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 $H:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ and also all weighted dyadic martingale transforms $T_{\sigma }:L{²}_W(ℝ,)\to L{²}_W(ℝ,)$ are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $r>\mathrm{𝚖𝚊𝚡}\{p^{\text{'}},2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

We prove an extension of a result by Peres and Solomyak on almost sure absolute continuity in a class of symmetric Bernoulli convolutions.

We study some operators originating from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G* and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on $L^p$. Moreover, we generalize a classical multiplier theorem by weakening its...

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.

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left(j^{-\alpha }\right)$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

We prove good-$\lambda$ inequalities for the area integral, the nontangential maximal function, and the maximal density of the area integral. This answers a question raised by R. F. Gundy. We also prove a Kesten type law of the iterated logarithm for harmonic functions. Our Theorems 1 and 2 are for Lipschitz domains. However, all our results are new even in the case of ${\mathbf{R}}_{+}^2$.

Upper bounds for GCD sums of the form ${\sum }_{k,\ell =1}^N\frac{{\left(\gcd (n_k,n_{\ell })\right)}^{2\alpha }}{{\left(n_k{n}_{\ell }\right)}^{\alpha }}$ are established, where ${\left(n_k\right)}_{1\le k\le N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish...

In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.

We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Békollé-Bonami weights in this setting. An application to bounded Toeplitz products is also given.

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...