Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L 2(ℝ+;dx/x)

Abstract

Abstract The Pseudo-differential operators (p.d.o.) L ⁢ ( x , A x ) {L(x,A_{x})} and ℒ ⁢ ( x , A x ) {\mathcal{L}(x,A_{x})} involving the Kontorovich–Lebedev transform are defined. An estimate for these operators in the Hilbert space L 2 ⁢ ( ℝ + ; d ⁢ x x ) {L^{2}(\mathbb{R}_{+};\frac{dx}{x})} is obtained. A symbol class Λ is defined and it is shown that the product of any two symbols from this class is again in Λ. At the end, commutators for the p.d.o. and their boundedness results are discussed.