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## Publications

Publications (87)

For appropriate domains \(\Omega _{1}, \Omega _{2}\), we consider mappings \(\Phi _{\mathbf {A}}:\Omega _{1}\rightarrow \Omega _{2}\) of monomial type. We obtain an orthogonal decomposition of the Bergman space \({\mathcal {A}}^{2}(\Omega _{1})\) into finitely many closed subspaces indexed by characters of a finite abelian group associated to the m...

For appropriate domains $\Omega_{1}, \Omega_{2}$ we consider mappings $\Phi_{\mathbf A}:\Omega_{1}\to\Omega_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(\Omega_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $\Phi_{\mathbf A}$. W...

The purpose of this paper is to study algebras of singular integral operators
on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the
composition of Calder\'on-Zygmund operators with different homogeneities, such
as operators that occur in sub-elliptic problems and those arising in elliptic
problems. For example, one would li...

This chapter discusses diagonal estimates for Bergman kernels in monomial-type domains. It begins by briefly recalling the definition and some of the elementary properties of the Bergman projection and Bergman kernel. The chapter then turns to a problem which requires obtaining uniform estimates for a Bergman kernel on the diagonal of a model monom...

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$.
We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra
under composition, and that such operators are bounded on $L^{p}(G)$ for
$1<p<\infty$.

We study the singularities of the Bergman and Szego{double acute} kernels for domains Ω={z1, z2}∈ C{double-struck}2 {pipe} Im z2 > b(Re z1)}. Here b is an even function in C∞(R{double-struck}) satisfying b(0)= b'(0)=0, b"(r) > 0 for r ≠ 0, and vanishing to infinite order at r = 0. A model example is b(r) = exp.(-{pipe}r{pipe}-a) for {pipe}r{pipe} s...

We establish the $L^{p}$ -boundedness, for $1 \lt p \leq \infty$ , of maximal averages over sets defined by a finite number of monomial inequalities. The proof requires a detailed study of the geometric properties of these monomial polyhedra

The purpose of this paper is to prove optimal estimates for solutions of the Kohn-Laplacian for certain classes of model domains in several complex variables. This will be achieved by applying a type of singular integral operator whose novel features (related to product theory and flag kernels) differ essentially from the more standard Calderon-Zyg...

We wish to acknowledge and correct an error in a proof in our paper On the product theory of singular integrals, which appeared in Revista Matematica Iberoamericana, volume 20, number 2, 2004, pages 531-561.

We wish to acknowledge and correct an error in a proof in our paper On the product theory of singular integrals, which appeared in Revista Matemática Iberoamericana, volume 20, number 2, 2004, pages 531-561.

We establish $L^p$-boundedness for a class of product singular integral operators on spaces $\widetilde{M} = M_1 \times M_2\times \cdots \times M_n$. Each factor space $M_i$ is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard sin...

We study a class of operators on nilpotent Lie groups G given by convolution with flag kernels. These are special kinds of product-type distributions whose singularities are supported on an increasing subspace (0)⊂V1⊂…⊂Vk⊂…⫋G. We show that product kernels can be written as finite sums of flag kernels, that flag kernels can be characterized in terms...

We show that for each control metric (i.e., Carnot-Caratheodory metric), there is an equivalent metric which has the maximal expected degree of smoothness. The equivalent metric satisfies the natural differential inequalities with respect to the vector fields used to define the metric. This generalizes the regularity of the usual Euclidean metric i...

. The following three classes of models of rigid submanifolds of higher type with CR dimension one are discussed: 1) a tube-like model that only depends on the real part of the holomorphic tangent coordinate; 2) a radial model that depends on the modulus of the holomorphic tangent coordinate and 3) a free model. The first and third models have a Li...

The purpose of this paper is to prove the L^p boundedness of singular Radon transforms and their maximal analogues. These operators differ from the traditional singular integrals and maximal functions in that their definition at any point x in R^n involves integration over a k-dimensional submanifold of R^n, depending on x, with k < n. The role of...

this paper, we generalize these ideas to the case of "wedge domains" with an"edge" which is a submanifold M ae C

We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in C n contains an open set in en which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is i...

The Raman spectra of A(BxW1-x)O3 (A: K, Rb, Cs; B: Mg, Al, Ga, Ti, V, Nb, Ta) with defect pyrochlore or hexagonal tungsten bronze (HTB) structure and of Cs(BMO3F3) (M: Mo, W; B: Ni, Cu, Zn) and AM2O5F (A: Rb, Cs; M: Nb, Ta) with defect pyrochlore structure are detected (synthesis conditions are given in a table).

Our purpose here is to announce results in harmonic analy sis related to a large class of hypoelliptic operators on arbitrary simply-connected nilpotent Lie groups. We find the asymptotic development of their fundamental solutions, both locally and at infinity, and study corresponding Riesz transforms and analogues of the Hardy-Littlewood-Sobolev...

Die Ramanspektren der gemischten Oxide Ay(BxW1−x)O3 (A = K, Rb, Cs; B = Mg, Al, Ga, Ti, V, Nb, Ta) sowie der Oxidfluoride Cs(BMO3F3) (M = Mo, W; B = Ni, Cu, Zn) und AM2O5F (A = Rb, Cs; M = Nb, Ta) mit Defektpyrochlor‐ bzw. hexagonaler Wolframbronzen(HTB)‐Struktur wurden gemessen. Alle Spektren enthalten Banden mit Wellenzahlen größer als 900 cm⁻¹,...

The purpose of this paper (some of whose conclusions were announced in [NRSW]) is to study the Bergman and Szegd projection operators on pseudoconvex domains Q of finite type in C2. The results we obtain are of three kinds: (i) The "size" estimates of the Bergman and Szeg6 kernels, and their derivatives. (ii) The "cancellation" properties of those...

We outline results obtained for the partial differential-Neumann problem for an arbitrary pseudoconvex domain in C(2) of finite type. We obtain an approximation to the Neumann operator. A number of sharp estimates for the solution of partial differentialu = f are a consequence; one of these is an extension of the L(1) estimate of Henkin and Skoda u...

We outline results obtained for the {partial}-Neumann problem for an arbitrary pseudoconvex domain in C2 of finite type. We obtain an approximation to the Neumann operator. A number of sharp estimates for the solution of {partial}u = f are a consequence; one of these is an extension of the L1 estimate of Henkin and Skoda used to characterize the ze...

Hilbert transforms and maximal functions along curves and surfaces, spectral synthesis problems, and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators. The problem of estimating such Fourier transforms has a long history. See for example Hlawka [3], Herz [2], Littman [4], Randol [9], [10], Sv...

We study several notions of distance and ball on the boundary of domains of finite type in C(n). We use these notions to develop a theory of maximal functions and area integrals for functions holomorphic in such domains.

Let D ⊂⊂ C ⁿ be a bounded domain with smooth boundary ∂ D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂ D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that...

Let {theta(j)} be a lacunary sequence going to zero. Let [Formula: see text]. Define [Formula: see text]. We prove [Formula: see text].

We describe a class of pseudo-differential operators and their singular integral realizations and show how these may be used to give precise estimates in various function spaces. Application will be given in particular to several situations in which subelliptic estimates arise for partial differential equations.

We show that pseudoconvex domains have the Mergelyan property if the Levi form is degenerate on a sufficiently small set in the boundary. This includes the case when the weakly pseudoconvex points all lie on a smooth curve.

Let ( t , γ ( t ) ) (t,\gamma (t)) be a plane curve. Set H γ f ( x , y ) = p.v. ∫ f ( x − t , y − γ ( t ) ) d t / t {H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t for f ∈ C 0 ∞ ( R 2 ) f \in C_0^\infty ({R^2}) . For a large class of curves, the authors prove ‖ H γ f ‖ p ⩽ A p ‖ f ‖ p , 5 / 3 > p > 5 / 2 {\left \| {{H_\gamma...

THE AUTHORS PROVE THAT [FORMULA: see text] is bounded from L(P)(R(2)) to L(P)(R(2)) for 1 < p </= infinity.

All functions mentioned in this paper will be real-valued. If f 1 , f 2 , g are nonnegative functions on a set S that satisfy g ≦ f 1 + f 2 , the Riesz decomposition problem associated with these data is to find functions g i on S such that
The formula
always furnishes a solution. The problem becomes more interesting if one asks under what condit...

We consider algebras A of continuous complex valued functions, which are given as the set of global sections of a sheaf y on a topological space X. Under the hypothesis that all the higher cohomology groups of the sheaf are zero, we investigate the relationship between ideals in A, kernels of algebra homomorphisms of A into the complex numbers C, a...

We consider algebras A of continuous complex valued functions, which are given as the set of global sections of a sheaf S \mathcal {S} on a topological space X . Under the hypothesis that all the higher cohomology groups of the sheaf are zero, we investigate the relationship between ideals in A , kernels of algebra homomorphisms of A into the compl...

This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in Cⁿ. Very general axioms for boundary behavior are given. This includes as special cases L& boundary behavior,¹ <p< oo; continuous boundary behavior; differentiable boundary behavior of order m, 0 < m <.

We give a proof, in a stronger form, of a conjecture proposed by A. Gleason. Assume D is a strongly pseudoconvex domain D ⋐ C2 with smooth boundary and let be any function continuous in D̄ and holomorphic in D. Assume further that (a, b) ϵD, h(a, b) = 0. Then , where h1 and h2 are continuous in D̄ and holomorphic in D. In addition we prove that if...