 21, 29, 48, 54, 55, 69, 72]. In this study we will use Theorem 5.3.1 to describe those positive Borel measures µ on D such that the differentiation operator defined by D (n) ( f ) = f (n) for n ∈ N ∪ { 0 } is bounded from A p,q ω to the Lebesgue space

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Our aim here is to obtain some results on unbounded **analytic** **functions** on some open subsets **of** infinite dimensional real Banach **spaces**. The proofs, which need some particular techniques, will follow some ideas used in the proofs **of** similar re- sults recently obtained in the complex case, most **of** **them** by the authors. See [1-3] **and** [8].

Proof **of** Theorem 1.8, necessary condition. We have already considered the case **of** growth **functions** satisfying the Dini condition in Remark 2.3. We assume now that ̺ is **of** upper type 1/n, so that we have the factorization **of** molecules. We adapt the proof given in [BGS]. Let b ∈ H 2 ( B n ). We assume that h

Let A be a selfadjoint operator on a complex Hilbert space (H ; h ., . i ) . The Gelfand map establishes a ∗ -isometrically isomorphism Φ **between** the set C (Sp (A)) **of** all continuous **functions** defined on the spectrum **of** A, denoted Sp (A) , an the C ∗ -algebra C ∗ (A) generated by A **and** the identity operator 1

In the following theorems we describe some basic properties **of** Paley-Wiener vectors **and** show that they share similar properties to those **of** the classical Paley- Wiener **functions**. The next theorem, whose proof can be found in [15], shows that the space P W ω (D) has properties (A) **and** (B). See also [14, 16]

ε}. In the case that (X, k · k) is a Banach space, then we will assume that the distance **between** two elements x, y ∈ X is given by d(x, y) := kx −yk. By L(X ) we denote the space **of** all continuous linear mappings from X into X. Let B be a fundamental family **of** bounded subsets **of** X. For every n ∈ N , B ∈ B let us define the continuous seminorm p n,B (T) := sup x∈B p n (T x) on L(X). Then the system

In this paper, we study the lineability **of** the set **of** unbounded holomorphic **functions** deﬁned on a Banach space. To begin with, let us explain the main notation used in our work. Throughout the article, E will denote a complex Banach space **and** U will be an open subset **of** E. If x ∈ E **and** r > 0, then B (x, r) represents the open ball in E with center x **and** radius r. For each n ∈ N, P ( n E) denotes the

In this paper we have analyzed several **functions** proposed in the literature to simultaneously generalize weighted means **and** OWA **operators**. On the one hand, OWAWA **operators** do not generalize weighted means **and** OWA **operators** in the usual sense. On the other hand, the HWA **operators** are neither idempotent nor compensative while WOWA **operators** do not always provide the expected result. Due to the questionable behavior **of** these **operators**, we have imposed a condition to maintain the relationship among the weights **and** we have characterized the **functions** that satisfy this condition. However, the obtained **functions** have some problems in their definition **and** they are not monotonic. So, we can conclude that none **of** the analyzed **functions** is fully convincing.

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13. Torra, V.: The weighted OWA operator. Int. J. Intell. Syst. 12(2), 153–166 (1997) 14. Torra, V.: On some relationships **between** the WOWA operator **and** the Choquet in- tegral. In: Proceedings **of** the Seventh International Conference on Information Pro- cessing **and** Management **of** Uncertainty in Knowledge-Based Systems (IPMU’98). pp. 818–824. Paris (France) (July 1998)

An important aspect **of** the DPG method is the inherent guaranteed error control by the computed residuals. For early computational experiments with adaptivity based on these residuals, see [10]. In [4], an a posteriori error analysis is given that includes data approximation errors. When coupling the DPG method with other discretizations, the inherent residuals for the DPG approximation have to be combined with appropriate estimators. For the case **of** boundary elements, see [13, 14], **and** for the DPG method dealing with contact conditions, see [15]. In this paper, we do not specifically deal with a posteriori error estimation. Having set our heterogeneous formulation, we proceed to rewrite it by using the so-called trial-to-test operator (which maps the test space to the ansatz space). This is only done for the ultra-weak formulation. The whole system then transforms into one where **spaces** on the ansatz **and** test sides are identical. In this way, our heterogeneous variational formulation fits the Lax–Milgram framework just as in [14]. We prove coercivity under the condition that the trial-to-test operator is weighted by a sufficiently large constant. Then quasi-optimal convergence **of** a discretized version follows by standard arguments. When proving coerciv- ity we follow steps that are similar to the ones when studying the coupling **of** DPG with boundary elements. But whereas [14] analyzes only the Laplacian, here we set up the scheme **and** prove coercivity for a gen- eral second-order equation **of** reaction-advection-diffusion type. Throughout, we assume that our problem is uniformly well posed, i.e., we do not study variations for singularly perturbed cases as in [13]. Also note that, since coefficients are variable, transmission problems can be treated the same way by selecting the sub- domains accordingly. One only has to move the possibly non-homogeneous jump data to the right-hand side functional.

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It must be emphasized that we are not requiring our harmonic mappings to be locally univalent. In other words, the Jacobian need not be **of** constant sign in the domain Ω. The orientation **of** the mapping may reverse, corresponding to a folding in the associated minimal surface. It is also possible for the minimal surface to exhibit several sheets above a point in the (u, v)–plane. Thus the lifted mapping f e may be locally or globally univalent even when the underlying mapping f is not.

Ahlfors L. V. (1979) Complex Analysis: An Introduction to the Theory **of** **Analytic** **Functions** **of** one Complex Variable, Third Edition, Mcgraw Hill, New York. Conwey, John B. (1987) **Functions** **of** One Complex Variable: Secon Edition Edition, Springer - Verlag, New York. Domínguez P., Contreras A. y Cano L. FCFM, BUAP (2017) Marsden Jerrold E. (1999) Basic Complex Analysis: Third Edition, W.H. Freeman, New York

The concept **of** bilinear isometry can be naturally extended to the context **of** **spaces** **of** vector-valued continuous **functions**. Examples **of** bilinear isometries defined on these **spaces** can be found, for instance, in [7, Proposition 5.2], where the author provide certain compact **spaces** X **and** Banach **spaces** E for which there exists a bilinear isometry T : C(X, E) × C(X, E) −→ C(Y, E).

This definition was rewritten in [6] considering only annihilation **operators** prov- ing they are enough to define the Hermite-Sobolev space. In the same work, the authors deal with weighted Sobolev **spaces** for weights in the Muckenhoupt class A p , defined for 1 < p < ∞ , as the set **of** weights (non-negative **and** locally integrable

The constant changes undergone by society lead us to propose simple, diverse **spaces** (flexible **and** customi-‐ zable) that can be adapted to each person's needs. In conceptual terms, tablets are a good example **of** a device that adapts to the needs **of** a changing society. The experience **of** uniting university **and** business has led to the evident break with the traditional classroom space as the absolute protagonist **of** learning **and** requires expan-‐ sion into other areas, transforming the new environments into living mutable elements, where exchanges **and** contributions **of** the diverse actors involved are constant.

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It has been pointed out that there is a need for a detailed geometrical characterization **of** soil pore structure in order to better understand its role in soil functioning, including its contribution to accumulation **and** protection **of** soil organic matter, to optimization **of** soil water **and** air regimes, **and** to storage **and** availability **of** plant nutrients [von Lutzow et al., 2006]. Mathe- matical morphology [Serra, 1982] provides a set **of** non-fractal parameters (in the realm **of** classical non-fractal geometry) that has been used to characterize soil structures that were a priori dierent [San José Martínez et al., 2013]. Fol- lowing the rationale **of** this theory, a set **of** fractals parameters beyond V F D **and** SF D is needed to built an exhaustive fractal description **of** soil structure. The same rationale suggests the utility **of** measures **of** soil pore connectiv- ity. Recent studies **of** intra-aggregate porosity with 2D images **of** soil sections seem to suggest that lacunarity could be a suitable fractal parameter choice [Kravchenko et al., 2011; Sukop, 2001; Sukop et al., 2002]. Therefore, our nd- ings could improve knowledge **of** the fractal geometry **of** soil pore structure with a comprehensive geometrical description with fractal attributes **of** soil pore space as a 3D shape. These fractals attributes should probably include, but not restricted to, V F D , SF D **and** lacunarity.

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