Inequalities related to spherical harmonics associated with the Weinstein operator. (English) Zbl 1527.33010
Summary: We establish a Babenko type inequality for the Weinstein transform which allows us to obtain, for \(2 \leq p < \infty\), a reverse Hölder inequality for the spherical harmonics associated with the Weinstein operator. Also, we give, for \(1 \leq p \leq 2\), another reverse Hölder inequality obtained by using the zonal. Next, we study the weighted \((L^p,L^q)\) Pitt inequalities for the Weinstein transform. The case \(p = 2\) is studied separately and as a consequence, we derive a logarithmic uncertainty principle for the Weinstein transform.
MSC:
| 33C55 | Spherical harmonics |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 44A20 | Integral transforms of special functions |
Keywords:
Weinstein transform; Hankel transform; Babenko inequality; spherical harmonics; Pitt inequalityReferences:
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