| 1. |
- Bénéteau, Catherine, et al.
(author)
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On the concept of inner function in Hardy and Bergman spaces in multiply connected domains
- 2019
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In: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 9:2, s. 839-866
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Journal article (peer-reviewed)abstract
- We discuss the notion of an inner function for spaces of analytic functions in multiply connected domains in C, giving a historical overview and comparing several possible definitions. We explore connections between inner functions, zero-divisors for Hardy spaces and Bergman spaces, and weighted reproducing kernels. After recording some obstructions and negative results, we suggest avenues for further research and point out several open problems.
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| 2. |
- Bergqvist, Linus
(author)
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A note on cyclic polynomials in polydiscs
- 2018
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In: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 8:2, s. 197-211
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Journal article (peer-reviewed)abstract
- We use methods from potential theory and harmonic analysis to show non-cyclicity of polynomials on a polydisc whose zero set meets the distinguished boundary along a hypersurface. We also generalize methods used for proving cyclicity for polynomials in two variables with small zero sets to arbitrary dimension. In doing so, we show that in higher dimension, the cyclicity properties of a function do not only depend on the codimension, but also on the orientation of the zero set. Furthermore, we illustrate our results by studying a special class of polynomials. Finally, we use methods from potential theory to prove that our estimates for non-cyclicity are in fact sharp.
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| 3. |
- Gravin, Nick, et al.
(author)
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On moments of a polytope
- 2018
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In: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 8:2, s. 255-287
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Journal article (peer-reviewed)abstract
- We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P subset of R-d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.
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| 4. |
- Gustafsson, Björn, 1947-, et al.
(author)
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Line bundles defined by the Schwarz function
- 2018
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In: Analysis and Mathematical Physics. - : SPRINGER BASEL AG. - 1664-2368 .- 1664-235X. ; 8:2, s. 171-183
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Journal article (peer-reviewed)abstract
- Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.
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| 5. |
- Gustafsson, Björn, 1947-
(author)
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The string equation for polynomials
- 2018
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In: Analysis and Mathematical Physics. - : Springer Basel. - 1664-2368 .- 1664-235X. ; 8:4, s. 637-653
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Journal article (peer-reviewed)abstract
- For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova–Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we show that the string equation makes sense and holds for general polynomials.
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| 6. |
- Petermichl, Stefanie, et al.
(author)
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A matrix weighted bilinear Carleson lemma and maximal function
- 2019
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In: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 9:3, s. 1163-1180
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Journal article (peer-reviewed)abstract
- We prove a bilinear Carleson embedding theorem with matrix weight and scalar measure. In the scalar case, this becomes exactly the well known weighted bilinear Carleson embedding theorem. Although only allowing scalar Carleson measures, it is to date the only extension to the bilinear setting of the recent Carleson embedding theorem by Culiuc and Treil that features a matrix Carleson measure and a matrix weight. It is well known that a Carleson embedding theorem implies a Doob’s maximal inequality and this holds true in the matrix weighted setting with an appropriately defined maximal operator. It is also known that a dimensional growth must occur in the Carleson embedding theorem with matrix Carleson measure, even with trivial weight. We give a definition of a maximal type function whose norm in the matrix weighted setting does not grow with dimension.
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| 7. |
- Shapiro, Boris
(author)
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On Evgrafov-Fedoryuk's theory and quadratic differentials
- 2015
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In: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 5:2, s. 171-181
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Journal article (peer-reviewed)abstract
- The purpose of this note is to recall the theory of the (homogenized) spectral problem for Schrodinger equation with a polynomial potential and its relation with quadratic differentials. We derive from results of this theory that the accumulation rays of the eigenvalues of the latter problem are in -correspondence with the short geodesics of the singular planar metrics induced by the corresponding quadratic differential. We prove that for a polynomial potential of degree the number of such accumulation rays can be any positive integer between (d - 1) and (d/2) .
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| 8. |
- Tkachev, Vladimir, 1963-
(author)
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Sharp pointwise gradient estimates for Riesz potentialswith a bounded density
- 2018
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In: Analysis and Mathematical Physics. - : Springer Berlin/Heidelberg. - 1664-2368 .- 1664-235X. ; 8:4, s. 711-730
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Journal article (peer-reviewed)abstract
- We establish sharp inequalities for the Riesz potential and its gradient inRnand indicate their usefulness for potential analysis, moment theory and other applications.
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