An inequality for the Hilbert transform

An inequality for the hilbert transform.

Wiener-wintner for Hilbert Transform

We prove the following extension of the Wiener–Wintner Theorem and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows (X,μ, Tt) and f ∈ L(X,μ), there is a set Xf ⊂ X of probability one, so that for all x ∈ Xf we have lim s↓0 ∫ s<|t|<1/s e f(Tt x) dt t exists for all θ. The proof is by way of establishing an appropriate oscillation inequality which ...

On an inequality for operators on Hilbert space

Oscillation and Variation for the Hilbert Transform

Hf (x)= lim →0+ H f (x). It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra information on their convergence as well as an estimate on the number of λ-jumps they can have. For earlier results on oscillation and variation operators in analysis and ergod...

Distributional Estimates for the Bilinear Hilbert Transform

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint e...