Modified Bessel ${\mathbf P}$-integrals and $\mathbf P$-derivatives and their properties

We study the modified Bessel ${\mathbf P}$-integral, whose properties are similar to those of the Bessel potential, and the modified Bessel ${\mathbf P}$-derivative. These operators are inverse to each other. We prove analogues of the embedding theorems of Hardy, Littlewood, Stein, Zygmund and Lizorkin concerning the images of $L^p(\mathbb R)$ under the action of Bessel potentials. We give applications of the Bessel integral and derivative to the integrability of the ${\mathbf P}$-adic Fourier transform and to approximation theory (an embedding theorem of Ul'yanov type).