Aspects of anisotropic harmonic analysis beyond Calderón-Zygmund Theory

Taylor, Brian John Sidney (2017). Aspects of anisotropic harmonic analysis beyond Calderón-Zygmund Theory. University of Birmingham. Ph.D.

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Abstract

We consider three major parts of Fourier analysis and their role in Fefferman-Stein inequalities. The three areas can be considered as three separate topics in their own right, or as three steps to proving certain \(L\)\(^p\)-\(L\)\(^q\) inequalities via the Fefferman-Stein inequalities of the form
\begin{align*}
\int_{\R^n} |T f|^2 w \lesssim \int_{\R^n}|f|^2 \mathcal{M}w.
\end{align*}
The first area discussed is that of maximal functions, specifically obtaining \(L\)\(^p\)-\(L\)\(^q\) inequalities on large classes of maximal functions. We then use a simple duality argument to transfer these to operators where we have a Fefferman-Stein inequality via
\begin{align*}
\|T\|_{p \to q} \lesssim \|\mathcal{M}\|^{1/2}_{(q/2)' \to (p/2)'}.
\end{align*}
The second area aims to control operators defined via multipliers by the previous section's geometrically defined maximal functions. In particular, we build up to a schema that can be used to prove Fefferman-Stein inequalities via the so called \(g\)-functions, originating in work of E. M. Stein* but having historic roots that can be easily seen by viewing \(g\)-functions as speciality square functions.

In the final section we consider some classes of operators with oscillatory kernels and obtain estimates on their multipliers, and by application of the previous two sections obtain some \(L\)\(^p\)-\(L\)\(^q\) inequalities.

[*Elias M Stein. Singular integrals and differentiability properties of functions (PMS-30), volume 30. Princeton University Press, 1970.]

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Thesis (Doctorates > Ph.D.)

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Doctorates > Ph.D.

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