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

Taylor17PhD.pdf
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Abstract
We consider three major parts of Fourier analysis and their role in FeffermanStein 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 FeffermanStein 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 FeffermanStein 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 FeffermanStein 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 (PMS30), volume 30. Princeton University Press, 1970.]
Type of Work:  Thesis (Doctorates > Ph.D.)  

Award Type:  Doctorates > Ph.D.  
Supervisor(s): 


Licence:  
College/Faculty:  Colleges (2008 onwards) > College of Engineering & Physical Sciences  
School or Department:  School of Mathematics  
Funders:  None/not applicable  
Subjects:  Q Science > QA Mathematics  
URI:  http://etheses.bham.ac.uk/id/eprint/7855 
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