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In Nagel et al. (J Funct Anal 181:29–118, 2001), Nagel–Ricci–Stein established the relationships between product kernels and flag kernels on the Euclidean space, that is, product kernels are finite sums of flag kernels. The main purpose of this paper is to characterize the product Hardy space as the intersection of flag Hardy spaces, and characterize the product Carleson measure space as the sum of flag spaces.

On the unit disk we introduce a new class of tent spaces \(T^q_{s,t}(\mu )\) for any positive Borel measure \(\mu \), consider a class of Mobius invariant spaces \(Z_p\) of analytic functions, and show that \(Z_p\) is contained in \(T^1_{p,1}\) if and only if \(\mu \) is a p-Carleson measure. This could be considered a continuation or variation of the work initiated by Xiao (Adv Math 217:2075–2088, 2008).

In this paper, we introduce a weighted Carleson measure \(d\nu _{\mathbb {E}, w}\) associated with the family \(\mathbb {E}\), where \(\mathbb {E}=\{E_r(x)\}_{r\in \mathcal {I}, x\in X}\) is a family of open subsets of a topological space X endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. Using Calderon–Zygmund theory, we show that the weighted BMO associated with the family \(\mathbb {E}\) can be characterized by the weighted Carleson measure \(d\nu _{\mathbb {E}, w}\).

In this paper we define a kind of vanishing Carleson measure on \(\mathbb {R}^{n+1}_+\) and give its characterization by the compact property of some convolution operator. We also investigate the construction of vanishing Carleson measures generated by a family of the multilinear operators \(\{\Theta _t\}_{t>0}\) and \(CMO\) functions. As some applications of our results, we also give the boundedness and compactness for the paraproduct \(\pi _{\vec {b}}\) associated with the family \(\{\Theta _t\}_{t>0}\) on \(L^2(\mathbb {R}^n)\), which is defined by $$\begin{aligned} \pi _{\vec {b}}(f)(x)= \int _0^\infty \eta _t*\big ((\varphi _t*f)\Theta _t(b_1,\ldots ,b_m)\big )(x)\; \frac{dt}{t}. \end{aligned}$$ Further, for the linear case (i.e., \(m=1\)), we show that the paraproduct $$\begin{aligned} B_b(f)(x)=\int _0^\infty (f*\varphi _t)(x)(b*\psi _t)(x)\frac{\alpha (t)}{t}dt, \end{aligned}$$ which was introduced by Coifman and Meyer, is also a compact operator on \(L^2(\mathbb {R}^n)\) if \(b\in CMO(\mathbb {R}^n)\) and \(\alpha \in L^\infty (\mathbb {R}^n)\).

Let L be the infinitesimal generator of an analytic semigroup on L2 (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L2 (ℝ). In this article, we introduce new function spaces H L 1 (ℝ × ℝ) and BMOL(ℝ × ℝ) (dual to the space H L* 1 (ℝ × ℝ) in which L* is the adjoint operator of L) associated with L, and they generalize the classical Hardy and BMO spaces on product domains. We obtain a molecular decomposition of function for H L 1 (ℝ × ℝ) by using the theory of tent spaces and establish a characterization of BMOL (ℝ × ℝ) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMOL (ℝ × ℝ). Applications include large classes of differential operators such as the magnetic Schrodinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.