\(L^{p(.)}-L^{q(.)}\) estimates for some convolution operators with singular finite measures (Q2166144)

The authors aim to prove necessary conditions on the exponent functions \(p(\cdot)\) and \(q(\cdot)\) for the boundedness of convolution operators from \(L^p(\cdot)\) to \(L^q(\cdot)\) in the entire space in dimension \(N+1\).\par These spaces are known as variable Lebesgue spaces and are a generalization of the classical \(L^p\) spaces. Among the different settings where these spaces are used, could be mentioned the mathematical modeling of electrorheological fluids, quasi-Newtonian fluids, magnetostatics, and image processing. \par To obtain the results the authors use embedding arguments and the boundedness of the convolution type-operators having a finite Borel measure, from \(L^p(\cdot)\) to \(L^q(\cdot)\) for some particular exponents \(p(\cdot)\) and \(q(\cdot).\)