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2015 | OriginalPaper | Chapter
Multilinear Calderón–Zygmund Singular Integrals
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It is quite common for linear operators to depend on several functions of which only one is thought of as the main variable and the remaining ones are usually treated as parameters. Examples of such operators are ubiquitous in harmonic analysis: multiplier operators, homogeneous singular integrals associated with functions Ω on the sphere, Littlewood–Paley operators, Calderón commutators, and the Cauchy integral along Lipschitz curves. Treating the additional functions that arise in these operators as frozen parameters often provides limited results that could be thought analogous to those that one obtains by studying calculus of functions of several variables by freezing variables. In this article, we advocate a more flexible point of view in the study of linear operators, analogous to that employed in pure multivariable calculus. Unfreezing the additional functions and treating them as input variables provides a more robust approach that often yields sharper results in terms of regularity of the input functions.