Linear response for systems with a cusp

The paper claims W^{1,p} and W^{2,p} spectral gaps and an L^p linear response for cusp maps under (A1)–(A9) with β in (−1,(1−2p)/(2p)), and gives Theorem 1.1 with q = PT0[A0 h0' + B0 h0] on [0,a0) (and 0 on [a0,1]) together with a two-norm Lasota–Yorke scheme (strong: W^{1,p}, weak: L^{2p}) and a second-derivative estimate (Lemma 2.5) for W^{2,p} regularity . However, in the displayed proof of the Lasota–Yorke inequality (Lemma 2.4), the Hölder step is misapplied: the text uses ∥(T''/(T')^2)f∥_{2p} ≤ ∥T''/(T')^2∥_{2p}∥f∥_{2p}, which is not a valid L^{2p} product estimate (one needs a different exponent split or an L^∞ bound) . More seriously, for the W^{2,p} step the coefficients g2,g3,g4 scale like |y−c|^{−2β−2}; membership in L^{2p} requires β < −1 + 1/(4p), whereas the paper assumes the weaker β < −1 + 1/(2p) when asserting g2,g3,g4 ∈ L^{2p} (see “Asymptotic Behavior of Coefficients”) . This leaves the W^{2,p} part of Proposition 2.3/Theorem 1.1 unproven under the stated β-range and forces a stricter cusp (e.g., in the example, k>4p rather than k>2p) . On the other hand, the candidate solution introduces a different integrability threshold β < −(p−1)/(2p−1) for W^{1,p}, and then claims a W^{2,p} bootstrap with exponents E1,E2 that it asserts are integrable “for any β∈(−1,0)”—but those exponents (e.g., E2=β(1−3p)−2p) are not >−1 for many (β,p), so the W^{2,p} step is generally invalid as written. The candidate’s final claim that the general case (with only β∈(−1,0)) was likely open therefore does not directly address the paper’s narrower claimed regime and overstates what their own sketch proves.