92 research outputs found
Capacity of shrinking condensers in the plane
We show that the capacity of a class of plane condensers is comparable to the
capacity of corresponding "dyadic condensers". As an application, we show that
for plane condensers in that class the capacity blows up as the distance
between the plates shrinks, but there can be no asymptotic estimate of the
blow-up
Carleson Measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on Complex Balls
We characterize the Carleson measures for the Drury-Arveson Hardy space and
other Hilbert spaces of analytic functions of several complex variables. This
provides sharp estimates for Drury's generalization of Von Neumann's
inequality. The characterization is in terms of a geometric condition, the
"split tree condition", which reflects the nonisotropic geometry underlying the
Drury-Arveson Hardy space
The characterization of the Carleson measures for analytic Besov spaces: a simple proof
We give a simple proof of the characterization of the Carleson measures for
the weighted analytic Besov spaces. Such characterization provides some
information on the radial variation of an analytic Besov function.Comment: 12 page
Potential Theory on Trees, Graphs and Ahlfors Regular Metric Spaces
We investigate connections between potential theories on a Ahlfors-regular
metric space X, on a graph G associated with X, and on the tree T obtained by
removing the "horizontal edges" in G. Applications to the calculation of set
capacity are given.Comment: 45 pages; presentation improved based on referee comment
Equilibrium measures on trees
We give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For p= 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders
Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group
We prove geometric versions of Hardy's inequality for the sub-elliptic
Laplacian on convex domains in the Heisenberg group ,
where convex is meant in the Euclidean sense. When and is the
half-space given by this generalizes an
inequality previously obtained by Luan and Yang. For such and the
inequality is sharp and takes the form \begin{equation}
\int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega}
\sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi),
\nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi,
\end{equation} where denotes the
Euclidean distance from .Comment: 14 page
Sharp estimates for second order Riesz transforms on multiply–connected Lie groups
We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component . We prove sharp estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set . The analysis shows that Itô integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local
Some notions of subharmonicity over the quaternions
This works introduces several notions of subharmonicity for real-valued
functions of one quaternionic variable. These notions are related to the theory
of slice regular quaternionic functions introduced by Gentili and Struppa in
2006. The interesting properties of these new classes of functions are studied
and applied to construct the analogs of Green's functions.Comment: 16 page
A note on completeness of weighted normed spaces of analytic functions
[EN] Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space H-v(infinity) (G)of all holomorphic functions on G such that the product v vertical bar f vertical bar is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.The first author was partially supported by MTM2013-43540-P and MTM2016-76647-P by MINECO/FEDER-EU and GVA Prometeo II/2013/013. The second author was partially supported by the MINECO/FEDER-EU Grant MTM2015-65792-P. Both authors were partially supported by Thematic Research Network MTM2015-69323-REDT, MINECO, Spain.Bonet Solves, JA.; Vukotic, D. (2017). A note on completeness of weighted normed spaces of analytic functions. Results in Mathematics. 72(1-2):263-279. https://doi.org/10.1007/s00025-017-0696-2S263279721-2Arcozzi, N., Björn, A.: Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces. Math. Proc. R. Ir. Acad. 102A, 175–192 (2002)Berenstein, C.A., Gay, R.: Complex Variables, An Introduction. Springer, New York (1991)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)Björn, A.: Removable singularities for weighted Bergman spaces. Czechoslov. Math. J. 56, 179–227 (2006)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Vogt, D.: Weighted spaces of holomorphic functions and sequence spaces. Note Mat. 17, 87–97 (1997)Conway, J.B.: Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics, vol. 11. Springer, New York (1978)Gaier, D.: Lectures on Complex Approximation. Birkhäuser, Boston (1987)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Philos. Soc. 136, 399–411 (2004)Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam (1979)Horváth, J.: Topological Vector Spaces and Distributions. Addison-Wesley, Reading (1966)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175, 19–45 (2006)Nakazi, T.: Weighted Bloch spaces which are Banach spaces. Rend. Circ. Mat. Palermo 62, 427–440 (2013)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971
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