In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at ±∞) and acting in the Hilbert space ℓ_{Ω}²(Z;C²) (Z:={0,±1,±2,...}) are considered. We consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. For each of these cases we establish a self...

In this work, we are interested in the small time local null controllability for the viscous Burgers’ equation yt − yxx + yyx = u(t) on the line segment [0, 1], with null boundary conditions. The second-hand side is a scalar control playing a role similar to that of a pressure. In this setting, the classical Lie bracket necessary condition [f1, [f1, f0]] introduced by Sussmann fails to conclude...

The mean operator R0 and its dual R0 play an important role and have many applications, for example, in image processing of the so-called synthetic aperture radar (SAR) data [11, 12] or in the linearized inverse scattering problem in acoustics [6]. Our purpose in this work is to define and study integral transforms which generalize the operators R0 and R0. More precisely, we consider the follow...

In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type (p, p) on Lp(Rn) if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery’s result is sharp in a certain sense. We also obtain a weighted analogue of Carbery’s result. Some applications of our results are also given.

Weighted L for p ∈ 1,∞ and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. Al...

In the present paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the ∂-equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local ∂-solution operator which involves only Cauch...

We analyze the spectral shift function (SSF) of a Schrödinger operator due to a compactly supported potential. We give a bound on the integral of the SSF with respect to a bounded compactly supported function. It is based on the control of the singular values of the difference of two Schrödinger semigroups. As an application we improve some earlier results on the regularity of the integrated de...

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly sing...

In this paper, we study the convexity of the integral operator

In this paper we pursue the study of the problem of controlling the maximal singular integral T ∗f by the singular integral Tf . Here T is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L2(Rn) norm and via pointwise estimates of T ∗f by M(Tf) or M2(Tf) , where M is the Hardy-Littlewood maximal operator and M2 = M ◦M ...