Analysis Seminar: "Singular integrals associated with Zygmund dilations"

Abstract: The theory of Calder\'on--Zygmund operators plays an important role in modern harmonic analysis. The core of this theory is that the regularity and cancellation conditions are invariant with respect to the one-parameter family of dilations on $\mathbb R^n$ defined by $$ \delta(x_1,\ldots,x_n)=(\delta x_1,\ldots,\delta x_n),\quad \delta>0, $$ in the sense that the kernel $\delta^n K(\delta x)$ satisfies the same conditions with the same upper bounds as $K(x)$ does. Indeed, the classical singular integrals, maximal functions, $A_p$ weights and multipliers are invariant with respect to such one-parameter dilations. On the other hand, the multi-parameter theory on $\mathbb R^n$ began with Zygmund�s study of the strong maximal function, and later has been extensively studied by Fefferman, Pipher, Ricci, Stein and others. It was point out by Fefferman and Pipher that the consideration of these operators associated with Zygmund dilations is a natural next step or the simplest case after those of the classical Calder\'on--Zygmund theory and the product space theory.

In this talk we provide our recent study on these multiparameter singular integral operators which commute with Zygmund dilations. We introduce a class of singular integral operators associated with the Zygmund's dilations and show the boundedness of these operators on $L^p$ for $1 < p < \infty$, which cover those concrete examples of such operators studied by Ricci--Stein, Fefferman--Pipher and Nagel--Wainger. We also establish the weighted Hardy and BMO spaces associated with the Zygmund's dilations and obtain the end point estimates of these singular integrals.

This is joint work with Y.S. Han, C.C. Lin and C.Q. Tan.