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Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.

Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $\gamma$. A surprising consequence of the analysis is that, whenever $\gamma$ is irrational, the Bergman projection is bounded only for $p=2$., Comment: 21 pages, 2 figures

A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.

A classical observation of Riesz says that truncations of a general $\sum_{n=0}^\infty a_n z^n$ in the Hardy space $H^1$ do not converge in $H^1$. A substitute positive result is proved: these partial sums always converge in the Bergman norm $A^1$. The result is extended to complete Reinhardt domains in $\C^n$. A new proof of the failure of $H^1$ convergence is also given., Comment: Polynomial density argument added. Several typos corrected

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.

Solution operators for the equation ∂ ¯ u = f are constructed on general product domains in C n . When the factors are one-dimensional, the operator is a simple integral operator: specific derivatives of f are integrated against iterated Cauchy kernels. For higher dimensional factors, the solution is constructed by solving sub- ∂ ¯ equations with modified data on the factors. Estimates in several norms are proved.

Let Ω ⋐ C n \Omega \Subset \mathbb {C}^{n} be a domain with smooth boundary, whose Bergman projection B B maps the Sobolev space H k 1 ( Ω ) H^{k_{1}}(\Omega ) (continuously) into H k 2 ( Ω ) H^{k_{2}}(\Omega ) . We establish two smoothing results: (i) the full Sobolev norm ‖ B f ‖ k 2 \|Bf\|_{k_{2}} is controlled by L 2 L^2 derivatives of f f taken along a single, distinguished direction (of order ≤ k 1 \leq k_{1} ), and (ii) the projection of a conjugate holomorphic function in L 2 ( Ω ) L^{2}(\Omega ) is automatically in H k 2 ( Ω ) H^{k_{2}}(\Omega ) . There are obvious corollaries for when B B is globally regular.

Let B be the Bergman projection associated to a domain Ω on which the \({\bar\partial}\) -Neumann operator is compact. We show that arbitrary L2 derivatives of Bf are controlled by derivatives of f taken in a single, distinguished direction. As a consequence, functions not contained in \({C^{\infty}(\overline{\Omega})}\) that are mapped by B to \({C^{\infty}(\overline{\Omega})}\) are explicitly described.

We give three proofs of the fact that a smoothly bounded, convex domain in ℝn has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.

Let $$\Omega \subset \subset \mathbb{C}^{n}$$ be a smoothly bounded domain. Suppose Ω has a defining function, such that the sum of any q eigenvalues of its complex Hessian is non-negative. We show that this implies global regularity of the Bergman projection, B j-1, and the $$\bar{\partial}$$ -Neumann operator, N j , acting on (0,j)-forms, for $$j\in\{q,\ldots,n\}$$ .

We establish a vanishing theorem on the square-integrable cohomology associated to the Cauchy-Riemann complex on some complete Kaehler manifolds. The hypothesis needed for this result is a growth condition on a primitive of the Kaehler form.

We show that the existence of a certain family of plurisubharmonic functions on a bounded domain in Cn implies that the ∂-Neumann operator associated to the domain is compact. Our condition generalizes previous work of Catlin on compactness of the ∂-Neumann operator.

First published in Proceedings of the American Mathematical Society in volume 130 and issue 1, published by the American Mathematical Society.

Emerging technologies that use spread spectrum and frequency hopping signals may enable more efficient use of the high-frequency (HF) band. High-powered HF radar is an example of this technology where channels are used that appear to be unoccupied and the frequency hopping signal of the HF radar has been remarkably free from interfering with conventional spectrum occupants. Considering future application of HF systems of this type, it is of interest to examine the availability of free channels in the frequency range between 3 and 30 MHz for bandwidth (BW) availability and time duration. This paper discusses the characteristics of channels that are available for use in the HF spectrum based on spectrum measurements by the Navy's Relocatable Over-the-Horizon Radar (ROTHR) located in the southeast section of Virginia. In particular, the ROTHR data being analyzed are from the spectrum monitor (SM). The SM is a subsystem of the ROTHR which measures the power level in 3-kHz intervals for frequencies between 5 and 28 MHz, the radar's operating frequency range. Measurements of the HF spectrum were analyzed for the fall and winter seasons at the radar's location. The analysis consisted of estimating the number of clear channels of various widths in selected megahertz bands over the ROTHR's operating frequency range. Characteristics of available channels are presented in various graphical forms which include clear channel availability, probability of at least one clear channel of a specified width, and duration of clear channels as a function of frequency. The Weibull distribution is proposed as an analytical model for investigating channel duration. Analysis results indicate which portion of the HF spectrum is most likely to have available bandwidths and what their expected durations are as a function of season and time of day.

A new proof of Oka's lemma is given for smoothly bounded, pseudoconvex domains $D\subset\mathbb{C}^n$. The method of proof is then also applied to other convexity-like hypotheses on the boundary of $D$.

Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for taking a log resolution of the monomial ideal.

We construct bounded, plurisubharmonic functions with maximally large Hessians near the boundary of a smoothly bounded convex domain in Cn. As a corollary, the equality of the order of contact of the boundary with complex analytic varieties (the D'Angelo type) and the order of contact with complex lines is demonstrated.

Toeplitz operators on strongly pseudoconvex domains in $\mathbb{C}^n$, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on $L^p$, as the power of the distance varies, are established.

We establish new results on weighted $L^2$ extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold., Comment: To Appear in Ann. Inst. Fourier

and declare the series to converge provided that limN-,N SN exists as a finite real number, which number is called the sum of the given series. This paradigm becomes so ingrained in most of us that we seldom question its appropriateness. To be sure, a frequently useful theorem of K. Weierstrass (1815-1897) tells us that an absolutely convergent series may be rearranged at will, with the convergence undisturbed and the resulting sum the same. A companion theorem due to G. F. B. Riemann (1826-1866), asserting that a conditionally convergent series may be rearranged to obtain any sum in the extended reals, seems to lay to rest any desire to rearrange conditionally convergent series. But that desire has blossomed forth in the authors of this article. One of us learned, in the course of teaching an analysis class, that there exist permutations a of the positive integers N with the following two properties

Acoustic intensity measurements using the two‐sensor cross‐spectral density technique are susceptible to errors when made in the presence of mean flow. These errors are predominantly due to turbulent pressure fluctuations acting on and near the sensors. To quantify these errors, intensity measurements were performed in a sound field that contained both an independent acoustic source and a mean, low Mach number (M

The holomorphic sectional curvatures in the Bergman metric of a smooth bounded pseudoconvex domain in C 2 {{\mathbf {C}}^2} are shown to be bounded in absolute value near a poinit of finite type in the boundary

We present the theory of twisted \(L^2\) estimates for the Cauchy–Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa–Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the \(\bar{\partial }\)-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.

Formic acid has been found in the urine of normal adults by several investigators. 1 With the intention of studying its presence in the urine of children, we examined fifty-six specimens from ten infants and nine older children for a period, in most cases, of three days each. The patients were suffering from various diseases (Tables 1 and 2). The urine in all cases was negative to copper reduction tests. The method of Autenrieth for the determination of formic acid in the urine was used. Three hundred c.c., or less, of urine, was distilled over a free flame, after the addition of 30 c.c. of 25 per cent, phosphoric acid, until about 75 c.c remained in the flask, which was then filled to the original volume by water from a separatory funnel. The distillation was continued until the distillate was no longer acid to litmus (about 1,200 c.c. in our