SINGULAR PARABOLIC EQUATIONS, MEASURES SATISFYING DENSITY CONDITIONS, AND GRADIENT INTEGRABILITY

Transcription

1 SIGULAR PARABOLIC EUATIOS, MEASURES SATISFYIG DESITY CODITIOS, AD GRADIET ITEGRABILITY PAOLO BAROI ABSTRACT. We consider solutions to singular arabolic equations with measurable deendence on the (x, t) variables and having on the right-hand side a measure satisfying a density condition. We rove that the less the measure is concentrated, the more the gradient is regular, in the Marcinkiewicz scale. We rovide local estimates and recover some classic results. To icola Fusco, on the occasion of his 60th birthday, with admiration and resect.. ITRODUCTIO, ASSUMPTIOS, STATEMETS The aim of this aer is to give a natural integrability result for solutions to singular arabolic equations with measure data: we consider roblems of the tye u t div a(x, t, Du) = µ in Ω T := Ω ( T, 0), (.) where Ω is a bounded oen set in R n, n 2, and the vector field a( ) satisfies only minimal growth and monotonicity assumtions: a(x, t, ξ ) a(x, t, ξ 2 ), ξ ξ 2 ν ( ξ + ξ 2 ) 2 ξ ξ 2 2, (.2) a(x, t, ξ) L ξ, for almost every (x, t) Ω T, every ξ, ξ 2, ξ R n, with 0 < ν L. The most rominent model we have in mind for (.) is the singular arabolic -Lalacian equation with measurable coefficients. Being := n + 2 the arabolic dimension, the exonent is assumed to satisfy 2 < < 2; (.3) we are hence considering singular arabolic equations, following DiBenedetto 7, Chaters IV, VII, VIII]. In this note µ will be a signed Borel measure with finite total mass, in general not belonging to the dual of the energy sace naturally associated to the oerator on the left-hand side: in view of this fact the lower bound in (.3) is natural in the whole theory (see 2, 3, 22, 25]) since it ensures the existence of a solution with Du L (Ω T ). The henomenon object of this investigation is the imrovement of integrability for the gradient of solutions of (.) in the case the measure on the right-hand side satisfies certain density-tye conditions. This is a general fact and it was first noted in 30] in the ellitic case, see the forthcoming lines for a detailed resentation of these results. The analog results in the arabolic direction can be found in 7, 6, 4]; the first two contributions include results in the non-degenerate case (that is, for vector fields satisfying (.2) with = 2) while the last contribution deals with the more difficult degenerate arabolic case Date: October 25, Mathematics Subject Classification. 35R06; 35K65, 35B65. Key words and hrases. Calderón-Zygmund estimates, measure data roblems, singular arabolic equations, Marcinkiewicz saces.

2 2 PAOLO BAROI ( > 2), which needs very different techniques with resect to both the ellitic and the non-degenerate cases. To be more secific, for a signed Borel measure µ of finite total mass, one is lead to consider a Morrey-tye condition on standard arabolic cylinders as follows: µ ( R (z 0 )) su R (z 0) Ω T R c d <, (.4) where R (z 0 ) = B R (x 0 ) (t 0 R 2, t 0 + R 2 ): note that R = c(n)r. This condition can be naturally extended to L functions by setting µ (A) := µ dx when A is a A measurable set. The imroved integrability of the gradient is formulated in terms of Marcinkiewicz saces, which are the otimal ones to be used when dealing with measure data roblems (this is natural in view of the behavior of the fundamental solution in the ellitic case), see for instance 8, 5, 30, 3]: summarizing the results of 7, 4], we have that µ satisfies (.4) for some 2 = Du M + loc (Ω T, R n ), (.5) where u is a solution to (.)-(.2) (in the sense described after Definition.2) with 2. We recall the reader that Marcinkiewicz saces M m (Ω T, R l ) are defined via the following decay condition on level sets (for f : Ω T R l a measurable ma): su m {z ΩT : f(z) > } =: f m M m (Ω T,R l ) <. >0 Their local variant is defined in the usual way. The imrovement of integrability in the case of Morrey data for the ellitic case has been resented in 30]: µ L, ( ) (Ω), n = Du Mloc (Ω, R n ). L, (Ω) is here the sace of signed Borel measures satisfying µ (B R (x 0 )) su B R (x 0) Ω R n c d <, (.6) ote that for < a classic result of Hedberg and Wolff states that L, embeds into the dual sace of W,. In the case = n, the results in 30] give back the classic, shar result µ L,n (Ω) M b (Ω) = Du M n( ) n (Ω, R n ), for which we refer to 8, 5, 8]; notice that the class of measures satisfying (.4) for = n is nothing else than the full sace of signed Borel measures with finite total mass M b (Ω T ). In this aer we rove the following singular counterart of the result described in (.5): Theorem.. If u V 2, (Ω T ) is a weak solution to (.) with µ L (Ω T ) satisfying (.4) for some, then: if n, then Du M m loc (Ω T, R n ) with m := ( ) ; if n <, then Du M m2 loc (Ω T, R n ) with m 2 := ( (2 )n ) 2.

3 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 3 Moreover, there exists a constant deending on n,, ν, L and c d such that the following local estimate holds for any arabolic cylinder 2R 2R (z 0 ) Ω T : ] R m Du M m ( R,R n ) c µ (2R ) ] m d ( Du + ) dz + ; (.7) 2R 2R the scaling deficit d > is defined as d := 2 2 n(2 ) (.8) and ( 2 ( m := + n )] ) 2 +. (.9) ote that m = m if n and m = m 2 in the remaining case > n. Several remarks are now in order. First of all, note that V 2, (Ω T ) is the energy sace for an oerator satisfying (.2) with arabolic -growth, see (2.5) and (2.7). As in Theorem., we are going to state every result as a riori estimate for energy solution in view of (art of) the existence theory for roblems as (.). Moreover we assume that µ L (Ω T ) for technical reasons; we will show that this is not restrictive. In the general case of true measure data roblems, one has the following definition: Definition.2. A very weak solution to (.), under the structural assumtions (.2)-(.3) and being µ a signed Borel measure, is a function u V, (Ω T ) (see the forthcoming (2.5)) such that the distributional formulation ] u ϕ t + a(x, t, Du), Dϕ dz = ϕ dµ, (.0) Ω T Ω T holds true for every ϕ C c (Ω T ). Since the seminal works of Boccardo and Gallouet 2, 3], a solution to (.) (couled with Cauchy-Dirichlet data) is usually found using an aroximation rocedure: one regularizes the datum µ (usually via mollification) and considers the energy solutions to the regularized roblems (note that the Morrey condition in (.4) remains essentially reserved under mollification, see 33]). Finally, one roves a.e. convergence of the gradients u to sub-sequences, and this allows to ass to the limit both in the weak formulation, obtaining a solution of (.) (see Definition.2), and in the estimates. This strategy leads to a articular tye of solution usually called Solution Obtained by Limits of Aroximations, SOLA in short. This is the reason why we can state our result in the form of a riori estimates; needless to say, all our estimates will still hold (in a slightly different form) for SOLAs in the case µ is a signed Borel measure satisfying (.4), see 24, Paragrah.4] or 27, Section.2] for the necessary adatations. In the general case, where µ is a signed Borel measure with finite mass, the archetyal result is the existence of a SOLA, distributional solution (in the sense of Definition.2) to (.), for > 2 /( ), such that Du L q (Ω T, R n ) for all q, m 0 ), m 0 := +. (.) ote that in general the Cauchy-Dirichlet roblem associated (.) does not have uniqueness; hence the SOLA is just one solution to the roblem considered. Uniqueness holds only in secial cases, as for instance = 2 or = n in the ellitic case; for a comrehensive account of several results in this field, we refer to 32]. Moreover, also in the class of SOLA uniqueness is not guaranteed, in the sense that different aroximation of µ could lead in general to different limit solutions. We will not go into details in those

4 4 PAOLO BAROI questions and we, again, refer to 20, 26, 32] for the ellitic case and to 4, 24, 25, 27] for the arabolic. ote that if d =, then the estimate in (.7) would be homogeneous and invariant under arabolic scaling. The fact that d > if 2 is a well-known henomenon that follows directly from the lack of natural homogeneous scalings for the equation. The exlicit value in (.8) is moreover characteristic when dealing with singular arabolic equations with measure data; for instance, observe that it also aears in 22, 25]. It is indeed naturally dictated by the lack of scaling of the equation together with the fact that we are forced to work below the energy level, and in articular with the L norm of the gradient, since we are considering the singular case; see (5.6). This fact is in turn symtomatic of the fact that we need to roduce estimates stable when assing to the limit, in view of what described above. We remark here that m m 2 if n while the inequality is reversed for > n, since here < 2. Moreover max{m, m 2 } m 0 being m 0 defined in (.), since < 2 and ; therefore the regularity of the gradient in the singular case is worse if comared with the one in the degenerate one. It is not clear if this fact is a consequence of the singular structure of the equation or it is a limit of the technique of our roof; this will be object of future investigations. One can on the other hand force the regularity Du M m0 loc (Ω T, R n ) in the singular case by artificially imosing an intrinsic condition of the tye su (z0) Ω T R>0, µ ( (z 0)) R ] c d, where the stretched cylinders (z 0) will be defined in (2.); comare with (.4). This condition is on the other hand difficult to verify in concrete cases and therefore we leave this remark just as a theoretical curiosity. ote that once one takes =, (.4) is satisfied for every signed Borel measure with finite mass. Thus this condition is essentially emty but we still get back (locally) the shar imrovement of the result in (.), which was already obtained in 3]; we also rovide a natural local estimate. Corollary.3. If u V 2, (Ω T ) is a weak solution to equation (.) with µ L (Ω T ), then Du M m0 loc (Ω T, R n ). The local estimate coincides with (.7) for = and m = m 0. Remark.4. We stress now that we restrict to the case for simlicity. One could consider also < but still close to (see (5.29): m should only be smaller than ( + η), with η > 0 deending on the data of the roblem); we refer to the results and the discussion in 4]. ote that as soon as <, then Du L loc (Ω T ). The fact that the natural lower bound from below for differs from that aearing for degenerate equations is liked to the fact that the energy sace in the singular case differs from that in the degenerate one, see 28, Page 66]. This will be object of future investigation. An imrovement of Theorem. can be obtained once considering more regular vector fields. In articular, if we relace the assumtion of simle measurability and boundedness with resect to the (x, t) variables with a sort nonlinear version of V MO regularity (considered for instance in, 0, 23]), then we can consider values of close to one and at the same time obtain integrability of Du in any Lebesgue sace. In articular, we consider now Carathéodory vector fields a(x, t, ξ) differentiable with resect to the variable ξ and

5 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 5 satisfying ξ a(x, t, ξ) ξ, ξ ν ξ 2 ξ 2, a(x, t, ξ) + ξ a(x, t, ξ) ξ L ξ, (.2) for all x Ω, t ( T, 0), ξ, ξ R n, with as in (.3) and structural constants 0 < ν L <. For what concerns the regularity with resect to the variable x, we assume that, denoting for R > 0 the nonlinear modulus of continuity we have ω(r) = su t ( T,0) su B r(x 0) Ω 0<r R a(x, t, ξ) (a) Br(x su 0)(t, ξ) ξ R n ξ, ξ 0 lim ω(r) = 0. (.3) R 0 (a) Br(x 0) : ( T, 0) R n R n denotes the averaged vector field (a) Br(x 0)(t, ξ) := a(x, t, ξ) dx B r(x 0) for t ( T, 0) and ξ R n. ote that a articular but fundamental instance of vector fields satisfying the assumtions in (.2)-(.3) are those V M O regular with resect to the sace variables but only measurable and bounded with resect to time: this is to say, those of the form a(x, t, ξ) = b(x)ã(t, ξ) with ã( ) a Carathéodory ma, with (t, ξ) ξ ã(t, ξ) Carathéodory regular too and satisfying ξ ã(t, ξ) ξ, ξ ν ξ 2 ξ 2, a(t, ξ) + ξ a(t, ξ) ξ (.4) L ξ, for all t ( T, 0), ξ, ξ R n, as in (.3) and with ν, L as in (.2) and b : Ω R bounded from zero and infinity and V MO regular, i.e. ν b( ) L and lim ω(r) = 0, where ω(r) := su b (b)br(x 0) dx. R 0 B r(x 0) Ω B r(x 0) 0<r R In this case we have the following Theorem.5. If u V 2, 0 (Ω T ) is a weak solution to equation (.) under the assumtions in (.2)-(.3), with µ L (Ω T ) satisfying the Morrey condition (.4) for some < n, then Du M m loc (Ω T, R n ). Slitting measures. We conclude this introduction with two results about slitting data. For the first one we assume that µ is of the following roduct-tye: µ = µ µ 2, µ L (Ω), µ 2 ( (τ R 2, τ + R 2 ) ) c d R 2 (.5) for every sub-interval (τ R 2, τ + R 2 ) ( T, 0) and for some < < 2. In this case we can deduce a stronger result: Theorem.6 (Ellitic/singular-arabolic regularity). If u V 2, (Ω T ) is a weak solution to equation (.) under the assumtions (.2)-(.3) with satisfying (.3) and µ being as in (.5), then Du M m3 loc (Ω T, R n ), where m 3 := 2.

6 6 PAOLO BAROI Again, for any 2R (z 0 ) Ω T a local estimate analogous to (.7), with m 3 relacing m, holds true. The constant here does also deends on the Morrey constant c d of µ 2 and on µ L. Finally, for comleteness, we resent the analog of Theorem.6 for degenerate equations, which was not included in 4]. We assume in this case µ = µ 3 µ 4, µ 3 L L, (Ω), < < n; µ 4 L ( T, 0), (.6) with L, (Ω) the ellitic Morrey sace defined in (.6). Theorem.7 (Ellitic/degenerate-arabolic regularity). If u V 2, (Ω T ) is a weak solution to equation (.) with 2 < n in (.2) and µ being as in (.6), then Du M m4 loc (Ω T, R n ), where m 4 := ( ) = m. A local estimate similar to that in Theorem.6 holds true for any 2R (z 0 ) Ω T, with the constant also deending on the Morrey constant of µ 3 and on µ 4 L. ote here in both cases µ satisfies (.5) or (.6) in articular we have that µ satisfies (.4); however, m 3 m ( < 2) and m 4 m 0 ( 2); thus in both cases a more careful analysis of the geometry of the roblem allows to obtain an imroved integrability for the gradient of the solution. Moreover note that in order to be able to consider nontrivial cases, in Theorem.6 we chose to consider more regular vector fields as in (.2), to allow for < 2; however also in this case Remark.4 alies. Finally, we stress that assuming (.5) in the degenerate case and (.6) in the singular one does not lead to any better estimates using our aroach, and this can be seen by our roofs. 2. OTATIO This section is devoted to fix the notation we will use in the rest of the aer. R n+ will always be thought as R n R, so a oint z R n+ will be often also denoted as (x, t), z 0 as (x 0, t 0 ), and so on. The cylinders (z 0), for a scaling arameter, are the natural cylinders associated to the equation in (.): they are defined as with (z 0 ) := B R (x 0 ) (t 0 R 2, t 0 + R 2 ) R := 2 2 R. = B R (x 0 ) ( t 0 2 R ] 2, t R ] 2) (2.) R is therefore the satial radius of ; observe that R (z 0) = c(n) 2 R ]. We recall the reader the different definition of the cylinders (z 0) for in the degenerate 2 case: we set (z 0 ) := B R (x 0 ) ( t 0 2 R 2, t R 2) (2.2) and here we have (z 0) = c(n) 2 R. ote that (in both cases), since we are always going to consider scaling arameters, then (z 0 ) R (z 0 ). (2.3) Moreover note that for fixed, scaled cylinders are the balls of the metric given by the distance d (z, z 2 ) := max { 2 2 x x 2, } t t 2. (2.4)

7 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 7 When dealing with a several cylinders or Euclidean balls in the same context, if not otherwise stated, they will all have the same vertex. For α > 0 we shall write α(x 0, t 0 ) := B αr (x 0 ) (t 0 (αr) 2, t 0 + (αr) 2 ). By arabolic boundary of a cylindrical set K := A I, with A Ω, I ( T, 0), we will mean P K := A {inf I} A I. Being C R m a measurable set with ositive measure and f : C R k an integrable ma, with m, k, we denote with (f) C the averaged integral (f) C := f(ξ) dξ := f(ξ) dξ. C C C Moreover for a measurable function g : K = A I R n R R k over a cylinder we will use the notation (g) A (t) := g(x, t) dx for all t I. A c will denote a generic constant greater than one, ossibly varying from line to line. Constants we need to recall will be denoted with secial symbols, such as c, c 2, c, c. Relevant deendencies will be highlighted between arentheses or after the equations; when non essential, the deendence on a arameter will be suressed. If for a constant we will not mention its deendencies, then it will be a numerical constant. and For a cylinder K := A I R n R, with V γ,r (K), γ, r, we denote the saces V γ,r (K) := L r (I; W,r (A)) C(I; L γ (A)). (2.5) V γ,r 0 (K) := L r (I; W,r 0 (A)) C(I; L γ (A)). ote that if g L (I, W, (A)),, then x g(x, t) W, (A) for a.e. t I. Hence we will be allowed to use Poincaré s inequality slice-wise. Finally, we introduce the auxiliary vector field V : R n R n by V (ξ) := ξ 2 2 ξ whenever ξ R n ; it turns out to be a bijection of R n and to encode the monotonicity roerties of the vector field in (.2): indeed it holds that c V (ξ ) V (ξ 2 ) 2 ( ξ + ξ 2 ) 2 ξ ξ 2 c 2 for c c() and for all ξ, ξ 2 R n not both zero if s = 0 (see, Lemma 2.3]); thus a(x, t, ξ ) a(x, t, ξ 2 ), ξ ξ 2 V (ξ ) V (ξ 2 ) 2. (2.6) c(ν, ) Moreover it holds a(x, t, ξ), ξ c ξ. (2.7) 3. ESTIMATES FOR HOMOGEEOUS PROBLEMS Let us consider v C 0 (I; L 2 (A)) L (I; W, (A)) a solution to the roblem v t div a(x, t, Dv) = 0 in A I Ω T, (3.) being A a domain, I an interval and with a : Ω T R n R the vector field aearing in (.), therefore satisfying (.2)-(.3) and (2.7). In this section we collect some regularity results for weak solutions to (3.). The next Lemma is the standard energy estimate for solutions to (3.); for its roof see 7, Chater II, Proosition 3.] or 2, Lemma 3.2].

8 8 PAOLO BAROI Lemma 3. (Cacciooli s inequality). Let v V 2, (A I) be a weak solution to (3.) and let ρ2,σ 2 ρ2,σ 2 (z 0 ) A I be a cylinder. Then su v(, t) k 2 dx + Dv dz t (t 0 σ,t 0+σ ) B ρ (x o) ρ,σ ] v k v k 2 c + dz (3.2) ρ2,σ 2 (ρ 2 ρ ) σ 2 σ holds for all concentric cylinders ρ,σ ρ,σ (z 0 ) ρ2,σ 2 (z 0 ) and for all k R; the constant deends only on n,, ν and L. The following is the su bound for solutions to singular arabolic equations. It can be found in 34]; the local estimate we use here is in 7, Chater V, Theorem 5.]. ote in articular that we have by arabolic Sobolev s embedding and our assumtion (.3) that w V 2, ( ) imlies w L ( ); moreover (.3) ensures that n( 2) + > 0. Recall that a sub-solution is a function such that the left-hand side of the weak formulation of (3.) is non-ositive, for every ositive test function. Proosition 3.2. Any ositive sub-solution w V 2, (A I) to (3.) in A I is locally bounded. Moreover, there exists a constant c deending only on n,, ν, L such that the quantitative estimate su w c 3R/4 ( (R 2 ) n 2 holds for every cylinder R (z 0) A I. ) n( 2)+ w dz + c R 2 ote that the lack of time regularity of solutions to (3.) denies the ossibility of using a standard Poincaré s inequality on cylinders. However, an amount of regularity can be retrieved by the equation itself; this allows to rove the following estimate. Similar estimates can be found in 4, 5, 35] and many other aers; for the roof, see the forthcoming Lemma 4.2 for µ 0. Lemma 3.3. Let v V 2, (A I) be a solution to (3.) in A I and let R (z 0) A I be a arabolic cylinder as in (2.). Then v (v) ( ) R R dz c Dv dz + c 2 Dv dz for a constant c deending only on n, and L. We now have the following simle corollary: Corollary 3.4. Let v be as in Lemma 3.3 and moreover suose that the intrinsic relation Dv dz κ holds for a constant κ. Then we have v (v) R R dz c(n,, L, κ). At a certain oint of the roof we will need to comare the intrinsic geometry for our solution u and the one for a certain comarison ma, solution to a homogeneous equation as the one in (3.). For the first one, the intrinsic relation will involve the L norm of the gradient, since we need estimates stables for very weak solutions. For the second one, the natural geometry will involve the L norm of the gradient instead. The following Proosition will show that the weak geometry for Dv is equivalent to the standard one (note indeed that the equivalent imlication for the bound from below is trivial in view of Hölder s inequality).

9 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 9 Proosition 3.5 (Intrinsic reverse Hölder s inequality). Let v be the weak solution to (3.). If Dv dz κ (3.3) holds for a constant κ, then Dv dz c, /2 with c deending on n,, ν, L and κ. Proof. From Cacciooli s inequality (3.2) we have ( v (v) Dv v (v) 3R/4 dz c + R R/2 3R/4 v (v) 3R/4 = c dz 3R/4 c R ] osc 3R/4 R + 2 v 3R/4 ] + 2 R ] 2 v (v) osc 3R/4 3R/4 2) ] dz 2 3R/4 R ] ] 2 v, where c c(n,, ν, L). To estimate the oscillation we now use Proosition 3.2: indeed both (v (v) 3R/4 ) + and (v (v) 3R/4 ) are ositive sub-solutions to (3.) and so osc v c 3R/4 = c c ( (R ) n 2 ( 2) ( (R ) n 2 ( 2)+ ( (R ) n 2 ) ( 2)+ n( 2)+ ) v (v) dz v (v) R R + R 2 n( 2)+ ) dz ] = c R 2 ] dz ] + R 2 n( 2)+ ] + R 2 with c c(n,, ν, L); in the second-last line we used Poincaré s inequality in its intrinsic form of Corollary 3.4 (note indeed that we are assuming (3.3) 2 ). We thus conclude with ] Dv dz c R ] R 2 ] ] R ] 2 R 2 c. /2 ext, a reverse-hölder s inequality for solutions to (3.). We stress that the imortant oint in (3.5) is not the fact that the gradient is highly integrable - this fact has been roven, in different forms, in several aers, as 9, 2, 35] just to mention only the ones including the singular case - but the recise form of the estimate. Proosition 3.6. Let v L (I; W, (A)) be a weak solution to (3.) and let (z 0) A I be a cylinder such that ( Dv κ) dz and Dv dz (κ) (3.4) /2

10 0 PAOLO BAROI hold for a constant κ. Then there exist an exonent η η(n,, ν, L) > 0 such that for any σ > 0 it holds ( ) ( ) Dv (+η) (+η) dz c Dv σ σ dz, (3.5) /2 with the constant c deending only on n,, ν, L, κ and σ. Proof. We will be a bit sloy in the following roof, in articular in the covering art, since the whole argument is quite standard: see its minor variants, for instance, in 2, 4, 5] and moreover the forthcoming Section 5. We fix two intermediate radii R < R 2, with R, R 2 R/2, R], and two further ones R r < r 2 R 2 and we fix ( ) µ d 0 := ( d) d Dv dz, B 0R2 d := r R 2 r 2 with the scaling deficit at the energy scale defined as (see 2, 5, 9, 35]) d = 2 (n + 2) 2n. We observe that we can build, for µ > Bµ 0 fixed, a covering of E( r, µ) = r { Dv(z) > µ, z is a Lebesgue s oint of Dv } consisting in a family of cylinders µ ( z), z E( ρ z r, µ) so that ρ z (r 2 r )/0. Indeed, defining CZ ( µ ρ(z) ) := Dv dz, µ ρ (z) if ρ ((r 2 r )/0, (r 2 r )/2) and µ > Bµ 0 we estimate enlarging the domain of integration CZ ( µ ρ(z) ) R 2 µ ρ(z) (d ) ( ) R2 (µ < ρ ( 0R2 ) ( µ r 2 r µ. d µ d 0 ) 2 2 n (d ) d µ d B d ) 2 2 n 2 2 n µ n+2 n ( ) n 0R2 r 2 r On the other hand, if z E( r, µ), then Dv( z) > µ and by Lebesgue s differentiation Theorem we have that CZ( µ ϱ( z)) > µ for small radii 0 < ϱ ; thus, by absolute continuity we find a critical, maximal radius ϱ z (r 2 r )/0 such that CZ( µ ( z)) = ϱ z µ ; moreover, by maximality we have µ c(n) µ αϱ z ( z) Dv dz µ for α, 0]. Thus we are in osition to use the reverse Hölder s inequality of 9, Lemma 3] to infer ( ) / Dv dz c Dv dz µ ϱ z ( z) µ 2ϱ z ( z) with = 2n/(n+2) < and a constant deending only on n,, ν, L and not on the energy of Du. ote that the cylinders in 9] differ from ours in the singular case < 2 but with

11 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA a simle change of variable (ϱ = ϱ z ( 2)/2 = ϱ z, where ϱ is aearing in 9]) we can recover our situation. This imlies, calling the constant aearing in the dislay above c, ( ) / µ Dv dz µ ϱ z ( z) c µ Dv dz ( z) 2ϱ z µ 2ϱ z ( z) { Dv µ/2 c]} ] + Dv dz µ 2ϱ z ( ( z) { Dv >µ/2 c]} µ ) c + 2 µ Dv dz; ( z) 2ϱ z µ 2ϱ z ( z) { Dv >µ/2 c]} thus µ c µ ϱ z ( z) µ 2ϱ z ( z) { Dv >µ/2 c]} Dv dz and as consequence, calling ς ς(n,, ν, L) := /2 c] Dv 5µϱ z dz µ c µ ( z) µ ϱ z ( z) E( µ 2ϱ z ( z),ςµ) Dv dz (3.6) for a constant c deending on n,, ν, L. ow we extract using Vitali s Lemma (recall that the cylinders µ ϱ z ( z) with fixed are the balls of the metric in (2.4)) a sub-collection {2 i } i I of { µ 2ϱ z ( z)} z E( r,µ), such that the 5-times enlarged cylinders 0 i cover almost all E( r, µ) and the cylinders are airwise disjoints. ote that 0 i r 2. We finally have, since the cylinders 2 i are disjoint, using (3.6) Dv dz Dv dz E( r,µ) i I 0 i c µ Dv dz i I E(2 i,ςµ) c µ Dv dz E( r,ςµ) 2 Finally, as in the usual roof of higher integrability estimates via Fubini s theorem, we have for η small enough, using the estimate above and calling µ := Bµ 0 Dv (+η) dz = η µ η Dv dz dµ r 0 E( r,µ) µ µ η Dv dz + c η µ η+ Dv dz dµ r µ E( r,ςµ) µ 2 µ η Dv dz + c η Dv (+η) dz; r r 2 note that we changed variable µ = ςµ, recalling that ς ς(n,, ν, L), to erform the estimate in the last line. ow we choose η small so that cη /2 and this yields, recalling the definition of µ Dv (+η) dz r 2 r Dv (+η) dz 2 ( ) dη R2 + c µ η 0 Dv dz. r 2 r r 2

12 2 PAOLO BAROI ow the standard iteration lemma 9, Lemma 6.] allows to reabsorb the L (+η) norm on the right-hand side and to deduce ( ) dη Dv (+η) R2 dz c Dv dz R 2 R R 2 η dη R2 Dv dz] dη. ote that the re-absortions does not cause roblems, since all the quantity in lay are finite; the gradient higher integrability is a well-established fact and what we are interested in are recise local estimates. At this oint notice that (3.4) imlies c(n,, κ) Dv dz and 2 Dv dz c(n,, κ) since R/2 R < R 2 R, and this imlies in turn ( ) dη ( Dv (+η) dz R c R Dv dz ) +η R 2 R 2 R. Recall that the revious inequality holds for every R, R 2 R/2, R] with R < R 2. We are now in osition to recisely aly 22, Lemma 5.] with dµ = dz/, which encodes the usual self-imroving roerty of reverse-hölder inequalities in a form fitting our context, to infer (3.5). 4. AUXILIARY RESULTS In this section we aroach the roof of Theorem., collecting some results: a comarison Lemma, a Poincaré-like inequality and a local energy estimate for measure data roblems. First of all, from now on we will choose as the set A I a cylinder (z 0) Ω T and we will introduce therein the comarison function solution to the Cauchy-Dirichlet roblem {vt div a(x, t, Dv) = 0 in R, v = u on, (4.) where u is a solution to (.). Recall we are dealing with aroximating, regular solutions u V 2, (Ω T ); therefore existence and uniqueness of v are well known arguments (see 7]) and so it is the fact that v u + V 2, 0 ( ). The following comarison result is a generalization of 25, Lemma 4.]. Lemma 4.. Let u be a weak solution to (.) and let v be the comarison function defined in (4.). Then ( ) ] Du Dv q q µ ( ( )( )+ dz c R ) + c µ ( R ) ( for every q, + ) and for a constant c c(n,, ν, q). Du q dz ) 2 q (4.2) (4.3)

13 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 3 Proof. For q as in the statement, we start from 24, Dislay (4.)], that reads as ( α := ) 2 u v q q( ) 2 dz c(n, q) µ () ] ( ) 2 Du Dv q dz q( ), (4.4) which actually holds in the full range > 2 /( ): indeed, the only ingredients needed to rove the estimate are the equations solved by both u and v, together with the arabolic Sobolev s embedding (taking into account that u v has trace zero on the lateral boundary). ote that we can assume without loss of generality α > 0; otherwise (4.2) would follow trivially. This imlies, see again 24, Proof of Lemma 4., Ste 2] or 25, Proof of Lemma 4.] 2q V (Du) V (Dv) dz µ ( c(n,, q) R ) α µ ( c R )] ( ] q ) 2 ] q Du Dv q q( ) dz ; (4.5) again the only things needed to deduce such an estimate are the weak formulations of the equations for u and v and the usual monotonicity estimate in (2.6). To conclude the roof, we ointwise bound ( Du ) ] 2 Du Du Dv = + Dv Dv 2 2 ( ) 2 Du + Dv 2 c(n, ) ( ) 2 V (Du) V (Dv) Du + Dv 2 c ( ) 2 V (Du) V (Dv) Du Dv + Du 2 c 2 V (Du) V (Dv) + Du Dv 2 + V (Du) V (Dv) Du 2 2, using also Young s inequality with conjugate exonents (2/, 2/(2 )). Thus, reabsorbing, taking the q-ower and averaging over Du Dv q 2q dz c(n,, q) V (Du) V (Dv) dz + 2q c(n,, q) V (Du) V (Dv) dz ( 2q + V (Du) V (Dv) dx = c I + II ]. If now I II, then using (4.5) Du Dv q dz c V (Du) V (Dv) q Du q 2 2 dz ) 2 ( R V (Du) V (Dv) 2q dz ] ) 2 ] Du q 2 dz

14 4 PAOLO BAROI and so µ ( c R )] µ ( Du Dv q dz c R )] on the other hand, if II I, then and thus µ ( Du Dv q dz c R )] µ ( Du Dv q dz c R )] ( ) 2 ] q Du Dv q q( ) dz ; ] q( ) ( )( )+ ; ( ) 2 ] q Du Dv q q( ) 2 dz ( ) 2 Du q 2 dz ] q ( ) (2 ) Du q dz. R ow it is the time for a Poincaré-tye inequality valid for solutions to (.). Lemma 4.2. Let u V 2, (Ω T ) be a solution to (.). Then, for every q, ], u (u) R R q dz c Du q dz ( + c 2 where the constant c deends only on n, and L. ) Du dz + c R2 µ ( ) ] q R, (4.6) Proof. We slit the integral in (4.6) as follows: u (u) R dz u (u) η B R(t) R dz R + t0+r 2 t0+r 2 R (u)η (u) B t 0 R R(t) η (τ) dτ B dt 2 t 0 R 2 R + t0+r 2 (u) η (τ) dτ (u) BR R = I + II + III. We used the notation (u) η := BR(t) η C c B R u(, t)η dx, R t 0 R 2 (BR ) being a ositive weight function satisfying η dx =, η(x) + R Dη(x) c(n). B R Using slice-wise a variant of Poincaré s inequality (see 29, Corollary.64]) we infer III I c(n) Du dz.

15 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 5 To estimate II we use the equation solved by u: testing the weak formulation with the test function η indeendent of time, we get for a.e. t, t 2 (t 0 R 2, t 0 + R 2 ) (u) η (t BR ) (u) η BR(t 2 ) t2 = t (u) η (t) ] t2 dt BR = t u η dx dt t c(n, L) R = c R2 R c R2 R t0+r 2 ( t 0 R 2 B R t B R Du dz + c(n)r ] n µ () Du dz + cr ] n µ () ) Du dz + c R 2 µ ( R ) ; we used the growth conditions in (.2) 2 and Hölder s inequality. ote that the revious estimate is just formal: a recise roof can be done using a regularizing in time rocedure (recall that we assume µ L (Ω T )); see for instance 9, Lemma 7]. ow a reverse Hölder s inequality for measure data roblems. The use of such inequalities is customary when dealing with regularity and, in articular, Calderón-Zygmund estimates for degenerate and non-uniform ellitic roblems, see for instance the use in 2, 4, 5, 0,, 6, 30]. Proosition 4.3. Let u be a weak solution to (.) under the assumtions (.2) with (.3) in force and with µ L (Ω T ); suose moreover that κ Du dz and Du dz κ (4.7) 2R hold in some cylinder R (z 0) such that (z 0) Ω T, for some constant κ. Then for every q as in (4.3), there exists a constant c c(n,, ν, q, κ) such that ( ) Du q q dz c Du dz + c 2R µ ( 2R ) 2R ] ( )( )+. Proof. Many of the forthcoming estimates are true for any cylinder = K I Ω T ; only at a certain oint we will secify the estimates to the situation we are considering in the statement of the Proosition. We test the weak formulation in (.0) with ϕ := ± min{, u ± /ɛ}ζ, for ɛ > 0 and with ζ C c (), ζ L. ote that this is ossible only at a formal level, due to the lack of time regularity of u; this choice can be anyway made rigorous by using Steklov averages - we will roceed formally. Following 24, Lemma 4.], we comute Dϕ = ɛ Du χ {0<u ±<ɛ}ζ ± min{, u ± /ɛ}dζ and thus we have (notice that ϕ L and recall (.2)) u t ϕ dz + a(x, t, Du), Du χ {0<u±<ɛ}ζ dz ɛ L Dζ L Du dz + µ ().

16 6 PAOLO BAROI We have, integrating twice by arts with resect to the time variable u t ϕ dz = ± t u min{, u ± /ɛ}ζ dz = = u± t min{, σ/ɛ} dσ ζ dz 0 u± 0 min{, σ/ɛ} dσ t ζ dz (see 24, (4.8)]). This leads to u± min{, σ/ɛ} dσ t ζ dz + a(x, t, Du), Du χ {0<u±<ɛ}ζ dz 0 ɛ c Dζ L Du dz + µ (); we note that by dominated convergence Theorem u± min{, σ/ɛ} dσ t ζ dz 0 as ɛ 0 and thus u t ζ dz + su ɛ>0 ɛ ow we choose u ± t ζ dz a(x, t, Du), Du χ {0<u±<ɛ}ζ dz c Dζ L Du dz + µ (). ζ(x, t) := ζ (x, t)ζ 2,ε,τ (t) (4.8) with ζ Cc () such that ζ and, for any τ I being fixed and ε small enough so that also τ + ε I, ζ 2,ε,τ is continuous and iecewise linear, with ζ 2,ε,τ on (, τ) and ζ 2,ε,τ 0 on (τ + ε, ). ote that R tζ 2,ε,τ dt = for any τ and ε and t ζ 2,ε,τ 0. With this choice we have u ( t (ζ ζ 2,ε,τ ) ) dz u (, τ) dx + u ( t ζ )χ R n (,τ) dz K as ε 0 for τ I fixed, so su u (, τ) dx + su a(x, t, Du), Du χ {0<u±<ɛ}ζ dz τ I B ɛ>0 ɛ t ζ L u dz + c Dζ L Du dz + c µ () := c E. (4.9) ow we fix two constants α > 0 and ξ > and we use in (.0) the test function ϕ 2 := ϕ (α + u ± ) ξ = ± min{, u ±/ɛ} (α + u ± ) ξ ζ, ɛ > 0, ζ as in (4.8); we use the weak formulation in the form ϕ (ξ ) a(x, t, Du), Du ± (α + u ± ) ξ dz dz = a(x, t, Du), Dϕ (α + u ± ) ξ u t ϕ 2 dz ϕ 2 dµ.

17 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 7 We estimate the terms on the right-hand side: firstly, using (4.9) dz a(x, t, Du), Dϕ (α + u ± ) ξ = a(x, t, Du), Du χ {0<u ±<ɛ}ζ dz ɛ (α + u ± ) ξ ± a(x, t, Du), Dζ min{, u ±/ɛ} dz (α + u ± ) ξ c α ξ E + α ξ Dζ L Du dz, so that dz su a(x, t, Du), Dϕ ɛ>0 (α + u ± ) ξ c α ξ E. Then we simly estimate the last term as follows: ϕ 2 dµ ϕ 2 d µ α ξ µ () c α ξ E. To estimate the arabolic term, we again use integration by arts and we deduce ϕ u± min{, σ/ɛ} u t ϕ 2 dz = t u dz = (α + u ± ) ξ (α + σ) ξ dσ tζ dz; again we choose ζ as above. We estimate the term containing ζ 2,ε,τ : u± min{, σ/ɛ} 0 (α + σ) ξ dσ ζ t ζ 2,ε,τ dz u±(,τ) α ξ su min{, σ/ɛ} dσ dx t ζ 2,ε,τ dt τ I K 0 I α ξ su τ I u ± (, τ) dx; thus, using (4.9), ( su ɛ>0 and in turn su ɛ>0 u± 0 K ) min{, σ/ɛ} (α + σ) ξ dσ ζ t ζ 2,ε,τ dz c α ξ E ( ) u t ϕ 2 dz c α ξ E 0 ( u± ) min{, σ/ɛ} + su ɛ>0 0 (α + σ) ξ dσ ζ 2,ε,τ t ζ dz c α ξ E ( u± ) min{, σ/ɛ} + su ɛ>0 0 (α + σ) ξ dσ ζ 2,ε,τ t ζ dz c α ξ E + α ξ t ζ L u dz c α ξ E. Merging all these estimates we obtain ϕ (ξ ) a(x, t, Du), ±Du ± (α + u ± ) ξ dz c α ξ E,

18 8 PAOLO BAROI that is a(x, t, Du), ±Du ± (α + u ± ) ξ min{, u ± /ɛ}ζ dz c α ξ ξ E. Letting ɛ 0 and erforming some algebraic maniulations yields a(x, t, Du), Du (α + u ) ξ ζ dz c α ξ E ξ. (4.0) ow we secialize the revious estimate: we take = 2R, intermediate radii R R < R 2 2R and ζ satisfying χ R ζ χ R2 in such a way that (R 2 R ) Dζ L + (R 2 R ) 2 t ζ L c; moreover we notice that if u solves (.), so does u (u) 2R ; thus we can aly (4.9) and (4.0) to u (u) R2. Under our assumtions, we have Dζ L and, using (4.6), t ζ L 2R 2R Du dz + µ ( 2R ) 2R c ( ) R R2 R R Du dz + µ ( 2R ) 2R 2R u (u) 2R dz ( R ) 2 2 c R2 R R ( R ) 2 2 c R 2 R R 2R 2R + ( R u (u) 2R 2R Du dz dz ) Du dz + µ ( 2R ) ] 2R. We hence come u, denoting R := R/(R 2 R ) with a(x, t, Du), Du (α + u (u) 2R 2R ) ξ ζ dz (4.) ( c α ξ R ) 2 2 ξ R 2 R R Du dz =: c α ξ ξ R2 Ẽ + R 2R ( 2R ) Du dz + µ ( 2R ) ] 2R and su u (u) 2R (, τ) dx c 2R R 2 Ẽ. (4.2) τ (t 0 R 2,t 0+R 2 ) B2R

19 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 9 Conclusion. ow we fix q, m 0 ) with m 0 defined in (.) and we take ξ = ( q) >, 2 αq 2 = ( ) u (u) ζ q 2 2R dz; 2R note that without loss of generality we can assume α 0, otherwise there would be nothing to rove. We estimate using arabolic Sobolev s embedding (see 7, Chater I, Proosition 3.] or 24, Lemma 4.]) ( α q 2 c(n, q) su τ (t 0 R 2,t 0+R 2 ) B R ) q 2 u (u) 2R (, τ) dx Du q dz + 2 2R c 2R R 2 Ẽ ] q u (u) 2 Du q dz + 2R 2 2R c 2R R 2 Ẽ ] q 2 Du q dz + 2 R Ẽ ] ] q, 2 after using (4.2). To conclude the roof, we comute using (2.7) and (4.) ( ) ( ) Du q q ] q q dz c a(x, t, Du), Du dz R ( ) a(x, t, Du), Du c (α + u (u) 2R ) ξ ζ dz 2R ( c 2R c R 2 α Ẽ c 2R 2 2 ( ( ] u (u) q Dζ 2R q dz ( ) ( ( ) ) qξ q α + u (u) ζ q dz 2R 2R a(x, t, Du), Du (α + u (u) 2R ) ξ ζ dz 2 R Ẽ 2 Du q dz ) ( ) α ξ 2 Du q dz ) 2 q + c R 2 2R 2 R ] q + c R ξ 2R 2 2 Ẽ 2 ( )( )+ Ẽ ( )( )+ 2 + c R 2 2 R ] 2R ( ) 2R 2 2 Du q dz ) q + c R ξ 2R ( )( )+ Ẽ q Ẽ 2 ( )( )+ + c, ] dz (4.3) for some ξ > 0 deending on n and, after using Young s inequality twice with conjugate exonents, resectively, ( ( ) ( ) ), ( )( ) + 2 and (if < 2) ( ( 2)(2 ), ) ; 2 ( )( ) +

20 20 PAOLO BAROI note that this choice is admissible since < < 2. Indeed 2 R ] 2 2R = 2 ( ) ] ( 2)(2 ) 2 R ] ] 2R 2 ( ) = c(n, ) 2 R ] 2 R ] ] 2 ( ) ] = c(n, ) = c(n, ). We conclude the roof first reabsorbing the L q average on the right-hand side using a standard iteration Lemma (see 9, Lemma 6.]) and then observing that by (4.7) we have E = 2 R Du dz + ( ) R Du dz + µ ( 2R ) 2R and thus 2R c(, κ) R + µ ( 2R ) 2R ( )( )+ 2R E ( )( )+ 2R c(n,, κ) 2R c 2 R µ ( c + c 2R ) R R + µ ( 2R ) 2R c Du dz + c + 2R µ ( 2R ) 2R 2R ] ( )( )+ µ ( 2R ) 2R ] ( )( )+ ] ( )( )+. ] ( )( )+ With exactly the same roof (excet for a different use of Young inequality in the last line of (4.3), namely with conjugate exonents (/2, /( 2))) we have the degenerate counterart of Proosition 4.3. ote however that we will not emloy this estimate anywhere in this aer; we include it for comleteness. Proosition 4.4. Let u be a weak solution to (.) under the assumtions (.2), with now 2; assume that the inequalities κ Du dz and Du dz κ hold in a cylinder R (z 0) as defined in (2.2), with (z 0) Ω T, for some κ. For every q as in (4.3), there exists a constant c c(n,, ν, q, κ) such that ( 2R ) Du q q dz c Du dz 2R ] µ ( ( )( )+ + c 2R ) µ ( + c 2R ) 2R (2R).

21 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 2 ote that the two terms taking into account the measure µ have the same dimensional character: µ ( 2R ) 2R ] ( )( )+ µ ( 2R ) R. ow, using the result in Proosition 4.3 we can deduce an imroved version of Lemma 4.: Corollary 4.5. Let u be a weak solution to (.); suose that 2R 2R (z 0) Ω T and let v be the solution to (4.) in. Assume that the inequalities κ Du dz, Du dz κ hold for some κ. Then ( 2R ) ] Du Dv q q µ ( ( )( )+ dz c 2R ) 2R ( + c µ ( 2R ) 2R 2R for every q as in (4.3) and a constant c deending only on n,, ν and κ. ) (2 ) Du dz (4.4) Proof. (4.4) follows simly combining Lemma 4. and the reverse Hölder s inequality of Proosition 4.3, taking into account that + (2 ) ( )( ) + = ( )( ) +. (4.5) 5. THE PROOF OF THEOREM. We finally have in our hands all the tools needed to rove Theorem.. As already mentioned, the roof will be based on a covering argument more-or-less standard in the singular and degenerate arabolic setting, see 2, 2], which the reader could have already met in the roof of Proosition 3.6; for the imlementation of the argument in the context of degenerate arabolic equations with measure data see our revious contribution in 4]. The weighted version we emloy here (the weight is clearly reresented by the large constant M, see few lines below, which we are going to choose later) has been develoed in 2] to rovide recise estimates of Calderón-Zygmund tye in the setting of energy solutions and it is, in some sense, the PDE version of the good--inequality rincile, see for instance 4]. Fix a arabolic cylinders 2R 2R (z 0 ) Ω T, R > 0 and let M be a free arameter to be chosen. Moreover we fix arbitrarily an exonent q such that < q < + = m 0 < (5.) so that it only deends on n and : for instance, we can choose the midoint of the interval, m 0 ]. We then define the functional ] CZ( r ) := Du dz + M µ ( m r ) (5.2) r r

22 22 PAOLO BAROI for cylinders r r ( z) 2R, where m m 0 > is as in (.9). We fix two intermediate radii R r < r 2 2R and we define n(2 ) d 2 0 = 0 := Du dz + M µ ( ] m r 2 ) + ; (5.3) r2 r2 recall the exression for d in (.8). Moreover we take satisfying > B 0 with B defined as ( ) d 40r2 B := (5.4) r 2 r and we consider radii satisfying r 2 r 40 r r 2 r. (5.5) 2 Observe that r ( z) r2 for any z r and for any radius r satisfying (5.5). Using Hölder s inequality, enlarging the domain of integration and using that µ is ositive we have, also taking (5.3), (5.4) and /m < into account CZ( r ( z)) r 2 r ( z) r2 Du dz + M µ ( ] r 2 ) ] m r2 r 2 r ( z) d 0 < 2 n( r ) 2 n(2 ) 2 2 B d r ( 40r2 ) B d. (5.6) r 2 r On the other hand, for > 0 and radii γ R, 2R], we define the level sets { } E(, γ ) := z γ (z 0 ) : Du(z) > and z is a Lebesgue s oint of Du ; then we take a oint z E(, r ) with > B 0. It holds lim r 0 CZ( r ( z)) lim Du dx >. r 0 r ( z) Hence for small radii we have CZ( r ( z)) >. Hence, from this consideration and (5.6), continuity imlies the existence of a maximal radius r z such that ] Du dx + M µ ( ( z)) m r z =, (5.7) r z ( z) ( z) r z that is CZ( ( z)) =. The word maximal refers to the fact that for all radii r r z (r z, (r 2 r )/2] the inequality CZ( r ( z)) holds. In articular we have 3 4 ] 40 Du dx + M µ ( jr z ( z)) m (5.8) jr z ( z) jr z ( z) for every j {,..., 40}; note that r z < (r 2 r )/40 and hence jr z ( z) r2. ow we single out one of the cylinders ( z) considered above and we note that one 2r z of the following two inequalities must hold true: ( ) m 4 4 µ ( Du dz or ( z)) M 2r z 2r z ( z) 4 4. (5.9) ( z) 2r z

23 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 23 First case: A comarison ma -close to u. Suose that the first case in (5.9) holds true. ow, thinking z E(4, r ) fixed and being 20r z ( z) one of the cylinders considered above, we will denote in short 20 := 20r z ( z). We introduce in 20 the function solution to the Cauchy-Dirichlet roblem { vt div a(x, t, Dv) = 0 in 20, v = u we observe that (5.8)-(5.9) imly 4 4 Du dz, 20 Thus Corollary 4.5 imlies ( Du Dv q dz 20 for q as in (4.3); c c(n,, ν). ) q 40 on (20); (5.0) Du dz. (5.) ] µ (40) ( )( )+ c 40 ( + c µ (40) 40 Du dz 40 We now distinguish the two cases considered in Theorem.: ) (2 ) (5.2) The good measure case. This is the case where n; using (5.8) and the value for m = m in this case we have µ (40) 40 ( ) M. (5.3) On the other hand, we notice that we can cover the cylinder 40 with at most 2 2 (ossibly overlaing) standard arabolic cylinders with radius 40r z (this exact number of cylinders is clearly not otimal): 40 = B 40r z ( x) ( t r 2 z, t + r 2 z) for oints t j ( t r 2 z, t + r 2 z). Thus using (.4) and in turn µ (40) µ (40) j= 2 2 j= 40r z ( x, t j ) µ ( 40r z ( x, t j ) ) 2 2 c d 40r z ] c(n, c d ) 2 40r z ] 2 40r z ] = c(n, c d)r z ]. We can thus estimate, for every ɛ (0, ) µ (40) 40 µ (40) = 40 c M ] ] µ (40) r z ] 2 40r z = c M ( )( )+.

24 24 PAOLO BAROI The bad measure case. This is the case where n < (and thus the measure might be more concentrated). As in (5.3) and noting that now m = m 2 we infer µ (40) 40 2 ( (2 )n ) ; M on the other hand, simly enlarging the cylinder (remember (2.3)), again using (.4) on the standard arabolic cylinder 40r z and taking in mind that = n + 2, we have µ (40) 40 µ ( 40r z ) 40 40r z] c(n, c d ) 2 40r z ] = c(n, c d ) 2 2 ( ) 40r z ] 2 40r z ] = c(n, c d ) 2 2 (n ) 40r z ]. So, similarly to above, we here have µ (40) µ (40) = c 2 M = c M Thus in both cases we have µ (40) c 40 M ] ] µ (40) (2 )n ( ) 2 2 ( n ) 40r z ] 2 2 c = ( )( )+ M. 40r z ( )( )+. (5.4) If now we consider (5.2), (5.4) and (5.) 2 imly that ( ) Du Dv q q ] dx c M ( )( )+ + M 20 =: c ζ(m) (5.5) for q as in (4.3), with the obvious definition of ζ; note that we used also the comutation in (4.5). ow we fix as M the large constant satisfying c ζ(m ) 4 8 ; note that we can take M M (n,, ν). If we take M M, we have, for q = and q = q as defined in (5.) 20 Du Dv dx 4 8, ( Du Dv q dx 20 ) q. (5.6) 48 First we have, considering the first of the revious inequalities and (5.8) Dv dz Du dz + Du Dv dz 2; (5.7) 20 using (5.), 5 20 ( 2 Dv dz 5) 2 Du dz Du Dv dz (5.8) too. Therefore, alying Proosition 3.5, we infer Dv q dz c q, Dv dz c ; (5.9) 0 0

25 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 25 we recall that q is defined in (5.) as a constant in (, ] deending only on n and. The constant c thus deends on n,, ν and L. Finally, we use (5.9) 2 and (5.8) that allow to aly Proosition 3.6: (3.5) reads here as ( Dv (+η) dz 5 ) (+η) c 0 Dv dz c (5.20) with c c(n,, ν, L), using (5.7). Finally, again using quasi-sub-additivity exactly as in (5.7), using this time (5.6) 2 together with (5.9), we finally conclude this aragrah with the last estimate we need: Du q dz c q. (5.2) 5 The constant still deends on n,, ν and L. First case: On the measure of the suer-level of Du. We briefly conclude here the study of the case where the first inequality in (5.9) holds. We slit the integral of Du over 2 and then use Hölder s inequality as follows, for ς to be chosen: Du dz ς 2 E(ς, r2 ) + Du dz 2 2 E(ς, r2 ) ς E(ς, r2 ) q ( Du q dz 2 ) q ( ) 2 E(ς, r2 ) ( q ς 2 + Du q dz 2 5 Dividing by 2 and using (5.9) and (5.2) yields ς + 44 ( 2 E(ς, r2 ) 2 ow we fix ς = 4 5 and thus we have 2 c 2 E(ς, r2 ) for a constant deending on n,, ν, L and c d. ) q ) q. Second case and conclusion. Clearly, if the second alternative (5.9) 2 holds, we have 2 c(n) M m µ (2) Thus merging those two cases we finally get the estimate for 2 we were looking for: 20 c(n) 2 c 2 E(, r 2 ) + c M µ (2). (5.22) m We recall that 2 2r z ( z).

26 26 PAOLO BAROI Covering and iteration. Summarizing what we have done u to now, we have that once we fix > B 0, with B and 0 defined in (5.3)-(5.4), then for every z E(, r ) we can find a cylinder r z ( z) such that (5.7) holds. Then we consider the collection of all such cylinders E := { 2r z ( z)} z E(,r ) and, by a Vitali-tye argument, we extract a countable sub-collection F E such that the 5- times enlarged cylinders cover almost all E(, r ) and the cylinders are airwise disjoints - note that we are working at fixed; thus those cylinders are metric balls (recisely, of the metric defined in (2.4)). This is to say, if we denote the cylinders of F by 0 i := 2r zi ( z i ), for i I, being ossibly I =, and with z i E(, r ), we have 0 i 0 j = whenever i j and E(, r ) i I i, (5.23) with = 0 and with i := 5 0 i = ( z 0r zi i ); note that using (5.5) we see that i r 2 for all i I. We now fix H to be chosen later and we estimate E(H, r ) i E(2H, r 2 ). (5.24) i I We slit every term in the following way: i E(H, r 2 ) = {z i : Du(x) > 2H} {z i : Du(x) Dv i (x) > H} + {z i : Dv i (x) > H} =: Ii + II i. (5.25) Here v i is the comarison function solution to (5.0) in 2 i ( z 20r zi i ) = 2 i. We estimate searately the two ieces: for the first one we take ɛ > 0 arbitrary and we use (5.5) and subsequently (5.22) to infer I i Du Dv i dz c ζ(m) H 2 H 2 i (5.26) i c ζ(m) 0 i E(, r2 ) + M µ (0 i ) ] H m, where ζ( ) is given in (5.5) and rovided M M. On the other hand we use the higher integrability (5.20) to get ( ) (+η) II i H i Dv i (+η) dz i ( ) (+η) c (H) (+η) 2 i Dv i dz 2 i c 0 H (+η) i E(, r2 ) + M µ (0 i ) ] m. (5.27) Connecting the two estimates (5.26) and (5.27) and lugging the result into (5.25), taking into account that H, gives E(2H, r2 ) c ζ(m) i H + c ] 0 H (+η) i E(, r 2 ) + c M µ (0 i ) m. At this oint, since the { 0 i } are disjoint, see (5.23), summing u and multilying both sides of the revious inequality by (2H) m, see also (5.24), gives (2H) m E(2H, r ) c ζ(m) H m + c H (+η) m ] m E(, r2 ) + c µ ( 2R ) (5.28)

27 SIGULAR PARABOLIC EUATIOS WITH MEASURE DATA 27 with c deending only on n,, ν, L, c d but nor on H neither on M; c instead deends also on H, M but this is not a roblem. Finally, recall that < m < χ; first choose H so large that c H (+η) m = 4. (5.29) At this oint, being now H H(n,, ν, L, c d ) fixed, we choose M 2 so large that c H m ζ(m 2 ) 4 and this fixes the value of M in (5.2) as max{m, M 2 }. This choice makes also M deend only on n,, ν, L and c d. Having such choices at hand, after taking the suremum with resect to > B 0, (5.28) rewrites as su m E(, r ) >2HB 0 2 su m E(, r2 ) + c µ ( 2R ) >B 0 and therefore by the definition of Marcinkiewicz quasi-norm 2 Du m M m ( r2 ) + c µ ( 2R) (5.30) Du m M m ( r ) 2 Du m M m ( r2 ) + c B 0 ] mr + c µ ( 2R ) for all R r < r 2 2R, since B 0. At this oint (.7) follows exactly as in 4], using for the last time 9, Lemma 6.]. ote that reabsortion is ossible since u is an aroximate energy solutions, thus since Du L χ loc (Ω T ), we have Du M m ( 2R ) < for m < χ. 6. PROOF OF THEOREMS.5,.6 AD.7 The roof of Theorem.5 is very similar to the roof of Theorem.: the changes concern the regularity of the reference roblem, that is the integrability of the comarison ma solution to (3.) and hence of (4.)-(5.0). If we consider vector fields a satisfying (.4), then the solution is indeed such that Du L q for every q ; this is to say, (3.5) holds for every η > (and clearly also the constant at this oint deends on η), see 2, Theorem ] and 0] taking into account that our right-hand side is zero, thus in L q for all q. The roof needed in order to obtain the exlicit local estimate (3.5) from the results in 2, 0] is exactly the same described in the roof of Proosition 3.6 to obtain (3.5) starting from 9, Lemma 3]; the changes only concern the range of exonents η considered. We can formally follow our roof, with the only difference that here η is not a constant deending on the data of the roblem, but a free arameter; the constants will thus deend also on η (and they all will show a critical deendence on η, in the sense that they will blow-u as η ). To conclude, given >, we will to follow the roof until (5.28) and there we will choose η η(, ) such that ( + η) = m + ; this will be sufficient to rove that Du M m loc (Ω T ). ote that our choice justifies the critical deendence of the constant uon, as. The aforementioned Theorems in 2, 0] also justify the reabsortion after (5.30): since the data of our aroximating roblems are regular, then the energy solutions u we consider are as integrable as needed. For the roofs of Theorems.6-.7, the only different oint is the treatment of the second term in µ (40) µ ( = ( z)) ] 40r z µ ( ] ( z)) 40r z 40 ( z) ( z) ( z) 40r z ; 40r z 40r z this will allow for a different value of the exonent m in the Calderón-Zygmund oerator in (5.2), used to estimate the first term in the quantity above if we want to mimic the algebraic

28 28 PAOLO BAROI comutations leading to (5.4) (this is the real core of the roof, since this is sufficient to give to the comarison estimates (5.5) their homogeneous form). The exonent m aearing in (5.2), in view of the rest of the roof, will become the Marcinkiewicz exonent aearing in the statements: resectively m 3 for Theorem.6 and m 4 for Theorem.7. In articular for Theorem.6 we can estimate µ (40) = µ (B 40r z ( x)) µ 2 ( ( t 40r z ] 2, t + 40r z ] 2 ) ) µ L c(n)40r z ] n c d 40r z ] 2 = c(n, µ L, c d ) 2 2 (2 ) 40r z ], while for Theorem.7 we have µ (40) = µ 3 (B 40r z ( x)) µ 4 ( ( t 40r z ] 2, t + 40r z ] 2 ) ) ow for Theorem.6 we have µ (40) m 3 c 40 M = c c d 40r z ] n µ 4 L 240r z ] 2 = c(n, µ 4 L, c d ) 2 40r z ]. M ] 2 2 (2 ) 40r z ] 2 40r z ] ] 2 40r z r z ] 2 40r z = c M 2 and (recall that here 2 and thus the intrinsic cylinders are different, see (2.2) and comare with 4]) µ (40) m 4 ] 2 40r z ] ] 2 c 40 M 2 40r z ] 40r z = c 40r z ] 2 40r z = c 2 M M for Theorem.7; both these exressions lead exactly to (5.4). ow the roofs roceed exactly as after (5.4), taking into account the first lines of this Section when dealing, if < 2, with vector field satisfying the assumtions in (.2)-(.3). Acknowledgements. The author is suorted by the Gruo azionale er l Analisi Matematica, la Probabilità e le loro Alicazioni (GAMPA) of the Istituto azionale di Alta Matematica (IdAM). The aer was finalized while the author was a guest of the FIM at ETH Zürich in the Fall 206; the suort and the hositality of the Institute is gratefully acknowledged. REFERECES ] E. ACERBI,. FUSCO: Regularity of minimizers of non-quadratic functionals: the case < < 2, J. Math. Anal. Al. 40: 5 35, ] E. ACERBI, G. MIGIOE: Gradient estimates for a class of arabolic systems, Duke Math. J., 36 (2): , ] F. ADREU, J. M. MAZÓ, S. SEGURA DE LEÓ, J. TOLEDO: Existence and uniqueness for a degenerate arabolic equation with L -data, Trans. Amer. Math. Soc., 35 (): , ] P. BAROI: Marcinkiewicz estimates for degenerate arabolic equations with measure data. J. Funct. Anal. 267 (204), no. 9, ] P. BAROI, V. BÖGELEI: Calderón-Zygmund estimates for arabolic (x, t)-lalacian systems. Rev. Mat. Iberoamericana 30, no. 4, , ] P. BAROI, A. DI CASTRO, G. PALATUCCI: Global estimates for nonlinear arabolic equations, J. Evol. Equ., 3 (): 63 95, ] P. BAROI, J. HABERMA: ew gradient estimates for arabolic equations, Houston J. Math., 38 (3): , 202.