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The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper we give a saturated fixed point principle for nonself operators on a large Kasahara space. Our results extend some results given by I.M Olaru and N.A. Secelean in A New Approach of Some Contractive Mappings on Metric Spaces, Mathematics, 2021 9(12). [ABSTRACT FROM AUTHOR]

This paper presents a series of results about the fuzzy degree of a hypergoup. It is considered a sequence of membership functions and of join spaces, obtained by starting with a hypergroupoid (H, ⊗), see Corsini's paper [Southeast Asian Bull. Math. 27 (2003) 221-229]. The fuzzy grade is the minumum natural number i such that two consecutive associated join spaces, of the above mentioned sequence, Hi and Hi+1 are isomorphic. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study the interpolation properties of anisotropic net spaces Np,q(M), where p = (p1; ...., pn), q = (q1 ...., qn). It is shown that, with respect to the multidimensional interpolation method, the following equality holds [ABSTRACT FROM AUTHOR]

In the paper, the Lorentz space Lq,τ (Tm) of periodic functions of several variables, the Nikol'skii--Besov class Sq,τ,θr B and the associated class Wq,τ, a,b,r 1q,τ, q, τ1 and Sq,τ,θr B in the norm of the space Lp,τ2(Tm) for different relations between the parameters q,τ1,p,τ1,a,θ.The proofs of the theorems are based on the constructive method developed by V.N. Temlyakov. [ABSTRACT FROM AUTHOR]

Let χ∈H1(Σh,Q)$\chi \in H^1(\Sigma _h,\mathbb {Q})$ denote a rational cohomology class, and let Hχ$\operatorname{H}_\chi$ denote its Hodge norm. We recover the result that Hχ$\operatorname{H}_\chi$ is a plurisubharmonic function on the Teichmüller space Th${\mathcal {T}}_h$, and characterize complex directions along which the complex Hessian of Hχ$\operatorname{H}_\chi$ vanishes. Moreover, we find examples of χ∈H1(Σh,Q)$\chi \in H^1(\Sigma _{h},\mathbb {Q})$ such that Hχ$\operatorname{H}_\chi$ is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering π:Σh→Σ2$\pi:\Sigma _{h}\rightarrow \Sigma _2$ such that the subgroup of H1(Σh,Q)$H_1(\Sigma _{h},\mathbb {Q})$ generated by lifts of simple curves from Σ2$\Sigma _2$ is strictly contained in H1(Σh,Q)$H_1(\Sigma _{h},\mathbb {Q})$. Finally, combining the characterization theorem with the Riemann–Roch, and the Li–Yau [Invent. Math. 69 (1982), no. 2, 269–291] gonality estimate, we show that geometrically uniform covers of Σg$\Sigma _g$ satisfy the Putman–Wieland Conjecture about the induced Higher Prym representations. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize the boundedness and compactness of the differences for composition operators on the weighted Bergman spaces with Békollé weights in the open unit ball and estimate the norm and essential norm of weighted composition operators. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]

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The flanged parallel-plate waveguide is analysed based on the method of Kobayashi potential (KP) using Fourier function space. The presentation of the method is free from intricate mathematics. Standard integral identities are used for problem formulation, without direct use of Weber-Schafheitlin (WS) integrals. The Fourier function space is exploited for the construction of the governing linear equations instead of Jacobi polynomials. A simple strategy is suggested for the evaluation of the required improper integrals. Near-field results are validated through convergence analysis. Far-field patterns are compared with predictions of the surface equivalence theorem (SET). [ABSTRACT FROM AUTHOR]

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of L p vector fields defined on a domain Ω that is either a bounded domain in R d or R d itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The L p compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of vector fields defined on a domain that is either a bounded domain in or itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. [ABSTRACT FROM AUTHOR]

Recently, Kayumov [Lobachevskii J. Math. 38 (2017), pp. 466–468] obtained a sharp estimate for the n-th truncated area functional for normalized functions in the Bloch space for n\le 5 and then, together with Wirths [Lobachevskii J. Math. 40 (2019), pp. 1319–1323], extended the result for n=6. We prove that for the functions with non-negative Taylor coefficients, the same sharp estimate is valid for all n. For arbitrary functions, we obtain an estimate that is asymptotically of the same order but slightly larger (roughly by a factor of 4/e). We also consider related weighted estimates for functionals involving the powers n^t, t>0, and show that the exponent t=1 represents the critical case for the expected sharp estimate. [ABSTRACT FROM AUTHOR]

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space. [ABSTRACT FROM AUTHOR]

To approximate a function, some simpler functions are used: simpler as definitory structure (formula) and also as regularity, continuity and/or smoothness. Any Lipschitz function is a uniformly continuous function, but conversely it not always true. We add to the usual examples some other, especially of the class of differentiable functions. [ABSTRACT FROM AUTHOR]

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [EMS Surv. Math. Sci. 7 (2020), 207–251] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces. [ABSTRACT FROM AUTHOR]

Let Sj(z)=εj+(z-εj)/2 be an iterated function system, where εj=e2jπi/3 for j=0,1,2. Then, there exists a uniform self-similar measure μ supported on a compact set K, which is the attractor of {Sj}²j=0. The Hausdorff dimension of the attractor K is α=log3/log2. Let F(z)=∫K(z-ω)-1dμ(ω) be the Cauchy transform of μ. In this paper, we consider the Hardy space and the multiplier property of F. We prove that F′ belongs to Hp fo 0

In this paper, we consider the multiplicity and concentration phenomenon of positive solutions to the following Choquard equation - ε 2 Δ u + V (x) u = ε - α Q (x) (I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ - 2 u + f (u) in R N , where N ≥ 3 , (N - 4) + < α < N , I α is the Riesz potential, ε is a small parameter, V (x) ∈ C (R N) ∩ L ∞ (R N) is a positive potential, f ∈ C 1 (R + , R) is a subcritical nonlinear term and 2 α ∗ = N + α N - 2 is the upper-critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. By means of variational methods and delicate energy estimates, we establish the relationship between the number of solutions and the profiles of potentials V and Q, and the concentration behavior of positive solutions is also obtained for ε > 0 small. [ABSTRACT FROM AUTHOR]

Let K and S be locally compact Hausdorff spaces and let T be a real linear isomorphism of an extremely regular subspace A of C_{0}(K) into C_{0}(S) satisfying \|T \| \ \|T^{-1}\|<2. Put L=T \ \|T^{-1}\| and pick 0 \leq \varepsilon <1 such that \|T \| \ \|T^{-1}\|=1+\varepsilon. We prove that there exist a subset S_0 of S, a proper map \varphi of S_{0} onto K and a map \lambda :S_{0} \to \{-1, 1\} such that \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For \varepsilon < 1/2 the maps \varphi and \lambda are continuous, and this approximation of L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where \varepsilon =0, K and S are compact and A=C_{0}(K). The approximation of L in the general case where 0 \leq \varepsilon <1 is an extension of a Benyamini's result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz's theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K is a continuous image not just of a subset of S but of a locally compact subset of S. Moreover, when K is compact, it is a continuous image not just a closed subset of S but of a compact subset of S, namely the set S_0 having the above properties. [ABSTRACT FROM AUTHOR]

We show that the norm in the Hardy space H^p satisfies \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f')(1-r)^{p\left (1-\frac 1q\right)}\,dr+|f(0)|^p\tag {\dagger } \end{equation} for all univalent functions provided that either q\ge 2 or \frac {2p}{2+p}

Let m ∈ N, uj, j = 1,m, be holomorphic functions on the open unit ball B ⊂ Cn, φ be a holomorphic self-map of B, and Dl be the partial derivative operator in the lth variable l ∈ {1,2,...,n}. We introduce here the following polynomial differentiation composition operator PD,φm f : = ∑j=1m ujCφDlj Dl1 f and give some necessary and sufficient conditions for the boundedness and compactness of the operator from the logarithmic Bloch spaces to weighted-type spaces of holomorphic functions on B. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

Certain convergence estimates for the semi-exponential operators associated with 1 + x² are established including weighted approximation, a direct result involving classical modulus of continuity and quantitative Voronovskaya-type theorem in weighted spaces. Initially, the moments are obtained following a recurrence relation for these operators and then some direct results are estimated. [ABSTRACT FROM AUTHOR]

V. Gupta introduced recently the Rathore type operators Rn,c. For them we obtain Voronovskaja type results. We extend the classical Szász-Mirakjan operator and compare the extension with Rn,c. [ABSTRACT FROM AUTHOR]

In this article, two unique common fixed point theorems for four self-maps on a metric-like space by using the concept of occasionally weakly biased maps of type (A) are proved. Our results represent an improvement and extension of some fixed point findings. We justify our results by giving two appropriate examples. [ABSTRACT FROM AUTHOR]

The composition Fn of Rathore and Gamma operators was considered in the literature. We introduce a generalization of Fn. For it we determine the eigenstructure and establish the corresponding Voronovskaja type formula. [ABSTRACT FROM AUTHOR]

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time the associated sequence of empirical measures converges in a renormalized H ˙ - 1 sense to a probability measure with density ω 0 ∈ L ∞ (R 2) and having finite energy as the number of point vortices N → ∞ , then the sequence converges in the weak-* topology for measures to the unique solution ω of the 2D incompressible Euler equation with initial datum ω 0 , locally uniformly in time. In contrast to previous results Schochet (Commun Pure Appl Math 49:911–965, 1996), Jabin and Wang (Invent Math 214:523–591, 2018), Serfaty (Duke Math J 169:2887–2935, 2020), our theorem requires no regularity assumptions on the limiting vorticity ω , is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty (J Am Math Soc 30:713–768, 2017) and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich (USSR Comput Math Math Phys 3:1407–1456, 1963) for global well-posedness of 2D Euler with vorticity in the scaling-critical function space L ∞ (R 2) . [ABSTRACT FROM AUTHOR]

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional Fp.(a:, t) l,p \ (b(x')t)q in the unit ball B of Rn, where 1 < p < q < 8 and b(·) is a non-negativc bounded function on B which is Holder continuous of order 0 (0,1]. [ABSTRACT FROM AUTHOR]

We give characterizations for the parabolicity of regular trees. [ABSTRACT FROM AUTHOR]

We contribute to the development of equivalence of fixed point iterative sequences for multivalued mappings in modular function spaces, by proving the equivalence of convergence of implicit Mann, implicit Ishikawa, implicit Noor, implicit multistep iterative sequences for multivalued ρ− quasi-contractive-like mapppings in modular function spaces. An example is provided to support the applicability of the results. This work is complementary to equivalence results on normed and metric spaces in the literature. [ABSTRACT FROM AUTHOR]

In this article we prove short time local well-posedness in low-regularity Sobolev spaces for large data general quasilinear Schrödinger equations with a nontrapping assumption. These results represent improvements over the small data regime considered by the authors in Marzuola et al. (Adv Math 231(2):1151–1172, 2012; Kyoto J Math 54(3):529–546, 2014), as well as the pioneering works by Kenig et al. (Invent Math 158:343–388, 2004; Adv Math 196(2), 402–433, 2005; Adv Math 206(2):373–486, 2006), where viscosity methods were used to prove the existence of solutions for localized data in high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors. [ABSTRACT FROM AUTHOR]

Let 0 < γ1 < ⋯ ≤ γk ≤ ⋯ be the sequence of imaginary parts of nontrivial zeros of the Riemann zetafunction. In [R. Macaitienė and D. Šiaučiūnas, Joint universality of Hurwitz zeta-functions and nontrivial zeros of the Riemann zeta-function, Lith. Math. J., 59(1):81–95, 2019] a joint universality theorem on the approximation of analytic functions by shifts of the Hurwitz zeta-functions ζ(s + ihγk, α1), ..., ζ(s + ihγk, αr) has been obtained. In the paper, we prove universality theorems for the compositions F(ζ(s + ihγk, α1), ..., ζ(s + ihγk, αr)) for some classes of operators in the r-dimensional space of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper we provide a complete description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X, E), where X is a compact subset of R with at least two points and E is a strictly convex normed space. Among the consequences of the main result, it is shown that generalized bi-circular projections are the only projections on AC(X) expressed as the convex combination of two surjective linear isometries, and an affirmative answer is given to the question in Botelho and Jamison (Canad Math Bull 53:398–403, 2010) for such spaces. [ABSTRACT FROM AUTHOR]

In this paper, we characterize spaces of L∞-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work is analogous to established results concerning invariant spaces of continuous and measurable functions on a compact Hausdorff space. The case for L∞-functions cannot be proved in the same way when endowed with the norm-topology, but a similar argument can be used when the space of L∞-functions is given the weak*-topology, as we show in this paper.

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G := ∥ x → ∥ 2 ∂ 0 + x → ∑ k = 1 3 x k ∂ k , according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory. 2019;13(6):2527–2539; Colombo F, González-Cervantes JO, Sabadini I. A non-constant coefficients differential operator associated to slice monogenic functions. Trans Am Math Soc. 2013;365:303–318]. This association allows to show a behavior of the theory of slice regular functions similar to the well known theories of the hypercomplex analysis. [ABSTRACT FROM AUTHOR]

We show that if the Hardy–Littlewood maximal operator is bounded on a separable Banach function space X (R) and on its associate space X ′ (R) , then the space X (R) has an unconditional wavelet basis. This result extends previous results by Soardi (Proc Am Math Soc 125:3669–3673, 1997) for rearrangement-invariant Banach function spaces with nontrivial Boyd indices and by Fernandes et al. (Banach Center Publ 119:157–171, 2019) for reflexive Banach function spaces. We specify our result to the case of Lorentz spaces L p , q (R , w) , 1 < p < ∞ , 1 ≤ q < ∞ with Muckenhoupt weights w ∈ A p (R) . [ABSTRACT FROM AUTHOR]

According to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d. [ABSTRACT FROM AUTHOR]

Hecke conjectured that an explicit set of theta series obtained from a quaternion algebra defined over ℚ ramified at a prime N is a basis of a space of holomorphicmodular forms of weight 2 for the Hecke congruence group Γ0(N). However, Eichler noticed that Hecke's conjecture is not true in general. Hence it is natural to ask the dimension of the subspace of M2(Γ0(N)) spanned by the theta series, and this question is called Hecke's basis problem, which we have shown an answer in [K. Sugiyama, On the space of theta functions for a prime level, Comment. Math. Univ. St. Pauli, 67(1):66–81, 2019]. In this paper, we generalize the results for a square-free positive integer N. [ABSTRACT FROM AUTHOR]

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in Rn for n≥2 on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov–Morrey and Herz spaces considered in known works on Navier–Stokes equations. Morrey–Sobolev and Besov–Morrey spaces based on weak-Herz space denoted as WK˙p,qαMμs and WK˙p,qαN˙μ,rs, respectively, represent new properties and interpolations. This class of spaces and its developed properties could also be employed to study elliptic, parabolic, and conservation-law type PDEs.

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conjecture. [ABSTRACT FROM AUTHOR]

In paper we prove that:

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field ϕ on a globally hyperbolic spacetime M with C1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both ϕ and □gϕ in order to ensure that □g◦G± and G±◦□g are the identity maps on those spaces. The causal propagator G = G+ − G− is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(□g). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C*-algebras. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by Dunkl (Am Math Soc 179:331–348, 1973), Jewett (Adv Math 18(1):1–101, 1975) and Spector (Apercu de la theorie des hypergroups, (French) Analyse harmonique sur les groupes de Lie (Sém. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier–Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier–Stokes equations with constant viscosity, the main difficulty here lies in the fact that the L 1 in time Lipschitz estimate of the velocity field can not be obtained by energy method (see Danchin and Mucha in The incompressible Navier–Stokes equations in vacuum; Liao and Zhang in Arch Ration Mech Anal 220:937–981, 2016; Commun Pure Appl Math 72:835–884, 2019 for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (Chemin in Invent Math 103:599–629, 1991; Ann Sci École Norm Sup 26(4):517–542, 1993), namely, striated regularity can help to get the L ∞ boundedness of the double Riesz transform, we derive the a priori L 1 in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of H 5 2 regularity of the interface between fluids with different densities and viscosities. [ABSTRACT FROM AUTHOR]

Quite recently, a new minimal structure m*H has been introduced in [12] by using a minimal structure m and a hereditary class H. In this paper, we introduce and investigate an operator Γ*mH, (*)-strongly m-codense hereditary class H and a minimal structure m which is said to be m-compatible with a hereditary class H in a hereditary m-space (X, m, H). [ABSTRACT FROM AUTHOR]

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces, e.g., smooth, analytic and Gevrey. Extending the methods of Greenfield and Wallach (Trans Am Math Soc 183:153–164, 1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of results due to Greenfield (Proc Am Math Soc 31:115–118, 1972), Greenfield and Wallach (Proc Am Math Soc 31:112–114, 1972), as well as global versions of a result of Caetano and Cordaro (Trans Am Math Soc 363(1):185–201, 2011) for involutive structures. [ABSTRACT FROM AUTHOR]

Let C(X) denote the ring of all real-valued continuous functions on a topological space X; and C∞(X) be the subring of all functions C(X) which vanish at infinity. In [2], the paper "Rings of continuous functions vanishing at infinity," (Comment. Math. Univ. Carolin.45(3) (2004), 519–533), by A.R. Aliabad, F. Azarpanah, and M. Namdari, it is shown that for every completely regular Hausdorff space X, whenever C∞(X) ≠ (0), then there exists a locally compact space Y such that C∞(X) ≅ C∞(Y). In fact, the space Y may be considered as a nonempty open locally compact subspace of X. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. [ABSTRACT FROM AUTHOR]

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space E is isomorphic to its sublattice of bounded functions if and only if E has dimension one. The lattice of Lipschitz functions on E carries a "hidden" f-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital f-ring structure is provided. [ABSTRACT FROM AUTHOR]

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.