Weighted Stein–Weiss and John–Nirenberg Inequalities for Fractional Singular Integral Operators on Fofana Spaces

Source abstract

The Stein–Weiss weighted inequality and the John–Nirenberg inequality are among the most important and widely applied results in harmonic analysis, underlying a broad range of developments in weighted norm estimates and the theory of bounded mean oscillation. Motivated by their central role, in this article we investigate new refinements of these two classical inequalities in the setting of variable exponent Fofana spaces. These results are established for the first time in this class of generalized function spaces, in which the exponent p(·) appearing in the norm is variable and the exponent η is constant. A further novelty is that, whereas such inequalities have previously been developed in other function spaces using the Riesz kernel, in the present study we instead use the more general Bessel–Riesz-type kernel, which additionally incorporates a decay parameter β controlling behavior at infinity and so yields sharper, more flexible estimates than the Riesz kernel alone. To verify the correctness of our results, we construct several nontrivial examples, establish some associated results, and give remarks recovering existing results under different settings; moreover, several of these results are new even in the constant-exponent case. In particular, we establish John–Nirenberg inequalities for functions of bounded mean oscillation in terms of the norm of variable exponent Fofana spaces on Rν, via an atomic Hardy space argument, recovering the classical and variable-Lebesgue-space John–Nirenberg inequalities as special cases.