An Endpoint Gradient Bound for the Centered Disk Maximal Operator

1 Introduction

For f\in L^1_{\mathrm{loc}}(\R^2), the centered disk maximal function is

Every averaging disk is centered at the point of evaluation, and all positive radii are allowed. Kinnunen proved that this operator is bounded on W^{1,p} for 1<p<\infty [10, Theorem 1.4]. The pointwise control of its gradient by M(|\nabla f|) gives the corresponding norm bound through the strong L^p maximal inequality. At p=1 that last step fails. Hajłasz and Onninen asked whether the gradient estimate itself nevertheless survives [8, Question 1]. We give a positive resolution of the planar centered-disk case of their endpoint question.

Here W^{1,1}(\R^2) is the inhomogeneous Sobolev space: the input and its weak first derivatives are globally integrable. Vector norms are Euclidean. No radiality or compact-support condition is imposed on the input. The maximal function itself need not be globally integrable, even for a nonzero nonnegative smooth function of compact support. The constant in (1.1) is finite and absolute; its optimal value is not determined. The argument below concerns disks in the plane.

1.1 The signed estimate behind the theorem

The intermediate estimate retains the sign of the input. For real g\in C_c^\infty(\R^2), put

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 1.2 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 1.2 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 [19, 20]; their role here is to permit this local subtraction.

1.2 Earlier endpoint results

In one dimension, Tanaka proved an L^1 derivative bound for the uncentered maximal operator [17, 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 [2, 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 [11, 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 [1, Theorem 11]. Luiro proved the L^1 gradient estimate for radial Sobolev input and the uncentered ball operator [13, Theorem 3.12]. Weigt established variation bounds for the dyadic maximal operator [19] and for the uncentered maximal operator over axis-parallel cubes [20, 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 [3, Theorem 1.2]. Weigt subsequently proved endpoint gradient estimates for both the centered and uncentered operators [18, 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 [12, 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 [4], and of González-Riquelme for the centered operator [6], concern this further issue. Theorem 1.1 asserts the planar norm estimate.

1.3 How the finite-band estimate is proved

Write

Scaling reduces Proposition 1.2 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.

Disk moments and concentration in a shell. At an interior maximizing radius, the radial derivative of u_t vanishes. A rotational moment vanishes on every disk. In complex notation these identities give a zero first moment, which is preserved by suitable positive mixtures of nearby disks. Variations of optimizing balls also produce weighted-gradient identities in Luiro’s work [13, Lemmas 2.2 and 2.4]; we derive the identities needed for centered signed averages directly.

Identify \R^2 with \mathbb C, let x be the disk center, and let z be a point of its thin boundary shell. The purpose of the moment identity is visible in the affine factor

This factor vanishes at z. Inserting it in a difference of balanced disk kernels therefore makes mass near z inexpensive. This leads to the geometric question: when is almost all relevant shell mass concentrated near one point? A sweep by nested disks tangent at that point answers it outside one exceptional set of centers, uniformly over every radius in the band. Crucially, it requires a bound on positive mass only in an inner disk; the error mass is controlled separately.

These observations yield a local directional estimate. On a square of side r, decompose h into positive measures with directions in one cone and localized errors. Errors smaller than the observation scale have zero total vector mass; errors at scale r are charged their full variation. An error of variation b supported at scale s\le r is charged b(s/r)^\alpha, with a fixed \alpha=1/100. The resulting sum controls the integral over the square of the supremum, over locally maximizing radii, of the gradient component pointing against the cone. Companion estimates handle band endpoints and the value oscillation

Those estimates include the endpoint arguments and the comparison of successively narrower kernels; shell concentration alone does not supply them.

Scalar cancellation and a common price for each cell. Let \phi_q be the product of the two one-dimensional tents at a vertex q of the level-k dyadic grid: each factor equals one at its vertex coordinate and decreases linearly to zero one mesh width away. Set h_q=\int\phi_q\,\dd h, a_q=|h_q|. The tents form positive partitions of unity and refine positively onto finer grids. We express vectors in each of finitely many bases formed by the two boundary directions of a cone. For each scalar coordinate, we start with the actual signed density at the finest level. Pair its positive and negative parts in equal masses, retain the constant-sign remainder, and repeat this operation on the weighted remainders at each parent. This bottom-up construction retains the support and birth level of every pair, including cancellation inside the finest tents. Its identity is an identity of measures, so there is no unrepresented terminal density.

The masses paired at successive levels form a telescoping sum. Charging each pair at coarser observation scales with the factor (s/r)^\alpha gives a nonnegative price D_Q for every dyadic cell, with

One price works for every direction in the fixed finite palette. The local estimates then cost D_Q together with appropriate nearby vector masses a_q at the observation scale. Multiscale oscillation budgets already play a role in Kurka’s one-dimensional proof [11, Lemma 1.5]; the scalar measure decomposition and its common prices are proved here.

Vector particles and the cost of choosing directions. The remaining vector masses require a different construction. Vector refinement makes their total increase with the level, while keeping it at most V. We realize these masses by a finite positive measure on refinement paths, adding mass when needed and recording its birth level. This construction preserves the complete path, including levels before birth, so that it can compare choices of direction made from the first scale onward. These vector particles differ from the scalar cancelling pairs: their role is to turn estimates along paths into sums of vector-mass charges over cells.

Two costs must be controlled. A change of direction starts a new run of radius scales and creates endpoint terms. Spatial variation of the choice of direction creates derivatives of its weights when we integrate by parts. Section 4 constructs a finite-state law at each point x and bounds sums of both costs, weighted by these vector masses, by CV, with the spatial suprema needed for the cell estimates. Vector-additivity defects control changes along particles, and comparison at nearby points transfers that control to the spatially varying law.

Assembly and cancellation of differentiated rows. After the common prices and the direction costs have been bounded, we assemble the radius bands. A run is a maximal consecutive interval of scales carrying the same direction. For each history, apply the local directional estimate to the supremum over the radii of every run, then average over histories. The common prices and the bounds for direction changes pay for the unfavorable components and the endpoints of runs.

For each run, subtract the average at its starting scale from its envelope; the endpoint estimates pay for the gradients of these baselines. The vector quantity to be integrated by parts is the sum of the resulting envelopes multiplied by their run directions. Its history weight depends on x. If \pi_k(x;n,n') is the probability of passing from direction n to direction n' at step k, normalization of each row gives

Before taking absolute values, we can therefore subtract the analogous vector sum for the preceding history, ending its last run at level k-1. That sum is independent of the next direction. The difference contains only extensions of the last run and new runs, and is bounded by C\sum_{l\ge k}v_l. The local value estimate contributes the scale r_l, while a differentiated transition row has scale 1/r_k. Their ratio r_l/r_k is exactly the geometric weight controlled by the direction-cost estimate. Together with the common-price budget, these bounds give (1.2).

1.4 From the finite-band estimate to a weak gradient

Approximation extends Proposition 1.2 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 1.1.

Lower-radius Lipschitz truncations appear in Hajłasz and Malý [7, Lemma 8], and local/nonlocal splitting with subtraction of an input mean appears in Saari [14, Section 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.

1.5 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 [10, (1.2)]; a general account is given in [16, Chapter I]. The analytic appendix records the remaining standard inputs in the forms used here.

2 Cancellation at disks and three local bounds

Throughout this section, g\in C^\infty(\R^2) is real valued, h=\nabla g\,dx, and u_t=A_tg. The averages are signed. No compact-support assumption on g is needed for the local estimates. Our goal is to control three quantities on a unit square: unfavorable directional gradients at locally maximizing radii, gradients at band maxima and endpoints, and differences of averages. The common input is a decomposition of the gradient into positive directions and error pieces, with small support and zero total mass below size one. The discount on the latter errors will make the scale sum in Section 3 finite. We fix

An equality of measures on an open set means equality of their restrictions there. The component measures themselves will always remain defined on the whole plane.

All the quantities in this statement are measurable. To see this, extend the displayed definitions of \mathcal I(x) and \mathcal J(x) to all x\in\R^2. Their graphs are closed in \R^2\times[1,2], by joint continuity of u_t(x) and continuity of a maximum over a compact interval. For either of the first two suprema, its superlevel set at a positive closed threshold is the projection of a closed set with the radius variable in a compact interval, and is therefore closed. Restricting these Borel functions to Q proves the claim also for half-open squares. The last supremum can be taken over positive rational t,t'\le4, by continuity in the radius variables.

The first estimate discounts cancellation without charging the positive mass S: its cone contribution is favorable at every tested radius. Comparisons of kernels will still require a bound on its weighted mass. The second estimate pays for S and also covers the two forced radii, so it can be used when a radius interval ends. The third controls values rather than gradients; its factor under rescaling will pay for derivatives of spatial weights. Section 3 gives those scaled applications.

We prove the first two bounds by comparing successively narrower disk kernels. Before estimating their changes, we explain why a maximizing radius supplies a useful moment identity and why concentration in a small cap makes that identity effective.

2.1 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 [13, 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,

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

The balanced kernels in Lemma 2.2 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(\R^2) would lose that information. The tangent-disk sweep below is what allows the inner mass bound to suffice.

2.2 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.

2.3 Cancellation of small error pieces

We now compare kernels at dyadic widths. Put

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 2.2 at width \delta_j. The zero-moment condition is not needed for these estimates. Write

Then

Indeed, for s_d<1, subtracting the value of K_j at the center of a supporting ball gives

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 \Lip(F)\le C/\delta_j, the pieces with s_d<\delta_j^2 satisfy

Only the Lipschitz bound is used here; constants cancel because each of these pieces has zero total vector mass. In particular, (2.22) 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

All these kernels are supported in \Omega when x\in Q. Thus (2.2) 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.

2.4 The directional estimate

We now prove (2.3). 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 2.4 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

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 2.2 to choose K_j for every j\ge j_0, and write

Call j active for (x,r) if \nu_j(D_j)\ge U_j. If it is inactive, (2.23) and 1-b>\eta give

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 2.4 with

It gives Borel sets E_j, independent of r, such that

To check the exponents, the first term of (2.11) 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 (2.26) 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

Before the first active index, all the increments of p_j obey (2.25). Hence, if j is the first active index, or if j=\infty and there are no active indices,

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 (2.21) at j and j+1, followed by the inactive increments between j+1 and l, gives

We need to estimate a change only when N_l>N_j. In that case e\cdot P_l<0, so (2.27) gives

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

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,

Writing p_* =\int K_*\,d\beta and using (2.29) and (2.27), we get

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

The affine identity in Lemma 2.3 gives

The difference of kernels is supported on D_j, and

The contribution of \sum_i v_i\sigma_i to (2.32) is therefore at most

by (2.31), |K_l-K_*|\le K_l+K_*, and (2.30).

For the error measures, K_l-K_* is a scalar multiple in [0,1] of K_l-K_j. Keep W fixed and telescope

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(\R^2)+CU_j by (2.30). The remaining pieces cost CA\delta_j^{1-b} by (2.22), since W(K_{j+1}-K_j) has Lipschitz constant at most C/\delta_j. For j<i<l, use

These are inactive steps, so their absolute values sum to at most C\lambda\delta_j^\eta by (2.25). Combining the estimates gives

Together with (2.33), and using \gamma-b=9/50>\eta, this yields

The same inequality is automatic when N_l\le N_j.

Sum (2.35) 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 (2.28) give

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 (2.26) and using (2.24), we have proved

Both powers exceed one. The layer-cake formula, the trivial bound |Q|=1 for smaller thresholds, and the substitution \lambda=As give

Finally, (2.7) and r\ge1 prove (2.3).

2.5 The size estimate, including band endpoints

To prove (2.4), place the measures v_i\sigma_i among the error measures with assigned size one. The augmented price is A_{\mathrm{tot}}=A+S(\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 (2.24). 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 \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 2.2 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 (2.21)–(2.23) 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:

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

If \dist(r,\{1,2\})<\delta_j, the shell D_j lies in one of the annuli in (2.38). Thus, outside F_j, an active index must satisfy \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 2.4 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 (2.26). 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

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

At an inactive step the same inequality follows directly from (2.23), without using a moment identity. Starting with |P_{j_0}|\le CA_{\mathrm{tot}}\delta_{j_0}^{-\alpha} and summing all increments, we obtain

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

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(\R^2), (2.7) proves (2.4).

2.6 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 2.1 by establishing (2.5). 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

For x\in E_Q, integrate \nabla g along the segments from x to points of \supp\psi and change variables to obtain

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 \supp\psi, and hence is supported in \Omega. It is smooth in y\ne x, with the uniform bounds

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 (2.42) implies

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

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 (2.42), applied on the segment from z to y, gives

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

The constant may depend on C_s. For a piece of assigned size one, and for the positive measures, use (2.43) directly. Minkowski’s inequality and the finite decomposition therefore give

These same estimates show that the integrals obtained by inserting (2.2) into (2.41) 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

where the maximal function on the right uses absolute values. The classical strong L^{3/2} maximal inequality [10, (1.2)], followed by Hölder’s inequality on the unit square and (2.45), proves (2.5). This completes the proof of Lemma 2.1.

3 Dyadic vector masses and common prices

Throughout this section, let g\in C_c^\infty(\R^2) be real valued, and write

The averages u_t are signed. Fix an integer m\ge1; all estimates below are uniform in m. Lemma 2.1 discounts a cancelling error according to the size of its support. We now decompose the gradient into errors to which that discount applies, and assign one summable price to each dyadic cell. The same price will work for every direction in a fixed finite palette. The construction starts with the actual density of each scalar coordinate of the gradient at the finest level and removes equal positive and negative masses as it moves toward coarser levels.

3.1 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

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

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 (3.1); 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

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

where \theta_p is an arbitrary fixed unit vector when a_p=0. Refinement gives

The triangle inequality and the column sums in (3.1) imply

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

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

satisfy

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 2.1.

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:

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 (3.5).

3.2 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

For s\in V_l, 1\le l\le m, define

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.

The equivalent path form. The price estimate below uses the vertex identity. The following equivalent form expresses shared-descendant coefficients as sums of path weights, using the same cancelling pieces. Section 4 will construct a separate positive measure on vector paths. Fix q\in V_k with 1\le k\le m, and let \xi range over all positive-weight paths \xi=(q_k,\ldots,q_l) with q_k=q and k\le l\le m, including the trivial path (q). Set

Composition gives \sum_{\xi:q_l=s}w_\xi=\lambda_{qs}, and therefore

Each path pair has mass w_\xi d_s^c in each sign and support in \supp\phi_s, where s=q_l. This recovers the path identity without a separate allocation along every path.

The cancellation amounts have an exact total budget. Summing (3.6) over a level and using the column normalization in (3.1) cancels intermediate absolute masses. The terminal partition of unity gives

Here \|c\| denotes the Euclidean operator norm, equal to 2/\sqrt3 for every c\in\mathcal C by (3.4). When m=1, the same formula is just the sum of the terminal defects. Zero masses introduce no exception to either the reconstruction or this telescoping identity.

3.3 One price at every observation scale

For Q=Q_p at level k, put

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 2.1 are

Also define

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.

4 Directions that change slowly in scale and position

The common prices in Proposition 3.3 pay for scalar cancellation. Its remaining terms are nearby vector masses a_q. We now choose directions so that these masses pay only a summable amount for changes of direction. A maximal consecutive interval of levels carrying the same direction will be called a run. Ending a run produces endpoint terms in the local estimates, so the first cost is the probability of a change between successive levels.

There is also a spatial cost. We will average signed directional estimates with weights depending on x and then integrate by parts. For any finite Lipschitz probability row (\omega_j(x))_j, normalization gives

almost everywhere, for arbitrary scalar values U_j and a common value U_*. Thus the cost of differentiating a row is controlled by the differences U_j-U_*, rather than the sizes of the values themselves. In Section 5, the differences will be bounded by sums of the small-radius oscillations v_l defined in Section 3. Their integral over a level-l cell costs r_l times its price and root masses, by (3.16). A row built from level-k tents has derivatives of size r_k^{-1}. The required joint bound therefore carries the ratio r_l/r_k=2^{k-l}, for k\le l. This explains both the row-derivative cost and its weight in the packing estimate below.

Retain g\in C_c^\infty(\R^2), h=\nabla g\,\dd x, V=|h|(\R^2), and the levels 1,\ldots,m, with m\ge1. We use the hats, vector masses h_q=a_q\theta_q, and refinement coefficients from Section 3, defined at every integer level. The coarser levels will be needed to compare nearby directions. The palette \mathcal N is the fixed \tau/20-net of the unit circle, where \tau=1/100. Recall that (q,n) is good when \theta_q makes an angle at most \pi/6 with n. All vertex and palette choices below use fixed tie rules, independently of the observation point x. Directions at zero vector masses retain the prescribed unit-vector convention.

4.1 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

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

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 (3.3), these total vector masses satisfy 0\le A_j\le A_{j+1}\le V. Their increase pays the weighted sum of the disagreements.

4.2 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\R^2 we define a probability law \Prob_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

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,

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 \Prob_x. We denote its expectation by \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

Since a trial counted by this expression changes the label, define

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

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

The norm inside the inner sum in (4.7) 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 (3.16). 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 (4.1), 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

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).

4.3 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 \Prob_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

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

Only a finite forest is needed. Choose all level-one roots whose closed tent supports meet \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 \supp h. Every ancestor tent then meets \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,

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.

4.4 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 4.3: 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.

4.5 The packing estimate

5 Signed finite-radius envelopes

The common prices control unfavorable directional components. The direction law controls the endpoints created when a direction changes. We now combine those estimates, keeping the remaining directional projections signed until integration by parts. The essential cancellation is that the derivatives of a probability row sum to zero. Subtracting the potential of the preceding history then leaves only oscillations at smaller radii.

Throughout this section, g\in C_c^\infty(\R^2) is real, u_t=A_tg, V=\int|\nabla g|, and r_k=2^{-k} for 1\le k\le m. Use the prices and label law of Sections 3 and 4. For 1\le i\le j\le m, put

The maximum is attained, since t\mapsto u_t(x) is continuous. These envelopes are compactly supported: all their radii are at most 2r_1=1, so all their averages vanish outside the closed 1-neighborhood of \supp g. They are Lipschitz, with common Lipschitz bound \|\nabla g\|_\infty.

Outside one null set all the finitely many F_{ij} are differentiable. At any such point x, every radius t attaining F_{ij}(x) satisfies

Indeed, with this radius held fixed, the differentiable function F_{ij}-u_t is nonnegative everywhere and is zero at x. Its gradient there is zero. This proves the assertion also when there are several attaining radii. The same touching principle underlies the optimizing-radius identity of Hajłasz and Malý [7, Theorem 2 and Remark 3]; here it applies directly to the smooth signed averages. No selected radius is differentiated.

To prove Proposition 1.2, it suffices first to bound the normalized band by an absolute constant independent of m:

6 Sobolev input and absolute continuity

The signed estimate now gives the endpoint theorem in two steps. First, approximation passes it to Sobolev inputs and yields a finite distributional derivative measure for the maximal function. Second, local mean subtraction shows that this measure gives no mass to a Lebesgue-null set. The second step is needed to obtain a weak gradient in L^1, and it is the reason the finite-band estimate was proved for signed inputs.

For real g\in W^{1,1}(\R^2) and 0<a\le b<\infty, continue to write

The disk integrals define these functions independently of the representative of g. The positive lower radius makes the supremum finite at every x, since |A_tg(x)|\le\|g\|_1/(\pi a^2). The elementary analytic facts used below, including the exact Euclidean variation formula, are proved in Appendix A.

This conclusion does not assert global integrability of Mf or absolute continuity of its derivative measure for arbitrary BV input.

A Analytic facts used in the limiting argument

We record the approximation, localization, and measure arguments used in Section 6. All gradient norms are Euclidean. The Sobolev space is inhomogeneous: the function and its distributional first derivatives belong to L^1(\R^2). General Sobolev calculus is treated in [5]; the exact chain, product, and Lipschitz facts used here also appear in [9, Chapter 3, Propositions 3.21–3.22 and Theorems 3.38–3.39]. We give the needed calculations, including the variation formula that preserves the Euclidean norm.

A.1 Approximation, translations, and equality sets

A.2 Square localization

A.3 Vector variation and local lower semicontinuity

For u\in L^1_{\mathrm{loc}}(\R^2) and open O\subset\R^2, define its distributional variation on O by

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.