Fourier-Bessel transformation of measures and singular differential equations

Abstract

Abstract This paper is devoted to the investigation of Fourier-Bessel transformation (see [2]) for non-negative f : f ˜ ( ξ , η ) = 1 η v ∫ 0 ∞ ∫ R n y v + 1 j v ( η y ) f ( x , y ) e − i ξ ⋅ x d x d y ; v > − 1 2 We apply the method of [5] which provides the estimate for weighted L ∞ -norm of the spherical mean of | f ^ | 2 via its weighted L 1 -norm (generally it is wrong without the requirement of the nonnegativity of f). We prove that (unlike in the classical case of Fourier transformation) a similar estimate is valid for the one-dimensional case too: a weighted L ∞ -norm of f ˜ is estimated by its weighted L 2 -norm. The obtained result and the estimates for the multidimensional case (see [6] and [7]) are applied to the investigation of singular differential equations containing Bessel operator (where the parameter at the singularity equals to 2 v + 1 ); equations of such kind arise in models of mathematical physics with degenerative heterogeneities and in axially symmetric problems. We obtain estimates for weighted L ∞ -norms of solutions (for ordinary differential equations) and of weighted hemispherical means of squared solutions (for partial differential equations) .