Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order $p$. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local H\"older continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for non-negative members of some super-class of De Giorgi, without any covering argument.


INTRODUCTION
De Giorgi classes consist of Sobolev functions in an open set Ω ⊂ R N satisfying a family of energy estimates, i.e. u ∈ W 1,p loc (Ω) and for some γ > 0, for all k ∈ R, and any pair of concentric cubes K ̺ (y) ⋐ K R (y) in Ω. The significance of De Giorgi classes lies in that they are general enough to include not only weak solutions to quasi-linear elliptic equation in divergence form (cf. [3,12]), but also local minima or quasi-minima of functionals that do not necessarily admit any Euler equations (cf. [8]). Formulated by Ladyzhenskaya and Ural'tseva (cf. [12]), it has been shown that functions in such classes (of elliptic nature) are locally Hölder continuous, using the beautiful ideas of De Giorgi in his celebrated work [1]. A probably even more striking discovery was made by DiBenedetto and Trudinger in [6] that non-negative members of De Giorgi classes actually satisfy Harnack's inequality, which is a typical property of harmonic functions. In addition to De Giorgi's techniques, the main new input of [6] includes realization of pointwise lower bound of non-negative members in De Giorgi classes with a power-like dependence on the measure distribution of their positivity. The proof uses a deep covering lemma due to Krylov and Safonov in [11].
The original consideration by De Giorgi in [1] was to obtain Hölder continuity of weak solutions to linear elliptic equations in divergence form with bounded and measurable coefficients. Later on, Moser invented a new approach in [16] to show the same kind of result. Moreover, he was able to obtain Harnack's inequality for such equations in [17]. A key idea of Moser's new proof in [16] is to show a certain logarithmic function of the solution is in fact a sub-solution and to formulate its energy estimates. The feature of Moser's approach is twofold: on the one hand, it simplifies the original proof of De Giorgi and gives a more intuitive method; on the other hand, it keeps referring We define also the class A p (E, γ) := A + p (E, γ) ∩ A − p (E, γ). Now suppose u ∈ C s − T, s; L p K R (y) ∩ L p s − T, s; W 1,p K R (y) We say u belongs to the parabolic De Giorgi class B ± p (E, γ) of order p, if u ∈ A ± p (E, γ) and in addition the following integral inequalities hold for any 0 < ̺ < R, 0 < τ < T and k ∈ R: Analogously we define the class B p (E, γ) := B + p (E, γ) ∩ B − p (E, γ). We remark that our definitions of De Giorgi classes mainly follow those in [13]. One difference is that we consider an arbitrary order p > 1, whereas p = 2 in [13]. Also, a certain non-homogeneous term is imposed in [13] for the inequalities (1.1) and (1.2). However, we decide to omit such a term for simplicity of presentation.
In the sequel, we refer to the set of parameters {γ, p, N } as the data and use C as a generic constant that can be quantitatively determined a priori only in terms of the data.
Here and in the sequel, we will use A(R, T, ̺, τ ) to denote a generic positive, homogeneous quantity in the sense that under the relation ̺ = σ 1 R, τ = σ 2 T and T = R p , it becomes a quantity of σ 1 and σ 2 , possibly also depending on the data. We will say u belongs to the generalized class A ± p , if (1.1) holds with γ replaced by A. Similar definition holds for B ± p .

LOCAL BOUNDEDNESS OF FUNCTIONS IN
In general the membership in A ± p (E, γ) does not guarantee continuity. A Heaviside function of the time variable would be an example. Nevertheless every function in A ± p (E, γ) is locally bounded from above or from below. Theorem 2.1. Suppose u ∈ A ± p (E, γ). Then there is a homogeneous quantity A, such that for any cube (y, s)+Q R,T ⊂ E and all k ∈ R. The same conclusion holds for members in the generalized classes A ± p . The proof is usually written using De Giorgi's iteration (cf. [13,15]). Nevertheless, we present here a proof based on Moser's iteration.
2.1. Proof by Moser's Iteration. Multiply both sides of (1.1) + by k β with β > −1 and integrate in dk from 0 to ∞ to get Fixing −τ < t < 0 and applying Fubini's theorem, the first term on the left-hand side is estimated bŷ One could verify that there exists an absolute constant C > 0, such that Similarly, the second term yields while the integral on the right-hand side is estimated from above by Combining the above calculation gives us that for all β > −1, Written in terms of w def = u p+β+1 p , the above estimate gives This is the starting point of Moser's iteration scheme. In order to use this energy estimate, we introduce for ̺, τ > 0, σ ∈ (0, 1) and n = 0, 1, · · · , Set ζ to be a standard cutoff function that vanishes on ∂ pQn and equals identity in Q n+1 , such that |Dζ| ≤ 2 n /̺. We apply the Sobolev imbedding (cf. [2, Chapter I, Proposition 3.1]), together with the energy estimate and the choice p o = pκ such that for some b, C > 1 depending only on the data. To simply the above iteration, we set , take the power p −1 n+1 on both sides, and rewrite it as Iterating this inequality yields Sending n → ∞ gives ess sup Then the above estimate yields An interpolation argument would give ess sup Fixing σ 1 , σ 2 ∈ (0, 1), it is not hard to see that there exists (y, s) ∈ Q σ 1 R,σ 2 T , such that ess sup Applying the above estimate to Q * to obtain ess sup Setting ̺ = σ 1 R and τ = σ 2 T , the desired conclusion follows.

Critical Mass Lemmas.
Assume a ∈ (0, 1) and M > 0 are parameters. The following lemma has been derived in [7]. It can be viewed as a direct consequence of the local boundedness estimate in Theorem 2.1.
There exists ν > 0 depending only on the data and a, such that if Proof. Assume (y, s) = (0, 0). We only treat the class A + p (E; γ). An application of As a result, we arrive at the desired conclusion ess sup

ADDITIONAL PROPERTIES OF FUNCTIONS IN
It is known that the convex, non-decreasing function of a sub-harmonic function yields another sub-harmonic function, whereas the concave, non-increasing function of a super-harmonic function gives another super-harmonic function. Similar conclusions hold for the heat operator, and even for more general linear parabolic operators with bounded and measurable coefficients. What we are concerned with next is to show analogous properties for members of A ± p (γ, E).
Lemma 3.1. Let ϕ : R → R be convex and non-decreasing and let u ∈ A + p (E, γ). Then ϕ(u) belongs to the generalized class A + p . Proof. For any such ϕ and h ≤ k, observe the following elementary identity where χ is the characteristic function of the indicated set. Moreover, by the convexity and monotonicity of ϕ, For I 1 , we estimate by using (3.1) and (3.2): For I 2 , we estimate by using (3.1), (3.2) and Theorem 2.1: Recalling A(R, T, ̺, τ ) represents a generic dimensionless quantity, we combine the above estimates to arrive at We now handle the part with the space gradient. From (3.1), taking the gradient of both sides, then taking the L p -norm over Q ̺,τ and applying the continuous version of Minkowski's inequality, we obtain One estimates I 3 using (1.2) and (3.2): One estimates I 4 by (1.2), (3.1), (3.2) and Theorem 2.1: Observe that the fractional with A is again a dimensionless quantity. Hence we havë Combining the above estimates gives the desired conclusion.
Lemma 3.2. Let ϕ : (a, ∞) → R, for some a < ∞ be convex and non-increasing, such that . Then ϕ(u) belongs to the generalized class A + p . Proof. Under the conditions of ϕ, one easily verifies , it is bounded from below by Theorem 2.1. Hence the above equation is well defined for such u and we may assume with no loss of generality that u ≥ 0.
First, we take L p -norm of both sides over K ̺ to obtain for all −τ < t < 0 The right-hand side is estimated by Minkowski's inequality and Theorem 2.1: As a result, Next, we take the spatial gradient of both sides of (3.4), then take the power p, and integrate over Q ̺,τ to obtain The right-hand side is estimated by If ϕ is convex, non-increasing and satisfying (3.3), then (ϕ − l) + verifies the same properties. Hence the desired conclusion is reached by replacing ϕ with (ϕ − l) + .  Proof. Assume (y, s) = (0, 0). According to (1.1), for all 0 < k < M , To proceed, we multiply both sides by k −p−1 and integrate from 0 to M . The left-hand side becomesˆM The integral on the right-hand side is estimated bŷ Hence combining the above two estimates we arrive aẗ Since u ∈ A − p (E, γ), the term in the curly bracket is non-positive and can be discarded.

TIME PROPAGATION OF POSITIVITY IN MEASURE
In this section, we examine the role of (1.2). First of all, we present a standard lemma which says (1.2) alone is sufficient to propagate positivity of u in measure for a short period of time (cf. citeLSU). Then there exist δ, ε ∈ (0, 1) depending only on the data and α, such that for all times s < t < s + δ̺ p .
Proof. Assume (y, s) = (0, 0). We may apply (1.2) − with k = M in the cylinders in such a case, we have for all 0 < t < δ̺ p , Set l = εM . The left-hand side of the above estimate can be bounded from below bŷ where we have defined for some ε to be chosen Notice that Collecting all the above estimates yields that Finally we may choose ε, σ and δ, such that Remark 4.1. One easily obtains the dependence of various constants on α from the above proof. Namely, ε ≈ α, σ ≈ α and δ ≈ α p+1 .
One wonders if the positivity in measure can be propagated further in time, i.e., δ can be made large by choosing a proper ε. It seems (1.2) − alone is insufficient. In the theory of parabolic equations, a standard tool to achieve this is a logarithmic estimate. See [3, Chapter 2, Section 3]. We do not know if such a logarithmic estimate holds for functions in parabolic De Giorgi classes. However we show in the following that a membership in u ∈ B − p (E, γ) still ensures that the measure information of positivity propagates further in time. Then there exist ε > 0 depending on the data and α, such that for all s < t < s + A̺ p .

4.1.
Shrinking the Measure of the Set [u ≈ 0]. We first prove the following shrinking lemma due to De Giorgi (cf. [1]).
There exists C > 0 depending only on the data, such that for any positive integer j * , we have Proof. We assume (y, s) = (0, 0) and set k j = 2 −j M for j = 0, 1, · · · , j * . Apply (1.1) − for the pair of cylinders such that Next, we apply [3, Chapter I, Lemma 2.2] to u(·, t) for t ∈ (0, δ̺ p ] over the cube K ̺ , for levels k j+1 < k j . Taking into account the measure theoretical information this gives and integrate the above estimate in dt over (0, δ̺ p ]; we obtain by using (4.2) Now take the power p p−1 on both sides of the above inequality to obtain Add these inequalities from 0 to j * − 1 to obtain

Proof of Proposition 4.2.
We come back at (4.1) and choose We choose δ and ε such that As a result, we obtain Having ε and δ determined in (4.3), we use (1.2) − again and repeat the above argument with where j 1 and n 1 are positive numbers to be determined. We may use the above measure theoretical information for t ∈ [s, s 1 ], and apply Lemma 4.1 to obtain a refined estimate: We choose j 1 and n 1 , such that .
As a result, we obtain that Now we may proceed by induction. Suppose the construction has been made up to the (i−1)-th step: the sequences {M i }, {n i } and {j i } have been chosen up to the (i−1)-th step, and we have the measure theoretical information where Setting , and using the above measure theoretical information at t = s i , we can repeat the above argument to obtain for all t ∈ [s i , s i+1 ] Assuming (i − 1)δ < A, we may choose ε and δ as in (4.3); this ensures

Now we set
where j i and n i are to be determined. Then we use the above measure theoretical information in Lemma 4.1 to obtain a refined estimate: for all t ∈ [s i , s i+1 ] We choose j i and n i , such that As a result, we obtain that for all times t ∈ [s i , s i+1 ] The above argument terminates if iδ ≥ A and we reach the desired conclusion with the choice εM = M i 2 n i +j i .

HÖLDER CONTINUITY FOR FUNCTIONS IN B p
Theorem 5.1. If u ∈ B p (E; γ), there are constants C > 0 and 0 < β < 1 depending only on the data, such that for every pair of cylinders (y, s) A similar adaption has been made to parabolic equations in [14], which however cannot be directly generalized to parabolic De Giorgi classes.
Without loss of generality, we may take (y, s) = (0, 0). For ease of notation, we write ω = ω(2̺). Let ε be the number determined in Proposition 4.1 with α = 1/2. We introduce two functions: We first apply Lemma 3.3 to w 1 and w 2 . Indeed, since u ∈ A p (E; γ) we have both µ + − u and u − µ − members of A − p (E; γ). Therefore, we may apply Lemma 3.3 to that is, in terms of w 1 , Similar inequality holds for w 2 . Now we go with two alternatives: either where δ is the constant appearing in Proposition 4.1 with α = 1/2. Let us suppose for instance the first case holds. According to Proposition 4.1, we have In terms of w 1 , this may be rephrased as We may employ the Poincaré type inequality (cf. [3, Chapter 10, Proposition 5.2]) for each time slice to w 1 (·, t), and then a time integration over (−δ̺ p , 0) on both sides, and the fact that w 1 ≥ − ln(2/ε) to obtain that The integral term on the right-hand side is estimated by Hölder's inequality, Young's inequality and (5.1) as Thus combining above estimates we obtain An interpolation argument (cf. [6, Theorem 1]) yields that An application of Lemma 3.1 gives that w 1+ belongs to the generalized A + p . As a result, Theorem 2.1 holds for w 1+ . The supreme estimate together with (5.2) yields that Therefore ess osc A standard iteration finishes the proof.

A Revisit to De
Giorgi's Approach. The purpose of this section is to point out that the Hölder regularity could be established with less assumptions. Namely, it suffices to assume u is a member of . The argument is modeled on the one in [2, Chapter III]. 5.2.1. Expansion of Positivity. Suppose K 4̺ (y) × (s, s + ̺ p ] ⊂ E. We show in the following that the measure information on the positivity of a non-negative member u of B − p (E; γ) at t = s translates into pointwise information forward in time and over a larger space cube.
p (E; γ) be non-negative. Suppose for some M > 0 and α ∈ (0, 1), Then there exist η, δ ∈ (0, 1) depending on the data and α, such that u(·, t) ≥ ηM a.e. in K 2̺ (y), for all Proof. Assume (y, s) = (0, 0). We rephrase the starting information in a larger cube: By Proposition 4.1, there exist δ, ε > 0 depending only on the data and α, such that for all 0 < t < δ(4̺) p . Next, by Lemma 4.1, there exists C > 0 depending only on the data, such that for any positive integer j * , we have Finally, let ν be the number claimed in Lemma 2.1. Choose j * so large that C Thus by Lemma 2.1 with µ − = 0, we conclude that u(·, t) ≥ εM 2 j * +1 a.e. in K 2̺ for all times The proof is finished by choosing η = ε2 −j * −1 .
The approach used in [7] is a direct one, thus by-passing a weak Harnack inequality. On the other hand, a weak Harnack inequality is established in [18] for p = 2 using the Krylov-Safonov covering argument (cf. [11]). Here we give a transparent proof of a weak Harnack inequality for the class B − p (E, γ), via a measure theoretical lemma in [5], thus avoiding the heavy covering argument.
p (E; γ) be non-negative. Then there exist δ o , q ∈ (0, 1) and C > 1 depending only on the data, such that A combination of Theorem 6.2 and Theorem 2.1 would give another proof of Theorem 6.1 (cf. [6,18]). The key to proving Theorem 6.2 is to show an expansion of positivity with a power-like dependence on the measure distribution of the positivity. The main tool is a certain clustering lemma from [5].
Finally we choose n so large that 2 n σ = 3, such that K 2̺ ⊂ K 2 n σ̺ (x o ). At the same time, taking into consideration of the power-like dependence on α of ε and σ, there exist η o ∈ (0, 1) and d > 1 depending only on the data, such that The time interval for such positivity is t n − 1 2δ (3̺) p = t n − 1 2δ (2 n σ̺) p < t < t n .
With no loss of generality, we assume σ < 1/4. In this way, it is not hard to see there exist δ o , η o ∈ (0, 1) depending only on the data, such that u(·, t) ≥ η o α d M a.e. in K 2̺ for all The qualitative location of t 1 ∈ (0, ̺ p ) may be removed by repeated applications of this conclusion. The proof is then finished by properly redefining δ o . u.
We first estimate the L q -norm of u(·, 0) by its measure distribution: By Theorem 6.2, there exist d > 1 and η o ∈ (0, 1) depending only on the data, such that Thus we may estimate the first term on the right-hand side of (6.4) by Now we stipulate to take q < 1/d, such that the improper integral on the right-hand side converges. In such a way, the right-hand side of the above inequality is bounded above by CI q |K ̺ |. Hence putting everything back in (6.4), we obtain the desired conclusion.