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.

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_p\left({ℝ}^d\right)$ (in the case $p>1$), but (in the case when $1/p(·)$ is log-Hölder continuous and $p_{-}=inf\{p\left(x\right):x\in {ℝ}^d\}>1$) on the variable Lebesgue spaces $L_{p(·)}\left({ℝ}^d\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma$-rectangles. We introduce a general maximal operator $M_s^{\gamma ,\delta }$ and with the help of generalized $\Phi$-functions, the strong- and weak-type...

We prove that, in arbitrary finite dimensions, the maximal operator for the Laguerre semigroup is of weak type (1,1). This extends Muckenhoupt's one-dimensional result.

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.

We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p,q}$, 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p,q,r}$ such that for any $\varphi \in L^{p,q}$, ${\left|\right|ℳ\varphi \left|\right|}_{r,\infty }\le C_{p,q,r}{\left|\right|\varphi \left|\right|}_{p,q}$.

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.