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Muckenhoupt conditions for piecewise-power weights in Euclidean space with Dunkl measure
D. V. Gorbachev, V. I. Ivanov Tula State University (Tula)
Abstract:
As a result of many years of research in the Fourier harmonic analysis, a class
of linear integral Calderon–Sigmund operators was defined that are bounded in
the spaces $L^p$ on $\mathbb{R}^d$ with the Lebesgue measure for $1<p<\infty$.
B. Muckenhoupt found conditions on weight that are necessary and sufficient for
the boundedness of the Calderon–Zygmund operators in $L^p$-spaces with one
weight. They are now known as the Muckenhoupt $A_p$-conditions. G.H. Hardy and
J.E. Littlewood $(d=1)$ and S.L. Sobolev $(d> 1)$ proved
$(L^p,L^q)$-boundedness of the Riesz potential $I_{\ alpha}$ for $1
<p<q<\infty$, $\alpha=d\Bigl(\frac{1}{p}-\frac{1}{q}\Bigr)$. B. Muckenhoupt and
R.L. Wheeden found $A_{p,q}$-weight condition for one weight
$(L^p,L^q)$-boundedness of the Riesz potential. An important generalization
of the Riesz potential has become the Dunkl–Riesz potential defined by
S. Thangavelu and Yu. Xu in Euclidean space with the Dunkl measure. For the
Dunkl–Riesz potential, we proved $(L^p,L^q)$-boundedness with two radial
piecewise-power weights. In this paper, we define the Muckenhoupt $A_p$ and
$A_{p,q}$-conditions for weights in Euclidean space with the Dunkl measure and
find out when they are satisfied for piecewise-power weights. The obtained
results show that the conditions of $(L^p,L^q)$-boundedness of the Dunkl–Riesz
potential with one piecewise-power weight can be characterized using the
$A_{p,q}$-condition. This suggests that the conditions of
$(L^p,L^q)$-boundedness of the Dunkl–Riesz potential with one arbitrary weight
can also be written using the $A_{p,q}$-condition.
Keywords:
weighted function, Muckenhoupt conditions, piecewise-power weight, Dunkl measure, Dunkl–Riesz potential.
Received: 18.05.2019 Accepted: 12.07.2019
Citation:
D. V. Gorbachev, V. I. Ivanov, “Muckenhoupt conditions for piecewise-power weights in Euclidean space with Dunkl measure”, Chebyshevskii Sb., 20:2 (2019), 82–92
Linking options:
https://www.mathnet.ru/eng/cheb754 https://www.mathnet.ru/eng/cheb/v20/i2/p82
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Abstract page: | 189 | Full-text PDF : | 54 | References: | 18 |
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