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Negative Powers of Laguerre Operators

  Published:2011-06-24
 Printed: Feb 2012
  • Adam Nowak,
    Instytut Matematyczny, Polska Akademia Nauk, Śniadeckich 8, 00-956 Warszawa, Poland
  • Krzysztof Stempak,
    Instytut Matematyki i Informatyki, Politechnika Wrocławska, 50-370 Wrocław, Poland
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Abstract

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 potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator
MSC Classifications: 47G40, 31C15, 26A33 show english descriptions Potential operators [See also 31-XX]
Potentials and capacities
Fractional derivatives and integrals
47G40 - Potential operators [See also 31-XX]
31C15 - Potentials and capacities
26A33 - Fractional derivatives and integrals
 

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