Weak variational lower bounds for a planar second-order Riesz kernel

Weak variational lower bounds for a planar second-order Riesz kernelZihuan Xu For the fixed planar second-order Riesz kernel K(x)=(x_1^2-x_2^2)/|x|^4, this manuscript constructs real compactly supported inputs f_N with L1 norm at most 16 sqrt(N) and weak q-variation ratio at least (pi/32)N^(1/q-1/2)sqrt(log log N), for every sufficiently large N and every finite q >= 2. The threshold N0 is absolute and asymptotic. Choosing N near exp(1/(q-2)) gives a sqrt(log(1/(q-2))) lower bound. An annular identity connects the truncations to semicircle averages. Separated binary digits transfer the square-variation asymptotic of Lewko and Lewko while retaining a positive-measure event and its input cost. Finite alternating paths illustrate the weak Lorentz sequence-to-path framework of Mirek, Stein and Zorin-Kranich. The final section formulates the uniform analytic estimates needed for matching convolution bounds. OpenAI Codex assisted with source retrieval, derivation checks, English drafting and LaTeX production. License: Creative Commons Attribution 4.0 International (CC BY 4.0).