Singular integrals and potentials in some Banach function spaces with variable exponent
In the paper under review, the authors define a weighted version of the rearrangement invariant Banach function spaces of Lorentz type \(\Lambda_{w} ^{p(\cdot)}(\Omega)\), defined by the norm \(\| f\| _{\Lambda ^{p(\cdot)}(\Omega)} = \| w f^{**}\| _{L^{p(\cdot)}(\Omega)}\), were \(f^{**}\) is the maximal function of the decreasing rearrangement \(f^*\) of \(f\) and \(L^{p(\cdot)}(\Omega)\) denotes the Lebesgue space of measurable functions with variable exponent on a bounded set \(\Omega\) from \( \mathbb{R}^{n}\). If the variable exponent \(p(t)\) satisfies the so-called logarithmic Dini conditions they prove the boundedness of singular integrals, fractional integrals and Cauchy singular operators associated to Lyapunov curves and to bounded rotation curves without cusps.
scientific article; zbMATH DE number 2111353