My research interests are in harmonic analysis, complex analysis in one and several variables, PDE, and their interplay; these have been primarily concerned with the study of singular integral operators, elliptic boundary value problems and Div-Curl inequalities. A leading theme is the use of boundary integral representation formulas, with main focus on the lack of boundary regularity of the ambient domain. More recently, I have made an effort to build parts of this program within the novel context of several complex variables — to develop a theory of Cauchy-like singular integrals with holomorphic kernel and for non-smooth domains in n-dimensional complex Euclidean space that effectively blends the complex structure of the ambient domain with the Calderòn-Zygmund theory for singular integrals on non-smooth domains in 2n-dimensional real Euclidean space. Applications to several complex variables include the regularity of orthogonal projections onto the Bergman and Hardy spaces of holomorphic functions on non-smooth domains.
Because of the rigidity imposed by the complex structure, the study of these problems requires new and different tools than in the classical setting.
18. Lanzani L., Myers J. and Raich S. A. Taylor Series of Conformal Mappings onto Symmetric Quadrilaterals, Cplx. Vbls & Ellipt. Eqns., to appear.
17. Lanzani L. and Stein E. M. , The Cauchy Integral in C^n for domains with minimal smoothness, Adv. Math. 264 (2014) 776 – 830.
16. Lanzani L. and Stein E. M. Cauchy-type integrals in several complex variables, Bull. Math. Sci. 3 (2) (2013), 241-285.
15. Lanzani L. Higher Order Analogues of Exterior Derivative, Bull. IMAS (New Series) 8 (3) (2013) 389 – 398.
14. Lanzani L. and Raich A. S. On Div-Curl for higher order, Advances in Analysis: the Legacy of E. M. Stein, Princeton U. Press (2013), ISBN: 9780691159416.
13. Lanzani L. and Stein E. M. The Bergman projection in L^p for domains with minimal smoothness, Illinois J. of Math. (invited submission) 56 (1) (2013) 127 – 154.
12. Barrett, D. E. and Lanzani L. The spectrum of the Leray transform on weighted boundary spaces for convex Reinhardt domains, J. Funct. Analysis 257 (9) (2009), 2780-2819.
11. Koenig, K. and Lanzani, L. Bergman vs. Szegö via Conformal Mapping, Indiana U. Math. J. 58, no. 2 (2009), 969-997.
10. Lanzani, L. Cauchy Integrals for non-smooth domains: C^n vs. C – the effect of dimension, Oberwolfach Reports 32 (2008), 55-60.
9. Brown, R., Capogna, L. and Lanzani, L. On the mixed boundary value problem in L^p for some two-dimensional Lipschitz domains, Math. Annalen, 342 (2008), 91-124.
8. Lanzani, L. and Mendez, O. The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains, Revista Mat. Iberoamer. 22 (2006) 181-204.
7. Lanzani, L. and Stein, E. M. A note on Div-Curl inequalities, Mathematical Research Letters 12 (2005), 57-61.
6. Lanzani, L. Shen, Z. On the Robin Boundary Condition for Laplace’s Equation in Lipschitz Domains Comm. Part. Diff. Eq. 29 (2004), 91-109.
5. Lanzani, L. and Stein, E. M. Szegö and Bergman projections on non-smooth planar domains, J. Geom. Anal. 14 (2004), 63-86.
4. Bernstein, S. and Lanzani, L. Szegö projections for Hardy spaces of monogenic functions and applications, Int. J. of Math. and Math. Sc., 29 (2002), 613-627.
3. Lanzani, L. The Cln-Valued Robin Bundary Value Problem on Lipschitz Domains in R^n, Clifford Analysis and its Appls. NATO ARW Series (2001), Kluwer, 183-192.
2. Lanzani, L. Cauchy Transform and Hardy Spaces for Rough Planar Domains, Contemp. Math., 251 (2000), 409-428.
1. Lanzani, L. Szegö Projection Versus Potential Theory For Non-Smooth Planar Domains, Indiana U. Math. J. 48 (1999), 537-556.
Capogna, L., Kenig, C., Lanzani, L. Recent Progress in Harmonic Measure: the Geometric and the Analytic Points of View, AMS University Lecture Series (book), 35 (2005), ISBN: 0821827286.
co-Editor, Harmonic Analysis and Boundary Value Problems: Selected Papers from the 25th University of Arkansas Spring Lecture Series, Contemp. Math. 277 (2001).