UMD Banach Space Research Articles

Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions

In this paper we consider the space \({{{BMO}_o(\mathbb{R}, X)}}\) of bounded mean oscillations and odd functions on \({{\mathbb{R}}}\) taking values in a UMD Banach space X. The functions in \({{{BMO}_o(\mathbb{R}, X)}}\) are characterized by Carleson type conditions involving Bessel convolutions and γ-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain γ-radonifying Carleson inequalities for Bessel–Poisson integrals of \({{{BMO}_o(\mathbb{R}, X)}}\) functions hold.

Vector-valued stochastic delay equations—a semigroup approach

Let E be a type 2 umd Banach space, H a Hilbert space and let p∈[1,∞). Consider the following stochastic delay equation in E: $$ \left\{\begin{array}{l}dX(t)= AX(t) + C X_t + B(X(t),X_t)dW_H(t),\quad t>0;\\[5pt]X(0)=x_0;\\[5pt]X_0=f_0,\end{array}\right. $$ (SDE) where A:D(A)⊂E→E is the generator of a C0-semigroup. The operator \(C\in\mathcal{L}(W^{1,p}(-1,0;E),E)\) is given by a Riemann-Stieltjes integral, and B:E×Lp(−1,0;E)→γ(H,E) is a Lipschitz function. Moreover WH is an H-cylindrical Brownian motion adapted to Open image in new window and Open image in new window, Open image in new window. We prove that a solution to (SDE) is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution to (SDE).

Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in UMD Banach Spaces

In a UMD Banach space E, we consider a boundary value problem for a second order elliptic differential-operator equation with a spectral parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator in E. A theorem on an isomorphism is proved and an appropriate estimate of the solution with respect to the space variable and the spectral parameter is obtained. In this way, Fredholm property of the problem is shown. Moreover, discreteness of the spectrum and completeness of a system of root functions corresponding to the homogeneous problem are established. Finally, applications of obtained abstract results to nonlocal boundary value problems for elliptic differential equations with a parameter in non-smooth domains are given.

Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces

In this work, we present a new result ofexistence, uniqueness and maximal regularity for the restriction solutions of the bilaplacian transmission problem set in the juxtaposition of two rectangular bodies. The study is performed in the space $L^{p}( (-1, 0) \cup(0, \delta) ;X),$ $1 < p < \infty,$ where $\delta $ is a small parameter which is destined to tend to zero and $X$ is a UMD Banach space. The geometry of the bodies allows us to find an explicit representation of the solutions in virtue of the operational Dunford calculus. We then use essentially the famous Dore-Venni theorem among others for the analysis of the solutions.

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.

Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space–time Hölder regularity for solutions of the parabolic stochastic evolution equation { d U ( t ) = ( A U ( t ) + F ( t , U ( t ) ) ) d t + B ( t , U ( t ) ) d W H ( t ) , t ∈ [ 0 , T 0 ] , U ( 0 ) = u 0 , where A generates an analytic C 0 -semigroup on a UMD Banach space E and W H is a cylindrical Brownian motion with values in a Hilbert space H. We prove that if the mappings F : [ 0 , T ] × E → E and B : [ 0 , T ] × E → L ( H , E ) satisfy suitable Lipschitz conditions and u 0 is F 0 -measurable and bounded, then this problem has a unique mild solution, which has trajectories in C λ ( [ 0 , T ] ; D ( ( − A ) θ ) ) ) provided λ ⩾ 0 and θ ⩾ 0 satisfy λ + θ < 1 2 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations.

Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces

We use a direct approach for proving an isomorphism result for a general higher order abstract ordinary differential equation in a UMD Banach space. In fact, it gives maximal $L_p$-regularity property. As a consequence, we get some interpolation theorem (about intermediate derivatives). A situation of a higher order equation generated by one operator is also treated. Finally, we present some application to elliptic PDEs.

A Simplified Approach in the Study of Elliptic Differential Equations in UMD Spaces and New Applications

This paper is devoted to provide some new clarifications in the study of complete abstract second order differential equations of elliptic type given in our recent papers [8], [9] and [10]. We improve the situation in the case of the space Lp(0,1;X) with a UMD Banach space X by considering an approach generalizing those used in [10]. On the other hand, we give some new examples to which our theory applies.

Continuous local martingales and stochastic integration in UMD Banach spaces

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Itô isometry, the Burkholder–Davis–Gundy inequality, the Itô formula and the martingale representation theorem.

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces L p 1 , … , p N ( R n 1 × ⋯ × R n N ; X ) where X is a UMD Banach space satisfying Pisier's property ( α ) . These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ Lp(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result.

Stochastic integration in UMD Banach spaces

In this paper we construct a theory of stochastic integration of processes with values in ℒ(H, E), where H is a separable Hilbert space and E is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an H-cylindrical Brownian motion. Our approach is based on a two-sided Lp-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of ℒ(H, E)-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the Itô isometry, the Burkholder–Davis–Gundy inequalities, and the representation theorem for Brownian martingales.

Pseudodifferential operators on Bochner spaces and an application

We consider pseudodifferential operators with operator valued symbols a(x,ξ) acting on a UMD Banach space X. Assuming some regularity of Holder type in x and Mihlin type in ξ we prove L p (ℝ n ;X) boundedness of such operators. This result is then applied to the study of L p -maximal regularity for nonautonomous parabolic evolution equations.

MAXIMAL FUNCTIONS AND SUBORDINATION FOR OPERATOR GROUPS

AbstractLet $E$ be a UMD Banach space, and $L$ a positive self-adjoint operator in $\mathrm{L}^2$ of Laplace type, for which the imaginary powers $L^{-\ri t}$ form a $C_0$-group of exponential growth $0\leq\alpha\lt \pi$ on $\mathrm{L}^p(E)$, where $1\lt p\lt\lt \infty$. Suppose $G(z)$ is holomorphic inside and on the boundary of the sector $\{z:z\neq0,\ |\arg z|\leq\phi\}$, and $z^\kappa G(z)\rightarrow0$ uniformly as $z\rightarrow\infty$ for some $\kappa\gt0$ and $\phi\gt\alpha$. Then $G(tL)$ $(t \gt0)$ defines a bounded family of linear operators on $\mathrm{L}^p(E)$; and the maximal operator $f\mapsto\sup_{t \gt 0}\|G(tL)f\|_E$ is bounded on the domain of $\log L$. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrödinger equations, and for Cesàro sums.AMS 2000 Mathematics subject classification: Primary 47D03; 42B25; 47D09

A New Approach to the Dore -Venni Theorem

The present paper works out the link between the Dore-Venni theorem and the theory of analytic generators developped by I. Ciornescu and L. Zsid. The main result is an inverse theorem: on an UMD-Banach space, analytic generators of C0-groups and operators with bounded imaginary powers are the same. The maximal regularity theorem of G. Dore and A. Venni appears as a corollary of this fact.

A New Approach to the Dore -Venni Theorem

The present paper works out the link between the Dore-Venni theorem and the theory of analytic generators developped by I. Ciornescu and L. Zsid. The main result is an inverse theorem: on an UMD-Banach space, analytic generators of C0-groups and operators with bounded imaginary powers are the same. The maximal regularity theorem of G. Dore and A. Venni appears as a corollary of this fact.

Counterexamples concerning sectorial operators

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L p-space, with \(1 \le p \neq 2 \le \infty \), and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L p-space; one acts on L p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on \(L^2 ({\Bbb R}^2;X)\) admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property \((\alpha )\) (geometric property introduced by G. Pisier).

Multipliers for semigroups

Let L be a positive invertible self-adjoint operator in L2(X;C). Using transference methods for locally bounded groups of operators we obtain multipliers for the group of complex powers Liu on vector-valued Lebesgue spaces. Using a Mellin inversion formula, we derive a sufficient condition for a function to be a multiplier of the semigroup e-tL on Lp(X;E) where E is a UMD Banach space and 1&lt;p&lt;∞.