Research
My main area of research interest involves spaces of analytic functions and their relations to operator theory and harmonic analysis. For example, I am interested in Hardy spaces, Bergman spaces, Fock spaces, and operators and linear functionals on these spaces. I am also interested in how these spaces relate to topics in harmonic analysis, and in spaces such as including real valued Hardy spaces, Littlewood-Paley theory, maximal functions, and square functions.
Much of my work relates to extremal problems. A typical problem is as follows: suppose that \(\phi\) is a bounded linear functional defined on some normed space of analytic functions. Maximize (the real part of) \(\phi(f)\) over all functions in the space of unit norm. For example, we could try to maximize the real part of \(f(0) + f(1/2) + f'(i/3)\) over all analytic functions in the unit disc that are bounded in absolute value by \(1\). Or we could maximize the same quantity over all functions analytic in the unit disc whose \(L^p\) norm (taken over the unit disc) is bounded by one.
I am also interested in finding upper and lower bounds for various operators on spaces of analytic function. For example, some coauthors and I have analyzed self commutators of Toeplitz operators in the Bergman space \(A^2\). For Hardy spaces, the norms of such operators are connected with the isoperimetric inequality, which is quite surprising! It turns out that these operators on Bergman spaces are connected with Saint-Venant's inequality for torsional rigidity.
Some other work of mine involves the order boundedness of weighted composition differentiation operators on Bergman spaces. This work is related to the solution of a certain interpolation problem on these spaces. I am interested in extending these results in several directions.
Another area of interest is in variable exponent Hardy and Bergman spaces, and their extension to variable Orlicz Hardy and Bergman spaces.
Much of my work relates to extremal problems. A typical problem is as follows: suppose that \(\phi\) is a bounded linear functional defined on some normed space of analytic functions. Maximize (the real part of) \(\phi(f)\) over all functions in the space of unit norm. For example, we could try to maximize the real part of \(f(0) + f(1/2) + f'(i/3)\) over all analytic functions in the unit disc that are bounded in absolute value by \(1\). Or we could maximize the same quantity over all functions analytic in the unit disc whose \(L^p\) norm (taken over the unit disc) is bounded by one.
I am also interested in finding upper and lower bounds for various operators on spaces of analytic function. For example, some coauthors and I have analyzed self commutators of Toeplitz operators in the Bergman space \(A^2\). For Hardy spaces, the norms of such operators are connected with the isoperimetric inequality, which is quite surprising! It turns out that these operators on Bergman spaces are connected with Saint-Venant's inequality for torsional rigidity.
Some other work of mine involves the order boundedness of weighted composition differentiation operators on Bergman spaces. This work is related to the solution of a certain interpolation problem on these spaces. I am interested in extending these results in several directions.
Another area of interest is in variable exponent Hardy and Bergman spaces, and their extension to variable Orlicz Hardy and Bergman spaces.
