Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the BeurlingAhlfors singular integral operator, acting on LpX(R ) with p ∈ (1,∞). Moreover, replacing equality by a linear equivalence, this is fo...

Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation. This volume integral equation, however, in general fails to feature a weakly singular integral operator. Nevertheless, after a suitable periodization, the involved integral operator can be efficiently evaluated on trigonomet...

We prove weighted inequalities for commutators of one-sided singular integrals (given by a Calderón-Zygmund kernel with support in (−∞, 0)) with BMO functions. We give the one-sided version of the results in [C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl., vol. 3 (6), 1997, pages 743–756] and [C. Pé...

The current on a linear strip or wire solves an equation governed by a linear integraldifferential operator that is the composition of the Helmholtz operator and an integral operator with logarithmically singular displacement kernel. We investigate the spectral behavior of this classical operator, particularly because various methods of analysis and solution rely on asymptotic properties of spe...

In these lectures we discuss examples that are relevant to two questions in the theory of singular integrals. The first question is the L boundedness of the maximal operator formed by dilates of Mikhlin-Hörmander multipliers, while the second concerns the L boundedness of a well-known object, the classical L-bounded Calderón-Zygmund homogeneous singular integral associated with an integrable fu...

A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur’s test, and a multilinear version of the T1 Theor...

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We analyze in classical L(R)-spaces, n = 2 or n = 3, 1 < q < ∞, a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating obstacle. The corresponding system of partial differential equations of second order involves an angular derivative which is not subordinate to the Laplacian. The main tools are Littlewood– Paley theory and a decomposition of the...