The Hardy space H ρr(R ) consists of all divergence free r-form distributions f whose non-tangential maximal functions are in L(R). We say that a system of singular integrals characterizes H ρr(R ) if this space consists precisely of those divergence-free r-form distributions f whose images under the singular integral operators are integrable. When the operators are determined by Fourier multip...

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a t−1 decay rate as an operator from the Hardy space H to BMO, the space of functions of bounded mean oscillation. This estimate, along with the L conservation law allows one to deduce a family of Strichartz estimates. We classify the structure o...

Abstract In this paper, we will discuss the space of functions weak bounded mean oscillation. particular, show that is dual special atom space, whose was already known to be derivative (in sense distribution) belonging Zygmund class functions. We show, in proves Hardy H 1 </m:ma...

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued $f$ on ${\mathbb R^n}$. This article devoted investigating the mapping properties of new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well its VMO CMO subspace...

We prove (see Theorem 1.3 below) that a generalized harmonic map into a round sphere, i.e. a map u ∈ W 1,1 loc ( , Sn−1) which solves the system div (ui∇uj − uj∇ui) = 0, i, j = 1, . . . , n, is smooth as soon as |∇u| ∈ L for any q > 1, and the norm of u in BMO is sufficiently small. Here, ⊂ R is open, and m, n are arbitrary. This extends various earlier results of Almeida [1], Ge [15], and R. M...

We present here a correction to an error in our paper [5]. We are grateful to Dachun Yang for bringing the error to our attention, and we thank Alan McIntosh for helpful discussions which have led to this correction of the error. In particular, our approach here follows that of Auscher, McIntosh and Russ [1], as explained to us by McIntosh. In [5] we develop a theory of H1 (Hardy type) and BMO ...

We prove an existence of a unique solution of an exponential martingale equation in the class of BMO martingales. The solution is used to characterize optimal martingale measures.

A dx. Given a Banach space (X, ‖ · ‖) and 1 ≤ p < ∞ we shall denote by L(R, X) the space of Bochner p-integrable functions endowed with the norm ‖f‖Lp(Rn,X) = ( ∫ Rn ‖f(x)‖ dx), by Lc (R , X) the closure of the compactly supported functions in L(R, X) and by Lweak,α(R , X) the space of measurable functions such that |{x ∈ R : ‖f(x)‖ > λ}| ≤ α(λ) where α : R → R is a non increasing function. We ...