The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman–Meyer–Stein tent spaces. Following recent work of Auscher–MIntosh–Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds, such that A always has a bounded H...

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-com...

This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space L(R,H) of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of L(R,H) and its bo...

In this note, we show that quasi-free Hilbert modules R defined over natural function algebras satisfying a certain positivity condition, defined via the hereditary functional calculus, admit a dilation (actually a co-extension) to the Hardy module over the polydisk. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. Some consequence...

In this paper we state the following weighted Hardy type inequality for any functions $ \varphi in a Sobolev space and weight \mu of quite general \begin{document}$ \begin{align*} c_{N, \mu} \int_{ \mathbb{R}^N}V\, \varphi^2\mu(x)dx\le \mathbb{R}^N}|\nabla \varphi|^2\mu(x)dx +C_\mu \mathbb{R}^N}W \varphi^2\mu(x)dx, \end{align*} $\end{document} where V is multipolar potential W bounded function ...

In this note, we show that quasi-free Hilbert modules R defined over the polydisk algebra satisfying a certain positivity condition, defined via the hereditary functional calculus, admit a unique minimal dilation (actually a co-extension) to the Hardy module over the polydisk. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. Some c...

The duality between Hl and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained . Using the atomic decomposition approach ([C], [L]) the author studied the problem of characterizing the dual space of Hl of vector-valued functions . In [B2] the author showed, for the case SZ = {Iz1 = 1}, tha...