Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a regularity property (e.g. continuity, differentiability, etc.) on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on $x$ systematically. Afterwards, we apply this theoretical framework to several different regularity properties that are of interest also in semigroup theory. In particular, a generalisation of the prior results on eventual differentiability of strongly continuous functions $u:\mathbb{R}_{\geq 0} \rightarrow \mathcal{L}(X;Y)$ obtained by Iley and Bárta follows as a special case of our method.
We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra $A$ and a compact space $X$, let $\mathcal{A}$ be a Banach $A$-valued function algebra on $X$ and let $\mathfrak{A}$ be the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra $\mathcal{A}$ and the associated algebras $\mathfrak{A}$ and $A$. That $\mathcal{A}$ inherits a certain regularity conditio…
In this paper we discuss on the vector valued version of Caristi's theorem. We show that the regularity of the cone is an essential condition to reach the vector valued version of Caristi's theorem in vector valued metric spaces. It is shown that how the absence of the regularity causes some previous Caristi type theorems and results fail to be correct. Our main result give a vector valued version of Caristi's theorem for correspondences with weakened hypotheses in comparison with previous results and generalize them.
The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions…
Denoting by $\mathbb{M}$ the complexification of the quaternionic algebra $\mathbb{H}$, we characterize the family of those $\mathbb{M}$-valued functions, defined on subsets of $\H$, whose values are actually quaternions, using an intrinsic approach. In particular, we show that the slice quaternionic regularity can be characterized via a Cauchy type transform acting on the space of analytic $\mathbb{M}$-valued stem functions.
On Matrix Valued Schrödinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, Hölder Regularity and Dispersive Estimates
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学应用数学与系统研究所）；Institute of Analysis and Algebra, Technische Universität Braunschweig（布伦瑞克工业大学分析与代数研究所）
Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.
Nonlinear and orbital directional regularities for set-valued mappings and applications to fixed points
Faculty of Mathematics, “Alexandru Ioan Cuza” University（亚历山德鲁·约安·库扎大学数学系）；“Octav Mayer” Institute of Mathematics, Iaşi Branch of Romanian Academy（罗马尼亚科学院雅西分院奥克塔夫·梅耶尔数学研究所）；Department of Mathematics and Statistics, Quy Nhon University（归仁大学数学与统计系）；XLIM UMR-CNRS 7252, Université de Limoges（利摩日大学XLIM UMR-CNRS 7252）
Ioffe's criterion and various reformulations of it have become a~standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland's variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semi-continuous env…
We show that, for the $C_0$-semigroups of scalar type spectral operators, a well-known necessary condition for the generation of eventually norm-continuous $C_0$-semigroups, formulated exclusively in terms of the location of the spectrum of the semigroup's generator in the complex plane, is also sufficient and, in fact, characterizes the generators of immediately norm-continuous such semigroups. Combining characterizations of the immediate differentiability and the Gevrey ultradifferentiability of scalar type spectral $C_0$-semigroups with the generation theorem, found earlier by the author, w…
We develop a maximal regularity approach in temporally weighted $L_p$-spaces for vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii-Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal wei…
In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces $F_1^{α,c}(\mathbb{R}^d,\mathcal{M})$ obtained in our previous paper, we establish the $F_1^{α,c}$-regularity of regular symbols for every $α\in \mathbb{R}$, and the $F_1^{α,c}$-regularity of forbidden symbols for $α>0$. As applications, we obtain the same results on the usual and quantum tori.
We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\E \sup_{n\geq 1}\|ξ_n\|$, where the $ξ_n$ are independent centered Gaussian random variables with va…
Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions $f\colon F\to Y$ with pre-assigned derivatives, with preservation of differentiability (at every point where the pre-assigned derivative is actually a derivative), preservation of continuity, preservation of (pointwise) Lipschitz property and (for finite dimensional domain $X$) preservation of strict differentiability and global (eventually local) Lipschitz continu…
We discuss Lebesgue spaces $\mathcal{L}^p([a,b],E)$ of Lusin measurable vector-valued functions and the corresponding vector spaces $AC_{L^p}([a,b],E)$ of absolutely continuous functions. These can be used to construct Lie groups $AC_{L^p}([a,b],G)$ of absolutely continuous functions with values in an infinite-dimensional Lie group $G$. We extend the notion of $L^p$-regularity of infinite-dimensional Lie groups introduced by Glöckner to this setting and adopt several results and tools.
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential abstract parabolic equation are established. Particularly, it is proven that the operators generated by these pseudo-differential equations are positive and aso are generators of analytic semigroups. As an application, the anisotropic parameter dependent pseudo-differential equations and the system of pseudo-differential equations are studied
The operator-valued multiplier theorems in weighted abstract Besov spaces are studied. These results permit us to show embedding theorems in weighted Besov-Lions type spaces. The most regular class of interpolation space is found such that the mixed differential operator is bounded from Besov-Lions space to this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results the separability properties of degenerate differential operators are studied. Especially, we prove that the associated differential operators are positive and also are generators of analytic se…
Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence form elliptic operator on $\R^n$ with bounded measurable real-valued coefficients and let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our main result implies that the stochastic convolution process $$ u(t) = \int_0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\ge 0,$$ satisfies, for all $1\le p<\infty$, a conical maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{ T_2^{p,2}(\R_+\times\R^n)}^p \le C_p^p \E \n g \n_{ T_2^{p,2}(\R_+\times\R^n;H)}^p.$$ Here, $T_2^{p,2}(\R_+\times\R^n)$ and $T_2^{p,2}(\R_+\times\R^n;H)$ ar…
We consider differential equations of the form y'(t)=f(t,y(t)) on a (possibly infinite-dimensional) Lie group G, for f : [0,1] x G -> TG a time-dependent left invariant vector field with measurable (but not necessarily continuous) dependence on t. If a solution Evol(c):=y on [0,1] starting at the neutral element e of G exists for each f corresponding to an L^1-curve c : [0,1] -> g in the Lie algebra g of G, and Evol is smooth as a map from L^1([0,1],g) to C([0,1],G), then G is called L^1-regular. We show that all Banach-Lie groups are L^1-regular, as well as all direct limits of finite-dimensi…
Extending the construction of the (intrinsically defined) full algebra of scalar valued Colombeau functions on a smooth manifold M (Grosser et al., Adv. Math. 166 (2002), 179-206) we present a suitable basic space for eventually obtaining tensor valued generalized functions on M, via the usual quotient construction. This basic space canonically contains the tensor valued distributions and permits a natural extension of the classical Lie derivative. Its members are smooth functions depending - via a third slot - on so-called transport operators, in addition to slots one (smooth n-forms on M) an…
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occu…
