Endpoint Bounds for an Analytic Family of Hilbert Transforms
نویسنده
چکیده
In R2, we consider an analytic family of operators Hz , z ∈ C, whose convolution kernel is obtained by taking −z − 1 derivatives of arclength measure on the parabola (t, t2) in a homogeneous way, defined in such a way so that H−1 be the standard parabolic Hilbert transform. For a fixed z, we study the set of p for which Hz is bounded on Lp(R2) and for the critical z that captures the degree of singularity of this operator on Lp(R2), we prove a positive endpoint result.
منابع مشابه
Uniform Bounds for the Bilinear Hilbert Transforms, Ii
We continue the investigation initiated in [8] of uniform L bounds for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. ∫ R f(x − αt)g(x − βt) dt t . In this work we show that Hα,β map L1(R) × L2(R) into L(R) uniformly in the real parameters α, β satisfying | β − 1| ≥ c > 0 when 1 < p1, p2 < 2 and 2 3 < p = p1p2 p1+p2 < ∞. As a corollary we obtain L × L∞ → L uniform bounds in the ...
متن کاملError bounds in approximating n-time differentiable functions of self-adjoint operators in Hilbert spaces via a Taylor's type expansion
On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in Hilbert Spaces via a Taylor's type expansion are given.
متن کاملRiesz Transforms for the Isotropic Estimation of the Local Phase of Moiré Interferograms
The estimation of the local phase and local amplitude of 1-D signals can be realized by the construction of the analytic signal. This includes the evaluation of the signal’s Hilbert transform, which performs a phase shift. In the past, different definitions of the analytic signal of multidimensional signals have been proposed, all of which are based on different combinations of partial and tota...
متن کاملPointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms
We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest.
متن کاملMultidimensional Analytic Signals and the Bedrosian Identity
The analytic signal method via the Hilbert transform is a key tool in signal analysis and processing, especially in the time-frquency analysis. Imaging and other applications to multidimensional signals call for extension of the method to higher dimensions. We justify the usage of partial Hilbert transforms to define multidimensional analytic signals from both engineering and mathematical persp...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1991