An inverse scattering procedure in Lebesgue spaces with non-constant exponents

Within the ever-growing field of electromagnetic imaging, inversion procedures are conventionally described in the mathematical framework of Hilbert spaces. Usually, the over-smoothing effects and oscillations that arise using a Hilbert-space formulation make the dielectric reconstruction of targets inaccurate. This problem is strongly reduced by the recent development of inversion techniques in Banach spaces. However, the selection of the Banach space norm parameter is critical for obtaining precise reconstructions, and no exact rules exist for this choice. To overcome this issue, an innovative approach in variable exponent Lebesgue spaces is proposed here, along with a preliminary numerical validation.