scholarly journals A Littlewood-Paley type inequality for harmonic functions in the unit ball of Rⁿ

Filomat ◽  
2006 ◽  
Vol 20 (2) ◽  
pp. 101-105
Author(s):  
Olivera Djordjevic
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Type Inequality

scholarly journals SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL

2017 ◽  
Vol 60 (1) ◽  
pp. 219-224 ◽  
Author(s):  
DAVID KALAJ
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Type Inequality ◽  
Holomorphic Mappings ◽  
Schwarz Lemma ◽  
Complex Spaces

AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

On q-Carleson Measures for Spaces of M-Harmonic Functions

1997 ◽  
Vol 49 (4) ◽  
pp. 653-674 ◽  
Author(s):  
Carme Cascante ◽  
Joaquin M. Ortega
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Sufficient Conditions ◽  
Carleson Measures

AbstractIn this paper we study the q-Carleson measures for a space of M-harmonic potentials in the unit ball of Cn, when q < p. We obtain some computable sufficient conditions, and study the relations among them.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

2019 ◽  
Vol 124 (1) ◽  
pp. 81-101
Author(s):  
Manfred Stoll
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Reproducing Kernel ◽  
Diagonal Function ◽  
Hyperbolic Harmonic Functions

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

scholarly journals Compact Toeplitz operators on weighted harmonic Bergman spaces

1998 ◽  
Vol 64 (1) ◽  
pp. 136-148 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Harmonic Bergman Spaces

AbstractWe consider the Bergman spaces consisting of harmonic functions on the unit ball in Rn that are squareintegrable with respect to radial weights. We will describe compactness for certain classes of Toeplitz operators on these harmonic Bergman spaces.

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