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In this paper, the maximal‐weighted Hölder regularity for the fractional abstract Cauchy problem is presented. First, two classes of novel interpolation spaces constructed by resolvent families are portrayed. Afterward, by applying the resolvent method and the properties of interpolation spaces, the maximal‐weighted Hölder regularity of the mild solution is established in weighted Hölder space C0β,γ(X)$C_0^{\beta ,\gamma }(X)$ (0≤γ≤β$0\le \gamma \le \beta$, 0<β<1$0<\beta <1$). Eventually, an instance for the fractional differential equation of 2m‐order multidimensional parabolic type is provided to verify the reasonability of main conclusions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1 / 3 , 1 / 2 ]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths. [ABSTRACT FROM AUTHOR]

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples. [ABSTRACT FROM AUTHOR]

We establish Arazy–Cwikel type properties for the family of couples (ℓ p , ℓ q) , 0 ≤ p < q ≤ ∞ , and show that (ℓ p , ℓ q) is a Calderón–Mityagin couple if and only if q ≥ 1 . Moreover, we identify interpolation orbits of elements with respect to this couple for all p and q such that 0 ≤ p < q ≤ ∞ and obtain a simple positive solution of a Levitina–Sukochev–Zanin problem, clarifying its connections with whether (ℓ p , ℓ q) has the Calderón–Mityagin property or not. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. [ABSTRACT FROM AUTHOR]

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains Ω ⊂ ℝ d with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by H 0 s (Ω ¯) , and Sobolev–Slobodeckij̆ spaces, denoted by H s (Ω) , where s > d 2 . Our goal is accomplished by obtaining the Green's solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green's kernels satisfying these problems are symmetric and positive definite reproducing kernels of H 0 s (Ω ¯) and H s (Ω) , respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains. [ABSTRACT FROM AUTHOR]

We investigate the existence, uniqueness and polynomial stability of the weighted pseudo almost automorphic solutions to a class of linear and semilinear parabolic evolution equations. The necessary tools here are interpolation spaces and interpolation theorems which help to prove the boundedness of solution operators in appropriate spaces for linear equations. Then for the semilinear equations the fixed point arguments are used to obtain the existence and stability of the weighted pseudo almost automorphic solutions. Lastly, our abstract results are applied to the Navier-Stokes equations (NSE) on some different circumstances such as the NSE on exterior domains, around rotating obstacles, and in Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems.

The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Moreover, we show that this property holds also for the class of all interpolation quasi-Banach spaces with respect to a quasi-Banach couple whenever all the couples involved have the uniform Calderón-Mityagin property. This complements and refines previous results by Arazy-Cwikel and Bykov-Ovchinnikov even in the case of Banach spaces. We apply results obtained to some classical families of spaces. [ABSTRACT FROM AUTHOR]

Abstract We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

In this paper, we first characterize the behaviour of fractional resolvent families on the real interpolation spaces (X , D (A)) θ , p , θ ∈ (0 , 1) , p ∈ [ 1 , ∞ ]. Second, we establish strict Hölder regularity in space and time to an abstract degenerate fractional order differential equations. We also provide an application to illustrate our results. [ABSTRACT FROM AUTHOR]

We study interpolation spaces obtained via a general discrete interpolation method based on a Banach space with an unconditional basis. We find conditions which guarantee that such interpolation spaces have the Opial property and the uniform Opial property. [ABSTRACT FROM AUTHOR]

We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP. [ABSTRACT FROM AUTHOR]

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14] ] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs. [ABSTRACT FROM AUTHOR]

We study kernel least-squares estimation under a norm constraint. This form of regularisation is known as Ivanov regularisation and it provides better control of the norm of the estimator than the well-established Tikhonov regularisation. Ivanov regularisation can be studied under minimal assumptions. In particular, we assume only that the RKHS is separable with a bounded and measurable kernel. We provide rates of convergence for the expected squared L² error of our estimator under the weak assumption that the variance of the response variables is bounded and the unknown regression function lies in an interpolation space between L² and the RKHS. We then obtain faster rates of convergence when the regression function is bounded by clipping the estimator. In fact, we attain the optimal rate of convergence. Furthermore, we provide a high-probability bound under the stronger assumption that the response variables have subgaussian errors and that the regression function lies in an interpolation space between L∞ and the RKHS. Finally, we derive adaptive results for the settings in which the regression function is bounded. [ABSTRACT FROM AUTHOR]

Abstract Let 1 ≤ p < ∞. A Banach lattice E is said to be disjointly homogeneous (resp. p -disjointly homogeneous) if two arbitrary normalized disjoint sequences from E contain equivalent in E subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in E to the unit vector basis of l p). Answering a question raised in the paper [11] , for each 1 < p < ∞ , we construct a reflexive p -disjointly homogeneous rearrangement invariant space on [ 0 , 1 ] whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to l p , 1 ≤ p < ∞ , or c 0. [ABSTRACT FROM AUTHOR]

Abstract We give a complete characterization of limiting interpolation spaces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. f ↦ f (t 2) ∕ t , or f ↦ f (t 1 ∕ 2)) acting on the underlying lattices that are used to control the K - and J -functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and characterized in this fashion. The resulting theory gives a unified roof to a large body of literature that, using ad-hoc methods, had covered only special cases of the results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain–Brezis–Mironescu–Maz'ya–Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided. [ABSTRACT FROM AUTHOR]

A new method is presented for simulation of nonlinear transmission line circuits based on proper orthogonal decomposition reduction techniques coupled with an efficient interpolatory algorithm. Evaluation of the nonlinear function and corresponding Jacobian is performed in the reduced domain. A key criterion is developed for a priori determination of the dimension of the interpolation space leading to a substantial reduction in the computational cost. The proposed algorithm is applicable to general nonlinear circuits and does not impose any constraints on the topology of the pertinent circuit or type of the nonlinear components. [ABSTRACT FROM AUTHOR]

It is a widely used method, for instance in perturbation theory, to associate with a given C 0 -semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L 2 ( R ) , the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff and already complete if we consider a C 0 -semigroup acting on a reflexive Banach space. If the space is Hilbert, then the inductive limit can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the latter duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space. [ABSTRACT FROM AUTHOR]

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider an infinite dimensional exponential family P of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space H, and show it to be quite rich in the sense that a broad class of densities on Rd can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in P. Motivated by this approximation property, the paper addresses the question of estimating an unknown density p0 through an element in P. Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between p0 and P, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. We propose an estimator pn based on minimizing the Fisher divergence, J(po||p) between p0 and p G P, which involves solving a simple finite-dimensional linear system. When p0 G P, we show that the pro- ...in Fisher divergence under the smoothness assumption that logp0 G R(Cß) for some ß > 0, where C is a certain Hilbert-Schmidt operator on H and R(Cß) denotes the image of Cß. We also investigate the misspecified case of p0 G P and show that J(p0||pn) ...infpep J(p0||p) as n...oe, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non-parametric kernel den. [ABSTRACT FROM AUTHOR]

Consider the abstract algebraic-delay differential system, xi (t) = Ax(t) + F(x(t), a(t)), a(t) = H(xt, at). Here A is a linear operator on D(A) ⊆ X satisfying the Hille-Yosida conditions, x(t) ∊ D(A) ⊆ X, a(t) ∊ Rn, and X is a real Banach space. Let C0 ⊆ D(A) be closed and convex, and K ⊆ Rn be a compact set contained in the ball of radius h > 0 centered at 0. Under suitable Lipschitz conditions on the nonlinearities F and H and a subtangential condition, the system generates a continuous semiflow on a subset of the space of continuous functions C([-h, 0], C0 ×Rn), which is induced by the algebraic constraint. The object of this paper is to find conditions under which this semiflow is also differentiable with respect to initial data. In the motivating example coming from modelling the dynamics of an age structured population, the nonlinearities F and H are not Fré-chet differentiable on the sets C0 × K and C([-h, 0], C0 × K), respectively. The main challenge of obtaining the differentiability of the semiflow is to determine the right type of differentiability and the right phase space. We develop a novel approach to address this problem which also shows how the spaces on which the derivatives of solution operators act reflect the model structure [ABSTRACT FROM AUTHOR]

We study two variants of measures of non-compactness of operators associated to a Banach operator ideal in the sense of Pietsch. These measures are motivated by the notions of surjective-ideal-compactness and injective-ideal-compactness, defined respectively by Carl and Stephani and by Stephani. Interpolation results on these measures in the cases of Banach couples generated by a single Banach space are given. As an application, we obtain interpolation theorems on p -compact operators and quasi p -nuclear operators. [ABSTRACT FROM AUTHOR]