The Kontorovich - Lebedev Transformation on Sobolev Type Spaces
DOI:
https://doi.org/10.5644/SJM.01.2.07Keywords:
Kontorovich-Lebedev transform, modified Bessel function, Sobolev spaces, Hardy inequality, Plancherel theorem, imbedding theoremAbstract
The Kontorovich-Lebedev transformation $$(KLf)(x)=\int_0^\infty
K_{i\tau}(x)f(\tau)d\tau, \;\, x \in {\mathbf R}_+$$ is considered
as an operator, which maps the weighted space $L_p(\mathbf R_+;$
$\omega(\tau)d\tau), \;\, 2 \le p \le \infty$ into the Sobolev
type space $S_p^{N, \alpha}({\mathbf R}_+)$ with the finite norm
$$||u||_{S_p^{N,\alpha}({\mathbf R}_+)}= \biggl( \sum_{k= 0}^N
\int_0^\infty |A_x^k u|^p x^{\alpha_k p -1} dx\biggr)^{1/p} <
\infty,$$
where $\alpha= (\alpha_0, \alpha_1, \dots, \alpha_N), \alpha_k \in
{\mathbf R}, k=0, \dots, N$, and $ A_x$ is the differential
operator of the form
$$A_x u= x^2u(x) - x\frac{d}{dx}\biggl[x\frac{du}{dx }\biggr], $$
and $A_x^k$ means $k$-th iterate of $A_x, \ A_x^0u= u$. Elementary properties for the space $S_p^{N, \alpha} ({\mathbf R}_+)$ are derived. Boundedness and inversion properties for the Kontorovich-Lebedev transform are studied. In the Hilbert case ($p=2$) the isomorphism between these spaces is established for the special type of weights and Plancherel's type theorem is proved.
2000 Mathematics Subject Classification. 44A15, 46E35, 26D10
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