The signed estimate behind the theorem

The intermediate estimate retains the sign of the input. For real \(g\in C_c^\infty(\mathbb R^2)\), put \[A_tg(x)=\frac1{\pi t^2}\int_{B(x,t)}g(y)\,\mathop{}\!\mathrm{d}y, \qquad S_{a,b}g(x)=\sup_{a\le t\le b}A_tg(x).\] For fixed \(0<a\le b<\infty\), this envelope is compactly supported and Lipschitz. The issue is to bound its gradient uniformly as the range of radii grows.

The proof of Proposition 2 is given in Section 5, after the local and multiscale estimates described below. A single radius, \(a=b\), is covered directly by convolution. The signed form of Proposition 2 has a specific purpose in the final regularity argument. After extension to Sobolev inputs, one applies it to \(g_Q=\chi_Q(g-c_Q)\), where \(c_Q\) is a local mean and \(\chi_Q\) equals one on the small disks being tested. On those disks, \(A_tg_Q=A_tg-c_Q\), so the corresponding envelopes have the same gradient. Poincaré’s inequality then bounds \(\|\nabla g_Q\|_1\) by the gradient of \(g\) on a slightly larger square. The function \(g_Q\) can take both signs. Signed averages also occur in Weigt’s dyadic and uncentered cube estimates (Weigt 2023, 2025); their role here is to permit this local subtraction.

Earlier endpoint results

In one dimension, Tanaka proved an \(L^1\) derivative bound for the uncentered maximal operator (Tanaka 2002, Theorem 1). Aldaz and Pérez Lázaro extended the result to inputs of bounded variation, established absolute continuity of the maximal function, and obtained the sharp variation constant \(1\) (Aldaz and Pérez Lázaro 2007, Theorem 2.5). Kurka proved a variation bound with an absolute constant for the centered operator on functions of bounded variation, together with local Sobolev regularity and an integrable derivative for Sobolev inputs (Kurka 2015, Theorem 1.2 and Corollary 1.4). The ordering of intervals used in these arguments does not directly organize a family of centered disks.

In higher dimensions, Aldaz and Pérez Lázaro proved regularity for block-decreasing inputs, invariant under coordinate sign changes and nonincreasing in each positive coordinate, and uncentered maximal operators over balls of unconditional norms; their weak-gradient conclusion applies, in particular, to block-decreasing \(W^{1,1}\) input and Euclidean balls (Aldaz and Pérez Lázaro 2009, Theorem 11). Luiro proved the \(L^1\) gradient estimate for radial Sobolev input and the uncentered ball operator (Luiro 2018, Theorem 3.12). Weigt established variation bounds for the dyadic maximal operator (Weigt 2023) and for the uncentered maximal operator over axis-parallel cubes (Weigt 2025, Theorem 1.1). Their geometry and indexing differ from centered disks. Moreover, a variation estimate controls a derivative measure, which may still have a singular part. For fractional orders \(0<\beta<1\), Beltran and Madrid established centered endpoint gradient bounds for radial \(W^{1,1}\) input (Beltran and Madrid 2020, Theorem 1.2). Weigt subsequently proved endpoint gradient estimates for both the centered and uncentered operators (Weigt 2022, Theorem 1.1). The positive order restricts overlap among optimizing balls in that proof; the mechanism degenerates at order zero.

Lahti and Weigt’s conditional regularity theorem upgrades local bounded variation of the centered maximal function to local Sobolev regularity when the input is Sobolev (Lahti and Weigt 2025, Theorem 1.2). It assumes the requisite variation bound. We prove that bound and then remove the singular part directly using the signed localization above. Finally, boundedness and continuity of the gradient map are distinct questions. The one-dimensional continuity results of Carneiro, Madrid and Pierce for the uncentered operator (Carneiro et al. 2017), and of González-Riquelme for the centered operator (González-Riquelme 2023), concern this further issue. Theorem 1 asserts the planar norm estimate.

How the finite-band estimate is proved

Write \[ u_t=A_tg,\qquad h=(\nabla g)\,\mathop{}\!\mathrm{d}x,\qquad V=|h|(\mathbb R^2)=\int|\nabla g|. \tag{3}\] Scaling reduces Proposition 2 to radii in \([2^{-m},1]\). The proof has one geometric input and two separate multiscale constructions. The first construction organizes scalar cancellation; the second chooses directions and pays for their changes in both scale and space.

From the finite-band estimate to a weak gradient

Approximation extends Proposition 2 to signed Sobolev inputs. Applying it to \(g=|f|\) and increasing the radius bands first gives a finite vector Radon measure representing the derivative of \(Mf\). To prove that this measure has no singular part, fix a compact null set and a positive radius threshold. The envelope over larger radii is Lipschitz, so its contribution vanishes as the neighborhood of the null set shrinks. For small radii, the signed cutoff and local mean subtraction described above reduce the cost to the integral of \(|\nabla g|\) within a fixed multiple of the threshold distance. After shrinking the neighborhood, let the radius threshold tend to zero. This gradient integral then vanishes as well. Inner regularity gives absolute continuity of the derivative measure and proves Theorem 1.

Lower-radius Lipschitz truncations appear in Hajłasz and Malý (Hajłasz and Malý 2010, Lemma 8), and local/nonlocal splitting with subtraction of an input mean appears in Saari (Saari 2019, sec. 3.1). The signed estimate above supplies the gradient-only localization used here.

Section 2 proves disk cancellation and the local estimates. Section 3 gives scalar refinement and common prices; Section 4 constructs the vector particles and spatial label law. Section 5 assembles the finite bands, and Section 6 passes to Sobolev input and removes the singular part. An analytic appendix records the Sobolev calculus, square localization, and vector-measure facts in the forms used in the final step.

Conventions

A constant denoted by \(C\) may increase between occurrences. All parameters, including cone apertures, finite direction nets and geometric neighborhood sizes, are fixed absolute choices. In a local lemma the explicitly listed cone and support constants are allowed dependencies. No constant depends on the number of radius scales.

For a vector measure \(\nu\), \(|\nu|\) denotes its total variation. For a real number \(a\), \(a_+=\max(a,0)\). A supremum of nonnegative quantities over an empty set is zero. When \(\nabla u_t\) is written, the radius is held fixed. We use the classical strong \(L^p\) maximal inequality for \(p>1\) (Kinnunen 1997, (1.2)); a general account is given in (Stein 1970, I). The analytic appendix records the remaining standard inputs in the forms used here.

Disk moments and an affine cancellation

We seek cancellation in a thin shell where two nearby disks differ. Estimating by the entire gradient mass on that shell would ignore the maximizing property of the radius. At an interior maximizing radius, a radial derivative vanishes; a rotational moment vanishes for every disk. We combine these identities using complex notation, then preserve them while smoothing the disk boundary.

Variations of optimizing balls also give weighted-gradient cancellation identities in Luiro’s work on the uncentered maximal operator (Luiro 2018, Lemmas 2.2 and 2.4). There the balls optimize averages of \(|f|\). We derive the identities needed here directly for centered signed averages.

We identify the plane with the complex numbers, using the usual positive orientation. A vector measure is then a complex measure, and \(\overline{y-x}\,dh(y)\) denotes complex multiplication. For every disk, \[ \operatorname{Im}\int_{B(x,t)}\overline{y-x}\,dh(y)=0, \qquad \partial_tu_t(x)=\frac1{\pi t^3} \operatorname{Re}\int_{B(x,t)}\overline{y-x}\,dh(y). \tag{10}\] The first identity follows by integrating the divergence of \(g(y)(y-x)^\perp\): the vector field \((y-x)^\perp\) is divergence free and tangent to the boundary circle. For the second, differentiate \(u_t(x)=\pi^{-1}\int_{B(0,1)}g(x+tw)\,dw\). The same change of variables gives \[ \nabla u_t(x)=\frac{h(B(x,t))}{\pi t^2}. \tag{11}\]

The balanced kernels in Lemma 4 smooth the disk boundary without losing its zero moment. Consider two such kernels, possibly with different widths. Their difference is supported in the larger shell. The following elementary observation explains what we want to know about the mass on that shell.

Thus concentration near \(z\) turns the factor \(1\) in a variation estimate into the small factor \(2\rho\). We will call \(D\cap B(z,\rho)\) a cap of the shell. The geometric question is now concrete: when can all but a controlled amount of shell mass be placed in one cap?

For the intended use, two positive measures play different roles. One, denoted by \(\nu\), detects whether the shell carries enough error mass to require this argument. The other, denoted by \(S\), describes positive contributions to the gradient. The conclusion must control both measures on the shell, but the available hypothesis bounds only \(S\) in the inner disk. A bound involving all of \(S(\mathbb R^2)\) would lose that information. The tangent-disk sweep below is what allows the inner mass bound to suffice.

One exceptional set for all radii

We use Lebesgue measure for the size \(|E|\) of a set of centers. The measures \(S\) and \(\nu\) in the next lemma are arbitrary finite positive Borel measures; they may have atoms. No maximizing radius is selected, and no regularity of these measures is assumed.

Under its shell and inner-disk mass hypotheses, the shell lemma supplies a cap for every radius at a center outside one Borel set. The affine identity then suppresses the mass on that cap. To obtain the discounted estimates, we still have to compare kernel widths, control the positive interior mass in the directional argument, and treat radii forced at band endpoints. We do this next; the value estimate will require a separate kernel.

Cancellation of small error pieces

We now compare kernels at dyadic widths. Put \[ b=2\alpha=\frac1{50},\qquad \eta=\frac1{10},\qquad \gamma=\frac15, \qquad \delta_j=2^{-j}. \tag{23}\] Choose a fixed \(\delta_*>0\) so small that \(\delta_*\le1/4\) and \(\delta\le\delta^{2\gamma}/1600\) whenever \(0<\delta\le\delta_*\).

We record estimates valid for any sequence of probability-mixture kernels \(K_j\), centered at the same \(x\) and \(r\in[1,2]\), with the support and Lipschitz properties of Lemma 4 at width \(\delta_j\). The zero-moment condition is not needed for these estimates. Write \[ \nu_j=\sum_{s_d\ge\delta_j^2}|\beta_d|, \qquad p_j=\int K_j\,d\beta, \qquad D_j=D(x,r,\delta_j). \tag{24}\] Then \[ \nu_j(\mathbb R^2)\le A\delta_j^{-b}, \qquad |p_j|\le C A\delta_j^{-\alpha}. \tag{25}\] Indeed, for \(s_d<1\), subtracting the value of \(K_j\) at the center of a supporting ball gives \[\left|\int K_j\,d\beta_d\right| \le C m_d\min(1,s_d/\delta_j).\] For \(s_d=1\), the same upper bound holds directly from \(|K_j|\le1\). Now use \(\min(1,a)\le a^\alpha\) for \(a>0\) and the definition of \(A\). The bound for \(\nu_j\) follows directly from \(s_d\ge\delta_j^2\).

If \(F\) is a scalar or complex test function with \(\mathop{\mathrm{Lip}}(F)\le C/\delta_j\), the pieces with \(s_d<\delta_j^2\) satisfy \[ \left|\sum_{s_d<\delta_j^2}\int F\,d\beta_d\right| \le \frac{C}{\delta_j}\sum_{s_d<\delta_j^2}s_dm_d \le C A\delta_j^{1-b}. \tag{26}\] Only the Lipschitz bound is used here; constants cancel because each of these pieces has zero total vector mass. In particular, (26) applies to \(K_{j+1}-K_j\) and to \(\overline{y-x}(K_{j+1}-K_j)\). These tests have the required global Lipschitz bounds, since their supports are in \(B(x,3)\). Their supports are also in \(D_j\). Consequently \[\begin{align*} |p_{j+1}-p_j| &\le C\bigl(\nu_j(D_j)+A\delta_j^{1-b}\bigr), \\ \left|\int\overline{y-x}(K_{j+1}-K_j)\,d\beta(y)\right| &\le C\bigl(\nu_j(D_j)+A\delta_j^{1-b}\bigr). \tag{27}\end{align*}\]

All these kernels are supported in \(\Omega\) when \(x\in Q\). Thus (6) can be used in every pairing below, even if some of its component measures have support outside \(\Omega\). For fixed \(x,r\), as \(j\to\infty\) the kernels converge off the circle to \(K_\infty=\mathbf1_{B(x,r)}\). Absolute continuity of all the component measures gives convergence of their integrals by dominated convergence. The same holds for the moment integrals. In particular, if every finite-width kernel is balanced, then \(K_\infty\) has zero moment as well. This is a statement at each fixed \((x,r)\): every circle has zero mass for every absolutely continuous component. There is no exceptional set depending on the radius to remove. Absolute continuity is used here, but was not needed in the Borel shell lemma.

The directional estimate

We now prove (7). If \(A=0\), all the error measures vanish, and \(e\cdot h(B(x,r))\ge0\) for every relevant disk; the conclusion follows. Assume \(A>0\).

We prove a superlevel-set estimate by separating widths at which the coarse error mass in the shell is small from those at which it is large. Small-mass widths have directly summable error increments. At the remaining widths we compare only increases of the negative projection: the later negative projection controls the interior positive mass, allowing Lemma 6 to be applied.

Let \(\lambda\ge L_0A\), where the fixed constant \(L_0\) will be chosen large enough in the course of the proof. Put \(q=A/\lambda\) and choose \(j_0\) so that \[ q/2<\delta_{j_0}\le q. \tag{28}\] We require \(L_0\ge1/\delta_*\), so all widths under consideration are at most \(\delta_*\). At a fixed \(x\in Q\) and \(r\in\mathcal I(x)\), use Lemma 4 to choose \(K_j\) for every \(j\ge j_0\), and write \[P_j=\int K_j\,dh, \qquad N_j=(-e\cdot P_j)_+, \qquad U_j=\lambda\delta_j^\eta.\] Call \(j\) active for \((x,r)\) if \(\nu_j(D_j)\ge U_j\). If it is inactive, (27) and \(1-b>\eta\) give \[ |p_{j+1}-p_j| +\left|\int\overline{y-x}(K_{j+1}-K_j)\,d\beta(y)\right| \le C\lambda\delta_j^\eta. \tag{29}\] Here and below we use \(A\le\lambda\).

Choose \(C_0\) sufficiently large depending only on \(C_s\) and \(\kappa\), and put \(T_j=C_0\lambda\delta_j^{-b}\). Apply Lemma 6 with \[\nu=\nu_j,\quad \delta=\delta_j,\quad \rho=\delta_j^\gamma,\quad U=U_j,\quad T=T_j.\] It gives Borel sets \(E_j\), independent of \(r\), such that \[ |E_j|\le C\left[ q^2\delta_j^{14/25}+q\delta_j^{9/25}\right]. \tag{30}\] To check the exponents, the first term of (15) gives \(1-\gamma-2b-2\eta=14/25\). The second term is a sum of powers \(1-2\gamma-b-\eta\) and \(1-2\gamma-2b-2\eta=9/25\); because \(\delta_j\le1\), both are bounded by a constant times the latter. Constants in (30) may depend on the already fixed \(C_0\).

Fix a center outside \(\bigcup_{j\ge j_0}E_j\), and keep any \(r\in\mathcal I(x)\) fixed. Positivity in the direction \(e\) implies \[ e\cdot P_j\ge\kappa\int K_j\,dS+e\cdot p_j, \qquad N_j\le|p_j|. \tag{31}\] Before the first active index, all the increments of \(p_j\) obey (29). Hence, if \(j\) is the first active index, or if \(j=\infty\) and there are no active indices, \[ N_j\le C\left(A\delta_{j_0}^{-\alpha} +\lambda\delta_{j_0}^\eta\right). \tag{32}\] The formula at infinity follows from the dominated-convergence observation above.

The first active width has therefore been controlled. To continue, we compare consecutive active widths, allowing arbitrarily many inactive widths between them. Only an increase of the negative projection needs a new estimate; that sign will supply the positive interior-mass bound required by the shell lemma.

Suppose now that \(j\) is active and \(l>j\) is the next active index; set \(l=\infty\) if there is no next one. The individual bound in (25) at \(j\) and \(j+1\), followed by the inactive increments between \(j+1\) and \(l\), gives \[ |p_j|+|p_l| \le C\left(A\delta_j^{-\alpha} +\lambda\delta_j^\eta\right) \le C\lambda\delta_j^{-b}. \tag{33}\] We need to estimate a change only when \(N_l>N_j\). In that case \(e\cdot P_l<0\), so (31) gives \[\int K_l\,dS\le\kappa^{-1}|p_l|.\] Since \(K_l=1\) on \(B(x,r-\delta_j)\), this proves \(S(B(x,r-\delta_j))\le T_j\) after fixing \(C_0\) large enough to dominate the constant just obtained. The shell lemma therefore supplies a point \(z\in D_j\) such that \[ (S+\nu_j)\bigl(D_j\setminus B(z,\delta_j^\gamma)\bigr) \le2U_j. \tag{34}\] This point is used only at the current center and radius; no measurable selection of \(z\) is required.

To estimate the positive mass on the cap, we need a weighted bound for both kernels being compared. The sign at \(l\) supplies one, but \(e\cdot P_j>0\) would give no corresponding bound for \(K_j\). We therefore replace \(K_j\), when necessary, by a kernel with zero projection and the same negative part. If \(e\cdot P_j\le0\), set \(K_*=K_j\). Otherwise, interpolate between \(K_j\) and \(K_l\) to obtain a convex combination \(K_*\) with \(e\cdot P_*=0\), where \(P_* =\int K_*\,dh\). This is possible because \(e\cdot P_l<0\). In either case, \[N_*:=(-e\cdot P_*)_+=N_j, \qquad e\cdot P_*\le0.\] Writing \(p_* =\int K_*\,d\beta\) and using (33) and (31), we get \[ \int(K_l+K_*)\,dS\le C\lambda\delta_j^{-b}. \tag{35}\]

All of \(K_j,K_l,K_*\) have zero moment, including when \(l=\infty\) by the preceding sharp-kernel limit. Since \(|z-x|\ge1/2\), define \[W(y)=1-\frac{\overline{y-x}}{\overline{z-x}}.\] The affine identity in Lemma 5 gives \[ P_l-P_* =\int W(y)(K_l-K_*)(y)\,dh(y). \tag{36}\] The difference of kernels is supported on \(D_j\), and \[|W|\le C\delta_j^\gamma\quad\hbox{on }B(z,\delta_j^\gamma), \qquad |W|\le C\quad\hbox{on }D_j.\] The contribution of \(\sum_i v_i\sigma_i\) to (36) is therefore at most \[ C\lambda\delta_j^{\gamma-b}+C U_j, \tag{37}\] by (35), \(|K_l-K_*|\le K_l+K_*\), and (34).

For the error measures, \(K_l-K_*\) is a scalar multiple in \([0,1]\) of \(K_l-K_j\). Keep \(W\) fixed and telescope \[ \int W(K_l-K_j)\,d\beta =\sum_{i=j}^{l-1}\int W(K_{i+1}-K_i)\,d\beta. \tag{38}\] For \(l=\infty\) this equality means the limit of the finite sums. We never apply a Lipschitz cancellation bound directly to the sharp kernel \(K_\infty\) or to its interpolant: only the finite consecutive differences in this sum need such a bound. On the first step, the pieces contributing to \(\nu_j\) cost at most \(C\delta_j^\gamma\nu_j(\mathbb R^2)+CU_j\) by (34). The remaining pieces cost \(CA\delta_j^{1-b}\) by (26), since \(W(K_{j+1}-K_j)\) has Lipschitz constant at most \(C/\delta_j\). For \(j<i<l\), use \[\int W(K_{i+1}-K_i)\,d\beta =p_{i+1}-p_i-\frac1{\overline{z-x}} \int\overline{y-x}(K_{i+1}-K_i)\,d\beta(y).\] These are inactive steps, so their absolute values sum to at most \(C\lambda\delta_j^\eta\) by (29). Combining the estimates gives \[\left|\int W(K_l-K_j)\,d\beta\right| \le C\left(A\delta_j^{\gamma-b} +A\delta_j^{1-b}+\lambda\delta_j^\eta\right).\] Together with (37), and using \(\gamma-b=9/50>\eta\), this yields \[ N_l\le N_j+|P_l-P_*|\le N_j+C\lambda\delta_j^\eta. \tag{39}\] The same inequality is automatic when \(N_l\le N_j\).

Sum (39) along the active indices. If there are finitely many, include the last comparison with \(l=\infty\); if there are infinitely many, pass to the limit along them. Dominated convergence and (32) give \[ (-e\cdot h(B(x,r)))_+ \le C\left(A\delta_{j_0}^{-\alpha} +\lambda\delta_{j_0}^\eta\right) \le C\lambda\left(q^{1-\alpha}+q^\eta\right). \tag{40}\] This holds for every \(r\in\mathcal I(x)\) at the fixed center outside the exceptional sets. Increase \(L_0\), depending only on \(C_s\) and \(\kappa\), so that the last bound is less than \(\lambda\) whenever \(q\le L_0^{-1}\).

Let \(Y(x)=\sup_{r\in\mathcal I(x)}(-e\cdot h(B(x,r)))_+\). Summing (30) and using (28), we have proved \[ |\{x\in Q:Y(x)>\lambda\}| \le C\left[ (A/\lambda)^{64/25}+(A/\lambda)^{34/25}\right] \qquad(\lambda\ge L_0A). \tag{41}\] Both powers exceed one. The layer-cake formula, the trivial bound \(|Q|=1\) for smaller thresholds, and the substitution \(\lambda=As\) give \[\int_QY\,dx \le L_0A+C A\int_{L_0}^{\infty} (s^{-64/25}+s^{-34/25})\,ds\le C A.\] Finally, (11) and \(r\ge1\) prove (7).

The size estimate, including band endpoints

To prove (8), place the measures \(v_i\sigma_i\) among the error measures with assigned size one. The augmented price is \(A_{\mathrm{tot}}=A+S(\mathbb R^2)\), where \(A\) and \(S\) refer to the original decomposition. The augmented positive part is zero; in this proof, \(\beta\) and \(\nu_j\) refer to the augmented error family. If \(A_{\mathrm{tot}}=0\), the result is immediate.

Take \(\lambda\ge L_0A_{\mathrm{tot}}\), put \(q=A_{\mathrm{tot}}/\lambda\), and choose \(j_0\) as in (28). Here \(L_0\) will depend only on \(C_s\). For \(x\in Q\) and \(r\in\mathcal J(x)\), choose the balanced kernel \(K_j\) when \(\mathop{\mathrm{dist}}(r,\{1,2\})\ge\delta_j\). Such an \(r\) is an actual maximizer on \([1,2]\), so the two neighboring radii needed in Lemma 4 are admissible. Otherwise choose the uniform probability mixture over \([r-\delta_j,r+\delta_j]\). The latter kernel need not have zero moment, but has all the same support, positivity and Lipschitz bounds. Thus (25)–(27) remain valid with \(A_{\mathrm{tot}}\) in place of \(A\). Write \(P_j=\int K_j\,dh=p_j\), and again call \(j\) active when \(\nu_j(D_j)\ge U_j=\lambda\delta_j^\eta\).

There is an additional Borel exceptional set for each width: \[ F_j=\bigcup_{a\in\{1,2\}} \left\{x:\nu_j\{y:\bigl||y-x|-a\bigr|\le2\delta_j\} \ge U_j\right\}. \tag{42}\] For each fixed \(y\), the areas of these two annuli of centers sum to \(24\pi\delta_j\). Tonelli’s theorem and Markov’s inequality give \[ |F_j|\le24\pi\delta_j\frac{\nu_j(\mathbb R^2)}{U_j} \le C(A_{\mathrm{tot}}/\lambda)\delta_j^{22/25}. \tag{43}\] If \(\mathop{\mathrm{dist}}(r,\{1,2\})<\delta_j\), the shell \(D_j\) lies in one of the annuli in (42). Thus, outside \(F_j\), an active index must satisfy \(\mathop{\mathrm{dist}}(r,\{1,2\})\ge\delta_j\), and both \(K_j\) and \(K_{j+1}\) have zero moment. In particular, the forced radii \(r=1,2\) are covered even if they are not maximizers.

Apply Lemma 6 with \(S=0\), \(T=0\), \(\nu=\nu_j\), \(\delta=\delta_j\), \(\rho=\delta_j^\gamma\), and \(U=U_j\). It supplies Borel exceptional sets \(E_j\), independent of \(r\), with bound \(C[q^2\delta_j^{14/25}+q\delta_j^{12/25}]\), which is at most the right side of (30). Its constants have no cone parameter: the positive measure and its interior-mass threshold are both zero. At an active step outside \(E_j\cup F_j\), its interior-mass hypothesis is automatic, and it gives a point \(z\in D_j\) with \(\nu_j(D_j\setminus B(z,\delta_j^\gamma))\le2U_j\). The zero moments allow the correction \[P_{j+1}-P_j=\int \left(1-\frac{\overline{y-x}}{\overline{z-x}}\right) (K_{j+1}-K_j)(y)\,d\beta(y).\] The coarse pieces cost at most \(C A_{\mathrm{tot}}\delta_j^{\gamma-b}+CU_j\), and the fine pieces cost at most \(C A_{\mathrm{tot}}\delta_j^{1-b}\), exactly as above. Hence \[|P_{j+1}-P_j|\le C\lambda\delta_j^\eta.\] At an inactive step the same inequality follows directly from (27), without using a moment identity. Starting with \(|P_{j_0}|\le CA_{\mathrm{tot}}\delta_{j_0}^{-\alpha}\) and summing all increments, we obtain \[|h(B(x,r))|\le C\lambda(q^{1-\alpha}+q^\eta)<\lambda\] after increasing \(L_0\). This holds simultaneously for all \(r\in\mathcal J(x)\) outside \(\bigcup_{j\ge j_0}(E_j\cup F_j)\). The measure bound is therefore \[\begin{align*} \left|\left\{x\in Q: \sup_{r\in\mathcal J(x)}|h(B(x,r))|>\lambda\right\}\right| \le C\bigl[&(A_{\mathrm{tot}}/\lambda)^{64/25} +(A_{\mathrm{tot}}/\lambda)^{34/25} \\[-1mm] &+(A_{\mathrm{tot}}/\lambda)^{47/25}\bigr]. \tag{44}\end{align*}\] Every exponent is larger than one. The same layer-cake computation as before gives the unnormalized bound by \(CA_{\mathrm{tot}}\). Since \(A_{\mathrm{tot}}=A+S(\mathbb R^2)\), (11) proves (8).

The value-oscillation estimate

This estimate involves all small radii and does not use a maximizing property. We express \(g\) minus a fixed mean through its gradient, prove an \(L^{3/2}\) cancellation bound for the resulting kernel, and then apply the strong maximal inequality.

We finish the proof of Lemma 3 by establishing (9). Choose a nonnegative smooth probability density \(\psi\) supported in the interior of \(Q\), with uniformly bounded derivatives after translating and rotating a fixed choice, and put \[c=\int g(z)\psi(z)\,dz, \qquad E_Q=\{x:\mathop{\mathrm{dist}}(x,Q)<4\}.\] For \(x\in E_Q\), integrate \(\nabla g\) along the segments from \(x\) to points of \(\mathop{\mathrm{supp}}\psi\) and change variables to obtain \[ g(x)-c=\int L(x,y)\cdot dh(y), \qquad L(x,y)=-(y-x)\int_0^1 \psi\bigl(x+(y-x)/t\bigr)t^{-3}\,dt. \tag{45}\] Indeed, this is the fundamental theorem of calculus in \(g(x)-g(z)=-\int_0^1(z-x)\cdot\nabla g(x+t(z-x))\,dt\), followed by \(y=x+t(z-x)\).

The displayed formula defines \(L(x,y)\) for every \(y\ne x\) in the plane, not only for \(y\in\Omega\). This global definition will allow cancellation against a support center outside \(\Omega\). For each fixed \(x\in E_Q\), the kernel is supported on segments joining \(x\) to \(\mathop{\mathrm{supp}}\psi\), and hence is supported in \(\Omega\). It is smooth in \(y\ne x\), with the uniform bounds \[ |L(x,y)|\le\frac{C}{|x-y|}, \qquad |\nabla_yL(x,y)|\le\frac{C}{|x-y|^2}. \tag{46}\] To verify them, if the integrand or one of its derivatives is nonzero, then \(|y-x|\le Ct\), since \(x\in E_Q\) and \(x+(y-x)/t\) lies in a fixed bounded neighborhood of \(Q\). The first bound follows by integrating \(t^{-3}\) from \(|y-x|/C\) to one and multiplying by \(|y-x|\). Differentiation in \(y\) produces either the same integral or \(|y-x|\int t^{-4}\,dt\), which gives the second bound.

Fix \(p=3/2\), so \(2/p-1=1/3>\alpha\). The first bound in (46) implies \[ \sup_{y\in\mathbb R^2}\|L(\cdot,y)\|_{L^p(E_Q)}\le C. \tag{47}\] This follows by integrating \(|x-y|^{-p}\) on a bounded set; \(p<2\) makes the singularity integrable, uniformly in \(y\).

Consider a piece \(\beta_d\) with \(s_d<1\), supported in \(B(z,a)\), where we may take \(a=C_s s_d\) after increasing \(C_s\) to at least one. On \(E_Q\cap B(z,2a)\), the first kernel bound and Minkowski’s inequality give \[\left\|\int L(\cdot,y)\cdot d\beta_d(y)\right\|_ {L^p(E_Q\cap B(z,2a))} \le C a^{2/p-1}m_d.\] In fact, for \(y\in B(z,a)\) the relevant \(x\) lie within \(3a\) of \(y\), and the \(L^p\) norm of \(|x-y|^{-1}\) there is \(C a^{2/p-1}\). For \(|x-z|>2a\), use the zero total vector mass to subtract \(L(x,z)\). The derivative bound in (46), applied on the segment from \(z\) to \(y\), gives \[\left|\int L(x,y)\cdot d\beta_d(y)\right| \le C a m_d|x-z|^{-2}.\] Its \(L^p\) norm on \(\{|x-z|>2a\}\) is at most \(C a^{2/p-1}m_d\), since \(p>1\). Combining the two regions yields \[ \left\|\int L(\cdot,y)\cdot d\beta_d(y)\right\|_{L^p(E_Q)} \le C s_d^{1/3}m_d\le C s_d^\alpha m_d. \tag{48}\] The constant may depend on \(C_s\). For a piece of assigned size one, and for the positive measures, use (47) directly. Minkowski’s inequality and the finite decomposition therefore give \[ \|g-c\|_{L^{3/2}(E_Q)}\le C\bigl(A+S(\mathbb R^2)\bigr). \tag{49}\] These same estimates show that the integrals obtained by inserting (6) into (45) exist for almost every \(x\); thus the use of the potentially singular kernel is justified.

Let \(f_Q=(g-c)\mathbf1_{E_Q}\), extended by zero to the plane. Every disk centered at \(x\in Q\) of radius at most four is contained in \(E_Q\). Consequently \[\sup_{0<t,t'\le4}|u_t(x)-u_{t'}(x)|\le2Mf_Q(x),\] where the maximal function on the right uses absolute values. The classical strong \(L^{3/2}\) maximal inequality (Kinnunen 1997, (1.2)), followed by Hölder’s inequality on the unit square and (49), proves (9). This completes the proof of Lemma 3.

Positive refinement and cone coordinates

For every integer \(k\), set \(r_k=2^{-k}\) and \(V_k=r_k\mathbb Z^2\). These definitions are not restricted to \(1,\ldots,m\): the neighboring direction estimate in Section 4 will also use level \(k-12\). A vertex includes its level, so vertices at different levels are distinct even when their spatial positions agree. For \(p\in V_k\), define \[Q_p=p+[0,r_k)^2, \qquad \phi_p(x)=\prod_{j=1}^2 \left(1-\frac{|x_j-p_j|}{r_k}\right)_+.\] For \(p\in V_i\) and \(q\in V_k\) with \(i<k\), put \(\lambda_{pq}=\phi_p(q)\), and at equal levels put \(\lambda_{pq}=\mathbf 1_{\{p=q\}}\). These nonnegative coefficients satisfy \[ \begin{gathered} \sum_{p\in V_i}\lambda_{pq}=1, \qquad \phi_p=\sum_{q\in V_k}\lambda_{pq}\phi_q \quad (i\le k), \\ \lambda_{ps}=\sum_{q\in V_j}\lambda_{pq}\lambda_{qs} \quad (i\le j\le k,\ p\in V_i,\ s\in V_k). \end{gathered} \tag{50}\] To verify these identities, first consider one coordinate. The hats on a fixed grid sum to one: between two consecutive vertices, the only nonzero hats have complementary affine values. A coarse hat is affine on every interval of any finer dyadic grid, since all its breakpoints belong to that grid. It therefore equals the piecewise affine interpolant of its values at the finer vertices. Taking products proves the first two identities in (50); evaluating the second identity at \(s\) proves the third. The sums are locally finite. Notice that the normalization is in the coarse index \(p\) for each fixed fine vertex \(q\). These coefficients are refinement weights, not stochastic transition rows.

If \(q\in V_{k+1}\) and \(\lambda_{pq}>0\), then in each coordinate \(q_j-p_j\in\{-r_k/2,0,r_k/2\}\). Consequently \[ \mathop{\mathrm{supp}}\phi_q\subseteq\mathop{\mathrm{supp}}\phi_p. \tag{51}\] Thus each vertex has nine positive-weight children, and the supports remain nested along every path of positive-weight edges.

Define the vector masses and their polar decompositions by \[h_p=\int\phi_p\,\mathop{}\!\mathrm{d}h, \qquad a_p=|h_p|, \qquad h_p=a_p\theta_p,\] where \(\theta_p\) is an arbitrary fixed unit vector when \(a_p=0\). Refinement gives \[h_p=\sum_{q\in V_k}\lambda_{pq}h_q \quad(p\in V_i,\ i\le k).\] The triangle inequality and the column sums in (50) imply \[ \sum_{p\in V_i}a_p \le\sum_{q\in V_k}a_q \le V \quad(i\le k). \tag{52}\] The last inequality follows directly from \(\sum_q\phi_q=1\). At each fixed level only finitely many of these masses can be nonzero, because \(h\) has compact support.

Fix \(\tau=1/100\) and a finite \(\tau/20\)-net \(\mathcal N\) of the unit circle, measured in chord distance. For \(n\in\mathcal N\), let \(n^\perp\) be its counterclockwise quarter turn and put \[e_+(n)=\frac{\sqrt3}{2}n+\frac12n^\perp, \qquad e_-(n)=\frac{\sqrt3}{2}n-\frac12n^\perp.\] These are unit vectors and \(e_+(n)\cdot e_-(n)=1/2\). Call \((q,n)\) good when \(\theta_q\) belongs to the closed sector between \(e_-(n)\) and \(e_+(n)\), and bad otherwise. The two scalar coefficient functionals \[ c_{n,+}(z)=\frac{n\cdot z}{\sqrt3}+n^\perp\cdot z, \qquad c_{n,-}(z)=\frac{n\cdot z}{\sqrt3}-n^\perp\cdot z \tag{53}\] satisfy \[z=e_+(n)c_{n,+}(z)+e_-(n)c_{n,-}(z), \qquad |c_{n,+}(z)|+|c_{n,-}(z)| =2\max\left\{\frac{|n\cdot z|}{\sqrt3}, |n^\perp\cdot z|\right\} \le2|z|.\] In particular, for a good pair \((q,n)\), both coefficients of \(h_q\) are nonnegative, including when \(h_q=0\). This sign information will supply the positive cone measures in Lemma 3.

For the finite-band assembly, we also need to bound a vector’s norm by its signed projection along \(n\) and the unfavorable projections controlled by the local estimate. The following inequality gives that bound, with an absolute constant: \[ |z|\le C\left(n\cdot z+ \sum_{\varepsilon\in\{+,-\}} (-e_\varepsilon(n)\cdot z)_+\right) \quad(z\in\mathbb R^2). \tag{54}\] Here the expression in parentheses is nonnegative. Indeed, writing \(a=n\cdot z\), its value is positive for every nonzero \(z\): if \(a>0\) this is immediate; if \(a=0\), one of the two negative parts equals \(|z|/2\); and if \(a<0\), their sum is at least \(-(e_+(n)+e_-(n))\cdot z=\sqrt3|a|\), so the expression is at least \((\sqrt3-1)|a|>0\). Its minimum on the unit circle is therefore positive by continuity. Rotation invariance makes this minimum independent of \(n\), and homogeneity proves (54).

Cancellation from the finest level upward

Let \(\mathcal C\) be the indexed finite family consisting of \(c_{n,+},c_{n,-}\) for every \(n\in\mathcal N\); repeated functionals, if any, retain their indices. For \(c\in\mathcal C\) write \[\nu^c=c(h)=c(\nabla g(x))\,\mathop{}\!\mathrm{d}x, \qquad c_s=c(h_s)=\int\phi_s\,\mathop{}\!\mathrm{d}\nu^c.\] For \(s\in V_l\), \(1\le l\le m\), define \[ d_s^c= \begin{cases} \displaystyle\frac12\left( \sum_{t\in V_{l+1}}\lambda_{st}|c_t|-|c_s|\right), & l<m,\\[6pt] \displaystyle\frac12\left( \int\phi_s(x)|c(\nabla g(x))|\,\mathop{}\!\mathrm{d}x-|c_s|\right), & l=m. \end{cases} \tag{55}\] These numbers are nonnegative by refinement and the triangle inequality. They measure the equal positive and negative masses that can be removed at \(s\). At level \(m\) the definition uses the positive and negative parts of the density itself. In particular, \(c_s=0\) need not imply \(d_s^c=0\).

We will realize this numerical cancellation by measures supported in the corresponding tent. The elementary operation is proportional removal; it does not require the two measures being paired to be mutually singular.

The measures are constructed once at each vertex. A descendant’s pair may appear in several coarse identities, with its specified coefficient; these identities do not assert a disjoint allocation.

One price at every observation scale

For \(Q=Q_p\) at level \(k\), put \[\mathcal R(Q)=\{q\in V_k:|q-p|_\infty\le20r_k\}, \qquad |z|_\infty=\max\{|z_1|,|z_2|\}.\] A pair created at level \(l\ge k\) has support of radius at most \(\sqrt2r_l\). At observation scale \(r_k\), the local estimate therefore charges it the relative factor \((r_l/r_k)^\alpha=2^{-\alpha(l-k)}\). We sum these charges over all the scalar coordinates so that the resulting price is independent of the label chosen later.

The scaled radius sets from Lemma 3 are \[\begin{align*} \mathcal I_k(x) &=\{r\in[r_k,2r_k]:u_r(x)\ge u_t(x) \text{ for every }t>0\text{ with }|t-r|\le r_k/4\},\\ \mathcal J_k(x) &=\{r\in[r_k,2r_k]:u_r(x)=\max_{r_k\le t\le2r_k}u_t(x)\} \cup\{r_k,2r_k\}. \end{align*}\] Also define \[v_k(x)=\sup_{0<t,t'\le4r_k}|u_t(x)-u_{t'}(x)|.\] An empty supremum of nonnegative quantities is zero, as before.

The prices now pay for every scalar cancellation and leave only nearby vector masses \(a_q\) in the local bounds. Controlling those remaining masses requires a choice of directions across scales and positions. The next section constructs a positive measure on complete refinement paths, recording a birth level, so that the mass of paths through a vertex \(q\) whose birth has already occurred is exactly \(a_q\). The scalar measures above supply neither this path measure nor the spatial estimates for the direction law; their role is the reconstruction and discounted budget proved in this section.

Directions in neighboring cells

At each vertex we first look for a nearby direction carrying maximal mass. This choice prevents a small neighboring mass from imposing an unpaid change of direction. For \(z\in V_k\), choose a vertex \(w(z)\in V_k\) maximizing \(a_w\) among \(|w-z|_{\infty}\le300r_k\), and set \[\Theta_z=\theta_{w(z)}.\] Choose \(n_z\in\mathcal N\) with \(|n_z-\Theta_z|\le\tau/20\). The level is part of the vertex label in this notation. Define \[E_k(q)= \max_{\substack{z\in V_k\\|z-q|_{\infty}\le200r_k}} |\Theta_z-\theta_q|^2, \qquad q\in V_k.\] In particular, \(0\le E_k(q)\le4\). The quantity \(E_k(q)\) measures how far the nearby chosen directions can depart from its own direction. Put \(A_j=\sum_{p\in V_j}a_p\) for every integer \(j\). By (52), these total vector masses satisfy \(0\le A_j\le A_{j+1}\le V\). Their increase pays the weighted sum of the disagreements.

The finite-state label law

The neighboring budget does not prescribe a direction at every point. We interpolate the fixed vertex choices by the tents, retaining the previous direction unless it is appreciably far from the sampled one. The interval on which the switching probability vanishes will keep small direction fluctuations from creating repeated run endpoints.

For each \(x\in\mathbb R^2\) we define a probability law \(\mathbb P_x\) on \(\mathcal N^m\). At the first step, sample \(z\in V_1\) with probabilities \(\phi_z(x)\) and set \(n_1=n_z\). At a step \(k\ge2\), condition on the old label \(n=n_{k-1}\), sample \(z\in V_k\) with probabilities \(\phi_z(x)\), and switch to \(n_z\) with probability \[ p_k(n,z)=P(|n-\Theta_z|), \qquad P(d)=\min\{1,\tau^{-2}(d-\tau)_+^2\}. \tag{73}\] Otherwise retain \(n\). These samples and switch trials are made independently of the past conditional on the old label. If \(p_k(n,z)>0\), then \(n_z\ne n\): the opposite equality would imply \(|n-\Theta_z|\le\tau/20<\tau\).

For a label \(n\), let \(\delta_n\) denote the probability vector concentrated at \(n\). The initial row and subsequent transition rows are, respectively, \[\begin{align*} \pi_1(x) &=\sum_{z\in V_1}\phi_z(x)\delta_{n_z}, \tag{74}\\ \pi_k(x;n,\cdot) &=\delta_n+\sum_{z\in V_k}\phi_z(x)p_k(n,z) (\delta_{n_z}-\delta_n),\quad k\ge2. \tag{75}\end{align*}\] Every entry is nonnegative and every row sums to one, as is also clear from the sampling rule. The rows are continuous, piecewise bilinear interpolations of bounded vectors, with Lipschitz constant at most \(C/r_k\). These formulas, together with the initial row, specify the probability of each label sequence as a product of entries; this is the law \(\mathbb P_x\). We denote its expectation by \(\mathbb E_x\) and the marginal law of \(n_k\) by \(\zeta_k(x)\). When convenient, the initial row is viewed as a transition from a single auxiliary state \(*\), with \(\zeta_0(x)=\delta_*\) for every \(x\).

The row switch probability for a prescribed old label is \[\rho_k(x\mid n)=\sum_{z\in V_k}\phi_z(x)p_k(n,z),\qquad k\ge2.\] Since a trial counted by this expression changes the label, define \[ \begin{aligned} R_k(x)&=\sum_{n\in\mathcal N}\zeta_{k-1}(x;n)\rho_k(x\mid n) =\mathbb P_x(n_k\ne n_{k-1}),\quad 2\le k\le m,\\ R_1(x)&=R_{m+1}(x)=1. \end{aligned} \tag{76}\] The two sentinel values count the beginning and end of a history; they apply also when \(m=1\). The functions \(R_k\) are continuous, since the rows are Lipschitz and each marginal is obtained by finitely many matrix products.

Let \(G\) be the union of the grid lines at levels \(1,\ldots,m\). It is a fixed Lebesgue-null set. Off \(G\) all the tent rows are differentiable. Define the scaled expected row-derivative costs there by \[ \begin{split} T_k(x)&=r_k\sum_{n\in\mathcal N}\zeta_{k-1}(x;n) \sum_{n'\in\mathcal N} |\nabla_x\pi_k(x;n,n')|,\quad 2\le k\le m,\\ T_1(x)&=r_1\sum_{n'\in\mathcal N} |\nabla_x\pi_1(x;n')|. \end{split} \tag{77}\] Set \(T_k=0\) on \(G\). These are measurable versions of the costs, and all later bounds on their suprema are essential-supremum bounds. The derivative in this definition acts on one transition row; it is not the derivative of the marginal \(\zeta_k\). Off \(G\), at most four level-\(k\) tents have a nonzero gradient, and \(|\nabla\phi_z|\le\sqrt2/r_k\). It follows from the row formulas that \[ 0\le R_k\le1, \qquad 0\le T_k\le C. \tag{78}\] The norm inside the inner sum in (77) is the Euclidean norm of the spatial gradient, whereas norms of probability vectors below are their \(\ell^1\) norms.

This lemma pays the bad-root terms in the first line of (65). If \(q\in\mathcal R(Q_p)\) at level \(k\), then \(|x-q|_\infty\le21r_k\) throughout \(Q_p\). Whenever \((q,n)\) is bad and \(E_k(q)\le c_*\), the marginal probability of \(n_k=n\) therefore vanishes on that whole cell. In the remaining case, \(a_q\le c_*^{-1}a_qE_k(q)\), whose sum is bounded by (71), with the fixed cell and palette multiplicities. It remains to pay root masses at the endpoints of runs and in the terms where spatial weights are differentiated.

For \(q\in V_l\) define the closed square \[D(q)=\{x\in\mathbb R^2:|x-q|_\infty\le25r_l\}.\] It contains every level-\(l\) pricing cell whose root neighborhood contains \(q\): indeed \(q\in\mathcal R(Q_p)\) and \(x\in Q_p\) give \(|x-q|_\infty\le21r_l\). We can now state the estimate that pays both types of cost on those whole cells.

We prove this proposition in three steps. First a positive measure on refinement paths will realize each \(a_q\) exactly. Along each path, vector additivity will bound switches at its terminal position. Finally comparison of the laws at nearby points will turn that fixed-position estimate into the suprema over \(D(q)\).

Particles and intrinsic direction changes

The scalar decomposition in Section 3 is an identity of signed measures with cancelling pairs. Here we need a different object: a positive measure on complete refinement paths, with exact occupancy \(a_q\) after birth. It must also retain the path before birth, because the label law has been evolving since level one. This path measure will not depend on \(x\). We will combine it with \(\mathbb P_x\) by evaluating that law at a path’s terminal vertex. A path node at level \(k\) is \(\xi=(q_1,\ldots,q_k)\), where \(q_j\in V_j\) and each consecutive refinement coefficient is positive. Its path weight and mass are \[b_\xi=\prod_{j=1}^{k-1}\lambda_{q_jq_{j+1}}, \qquad M_\xi=b_\xi a_{q_k}.\] The empty product at a root is one. The associated vector is \(b_\xi h_{q_k}\). Vectors add from the children to their parent, and therefore \[ M_\xi\le\sum_{\eta\text{ child of }\xi}M_\eta. \tag{80}\] Only a finite forest is needed. Choose all level-one roots whose closed tent supports meet \(\mathop{\mathrm{supp}}h\), and retain all their positive-weight paths through level \(m\). There are finitely many roots and each node has finitely many children. If \(a_q>0\), the support of \(\phi_q\) meets \(\mathop{\mathrm{supp}}h\). Every ancestor tent then meets \(\mathop{\mathrm{supp}}h\) by support nesting, so every positive-weight path to \(q\) starts among the chosen roots. In particular no contribution to a nonzero mass is lost by this finite restriction. If \(h=0\), the forest may be empty. Consequently, at each level \(k\), \[ \sum_{\xi:\,q_k=q}b_\xi=1\quad\text{if }a_q>0, \qquad \sum_{|\xi|=k}M_\xi=A_k. \tag{81}\]

The particle measure now realizes every vector mass exactly at its level. It lets us prove a pathwise bound and then recover a sum over all cells by integration. We next mark the genuine direction changes along a path. The sum of the norms of descendant vectors may exceed the norm of their sum at the ancestor; this nonnegative difference will pay for the marks.

Mark every root of the path forest. Below a mark, mark the first nodes whose direction \(\theta_{q_s}\) has distance greater than \(\tau/8\) from the direction at that mark, and repeat the rule below each new mark. These intrinsic marks depend on the path forest and the vector masses; they do not depend on the label process at any point \(x\).

Comparison of label laws at nearby points

We have bounded the expected switches at the single point \(y\) attached to each particle. This does not yet give Proposition 12: the local prices are integrated on cells, so their coefficients must be controlled uniformly there. We also still need the row-derivative cost. The same comparison of nearby rows supplies both missing bounds.

We compare the law at any point of \(D(q_l)\) with the law at the terminal position of a particle. The comparison uses the complete particle path, including its levels before birth.

The packing estimate

Approximation, translations, and equality sets

Square localization

Vector variation and local lower semicontinuity

For \(u\in L^1_{\mathrm{loc}}(\mathbb R^2)\) and open \(O\subset\mathbb R^2\), define its distributional variation on \(O\) by \[ \mathcal V(u;O)= \sup\left\{-\int u\,\operatorname{div}\varphi: \varphi\in C_c^1(O;\mathbb R^2),\ |\varphi(x)|\le1\right\}. \tag{123}\] The test-field norm is Euclidean. The class is closed under negation, so the same supremum results if the tested integral is replaced by its absolute value.

In Section 6, this lemma first yields a finite derivative measure from the expanding bands. The signed cutoff argument then proves that it vanishes on compact null sets; the lemma’s final assertion identifies the resulting weak gradient.