2024/08/22 by David Cruz-Uribe, Cruz-Uribe, David, Troy Roberts +1
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2408.12745
In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function \pp for a fractional maximal operator Mα or a non-degenerate fractional singular integral operator Tα, 0 ≤ α< n, to satisfy weak (\pp,\qq) inequalities or strong (\pp,\qq) inequalities, with \qq being defined pointwise almost everywhere by % (1)/(p(x)) - (1)/(q(x)) = \fracαn. % We first prove preliminary results linking fractional averaging operators and the K0α condition, a qualitative condition on \pp related to the norms of characteristic functions of cubes, and show some useful implications of the K0α condition. We then show that if Mα satisfies weak (\pp,\qq) inequalities, then \pp ∈ K0α(\Rn). We use this to prove that if Mα satisfies strong (\pp,\qq) inequalities, then p->1. Finally, we prove a powerful pointwise estimate for Tα that relates Tα to Mα along a carefully chosen family of cubes. This allows us to prove necessary conditions for fractional singular integral operators similar to those for fractional maximal operators.