We survey results concerning the L2 boundedness of oscillatory and Fourier integral operators and discuss applications. The article does not intend to give a broad overview; it mainly focuses on topics related to the work of the authors.[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
We consider a convolution operator Tf = p.v. Ω ⁎ f with , where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued function on . We give a criterion for such an operator to be bounded from the space into itself.
Let , where P is a real polynomial on ℝ. It is proved that T is bounded on the weighted H¹(wdx) space with w ∈ A₁.
In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+.
G. David, J.-L. Journé and S. Semmes have shown that if b1 and b2 are para-accretive functions on Rn, then the Tb theorem holds: A linear operator T with Calderón-Zygmund kernel is bounded on L2 if and only if Tb1 ∈ BMO, T*b2 ∈ BMO and Mb2TMb1 has the weak boundedness property. Conversely they showed that when b1 = b2 = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b1 and b2 is necessary for the Tb Theorem to hold in general. In...
In this paper, we study the the parabolic Marcinkiewicz integral [...] MΩ,hρ1,ρ2 on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.
Parabolic wavelet transforms associated with the singular heat operators and , where , are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderón reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.
In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which...