We extend the classical theory of the continuous and discrete wavelet transform to functions with values in UMD spaces. As a by-product we obtain equivalent norms on Bochner spaces in terms of g-functions.

We prove weak type (1,1) estimates for a special class of Calderón-Zygmund homogeneous kernels represented as l¹ sums of "equidistributed" H¹ atoms on 𝕊¹.

We derive two-weight weak type estimates for operators of potential type in homogeneous spaces. The conditions imposed on the weights are testing conditions of the kind first studied by E. T. Sawyer [4]. We also give some applications to strong type estimates as well as to operators on half-spaces.

In this paper we will study the behavior of the Riesz transform associated with the Gaussian measure γ(x)dx = e-|x|2dx in the space Lγ1 (Rn).

Some weighted sharp maximal function inequalities for the Toeplitz type operator $T_b={\sum }_{k=1}^m{T}^{k,1}{M}_b{T}^{k,2}$ are established, where $T^{k,1}$ are a fixed singular integral operator with non-smooth kernel or ±I (the identity operator), $T^{k,2}$ are linear operators defined on the space of locally integrable functions, k = 1,..., m, and $M_b\left(f\right)=bf$. The weighted boundedness of $T_b$ on Morrey spaces is obtained by using sharp maximal function inequalities.

For 1 < p < ∞ and for weight w in $A_p$, we show that the r-variation of the Fourier sums of any function f in $L^p\left(w\right)$ is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality...

Given $\alpha$, $0<\alpha <n$, and $b\in \mathrm{B}MO$, we give sufficient conditions on weights for the commutator of the fractional integral operator, $[b,I_{\alpha }]$, to satisfy weighted endpoint inequalities on ${ℝ}^n$ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on ${ℝ}^n$.

Let $m$ be a positive integer, $0<\alpha <mn$, $\overrightarrow{b}=(b_1,\cdots ,b_m)\in {\mathrm{BMO}}^m$. We give sufficient conditions on weights for the commutators of multilinear fractional integral operators ${ℐ}_{\alpha }^{\overrightarrow{b}}$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional type...

We study boundedness properties of commutators of general linear operators with real-valued BMO functions on weighted $L^p$ spaces. We then derive applications to particular important operators, such as Calderón-Zygmund type operators, pseudo-differential operators, multipliers, rough singular integrals and maximal type operators.

The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple weights,...

An improved multiple Cotlar inequality is obtained. From this result, weighted norm inequalities for the maximal operator of a multilinear singular integral including weak and strong estimates are deduced under the multiple weights constructed recently.