A class of index transforms with general kernels

This paper deals with a class of integral transforms of the non - convoluti on type involving sufficiently general kernels, which depend upon two essen tially independent arguments. One of them, in various particular cases, is a parameter or index of the corresponding special functions. This class of integral transforms comprises the famous Kontorovich-Lebedev and Mehler-Foc k transforms. We study here the mapping properties and give also inversion theorems of the general index transforms on the space L-p(R), p greater tha n or equal to 1, that covers the respective measurable functions on the who le real axis with the norm parallel to f parallel to(p) = (integral(-infinity)(infinity)\f(t)\(p) dt)( 1/p) < infinity It is shown that the images of the transforms belong to the space L-v,L-p(R+), v is an element of (R+) 1 less than or equal to R less than or equal to infinity of functions normed by parallel to f parallel to(v,p) = (integral(0)(infinity)t(vp-1)\f(t)\(p) dt) (1/p) < infinity In particular, when v = 1/p we get the usual L-p(R+) space. We also direct our attention to the case of the Hilbert space and give certain interesting examples of these transforms.