We establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

We present a number of new solutions to an integral equation arising in the limiting theory of Bellman-Harris processes. The argument proceeds via straightforward analysis of Mellin transforms. We also derive a criterion for the analyticity of the Laplace transform of the limiting distribution on Re(u) ≥ -c for some c > 0.

Let $J_\mu$ denote the Bessel function of order $\mu$. The functions $x^{-\alpha-1} J_{\alpha+2n+1}(x), n=0,1,2,\ldots,$ form an orthogonal system in the space $L^2((0,\infty), x^{2\alpha+1}dx)$ when $\alpha >{-}1$. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if $\alpha\,{\ge}\,0$.

Let ψ be a positive function defined near the origin such lim1 →0+ ψ(t) = 0. We consider the operator Tzƒ, defined as the pricipal value of the convolution of function ƒ and a kernel K(t) = eiy(t)t−z /ψ(t)1−z, where z is a complex number, 0 ≤ Re(z) ≤ 1, 0 < t ≤ 1 and γ is a real function. Assuming certain regularity conditions on ψ and γ and certain relations between ψ and γ we show that Tθ is a bounded operator on Lp (R) for 1/p = (1+ θ) /2 and 0 ≤ θ < 1, and T1 is bounded from H1 (R) to L1 (R).