Functional Analysis and Applications Seminar 

15/11/2002, 15:00 — 16:00 — Room P3.10, Mathematics Building
Stefan Samko, Universidade do Algarve, Faro

Sonine Integral Equations of the First Kind

An integral equation of the first kind \[ K\phi(x):\equiv\int_{-\infty}^x k(x-t)\phi(t)\, dt = f(x)\] with a locally integrable kernel $k(x)\in L_1^{loc}(\mathbb{R}_+^1)$ is called Sonine equation if there exists another locally integrable kernel $\ell(x)$ such that \[\int_0^x k(x-t)\ell(t)\, dt \equiv 1\] (locally integrable divisors of the unit, with respect to the operation of convolution). In this case the formal solution of the equation is $\phi(x)=\frac{d}{dx}\int_0^x \ell(x-t)f(t)\, dt$. However, this inversion operator is formal: it does not work, for example, for solutions in the spaces $X=L_p(\mathbb{R}^1)$ and is not defined on the whole range $K(X)$.

We develop many properties of Sonine kernels which allow us — in a very general case — to construct the real inverse operator, within the framework of the spaces $X=L_p(\mathbb{R}^1)$, in Marchaud form:\[K^{-1}f(x)= \int_0^\infty \ell'(t)[f(x-t)-f(x)]\, dt\] with the interpretation of the convergence of this “hypersingular” integral in $L_p$-norm. The description of the range $K(X)$ is given; it requires already the language of Orlicz spaces even in the case when $X$ is the Lebesgue space $L_p(\mathbb{R}^1)$.

Some examples are considered.