We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. ...

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space B2 of functions whose Hardy-Littlewood maximal function converges ...

Conditions on weights u(·), v(·) are given so that a classical operator T sends the weighted Lorentz space Lrs(vdx) into Lpq(udx). Here T is either a fractional maximal operator Mα or a fractional integral operator Iα or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for Mα and Iα when the weights have some usual properties and max(r, s) ≤ min(p, q). § 0. Introd...

We prove that any distribution q satisfying the equation ∇q = div f for some tensor f = (f i j), f i j ∈ h (U) (1 ≤ r < ∞) -the local Hardy space, q is in h, and is locally represented by the sum of singular integrals of f i j with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incomp...

In this paper, the authors study the boundedness of multilinear Calderón-Zygmund singular integral operators and their commutators in generalized Morrey spaces.

The purpose of this paper is to provide a more general Cameron-Storvick theorem for the generalized analytic Feynman integral associated with Gaussian process $\mathcal Z_k$ on very Wiener space $C_{a,b}[0,T]$. $C_{a,b}[0,T]$ can be considered as set all continuous sample paths Brownian motion determined by functions $a(t)$ and $b(t)$ $[0,T]$. As an interesting application, we apply evaluate ce...

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general which admit Markov semigroup purely algebraic assumptions. We shall construct an abstract form ‘Markov metric’ governing process and naturally associated BMO spaces, interpolate with Lp-scale endpoint ...