When are nonlocal divergence-form operators with measurable kernels coercive and regular?

A negative answer to the (L)-(U1) regularity question

The paper studies the nonlocal divergence-form equation PV∫_{R^n}(u(x)-u(y))K(x,y)dy=0 in B_2, where the kernel K is measurable, symmetric (K(x,y)=K(y,x)), and nonnegative, and solutions are understood in the weak sense. The motivating question, attributed to Fernández-Real and Ros-Oton, asked whether a weak solution must be locally Hölder continuous when K satisfies two natural conditions: a lower bound (L) on the second moment of K over balls in every direction, and an upper bound (U1) on the integral of K over annuli.

The paper answers that question negatively. Example 2.1 takes K(x,y)=|x-y|^{-n-2s}·1{x_1 y_1 ≥ 0}, which is symmetric and satisfies (U1) with Λ=|S^{n-1}|(1-4^{-s})/(2s), and in fact satisfies the stronger annulus condition (U_p) for every p in (1,∞]. The author verifies (L) with λ=|S^{n-1}|/(2n(2-2s)). For this kernel, the bounded jump function u(x)=1{x_1>0} is a weak solution: the product |u(x)-u(y)|K(x,y) is nonzero only on the measure-zero set {y_1=0}∪{x_1=0}, so the double integral defining the weak formulation vanishes. Since u is discontinuous, (L) and (U1) cannot by themselves force local Hölder continuity.

The counterexample fails the one-sided lower bound (L+)

The distinction that matters is between (L) and the stronger one-sided condition (L+), which replaces the full second moment by its positive part: inf over unit vectors e of ∫_{B_r(x)}((e·(y-x))_+)^2 K(x,y)dy ≥ λ r^{2-2s}. Remark 2.1 shows that the Example 2.1 kernel fails (L+). Taking x with x_1<0 and e=e_1, the integrand ((e_1·(y-x))_+)^2 K(x,y) is nonzero only on the slab x_1<y_1≤0, of width |x_1|. On that slab (y_1-x_1)^2≤x_1^2, so the integral is bounded by a term of order |x_1|^{2-2s}, which tends to 0 as x_1→0. Thus (L+) fails, and the counterexample does not contradict the paper's positive theorem.

This is the paper's key structural point: the negative example lives in the gap between (L) and (L+). The positive result is therefore not a repair of (L) but a statement about a strictly smaller class of kernels, and the reader should not read the counterexample as evidence against the coercivity conjecture, which is stated under (L+).

An L^p annulus bound replaces the pointwise upper bound

Beyond (L+), the theorem assumes an L^p bound on K over annuli. Writing A_r(x)=B_{2r}(x)\B_r(x) for the annulus of width r at distance r from x, condition (U_p) requires that for some p in (1,∞], the normalized L^p norm (|A_r(x)|^{-1}∫_{A_r(x)}K(x,y)^p dy)^{1/p} be at most Λ_p r^{-n-2s} for all r>0 and all x. For p=1 this reduces to (U1) with Λ=(2^n-1)|B_1|Λ_1, so (U_p) is a genuine strengthening for p>1.

The role of p>1 is not cosmetic. In the proof of the key lemma, the annulus bound is combined with Hölder's inequality to convert an integral of K over a subset of an annulus into a bound involving the (p-1)/p power of that subset's measure. This is exactly the step where p>1 becomes crucial: it produces the concavity estimate that yields a positive lower bound on the measure of the good set. The paper also notes that the conditions in Theorem 1.1 are optimal and cannot be reduced to p=1, because Fernández-Real constructed a counterexample to the coercivity estimate and to regularity under (U1)-(L+) for p=1.

A positive-measure set where K is large and aligned

The technical heart is Lemma 2.1. Assuming (L+) and (U_p) for some p in (1,∞], it produces constants κ, λ', μ in (0,1), depending only on n, s, p, λ, Λ_p, such that for every x, every r>0, and every unit vector e, the set N_e(x,r) of points y in B_r(x)\B_{κr}(x) satisfying both K(x,y)≥λ'|x-y|^{-n-2s} and e·(y-x)≥κ|y-x| has measure at least μ|B_r|.

The proof decomposes B_r(x) into the small ball B_{κr}(x) plus three sets: D_1 where K is below the threshold λ'|x-y|^{-n-2s}, D_2 where the directional projection is too small, and the good set N_e(x,r). Using (U_p), the contribution of B_{κr}(x) to the weighted second moment is bounded by C_{n,s}Λ_p(κr)^{2-2s}; the contributions of D_1 and D_2 are bounded by λ'C̃_{n,s}r^{2-2s} and κ^2C_{n,s}Λ_p r^{2-2s}. Choosing κ and λ' small enough makes the total at most (3λ/4)r^{2-2s}, so (L+) forces the remaining good set to carry at least (λ/4)r^{2-2s} of the weighted second moment. Hölder's inequality over the annuli covering B_r(x)\B_{κr}(x) then converts this into the measure lower bound |N_e(x,r)|≥μ|B_r|.

Coercivity follows from a uniform density of large-K points

Lemma 2.2 upgrades the good-set estimate into a statement about every ball. For every ball B=B_R(z) and every x in B, the set of y in B with K(x,y)≥λ'|x-y|^{-n-2s} has measure at least μ_{CS}|B|, where μ_{CS}=μκ^n2^{-n}. The proof splits on whether x is within R/2 of the center z: in the near case, Lemma 2.1 applies directly with r=R/2; in the far case, one takes e=(z-x)/|z-x| and r=κR and shows that N_e(x,κR) lies inside B, giving the same type of bound.

This uniform density is exactly the sufficient condition identified by Chaker and Silvestre, so the coercivity estimate (1.3) follows from their theorem. Concretely, the paper obtains ∫_B∫_B (w(x)-w(y))^2/|x-y|^{n+2s} dxdy ≤ C∫_B∫_B (w(x)-w(y))^2 K(x,y) dxdy for all measurable w, with C depending only on n, s, p, λ, Λ_p. A scaling argument with K_ρ(x,y)=ρ^{n+2s}K(x_0+ρx, x_0+ρy), which satisfies (L+) and (U_p) with the same constants, extends the estimate to arbitrary balls B_ρ(x_0), and a covering argument yields the global form (2.7).

Interior Hölder regularity with constants fixed by the kernel data

Theorem 1.1 states that if (L+) and (U_p) hold for some p in (1,∞], then the coercivity estimate holds and there exist β in (0,1) and C>0, both depending only on λ, Λ_p, s, n, p, such that every bounded solution u in V_loc(B_2|R^n) of the equation satisfies |u(x)-u(y)|≤C‖u‖_{L^∞(R^n)}|x-y|^β for all x,y in B_1. The Hölder exponent and constant are thus controlled by the same structural parameters that appear in the kernel assumptions, not by any additional regularity of K.

The regularity proof proceeds by verifying the hypotheses of a known interior Hölder result for nonlocal equations with rough kernels. From (U_p) one checks the required tail condition by summing over dyadic annuli, obtaining ∫_{B_r^c(x)}K(x,y)dy≤Λ_p C_n r^{-2s}Σ_j(4^{-s})^j. The coercivity estimate supplies the two-sided energy comparison A^{-1}‖w‖_{Ḣ^s(B_ρ(x_0))}^2 ≤ ∫∫(w(x)-w(y))^2K(x,y)dxdy ≤ A‖w‖_{Ḣ^s(B_ρ(x_0))}^2 for all w in H^s(B_ρ(x_0)), with A depending on n, s, p, λ, Λ_p, and a Lipschitz cut-off computation gives the required bound on the kernel energy of the cut-off. Finally, the coercivity estimate implies that any test function in the natural energy space V_0(B_2) lies in H^s(R^n) and can be approximated by smooth compactly supported functions, so the weak formulation extends to that class.

The p>1 threshold is the paper's operative dividing line

The paper's own comparison is between two regimes. Under (L) and (U1) alone, the Example 2.1 kernel admits a bounded discontinuous weak solution, so no regularity conclusion is available. Under (L+) and (U_p) with p>1, coercivity and interior Hölder regularity hold, with constants depending only on n, s, p, λ, Λ_p. The author further reports that Fernández-Real constructed a counterexample to the coercivity estimate and to regularity under (U1)-(L+) for p=1, which is why the conditions in Theorem 1.1 are described as optimal and not reducible to p=1.

For a reader deciding how to use this result, the practical reading is that the decisive hypothesis is not the symmetry of K, nor the lower bound (L) alone, but the combination of the one-sided lower bound (L+) with an annulus L^p bound for some p strictly greater than 1. The counterexample shows that weakening either ingredient back toward (L) or p=1 can destroy the conclusion, while the theorem shows that this particular strengthening is enough to recover both the coercivity estimate and Hölder continuity of bounded weak solutions.

Sources used in this answer