### A correction to the paper "A multiplier theorem for Jacobi expansion" Studia Math. 52 (1975), pp. 234-261

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Under certain conditions on a function space X, it is proved that for every ${L}^{\infty}$-function f with $\parallel f{\parallel}_{\infty}\le 1$ one can find a function φ, 0 ≤ φ ≤ 1, such that φf ∈ X, $mes\phi \ne 1\le \varepsilon \parallel f{\parallel}_{1}$ and $\parallel \phi f{\parallel}_{X}\le const(1+log{\varepsilon}^{-1})$. For X one can take, e.g., the space of functions with uniformly bounded Fourier sums, or the space of ${L}^{\infty}$-functions on ${\mathbb{R}}^{n}$ whose convolutions with a fixed finite collection of Calderón-Zygmund kernels are also bounded.

We consider a problem of intervals raised by I. Ya. Novikov in [Israel Math. Conf. Proc. 5 (1992), 290], which refines the well-known theorem of J. Marcinkiewicz concerning structure of closed sets [A. Zygmund, Trigonometric Series, Vol. I, Ch. IV, Theorem 2.1]. A positive solution to the problem for some specific cases is obtained. As a result, we strengthen the theorem of Marcinkiewicz for generalized Cantor sets.

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space . We show that if W and its inverse ${W}^{-1}$ both satisfy a matrix reverse Hölder property introduced by Christ and Goldberg, then the weighted Hilbert transform $H:L{\xb2}_{W}(\mathbb{R},)\to L{\xb2}_{W}(\mathbb{R},)$ and also all weighted dyadic martingale transforms ${T}_{\sigma}:L{\xb2}_{W}(\mathbb{R},)\to L{\xb2}_{W}(\mathbb{R},)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

There is a one parameter family of bilinear Hilbert transforms. Recently, some progress has been made to prove Lp estimates for these operators uniformly in the parameter. In the current article we present some of these techniques in a simplified model...

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in {A}_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,{I}_{\alpha}]:{L}^{p}\left({\mu}^{p}\right)\to {L}^{q}\left({\lambda}^{q}\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda}^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.