The Kontorovich - Lebedev Transformation on Sobolev Type Spaces

The Kontorovich-Lebedev transformation (KLf)(x)=∫₀ⁱτ(x)f(τ)dτ, \;\, x ∈ R₊ is consideredas an operator, which maps the weighted space Lₚ( R₊;ω(τ)dτ), \;\, 2 ≤ p ≤ ∞$ into the Sobolevtype space $S_pN, α({ R}_+)$ with the finite norm$$||u||_{S_pN,α({ R}_+)}= ( ∑k= 0^N∫_0^∞ |A_x^k u|^p xα_k p -1 dx)1/p <∞,$$where $α= (α_0, α_1, , α_N), α_k ∈{ R}, k=0, , N$, and $ A_x$ is the differentialoperator of the form$$A_x u= x^2u(x) - xd/dx[xdu/dx ], $$and $A_x^k$ means $k$-th iterate of $A_x, \ A_x^0u= u$. Elementary properties for the space $S_pN, α ({ R}_+)$ 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