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Search: MSC category 31C15 ( Potentials and capacities )

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1. CJM 2011 (vol 64 pp. 183)

Nowak, Adam; Stempak, Krzysztof
Negative Powers of Laguerre Operators
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator
Categories:47G40, 31C15, 26A33

2. CJM 1997 (vol 49 pp. 653)

Cascante, Carme; Ortega, Joaquin M.
On $\lowercase{q}$-Carleson measures for spaces of ${\cal M}$-harmonic functions
In this paper we study the $q$-Carleson measures for a space $h_\alpha^p$ of ${\cal M}$-harmonic potentials in the unit ball of $\C^n$, when $q
Categories:32A35, 31C15

3. CJM 1997 (vol 49 pp. 568)

Mateu, Joan
A counterexample in $L^p$ approximation by harmonic functions
For ${n \over {n-2}}\leq p<\infty$ we show that the conditions $C_{2,q}(G\setminus \dox)=C_{2,q}(G \setminus X)$ for all open sets $G$, $C_{2,q}$ denoting Bessel capacity, are not sufficient to characterize the compact sets $X$ with the property that each function harmonic on $\dox$ and in $L^p(X)$ is the limit in the $L^p$ norm of a sequence of functions which are harmonic on neighbourhoods of $X$.

Categories:41A30, 31C15

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