Remarks on localized sharp functions on certain sets in $${\mathbb {R}}^n$$Rn

The aim of this note is to define localized sharp functions on certain domains in $${\mathbb {R}}^n$$Rn and prove $$L^p$$Lp estimates analogue to that of Fefferman–Stein. The proofs go by modifications of the good lambda inequality.


Introduction
On R n let f # Δ (x) and M Δ f (x) denote the classical dyadic sharp function and dyadic maximal function respectively, that is, where here and subsequently, Δ denotes the collection of all dyadic cubes in R n and Suppose that f ∈ L p 0 (R n ) for some p 0 . The well-known Fefferman-Stein inequality asserts that if 1 < p < ∞, 1 ≤ p 0 ≤ p, and f # Δ ∈ L p (R n ), then M Δ f ∈ L p (R n ) and (see [2,Sect. 3], [4,Chapter 4]). The inequality (1.1) implies that for every 1 < p < ∞ one has The estimate (1.1) is a consequence of the following good lambda distributional inequality (1.2) where λ > 0, c > 0, 0 < b < 1, a = 2 n c/ (1 − b), and f ∈ L 1 loc (R n ) (see [4]). Let Ω be a domain in R n . Our goal is to define for f ∈ L 1 loc (Ω) a localized version f # loc of the sharp function which will satisfy By localized we mean that the cubes which are taken in the definition of f # loc (x) are contained in a bounded set B x ⊂ Ω. So one possible definition can be taken as follows. Let τ : Ω → (0, ∞). For f ∈ L 1 loc (Ω) we set where Q is any cube (not necessarily dyadic) and (Q) denotes its side-length. Note that τ cannot be taken arbitrarily. For example, if Ω = (0, ∞) and τ is such that On the other hand, we shall show that for certain sets Ω in R n if τ (x) behaves like 1 2 dist(x, ∂Ω), then f # loc, τ satisfies (1.3). Moreover, the inequality (1.3) holds for p = 1, provided f is supported by a bounded set and | f | log(2 + | f |) is integrable. These will be obtained by proving modifications of the good lambda inequality (see Propositions 2.4, 3.7, and 4.5).

Localized sharp function on R n \{0}
Let Ω = R n \{0}. We define the localized sharp function on Ω as where the supremum is taken over all cubes K (not necessarily dyadic) contained in the set We now turn to define the local dyadic maximal function associated with a Whitney decomposition of Ω. For this purpose, put The set L forms a Whitney covering of Ω. For every integer k we define the k-th layer L k of L as Clearly, Q ∈ L k if and only if 2 −m Q ∈ L k+m . Figure 1 shows three k-layers for n = 2. Here and subsequently, α Q = {αx : x ∈ Q}, α > 0. For every positive integer m the partition L m of Ω is obtained by dividing each cube Q from L into 2 nm dyadic cubes each of side-length 2 −m (Q). Let The local dyadic maximal function associated with the Whitney covering L of Ω is defined by Our goal of this section is to prove the following theorem.

Corollary 2.3
There is a constant C > 0 such that if f is supported by a bounded set and | f | log(2 + | f |) is integrable, then There is no loss of generality if we assume that all the functions under consideration take values in R. Clearly, for almost every x ∈ Ω there is a unique cube Q ∈ L such that x ∈ Q. For such an x let The proof of Theorem 2.1 is a consequence of the following modified version of the good lambda inequality, which is stated in the proposition below.
where here and subsequently, Proof of Theorem 2.1 If we assume Proposition 2.4, the proof of the theorem is a slight modification of that in the classical case (see [2][3][4]). For the convenience of the reader, we provide details. We may assume that f # loc, Ω L p (Ω) is finite. Then, by the Minkowski inequality, Applying Proposition 2.4, we obtain Clearly, I R < ∞, since, by assumption, M D f ∈ L p 0 (Ω) and 0 < p 0 ≤ p. Moreover, Taking c small enough such that Ccb − p < 1 we obtain Letting R → ∞, we conclude where in the last inequality we have used (2.2).
The remaining part of the section is devoted to proving Proposition 2.4. The following two lemmas will play a crucial role in the proof.

Lemma 2.5 For every locally integrable function f and almost every x
Proof It suffices to prove the lemma for x = (x 1 , x 2 , . . . , x n ) such that x j > 0 for every j = 1, 2, . . . , n. Let Q 1 be the unique dyadic cube from L which contains x.
Let k be such that 2 −(k+1) = (Q 1 ). Set Q 2 = 2Q 1 . Let Then there is the unique vector p = ( p 1 , p 2 , . . . , p n ) = 0, p j ∈ {0, 1}, such that We shall call the set Let us remark that for n = 1 the complementary neighborhood of two intervals is the empty set. For n = 2 the complementary neighborhoods are presented in Fig. 2.
We have We consider two cases.
The proof in this case is similar to that in Case 1. The only difference is that we diminish the area of integration to Q 2 instead of Q 1 . We omit the details. If n = 1, then M = ∅. In this case we set f M = 0 and proceed as in Case 1.

Remark 2.6
If we apply the lemma to the function − f , we obtain the inequality

Lemma 2.7 For every locally integrable function f and almost every x ∈ Ω one has
Iterating the inequality (2.5) we obtain the following corollary.

Corollary 2.8 Assume that a locally integrable function f on Ω satisfies
Proof of Proposition 2.4 The proof of the proposition is a modification of that of the classical good lambda inequality (cf. [1,4]). Let {Q j } be the partition of the set Obviously, the cubes Q j have disjoint interiors. Further, Thus, it suffices to show that either Assume that the set A j = {x ∈ Q j : M D f (x) > α, f # loc, Ω (x) < cα} is not empty, otherwise there is nothing to prove. Fix x 0 ∈ A j . We consider two cases.
for every y ∈ Q j , where in the last inequality we have used Corollary 2.8. Thus Q j ⊂ {x ∈ Ω : S f (x) > bα} and (2.7) holds in this case.
Case 2 Q j ∈ L m for m ≥ 1. In this case the proof follows the pattern from [4]. Indeed, first observe that for every Q ∈ D such that Q j Q one has | f | Q ≤ bα. Thus, for every Let Q j be the parent of Q j . Clearly, Q j ∈ D and Since M D satisfies the weak type (1,1) inequality with the constant C = 1, we have so (2.6) holds in this case with C = 2 n (1 − b) −1 . Clearly, Similarly to the previous section, for every positive integer m the partition L m consists of dyadic cubes which are obtained by dividing each cube Q from L into 2 mn dyadic cubes each of the side-length 2 −m (Q) (Fig. 3). Set Define the local maximal dyadic function M D and localized sharp function f # loc, Ω associated with the Whitney covering L of Ω as where the supremum is taken over all cubes (not necessarily dyadic). It turns out that the following theorem analogue to Theorem 2.1 holds.

Corollary 3.2 For every
The remaining part of this section is devoted to proving Theorem 3.1.
Similarly to the previous section [see (2.1)] we set where Q is the unique cube from L which contains x (such a Q is well-defined for almost every x). Let k be such that Q ∈ L k . Our goal is to define the successors x and Q ∈ L k−1 of x and Q respectively in such a way that x ∈ Q and To this end, observe that there is a unique vector q = (q 1 , q 2 , . . . , q n ), where q j are non-negative integers such that at least one q j equals 0, and Q = 2 −k Q 0 +2 −k (q +1), where here and subsequently, 1 = (1, 1, . . . , 1). Consider the coordinates x j of x for which q j = 0. There is no loss of generality if we assume these are the first m coordinates, m ∈ {1, 2, . . . , n}. (1, 1, . . . , 1, 1 + q m+1 , . . . , 1 + q n ), q m+1 , . . . , q n ≥ 1. Define Then, for almost every x, the point x belongs to the unique Q ∈ L k−1 and = (0, . . . , 0, q m+1 , . . . , q n ), q j = (q j − 1)/2 , j = m + 1, . . . , n.
hence the proof of the lemma is the same as those of Lemmata 2.5 and 2.7.

Corollary 3.4
Assume that f ∈ L 1 loc ( Ω) and lim m→∞ S f (F m (x)) = 0 for almost every x ∈ Ω, where F m (x) = F(F m−1 (x)) (Fig. 4). Then Proof It suffices to apply Lemma 3.3 and note that Remark 3.5 Let us note that lim m→∞ S f (F m (x)) = 0 for f ∈ L p ( Ω). This is a consequence of the fact that ρ( Proof This follows from the Minkowski inequality and the summability of the series

Proposition 3.7
For every constant 0 < b < 1 there is a constant C > 0 such that for all c, α > 0, and every f ∈ L 1 loc ( Ω) satisfying Proof The proof is identical to that of Proposition 2.4, and uses Corollary 3.4 instead of Corollary 2.8.

Proof of Theorem 3.1
The theorem follows from Lemma 3.6 and Proposition 3.7. Its proof is identical to that of Theorem 2.1.

Localized sharp function for cube
In this section we consider the cube (0, 2) n ⊂ R n and its Whitney decomposition L which is defined in the following way. Let L be the restriction of the decomposition L defined in the previous section into the unit cube (0, 1] n . Let us denote by L the set of cubes which is obtained from L under the action of the group G of transformation generated by the reflections with respect to planes x j = 1. Let L k be the set of cubes from L of the side-length 2 −k . Clearly, L = L 1 ∪ L 2 ∪ · · · (Fig. 5).
We define the partition L 1 by dividing each cube K from L into 2 n dyadic cubes each of the side-length 2 −1 (K ). Inductively, L m+1 is defined by dividing each cube K from L m into 2 n dyadic cubes of side-length 2 −1 (K ). Set The localized sharp function is defined by where B x = y ∈ (0, 2) n : and the supremum is taken over all cubes K not necessarily dyadic. Our aim of this section is to prove the following theorem. Pre(K 1 ) The proof requires preparations. For each K ∈ L there is a unique σ ∈ G such that σ (K ) ⊂ [0, 1] n . Therefore in our considerations we shall deal with cubes contained in [0, 1] n and then use the group action for other cubes.
From now on, let Q 1 = 1 2 Q 0 + 1 2 1. For x ∈ (0, 1] n let F(x) be defined by (3.2). Clearly, for every R ∈ L ∩ L k with k ≥ 2 there is a unique K ∈ L ∩ L k−1 such that F(R) ⊂ K . For K ∈ L we set Pre(K ) = {R ∈ L : F j (R) ⊂ K for a certain positive integer j}.