Bilinear Hilbert Transform on Measure Spaces

2008

Annals of Mathematics, 2004

We continue the investigation initiated in [8] of uniform L p bounds for the family of bilinear Hilbert transforms H α,

Advanced Courses of Mathematical Analysis III, 2008

Journal of Geometric Analysis, 2006

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 p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L p 1 × L p 2 when either 1/p1 + 1/p2 = 3/2 or one of p1, p2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

Revista Matemática Iberoamericana, 2000

We continue the investigation initiated in [8] of uniform L p bounds for the family of bilinear Hilbert transforms H α,

2017

In this note, the unboundedness of the bilinear Hilbert transform from products of Hardy spaces H^{p} \cross H^{q} to L^{r}, 0<p\leq 1, 0<q\leq 1, 1/p+1/q= 1/r , is considered. §1. Introduction The bilinear Hilbert transform H is defined by H (f ,) (x) = \l i m_{ \epsi l onar r ow 0} | >\epsi l on f (x+y) g(x-y) \f r ac{ dy} { y} = \f r a c { 1} { (2\pi){ 2} } \ma t hbb{ R} \ma t hbb{ R}{ ê{ i x (\x i +\e t a) } } (i \pi s g n(\x i-\e t a)) \ha t { f } (\x i) g (\e t a) d\x i d\e t a for f, \in S , where sgn\xi is the signum function. In the study of the Cauchy integral along Lipschitz curves, A.P. Calderón raised the problem whether the boundedness 0 H from L^{2} \cross L^{2} to L^{1} holds. After some 30 years, this problem was solved positively by Lacey-Thiele [4, 5]. More precisely, they proved that H is bounded from L^{p}\cross L^{q} to L^{r} for 1 <p, q \l eq 1 and 2/3 < r < 1 satisfying 1/p+1/q= 1/r. However, it is still open whether we can remove the restriction r>2/3. The purpose of this note is to consider the endpoint cases. In particular, we can prove the unboundedness of H from H^{1} \cross H^{1} to L^{1/2} , even though we do not know whether H is bounded from L^{1}\cross L^{1} to L^{1/2,\infty} , where H^{1} is the Hardy space and L^{1/2,\infty} is the weak L^{1/2}-space. More generally, we can prove

Colloquium Mathematicum

Let m:ℝ→ℝ be a function of bounded variation. We prove the L p (ℝ)-boundedness, 1&lt;p&lt;∞, of the one-dimensional integral operator defined by T m f(x)=p.v.∫k(x-y)m(x+y)f(y)dy, where k(x)=∑ j∈ℤ 2 j φ j (2 j x) for a family {φ j } j∈ℤ of functions in L 1 (ℝ) satisfying, for all j∈ℤ, ∫φ j (x)dx=0, suppφ j ⊆{x∈ℝ:1/2≤|x|≤2}, and for some c&gt;0, 0&lt;ε&lt;1 and for all j∈ℤ, ∫|φ j (x+y)-φ j (x)|dx≤c|y| ε ·

arXiv: Functional Analysis, 2020

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1\pi\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $\lambda$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues...

Georgian Mathematical Journal, 2009

It is proved that the Hilbert transform defined on a finite interval is bounded in the grand Lebesgue space L p) w (1 < p < ∞) if and only if w satisfies the Muckenhoupt condition A p .

1982