Fourier–Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations

Abstract

In this paper, we review a number of results about the Fourier–Bessel transformation of nonnegative functions. For the specified case, weighted L _∞-norms of the spherical mean of are estimated by its weighted L ₁-norms; note that such a phenomenon does not take place in the general case, i.e., without the requirement of the nonnegativity of f. Moreover, unlike the classical case of the Fourier transform, this phenomenon takes place for one-variable functions as well: weighted L _∞-norms of the Fourier–Bessel transform are estimated by its weighted L ₂-norms. Those results are applied to the investigation of singular differential equations containing Bessel operators acting with respect to selected spatial variables (the so-called special variables); equations of such kind arise in models of mathematical physics with degenerative heterogeneities and in axially symmetric problems. The proposed approach provides a priori estimates for weighted L _∞-norms of the solutions (for ordinary differential equations) and for weighted spherical means of the squared solutions (for partial differential equations).

This work was financially supported by the Ministry of Education and Science of the Russian Federation on the program to improve the competitiveness of Peoples’ Friendship University (RUDN University) among the world’s leading research and education centers in the 2016–2020, by the Russian Foundation for Basic Research (grant No. 17-01-00401), and by the President Grant for the Government Support of the Leading Scientific Schools of the Russian Federation, No. 4479.2014.1.