Overview
My work develops analytic tools for understanding linear and nonlinear partial differential equations, especially through harmonic analysis and spectral theory. A central theme is the study of how geometric and functional-analytic properties of differential operators influence heat kernels, wave propagation, spectral multipliers, Bochner--Riesz means and Riesz transforms. This includes work on elliptic and sub-elliptic operators, manifolds with ends, Lie groups, functional calculi, restriction-type estimates and Schrödinger and wave equations. Alongside my research, I am committed to strengthening inclusion and gender equality in mathematics and to supporting a more diverse mathematical community.
Research Focus
Partial differential equations and harmonic analysis; mathematical physics; analysis on Lie groups; singular integrals; spectral and Fourier multiplier theorems; Bochner--Riesz summability; restriction-type theorems; functional calculi and spectral analysis of elliptic, sub-elliptic and degenerate elliptic operators; Riesz transforms; semigroups of operators and heat kernels; wave equations; Schrödinger propagators; nonlinear heat and wave equations.
Publications
- Sikora, A. (2004). Riesz transform, Gaussian bounds and the method of wave equation. Mathematische Zeitschrift, 247(3), 643--662.
- Duong, X. T., Ouhabaz, E. M., & Sikora, A. (2002). Plancherel-type estimates and sharp spectral multipliers. Journal of Functional Analysis, 196(2), 443--485.
- Coulhon, T., & Sikora, A. (2007). Gaussian heat kernel upper bounds via the Phragmén--Lindelöf theorem. Proceedings of the London Mathematical Society, 96(2), 507--544.
- Guillarmou, C., Hassell, A., & Sikora, A. (2013). Restriction and spectral multiplier theorems on asymptotically conic manifolds. Analysis & PDE, 6(4), 893--950.
- Chen, P., Lee, S., Sikora, A., & Yan, L. (2020). Bounds on the maximal Bochner--Riesz means for elliptic operators. Transactions of the American Mathematical Society, 373(6), 3793--3828.
- Chen, P., Ouhabaz, E. M., Sikora, A., & Yan, L. (2016). Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner--Riesz means. Journal d’Analyse Mathématique, 129(1), 219--283.
- Hassell, A., & Sikora, A. (2019). Riesz transforms on a class of non-doubling manifolds. Communications in Partial Differential Equations, 44(11), 1072--1099.
- Chen, P., Hebisch, W., & Sikora, A. (2016) Bochner–Riesz profile of anharmonic oscillator L=− dx^2+| x|. Journal of Functional Analysis, 271 (11), 3186--3241
- Sikora, A., & Wright, J. (2001). Imaginary powers of Laplace operators. Proceedings of the American Mathematical Society, 129(6), 1745--1754.
- Hebisch, W., & Sikora, A. (1990). A smooth subadditive homogeneous norm on a homogeneous group. Studia Mathematica, 96(3), 231--236.
- Cowling, M., & Sikora, A. (2001). A spectral multiplier theorem for a sublaplacian on SU(2). Mathematische Zeitschrift, 238(1), 1--36.