The heat kernel associated with generalized Kontorovich–Lebedev transform

ABSTRACT

We study some properties of the eigenfunction ${\phi }_{\alpha ,\mathrm{i\tau }}^{\mu ,\beta }$ of the following second-order differential operator defined on $(0,\mathrm{\infty })$, by ${\mathrm{\Delta }}_{x,\alpha }^{\mu ,\beta }=x^2\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+(1-2\alpha )x\frac{\mathrm{d}}{\mathrm{d}x}-{\beta }^2{\mu }^2{x}^{2\mu };\alpha ,\beta ,\mu \in (0,\mathrm{\infty }).$ We obtain estimates of the integral of the kernel $w_{\alpha ,\beta ,\mu }$, translation and convolution product related to ${\phi }_{\alpha ,\mathrm{i\tau }}^{\mu ,\beta }$. We define and explore a generalization of the Kontorovich–Lebedev transform. The heat kernel and Weierstrass transform tied to ${\mathrm{\Delta }}_{x,\alpha }^{\mu ,\beta }$ are considered.

KEYWORDS:

• Kontorovich–Lebedev transform
• convolution product
• translation
• heat kernel
• Weierstrass transform

AMS CLASSIFICATIONS:

• 33C10
• 42A38
• 44A20
• 44A35

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions to improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).