We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The a(x,η) are elements C * S 1, classes that have limited regularity in the x variable. show associated operator a(x, D) maps between Sobolev ℌ FIO (ℝn) and over space (ℝn). Our main result implies m = 0, δ =l/2 r > n − acts boundedly all p ∈ (1, ∞).

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on L, 1 < p < ∞, and of a perfect dyadic singular integral operator. This allows to deduce a local T (b) theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Tao, Thi...

In this set of notes we begin the theory of singular integral operators operators which are almost integral operators, except that their kernel K(x, y) just barely fails to be integrable near the diagonal x = y. (This is in contrast to, say, fractional integral operators such as Tf(y) := ∫ Rd 1 |x−y|d−s f(x) dx, whose kernels go to infinity at the diagonal but remain locally integrable.) These ...

We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an L2 space weighted by the weight function of the continuous q-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a q-analog of the expansion of eixy in Jacobi polynomials of argument x. We also outline a general procedure of finding integral repre...

We prove a local T (b) theorem for singular integral operators in the spirit of the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Thiele, Tao and the first author. The method consists in reducing one situation to the other by representing via the Beylkin-Coifman-Rokhlin algorithm any singular integral operator with truncated kernel as the sum of a bounded oper...

In recent years, a remarkably large number of inequalities involving the fractional q-integral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractional q-integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kobe...