The Kontorovich - Lebedev Transformation on Sobolev Type Spaces - Yakubovich, Semyon B. (Sarajevo Journal of Mathematics, 2024)
The Kontorovich-Lebedev transformation (KLf)(x)=_0^∞ K_iτ(x)f(τ)dτ, x R_+ is considered as an operator, which maps the weighted space L_p(R_+; ω(τ)dτ), 2 p ∞ into the Sobolev type space S_p^N, α(R_+) with the finite norm ||u||_S_p^N,α(R_+)= ( _k= 0^N _0^∞ |A_x^k u|^p x^α_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 differential operator of the form A_x u= x^2u(x) - xddx[xdudx ], and A_x^k means k-th iterate of A_x, A_x^0u= u. Elementary properties for the space S_p^N, 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