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We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an involution, and polarization is viewed as an averaging operation over a compact multiplicative subgroup of the scalars. Using this we prove a general form of the Jordan-von Neumann theorem on characterizing inner product spaces among normed linear spaces, when the scalars are taken in an associative algebra., Comment: Typos corrected. References to old work on polarization identities and the Jordan-von Neumann theorem over noncommutative scalars added. To appear in Linear Algebra and its Applications

We study Sobolev estimates for solutions of the inhomogenous Cauchy–Riemann equations on annuli in $${\mathbb {C}}^n$$ , by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescribed support and estimates in Sobolev spaces using our method.

We prove a modified form of the classical Morrey-Kohn-Hormander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in $${{\mathbb {C}}}^n$$ , where the inner domain has $${\mathcal {C}}^{1,1}$$ boundary, we show that the $$L^2$$ Dolbeault cohomology group in bidegree (p, q) vanishes if $$1\le q\le n-2$$ and is Hausdorff and infinite-dimensional if $$q=n-1$$ , so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the $$L^2$$ Sobolev space $$W^1$$ on any pseudoconvex domain with $${\mathcal {C}}^{1,1}$$ boundary. We also generalize our results to annuli between domains which are weakly q-convex in the sense of Ho for appropriate values of q.

It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin., Comment: Updated version; to appear in the Proceedings of the AMS

We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases., New main theorem and two new co-authors. The new results are applicable to a much more general class of domains

We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series., Typos corrected. To appear in Pacific Journal of Mathematics

We show that holomorphic functions of polynomial growth on domains with corners have distributional boundary values in an appropriate sense, provided the corners are generic CR manifolds. We also prove an analog of the Bochner–Hartogs theorem for these boundary values for the simplest such domains, the product domains.

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.

We explicitly compute the Dolbeault cohomologies of certain domains in complex space generalizing the classical Hartogs figure. The cohomology groups are non-Hausdorff topological vector spaces, and it is possible to identify the reduced (Hausdorff) and the indiscrete part of the cohomology.

We give an example of a pseudoconvex domain in a complex manifold whose \(L^2\)-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in \({\mathbb {C}}^2\), hence Stein. This implies that for \(q>0\), the usual Dolbeault cohomology with respect to smooth forms vanishes in degree \((p,q)\). But the \(L^2\)-Cauchy–Riemann operator on the domain does not have closed range on \((2,1)\)-forms and consequently its \(L^2\)-Dolbeault cohomology is not Hausdorff.

Expected duality and approximation properties are shown to fail on Bergman spaces of domains in C n , via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Reinhardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc.

We consider a class of domains, generalizing the upper half-plane, and admitting rotational, translational and scaling symmetries analogous to the half-plane. We prove Paley-Wiener type representations of functions in Bergman spaces of such domains with respect to each of these three groups of symmetries. The Fourier series, Fourier integral and Mellin integral representations so obtained may be used to give representations of the Bergman kernels of these domains.

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli., Small typos corrected; Final Version; To appear in Indiana University Mathematics Journal

The ∂ ¯ \overline {\partial } -Neumann operator (the inverse of the complex Laplacian) is shown to be noncompact on certain domains in complex Euclidean space. These domains are either higher-dimensional analogs of the Hartogs triangle or have such a generalized Hartogs triangle imbedded appropriately in them.

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion., Comment: 2 figures

The regularity of the \(\overline{\partial }\)-problem on the domain \(\{\left|{z_1}\right|\!

In this popular expository article, we discuss some important ways in which complex analysis in more than one variable is different from complex analysis in one variable. Analytic continuation in several variables is contrasted with that in one variable, and the notion of psuedoconvexity is defined. Hartogs phenomenon and the Levi problem are also discussed in an informal way.

We establish the L 2 theory for the Cauchy–Riemann equations on product domains provided that the Cauchy–Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p, 1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at p of a smooth CR function f that cannot be holomorphically extended to either side of M. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface M, which guarantees one-sided holomorphic extension of CR functions on M, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case., Comment: Minor typos fixed. Final version. To appear in the Indiana University Mathematics Journal

We give examples of non-smooth sets in the complex plane with the property that every holomorphic map continuous to the boundary from these sets into any complex manifold may be uniformly approximated by maps holomorphic in some neighborhood of the set (Mergelyan-type approximation for manifold-valued maps.) Similar results are proved for sections of complex-valued holomorphic submersions from complex manifolds.

We discuss without proof some recent advances in the theory of distributional boundary values, and give an example to show that the condition that a piecewise smooth domain has generic corners is necessary for the existence of boundary values., To appear in Contemporary Mathematics ( Proceedings of the Conference on Analysis and Geometry in Several Complex Variables, Doha, Qatar, January 2015)

We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted $L^p$ spaces when $p>\frac{4}{3}$, where the weight is a power of the distance to the singular boundary point. For $1, Comment: Small typos corrected. To appear in the Proceedings of the AMS

We show that the $$\overline{\partial }$$ -problem is globally regular on a domain in $${\mathbb {C}}^n$$ , which is the $$n$$ -fold symmetric product of a smoothly bounded planar domain. Remmert–Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from Cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary.

We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains., Some typos fixed in this final version. To appear in {\em Advances in Mathematics.}

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate., Comment: Final version: to appear in Illinois Journal of Mathematics

A simple proof of Ramanujan's formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by solving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis., Comment: To appear in Archiv der Mathematik

We show that the spectrum of the complex Laplacian on a product of hermitian manifolds is the Minkowski sum of the spectra of the complex Laplacians on the factors. We use this fact to show that the range of the d-bar operator on a product manifold is closed, provided it is closed in each factor., Minor typos fixed; to appear in the Proceedings of the AMS.

We study the approximation of J-holomorphic maps continuous to the boundary from ma domain in the complex plane into an almost complex manifold by maps J-holomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application, we obtain $\mathcal{C}^k$ approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough., Comment: Based on the author's doctoral dissertation/

It is proved that CR functions on a quadratic cone M in $\C^n$, n>1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in $\C^2$ is also given., A previous version of this article appeared in Mathematische Annalen, and subsequently a short erratum appeared. This version incorporates the (minor) changes given in the erratum