Alexander Nagel

Alexander Nagel
University of Wisconsin–Madison | UW · Department of Mathematics

About

87
Publications
5,017
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3,546
Citations
Citations since 2016
3 Research Items
885 Citations
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150

Publications

Publications (87)
Article
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...
Preprint
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...
Article
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...
Chapter
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...
Article
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 $1<p<\infty$.
Article
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...
Article
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
Article
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...
Article
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.
Article
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.
Article
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...
Article
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...
Article
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...
Article
. 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...
Article
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...
Article
this paper, we generalize these ideas to the case of "wedge domains" with an"edge" which is a submanifold M ae C
Article
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...
Article
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).
Article
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...
Article
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,...
Article
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...
Article
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...
Article
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...
Article
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...
Article
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.
Article
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...
Article
Let {theta(j)} be a lacunary sequence going to zero. Let [Formula: see text]. Define [Formula: see text]. We prove [Formula: see text].
Article
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.
Article
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.
Article
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...
Article
THE AUTHORS PROVE THAT [FORMULA: see text] is bounded from L(P)(R(2)) to L(P)(R(2)) for 1 < p </= infinity.
Article
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...
Article
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...
Article
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...
Article
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 <.
Article
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...

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