In recent time, wavelet transforms are used extensively for efficient storage, transmission and representation of multimedia signals. Hilbert transform pairs of wavelets is the basic unit of many wavelet theories such as complex filter banks, complex wavelet and phaselet etc. Moreover, Hilbert transform finds various applications in communications and signal processing such as generation of sin...

A classical theorem of C. Fefferman [3] says that the characteristic function of the unit disc is not a Fourier multiplier on L(R) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit disc in R is the Fourier multiplier of a bounded bilinear operator from L1(R) × L2(R) into L(R), when 2 ≤ p1, p2 <...

C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in [27] that the standard bilinear and bi-parameter Hilbert transform does not satisfy any L estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by Tm(f1, f2)(x) := ∫ R m(ξ, η)f̂1(ξ1, η1)f̂2(ξ2, η2)e 1122dξdη satisfies any L estimates, where the symbol m satisfies |∂ ξ ∂ ηm(ξ, η)| . 1 dist(ξ,Γ1...

In this paper, we determine the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of bilinear Hilbert transform $H_{\gamma}(f,g)$ along a convex curve $\gamma$ $$H_{\gamma}(f,g)(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x-t)g(x-\gamma(t)) \,\frac{\textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, $r>\frac{1}{2}$, $p>1$, $q>1$...