BOUNDEDNESS OF THE HILBERT TRANSFORM FROM ONE ORLICZ SPACE TO ANOTHER - Nessipbayev, Y.; Tulenov, K. (Academic Scientific Journal of Computer Science, 2020)
Towards these goals we also investigate boundedness of the Calderón operator from one rearrangement-invariant Banach function space to another. Such questions have been attracting a great deal of attention for many years, in particular in connection with embeddings of Sobolev spaces. In the present paper we discuss such boundedness problems for classical operators of great interest in analysis and its applications, namely the Hilbert transform and the Calderón operator. The action of these operators on specific classes of function spaces has been extensively studied over the several decades. Classical results are available for example in connection with familiar function spaces. Besides the importance of these operators is very well known, and their properties have been deeply studied.Classical Lorentz spaces which originated in 1950s and have been occurring occasionally later became extremely fashionable in 1990s when the fundamental papers appeared. In this paper we study the boundedeness of such classical operators on rearrangement-invariant spaces, a class of function spaces that includes for example all Lebesgue, Lorentz, Orlicz, Marcinkiewicz spaces and more. Our focus is mainly on boundedness of the Hilbert transform from one Orlicz space to another. We also give examples of particular rearrangement-invariant spaces on which the Hilbert transform acts boundedly.