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.

**Articles**

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.

**Books**

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).