We consider maximal operators acting on vector valued functions, that is, functions taking values on ℂ^d, that incorporate matrix weights in their definitions. We show vector valued estimates, in the sense of Fefferman–Stein inequalities, for such operators. These are proven using an extrapolation result for convex body valued functions due to Bownik and Cruz-Uribe. Finally, we show an H^1-BMO duality for matrix valued functions and we apply the previous vector valued estimates to show upper bounds for biparameter paraproducts. For the reader's convenience, we include an appendix explaining how to adapt the extrapolation for real convex body valued functions of Bownik and Cruz-Uribe to the setting of complex convex body valued functions that we treat.

Consider a finite Blaschke product $f$ with $f(0) = 0$ which is not a rotation and denote by $f^n$ its $n$-th iterate. Given a sequence $\{a_n\}$ of complex numbers, consider the series $F(z) = \sum_n a_n f^n(z).$ We show that for any $w \in \mathbb{C},$ if $\{a_n\}$ tends to zero but $\sum_n |a_n| = \infty,$ then the set of points $\xi$ in the unit circle for which the series $F$ converges to $w$ has Hausdorff dimension $1.$ Moreover, we prove that this result is optimal in the sense that the conclusion does not hold in general if one considers Hausdorff measures given by any measure function more restrictive than the power functions $t^\delta,$ $0 < \delta < 1.$

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journé operators as well as other central operators under the assumption of the product matrix Muckenhoupt condition. In particular, we provide a complete theory for biparameter Journé operator bounds on matrix-weighted L2 spaces. We also achieve bounds in the general case of matrix-weighted Lp spaces, for 1<p<∞ for paraproduct-free Journé operators. Finally, we expose an open problem involving a matrix-weighted Fefferman–Stein inequality, on which our methods rely in the general setting of matrix-weighted bounds for arbitrary Journé operators and p≠2.

We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case as well as with Bloom weights. Moreover, we address the flexibility of one of our methods.

In this paper we set up a theory of two-matrix weighted little BMO in two parameters. We prove that being a member of this class is equivalent to belonging uniformly in each variable to two-matrix weighted (one-parameter) BMO, a class studied extensively by J. Isralowitz, S. Pott, S. Treil and others. Using this equivalence, we deduce lower and upper bounds in terms of the two-matrix weighted little BMO norm of the symbol for the norm of commutators with Journé operators.

We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda_{\ast}(\mathbb{R}^n)$ to the subspace $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image under the Bessel potential $J := (1-\Delta)^{-1/2}$ of the space $\mathbf{bmo}(\mathbb{R}^n),$ which is a non-homogeneous version of the classical $\mathrm{BMO}.$ Locally, $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n)$ consists of functions that together with their first derivatives are in $\mathbf{bmo}(\mathbb{R}^n).$ More generally, we consider the same question when the Zygmund class is replaced by the H\"older space $\Lambda_{s}(\mathbb{R}^n),$ with $0<s \leq 1$ and the corresponding subspace is $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n),$ the image under $(1-\Delta)^{-s/2}$ of $\mathbf{bmo}(\mathbb{R}^n).$ One should note here that $\Lambda_{1}(\mathbb{R}^n) = \Lambda_{\ast}(\mathbb{R}^n).$ Such results were known earlier only for $n = s = 1$ with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from $f \in \Lambda_{s}(\mathbb{R}^n)$ to $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of $f$ on the upper half-space $\mathbb{R}^{n+1}_{+}.$

O. Blasco and S. Pott showed that the supremum of operator norms over L-2 of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < infinity. The main tool is a new characterization in terms of paraproducts and two-weight John-Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.

A Central Limit Theorem for linear combinations of iterates of an inner function is proved. The main technical tool is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures.

We study the distance in the Zygmund class ?* to the subspace I( BMO ) of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of I( BMO ) in the Zygmund seminorm. We also generalise this result to Zygmund measures on Rd. Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in ?* that are also in the classical Sobolev space W1,p, for 1.

We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda_{\ast}(\mathbb{R}^n)$ to the subspace $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image under the Bessel potential $J := (1-\Delta)^{-1/2}$ of the space $\mathbf{bmo}(\mathbb{R}^n),$ which is a non-homogeneous version of the classical $\mathrm{BMO}.$ Locally, $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n)$ consists of functions that together with their first derivatives are in $\mathbf{bmo}(\mathbb{R}^n).$ More generally, we consider the same question when the Zygmund class is replaced by the Holder space $\Lambda_{s}(\mathbb{R}^n),$ with $0 < s \leq 1$ and the corresponding subspace is $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n),$ the image under $(1-\Delta)^{-s/2}$ of $\mathbf{bmo}(\mathbb{R}^n).$ One should note here that $\Lambda_{1}(\mathbb{R}^n) = \Lambda_{\ast}(\mathbb{R}^n).$ Such results were known earlier only for $n = s = 1$ with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from $f \in \Lambda_{s}(\mathbb{R}^n)$ to $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of $f$ on the upper half-space $\mathbb{R}^{n+1}_{+}.$

It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied by Fernández and Pestana. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. We use our results to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half-plane.