Harmonic analysis, partial differential equations, geometric analysis, geometric computation
My primary research interests lie in harmonic analysis, partial differential equations, and geometric analysis. For example, Strichartz estimates for the solution of various evolution equations, including dispersive equations and transport equations. For dispersive equations, such estimates are directly related to questions in the context of the Stein restriction conjecture, which is a central problem concerning the Fourier transform in euclidean harmonic analysis, and enjoys further connections to problems in geometric analysis, such as the Kakeya conjecture and, via the multilinear perspective, to various generalisations of the Brascamp-Lieb inequality. Recently, I have been particularly interested in establishing optimal versions of such estimates, seeking sharp constants and information about the nature of maximising input functions when they exist, for example, via heat flow methods. In a different direction, I am also interested in geometric computation, including the analysis of Bézier curves and subdivision of curves and surfaces.
The nonlinear Brascamp-Lieb inequality for simple data
with J. Bennett, S. Buschenhenke, T. Flock, submitted.
On the Strichartz estimates for orthonormal systems of initial data with regularity
with Y. Hong, S. Lee, S. Nakamura, Y. Sawano, submitted.
Smoothing estimates for the kinetic transport equation at the critical regularity
with J. Cunanan, S. Lee, to appear in SIAM Journal on Mathematical Analysis.
A sharp k-plane Strichartz inequality for the Schrodinger equation
with J. Bennett, T. Flock, S. Gutierrez, M. Iliopoulou, to appear in Transactions of the American Mathematical Society.
Stability of trace theorems on the sphere
with C. Jeavons, T. Ozawa, M. Sugimoto, to appear in Journal of Geometric Analysis.
Estimates for the kinetic transport equation in hyperbolic Sobolev spaces
with J. Bennett, S. Gutierrez, S. Lee, to appear in Journal de Mathematiques Pures et Appliquees.
Generating monotone quantities for the heat equation
with J. Bennett, to appear in Crelle's journal.
Stability of the Brascamp-Lieb constant and applications
with J. Bennett, T. Flock, S. Lee, to appear in American Journal of Mathematics.
Remarks on the Mizohata-Takeuchi conjecture and related problems
with M. Sugimoto, to appear in Asympototic Analysis for Nonlinear Dispersive and Wave Equations, Advanced Studies in Pure Mathematics.
Behaviour of the Brascamp-Lieb constant
with J. Bennett, M. Cowling, T. Flock, Bulletin of the London Mathematical Society, 49 (2017), 515-518.
Sharpness of the Brascamp-Lieb inequality in Lorentz spaces
with S. Lee, S. Nakamura, Y. Sawano, Electronic Research Announcements in Mathematical Sciences, 24 (2017), 53-63.
A conjecture regarding optimal Strichartz estimates for the wave equation
with C. Jeavons, T. Ozawa, H. Saito, New Trends in Analysis and Interdisciplinary Applications, Trends in Mathematics, Springer (2017), 293-300.
Some sharp bilinear space-time estimates for the wave equation
with C. Jeavons, T. Ozawa, Mathematika, 62 (2016), 719-737.
A survey on optimal smoothing estimates and trace theorems
with M. Sugimoto, Advances in Mathematics (China), 45 (2016), 801-816.
Extremisers for trace estimates on the sphere
with S. Machihara, M. Sugimoto, Mathematical Research Letters, 23 (2016), 633-647.
On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type
with J. Bennett, C. Jeavons, N. Pattakos, Journal of the Mathematical Society of Japan, 69 (2017), 459-476.
Optimal constants and extremisers for some smoothing estimates
with M. Sugimoto, Journal d'Analyse Mathematique, 131 (2017), 159-187.
Flow monotonicity and Strichartz inequalities
with J. Bennett, M. Iliopoulou, International Mathematics Research Notices, 19 (2015), 9415-9437.
Some recent progress on sharp Kato-type smoothing estimates
with M. Sugimoto, Israel Mathematical Conference Proceedings, Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, Radon Transform, Contemporary Mathematics, 653 (2015), 41-50.
Applications of the Funk-Hecke theorem to smoothing and trace estimates
with H. Saito, M. Sugimoto, Advances in Mathematics, 285 (2015), 1767-1795.
Optimal forward and reverse estimates of Morawetz and Kato--Yajima type with angular smoothing index
with M. Sugimoto, Journal of Fourier Analysis and Applications, 21 (2015), 318-341.
A sharp Sobolev-Strichartz estimate for the wave equation
with C. Jeavons, Electronic Research Announcements in Mathematical Sciences, 22 (2015), 46-54.
Sharp Sobolev-Strichartz estimates for the free Schrodinger propagator
with C. Jeavons, N. Pattakos, Current Trends in Analysis and Its Applications, Proceedings of the 9th ISAAC Congress, Krakow, Birkhauser Basel, 2015, pages 281-288.
A note on magnitude bounds for the mask coefficients of the interpolatory Dubuc-Deslauriers subdivision scheme
with H. E. Bez, LMS Journal of Computation and Mathematics, 17 (2014), 226-232.
On the Strichartz estimates for the kinetic transport equation
with J. Bennett, S. Gutierrez, S. Lee, Communications in Partial Differential Equations, 39 (2014), 1821-1826.
Optimal constant for a smoothing estimate of critical index
with M. Sugimoto, Fourier Analysis, Trends in Mathematics, Birkhauser/Springer Basel AG, Basel, 2014, pages 1-7.
A sharp Strichartz estimate for the wave equation with data in the energy space
with K. Rogers, Journal of the European Mathematical Society, 15 (2013), 805-823.
On derivative bounds for the rational quadratic Bezier paths
with H. E. Bez, Computer Aided Geometric Design, 30 (2013), 254-261.
New minimal bounds for the derivatives of rational Bezier paths and rational rectangular Bezier surfaces
with H. E. Bez, Applied Mathematics and Computation, 225 (2013), 475-479.
Transversal multilinear Radon-like transforms: local and global estimates
with J. Bennett, S. Gutierrez, Revista Matematica Iberoamericana, 29 (2013), 756-788.
Global nonlinear Brascamp-Lieb inequalities
with J. Bennett, S. Gutierrez, Journal of Geometric Analysis, 23 (2013), 1806-1817.
A majorant problem for the periodic Schrodinger group
with J. Bennett, Symposium on Harmonic Analysis and Nonlinear PDE (RIMS, Kyoto, 2011), RIMS Kokyuroku Bessatsu B33, pages 1-10.
Some nonlinear Brascamp-Lieb inequalities and applications to harmonic analysis
with J. Bennett, Journal of Functional Analysis, 259 (2010), 2520-2556.
Maximal operators and Hilbert transforms along flat curves near L^1
Journal of the Australian Mathematical Society, 87 (2009), 311-323.
Maximal operators along piecewise linear curves near L^1
Indiana University Mathematics Journal, 58 (2009), 1639-1658.
Heat-flow monotonicity of Strichartz norms
with J. Bennett, A. Carbery, D. Hundertmark, Analysis & PDE 2 (2009), 147-158.
Heat-flow monotonicity underlying some sharp inequalities in geometric and harmonic analysis
Symposium on Harmonic Analysis and Nonlinear PDE (RIMS, Kyoto, 2008), RIMS Kokyuroku Bessatsu B14, pages 1-16.
Heat-flow monotonicity related to the Hausdorff-Young inequality
with J. Bennett, A. Carbery, Bulletin of the London Mathematical Society 41 (2009), 971-979.
Closure properties of solutions to heat inequalities
with J. Bennett, Journal of Geometric Analysis, 19 (2009), 584-600.
Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms
Journal of Functional Analysis, 255 (2008), 3281-3302.
L^p-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves
Proceedings of the American Mathematical Society, 135 (2007), 151-161.