向量值Musielak-Orlicz空间的两种几何性质(英)

设Ｘ是复Ｂａｎｃｈ空间,Ｍ（ｔ,ｕ）是以ｔ为参数的满足某些通常条件的Φ－函数．我们证明了;（ｉ）Ｍｕｓｉｅｌａｋ－Ｏｒｌｉｃｚ空间Ｌ_Ｍ（Ｘ）具有解析ＵＭＤ性质当且仅当Ｘ具有;（ｉｉ） Ｌ_Ｍ（Ｘ）具有解析ＲＮ性质当且仅当Ｘ具有．     

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.     

Anisotropic Fourier multipliers and singular integrals for vector-valued functions

We show that an anisotropic Mihlin-type condition on the symbol guarantees the boundedness of the associated Fourier multiplier operator on L ^p (R ⁿ ,X) for 1 < p < ∞ and an arbitrary UMD space X. In many cases, this result can be used as a substitute for the Marcinkiewicz–Lizorkin multiplier theorem, which is invalid in general UMD spaces. An application to anisotropic singular integrals is given. Mathematics Subject Classification (2000) 42B15, 42B20, 46E40     

Operator-valued Fourier Haar multipliers

Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from Lp([0,1],X) to Lq([0,1],Y) where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.     

Linear bounds for Calderón–Zygmund operators with even kernel on UMD spaces

It is well-known that several classical results about Calderón–Zygmund singular integral operators can be extended to X-valued functions if and only if the Banach space X has the UMD property. The dependence of the norm of an X-valued Calderón–Zygmund operator on the UMD constant of the space X is conjectured to be linear. We prove that this is indeed the case for sufficiently smooth Calderón–Zygmund operators with cancellation, associated to an even kernel. Our method uses the Bellman function technique to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.     

Vector-valued singular integral operators with rough kernels

In this paper, we establish a weak-type (1,1) boundedness criterion for vector-valued singular integral operators with rough kernels. As applications, we obtain weak-type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, $Y={[H,X]}_{\theta }$ is a complex interpolation space between a Hilbert space H and a UMD space X.