On the weighted fractional Poincare-type inequalities
OON THE WEIGHTED FRACTIONAL POINCAR ´E-TYPEINEQUALITIES
RITVA HURRI-SYRJ ¨ANEN AND FERNANDO L ´OPEZ-GARC´IA
Abstract.
Weighted fractional Poincar´e-type inequalities are proved on John do-mains whenever the weights defined on the domain depend on the distance to theboundary and to an arbitrary compact set in the boundary of the domain. Introduction
In this article we study a version of the classical fractional Poincar´e-type inequalitywhere the domain in the double integral in the Gagliardo seminorm is replaced by asmaller one: (cid:18)(cid:90) Ω | u ( x ) − u Ω | p d x (cid:19) /p ≤ C (cid:18)(cid:90) Ω (cid:90) B ( x,τd ( x )) | u ( x ) − u ( y ) | p | x − y | n + sp d y d x (cid:19) /p . (1.1)The parameter τ in the double integral belongs to (0 ,
1) and d ( x ) denotes the distancefrom x to ∂ Ω. The inequality (1.1) was introduced in [4]. It is well-known that thefractional classical Poincar´e inequality is valid for any bounded domain, while this newversion (1.1) depends on the geometry of the domain. In [4] it was proved that theinequality (1.1) is valid on John domains and, hence, in particular on Lipschitz domains.An example of a domain where the inequality (1.1) is not valid was also given. Werefer the reader to [5] and [2] where the fractional Sobolev-Poincar´e versions of (1.1)are considered. For a weighted version of (1.1) where weights are power functions tothe boundary we refer to [3].The main result of our paper is the following theorem where the distance to anarbitrary set of the boundary has been added as a weight.
Theorem 1.1.
Let Ω in R n be a bounded John domain and < p < ∞ . Givena compact set F in ∂ Ω , and the parameters β ≥ and s, τ ∈ (0 , , there exists aconstant C such that (cid:18)(cid:90) Ω | u ( x ) − u Ω ,ω | p d pβF ( x )d x (cid:19) /p ≤ C (cid:18)(cid:90) Ω (cid:90) B ( x,τd ( x )) | u ( x ) − u ( y ) | p | x − y | n + sp d ps ( x ) d pβF ( x )d y d x (cid:19) /p (1.2) for all functions u ∈ L p (Ω , d ( x ) pβ ) , where d ( x ) and d F ( x ) denote the distance from x to ∂ Ω and F respectively, and u Ω ,ω is the weighted average d pβF (Ω) (cid:82) Ω u ( z ) d pβF ( z )d z . Date : July 23, 2018.2010
Mathematics Subject Classification.
Primary: 46E35 ; Secondary: 26D10.
Key words and phrases.
Fractional Poincar´e inequalities, Hardy-type operator, Tree covering,Weights. a r X i v : . [ m a t h . F A ] D ec In addition, the constant C in (1.2) can be written as C = C n,p,β τ s − n K n + β , where K is the geometric constant introduced in (5.1). We would like to emphasize two points in this result: The first one is that no extraconditions are required for the compact set F in ∂ Ω. The second point is that theestimate shows how the constant depends on the given τ and a certain geometriccondition of the domain.Some of the essential auxiliary parts for the proofs for weighted inequalities arefrom [7] and [8] where a useful decomposition technique was introduced by the secondauthor. Our work was stimulated by the papers of Augusto C. Ponce, [10], [11], [12],where more general fractional Poincar´e inequalities for functions defined on Lipschitzdomains were investigated.The paper is organized as follows: In Section 2, we introduce some definitions andpreliminary results. In Section 3, we show how to use decompositions of functions toextend the validity of certain inequalities on “simple domains”, such as cubes, to morecomplex ones. We are interested in extending the results from cubes to John domains.In Section 4, we apply the results obtained in the previous section to estimate theconstant in the unweighted version of (1.2) on cubes. Especially we are interested inhow the constant depends on τ . This result is auxiliary of our main theorem but itmight be of independent interest. In Section 5, we show the validity of the weightedfractional Poincare inequality studied in this paper with the estimate of the constantand a generalization to the type of inequalities considered by Ponce.2. Notation and preliminary results
Throughout the paper Ω in R n is a bounded domain with n ≥
2, 1 < p < ∞ , and1 < q < ∞ with p + q = 1, unless otherwise stated. Moreover, given η : Ω → R aweight (i.e., a positive measurable function) and 1 ≤ r ≤ ∞ , we denote by L r (Ω , η )the space of Lebesgue measurable functions u : Ω → R equipped with the norm (cid:107) u (cid:107) L r (Ω ,η ) := (cid:18)(cid:90) Ω | u ( x ) | r η ( x ) d x (cid:19) /r if 1 ≤ r < ∞ , and (cid:107) u (cid:107) L ∞ (Ω ,η ) := ess sup x ∈ Ω | u ( x ) η ( x ) | . Finally, given a set A we denote by χ A ( x ) its characteristic function. Definition 2.1.
Let C be the space of constant functions from R n to R and { U t } t ∈ Γ a collection of open subsets of Ω that covers Ω except for a set of Lebesgue measurezero; Γ is an index set. It also satisfies the additional requirement that for each t ∈ Γthe set U t intersects a finite number of U s with s ∈ Γ. This collection { U t } t ∈ Γ is calledan open covering of Ω. Given g ∈ L (Ω) orthogonal to C (i.e., (cid:82) g ϕ = 0 for all ϕ ∈ C ),we say that a collection of functions { g t } t ∈ Γ in L (Ω) is a C - orthogonal decompositionof g subordinate to { U t } t ∈ Γ if the following three properties are satisfied:(1) g = (cid:80) t ∈ Γ g t . (2) supp( g t ) ⊂ U t . (3) (cid:82) U t g t ϕ = 0, for all ϕ ∈ C and t ∈ Γ. We also refer to this collection of functions by a C - decomposition . We say that { g t } t ∈ Γ is a finite C -decomposition if g t (cid:54)≡ t ∈ Γ.Inequality (1.2), and similar Poincar´e type inequalities, can be written in terms of adistance to the space of constant functions C by replacing its left hand side byinf α ∈C (cid:18)(cid:90) Ω | u ( x ) − α | p d pβF ( x )d x (cid:19) /p . The technique used in this paper may also be considered when the distance to othervector spaces V are involved, in which case, a V -orthogonal decomposition of func-tions is required. We direct the reader to [9] where a generalized version of the Korninequality is studied by using decomposition of functions.Let us denote by G = ( V, E ) a graph with vertices V and edges E . Graphs in thispaper have neither multiple edges nor loops and the number of vertices in V is at mostcountable.A rooted tree (or simply a tree) is a connected graph G in which any two verticesare connected by exactly one simple path, and a root is simply a distinguished vertex a ∈ V . Moreover, if G = ( V, E ) is a rooted tree with a root a , it is possible to definea partial order “ (cid:22) ” in V as follows: s (cid:22) t if and only if the unique path connecting t with the root a passes through s . The height or level of any t ∈ V is the number ofvertices in { s ∈ V : s (cid:22) t with s (cid:54) = t } . The parent of a vertex t ∈ V is the vertex s satisfying that s (cid:22) t and its height is one unit smaller than the height of t . We denotethe parent of t by t p . It can be seen that each t ∈ V different from the root has aunique parent, but several elements in V could have the same parent. Note that twovertices are connected by an edge ( adjacent vertices ) if one is the parent of the other. Definition 2.2.
Let Ω be in R n be a bounded domain. We say that an open covering { U t } t ∈ Γ is a tree covering of Ω if it also satisfies the properties:(1) χ Ω ( x ) ≤ (cid:80) t ∈ Γ χ U t ( x ) ≤ N χ Ω ( x ), for almost every x ∈ Ω, where N ≥ , E ) with a root a .(3) There is a collection { B t } t (cid:54) = a of pairwise disjoint open cubes with B t ⊆ U t ∩ U t p . Definition 2.3.
Given a tree covering { U t } t ∈ Γ of Ω we define the following Hardy-typeoperator T on L -functions: T g ( x ) := (cid:88) a (cid:54) = t ∈ Γ χ t ( x ) | W t | (cid:90) W t | g | , (2.1)where W t := (cid:91) s (cid:23) t U s , (2.2)and χ t is the characteristic function of B t for all t (cid:54) = a .We may refer to W t by the shadow of U t .Note that the definition of T is based on the a-priori choice of a tree covering { U t } t ∈ Γ of Ω. Thus, whenever T is mentioned in this paper there is a tree covering { U t } t ∈ Γ ofΩ explicitly or implicitly associated to it.The following fundamental result was proved in [8, Theorem 4.4], which shows theexistence of a C− decomposition of functions subordinate to a tree covering of thedomain. Theorem 2.4.
Let Ω in R n be a bounded domain with a tree covering { U t } t ∈ Γ . Given g ∈ L (Ω) such that (cid:82) Ω gϕ = 0 , for all ϕ ∈ C , and supp( g ) ∩ U s (cid:54) = ∅ for a finite numberof s ∈ Γ , there exists a C -decompositions { g t } t ∈ Γ of g subordinate to { U t } t ∈ Γ (refer toDefinition 2.1).Moreover, let t ∈ Γ . If x ∈ B s where s = t or s p = t then | g t ( x ) | ≤ | g ( x ) | + | W s || B s | T g ( x ) , (2.3) where W t denotes the shadow of U t defined in (2.2) . Otherwise | g t ( x ) | ≤ | g ( x ) | . (2.4) Remark 2.5.
The C -decomposition stated in Theorem 2.4 is finite. This fact is not inthe statement of [8, Theorem 4.4] but it is easily deduced from its proof.In the next lemma, the continuity of the operator T is shown. We refer the readerto [7, Lemma 3.1] for its proof. Lemma 2.6.
The operator T : L q (Ω) → L q (Ω) defined in (2.1) is continuous for any < q ≤ ∞ . Moreover, its norm is bounded by (cid:107) T (cid:107) L q → L q ≤ (cid:18) qNq − (cid:19) /q . Here N is the overlapping constant from Definition 2.2. If q = ∞ , the previous inequality means (cid:107) T (cid:107) L ∞ → L ∞ ≤
2. Actually, for being T an averaging operator, it can be easily observed that (cid:107) T (cid:107) L ∞ → L ∞ = 1, but it does notaffect our work. Notice that L q (Ω , ω − q ) ⊂ L (Ω) if the weight ω : Ω → R > satisfiesthat ω p ∈ L (Ω). Then, the operator T introduced in Definition 2.3 for functions in L (Ω) is well-defined in L q (Ω , ω − q ). Lemma 2.7.
Let Ω in R n be a bounded domain, { U t } t ∈ Γ a tree covering of Ω and ω : Ω → R a weight which satisfies ω p ∈ L (Ω) . If ω satisfies that ess sup y ∈ W t ω ( y ) ≤ C ess inf x ∈ B t ω ( x ) , (2.5) for all a (cid:54) = t ∈ Γ , then the Hardy-type operator T defined in (2.1) and subordinate to { U t } t ∈ Γ is continuous from L q (Ω , ω − q ) to itself. Moreover, its norm for < q < ∞ isbounded by (cid:107) T (cid:107) L → L ≤ (cid:18) qNq − (cid:19) /q C , where L denotes L q (Ω , ω − q ) , and N is the overlapping constant from Definition 2.2.Proof. Given g ∈ L q (Ω , ω − q ) we have (cid:90) Ω | T g ( x ) | q ω − q ( x ) d x = (cid:90) Ω ω − q ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) a (cid:54) = t ∈ Γ χ t ( x ) | W t | (cid:90) W t | g ( y ) | d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q d x = (cid:90) Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) a (cid:54) = t ∈ Γ χ t ( x ) | W t | ω − ( x ) (cid:90) W t | g ( y ) | ω − ( y ) ω ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q d x. Now, condition (2.5) implies that ω ( y ) ≤ C ω ( x ) for almost every x ∈ B t and y ∈ W t .Thus, (cid:90) Ω | T g ( x ) | q ω − q ( x ) d x ≤ (cid:90) Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) a (cid:54) = t ∈ Γ χ t ( x ) | W t | ω − ( x ) C ω ( x ) (cid:90) W t | g ( y ) | ω − ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q d x = C q (cid:90) Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) a (cid:54) = t ∈ Γ χ t ( x ) | W t | (cid:90) W t | g ( y ) | ω − ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q d x = C q (cid:90) Ω (cid:12)(cid:12) T ( gω − ) (cid:12)(cid:12) q d x. Finally, gω − belongs to L q (Ω) and T is continuous from L q (Ω) to itself; we refer toLemma 2.6, hence (cid:90) Ω | T g ( x ) | q ω − q ( x ) d x ≤ (cid:18) q qNq − (cid:19) C q (cid:107) g (cid:107) qL q (Ω ,ω − q ) . (cid:3) A decomposition and Fractional Poincar´e inequalities
Let Ω in R n be an arbitrary bounded domain and { U t } t ∈ Γ an open covering of Ω.The weight ω : Ω → R > satisfies that ω p ∈ L (Ω). In addition, u Ω denotes the average | Ω | (cid:82) Ω u ( z )d z . For weighted spaces of functions, u Ω ,ω represents the weighted average ω (Ω) (cid:82) Ω u ( z ) ω ( z )d z , where ω (Ω) := (cid:82) Ω ω ( z )d z .Now, given a bounded domain U in R n and a nonnegative measurable function µ : U × U → R we introduce the following Poincar´e type inequalityinf c ∈ R (cid:107) u − c (cid:107) L p ( U,ω p ) ≤ C (cid:18)(cid:90) U (cid:90) U | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:19) /p , (3.6)where u ∈ L p ( U, ω p ). Notice that the right hand side in this inequality might beinfinite. The validity of (3.6) depends on U , p , µ and ω . The function µ ( x, y ) mightbe zero, however, ω ( x ) is strictly positive almost everywhere in Ω.Let us mention three examples. Examples 3.1. (1)
The weighted fractional Poincar´e inequality with µ ( x, y ) = | x − y | n + sp , where s ∈ (0 , , is the classical fractional Poincar´e inequality whichis clearly valid for any arbitrary bounded domain. (2) If µ ( x, y ) = χ Bx ( y ) | x − y | n + sp , where B x is the ball centered at x with radius τ d ( x ) for s, τ ∈ (0 , , then the inequality represents a more recently studied fractionalPoincar´e inequality whose validity depends on the geometry of the domain (re-fer to [4] for details). (3) Finally, µ ( x, y ) = ρ ( | x − y | ) | x − y | p , where ρ is a certain nonnegative radial function, isanother inequality which has also been studied recently (refer to [10] for details). Inequality (3.6) deals with an estimation of the distance to C of an arbitrary function u in L p (Ω , ω p ). The local-to-global argument used in this paper to study this Poincar´etype inequalities is based on the fact that L p (Ω , ω p ) is the dual space of L q (Ω , ω − q )and the existence of decompositions of functions in L q (Ω , ω − q ) orthogonal to C . Let usproperly define this set and a subspace: W := { g ∈ L q (Ω , ω − q ) : (cid:90) gϕ = 0 for all ϕ ∈ C} (3.7) W := { g ∈ W : supp( g ) intersects a finite number of U t } . (3.8)The integrability of ω p implies that L q (Ω , ω − q ) ⊂ L (Ω), then W and W are well-defined. Following Remark 2.5, the C -decomposition of functions in W stated inTheorem 2.4 is finite, which is not valid in general for functions in W . This propertyverified by the functions in W simplifies the proof of Lemma 3.3, which motivates thedefinition of this space.Now, we introduce the spaces W ⊕ ω p C = { g + αω p / g ∈ W and α ∈ C}S := W ⊕ ω p C = { g + αω p / g ∈ W and α ∈ C} . (3.9)It is not difficult to observe that L q (Ω , ω − q ) = W ⊕ ω p C and S is a subspace of L q (Ω , ω − q ). The following lemma, which was proved in [8, Lemma 3.1], states that S is also dense in L q (Ω , ω − q ) and uses in its proof the requirement that says that for each t ∈ Γ the set U t intersects a finite number of U s with s ∈ Γ. Lemma 3.2.
The space S is dense in L q (Ω , ω − q ) . Moreover, if g + αω p is an elementin S then (cid:107) g (cid:107) L q (Ω ,ω − q ) ≤ (cid:107) g + αω p (cid:107) L q (Ω ,ω − q ) . Lemma 3.3.
If there exists an open covering { U t } t ∈ Γ of Ω such that (3.6) is valid on U t for all t ∈ Γ , with a uniform constant C , and there exists a finite C -orthogonaldecomposition of any function g in W subordinate to { U t } t ∈ Γ , with the estimate (cid:88) t ∈ Γ (cid:107) g t (cid:107) qL q ( U t ,ω − q ) ≤ C q (cid:107) g (cid:107) qL q (Ω ,ω − q ) , then, there exists a constant C such that (cid:107) u − u Ω ,ω (cid:107) L p (Ω ,ω p ) ≤ C (cid:32)(cid:88) t ∈ Γ (cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:33) /p (3.10) is valid for any u ∈ L p (Ω , ω p ) . Moreover, the constant C = 2 C C holds in (3.10).Proof. Without loss of generality we can assume that u Ω ,ω = 0. We estimate thenorm on the left hand side of the inequality by duality. Thus, let g + ω p ψ be anarbitrary function in S , we refer to Lemma 3.2. Then, by using the finite C -orthogonaldecomposition of g we conclude that (cid:90) Ω u ( g + αω p ) = (cid:90) Ω ug = (cid:90) Ω u (cid:88) t ∈ Γ g t = (cid:88) t ∈ Γ (cid:90) U t ug t = (cid:88) t ∈ Γ (cid:90) U t ( u − c t ) g t . (3.11)Notice that the identity in the second line is valid for any t ∈ Γ and c t ∈ R . Next, by using the H¨older inequality in (3.11), the fact that (3.6) is valid on U t witha uniform constant C and, finally, the H¨older inequality over the sum, we obtain (cid:90) Ω u ( g + αω p ) ≤ (cid:88) t ∈ Γ inf c ∈ R (cid:107) u − c (cid:107) L p ( U t ,ω p ) (cid:107) g t (cid:107) L q ( U t ,ω − q ) ≤ C (cid:88) t ∈ Γ (cid:18)(cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:19) /p (cid:107) g t (cid:107) L q ( U t ,ω − q ) ≤ C (cid:32)(cid:88) t ∈ Γ (cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:33) /p (cid:32)(cid:88) t ∈ Γ (cid:107) g t (cid:107) qL q ( U t ,ω − q ) (cid:33) /q ≤ C C (cid:32)(cid:88) t ∈ Γ (cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:33) /p (cid:107) g (cid:107) L q ( U,ω − q ) ≤ C C (cid:32)(cid:88) t ∈ Γ (cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p µ ( x, y ) d y d x (cid:33) /p (cid:107) g + αω p (cid:107) L q ( U,ω − q ) . Finally, as S is dense in L q (Ω , ω − q ), by taking the supremum over all the functions g + αω p in S with (cid:107) g + αω p (cid:107) L q (Ω ,ω − q ) ≤ (cid:3) On fractional Poincar´e inequalities on cubes
In this section, we use the results stated in the previous two sections to show acertain fractional Poincar´e inequality on an arbitrary cube Q . Thus, in order to showthe existence of the C -decomposition, which is used later to apply Lemma 3.3, we definea tree covering { U t } t ∈ Γ of Q . This covering is only used in this section and for cubes. Inthe following section, we work with a different bounded domain, an arbitrary boundedJohn domain, which requires a different covering. However, let us warn the reader thatwe will keep the notation { U t } t ∈ Γ used in Section 3.The validity of the local inequality stated in the following proposition is well-known.We refer the reader to [3] for its proof. Proposition 4.1.
The fractional Poincar´e inequality inf c ∈ R (cid:107) u ( x ) − c (cid:107) L p ( U ) ≤ (cid:18) diam ( U ) n + sp | U | (cid:90) U (cid:90) U | u ( y ) − u ( x ) | p | y − x | n + sp d y d x (cid:19) /p holds for any bounded domain U in R n and ≤ p < ∞ . The following proposition is a special case of [4, Lemma 2.2]. In the present paper,we give a different proof which let us estimate the dependance of the constant withrespect to τ . Proposition 4.2.
Let Q in R n be a cube with side length l ( Q ) = L , < p < ∞ and τ ∈ (0 , . Then, the following inequality holds inf c ∈ R (cid:107) u ( x ) − c (cid:107) L p ( Q ) ≤ C n,p τ s − n L s (cid:18)(cid:90) Q (cid:90) Q ∩ B ( x,τL ) | u ( y ) − u ( x ) | p | y − x | n + sp d y d x (cid:19) /p , where C n,p depends only on n and p . Proof.
This result follows from Lemma 3.3 on the cube Q , where µ ( x, y ) = | x − y | n + sp and ω ≡
1. So, let us start by defining an appropriate tree covering of Q to obtain,via Theorem 2.4 and Remark 2.5, a finite C -decomposition of any functions in W . Let m ∈ N be such that √ n +3 τ < m ≤ √ n +3 τ and { A t } t ∈ Γ the regular partition of Q with m n open cubes. The side length of each cube is l ( A t ) = Lm . In the example shown inFigure 1, m = 4 and the index set Γ has 16 elements. Figure 1.
A tree covering of Q The tree covering of Q that we are looking for will be defined by enlarging the sets inthe covering { A t } t ∈ Γ in an appropriate way but keeping the tree structure of Γ, whichis introduced in the following lines. Indeed, we pick a cube A a , whose index will be theroot, and inductively define a tree structure in Γ such that the unique chain connecting t with a is associated to a chain of cubes connecting Q t with Q a , with minimal numberof cubes, such that two consecutive cubes share a n − A a is in the lower left corner and the tree structure is represented using blackarrows that“descend” to the root. Now that Γ has a tree structure, we define the treecovering { U t } t ∈ Γ of Q with the rectangles U t := ( A t ∪ A t p ) ◦ if t (cid:54) = a and U a := A a . Inorder to have a better understanding of the construction, notice that U t ∩ U t p = A t p for all t (cid:54) = a . Moreover, the index set Γ in the example with its tree structure has 7levels, from level 0 to level 6 (refer to page 3 for definitions), with only one index oflevel 6, whose rectangle U t appears in Figure 1 in a different color.Now, let us define the collection { B t } t (cid:54) = a of pairwise disjoint open cubes B t ⊆ U t ∩ U t p or equivalently B t ⊆ A t p . Given t (cid:54) = a , we split A t p into 3 n cubes with the same size.The open set B t is the cube in the regular partition of A t p whose closure intersects the n − A t p in the intersection ( A t ∩ A t p ). There are 3 n − cubes withthat property but we pick B t to be the one which does not share any part of any other n − A t p .The cubes in { B t } t (cid:54) = a have side length equal to L m and are represented in Figure1 by the 15 grey gradient small cubes. By its construction, it is easy to check that { B t } t (cid:54) = a is a collection of pairwise disjoint open cubes B t ⊆ U t ∩ U t p , hence, { U t } t ∈ Γ isa tree covering of Q with N = 2 n (it could also be less).By Theorem 2.4, there is a finite C -decomposition of functions { g t } t ∈ Γ subordinateto { U t } t ∈ Γ which satisfies (2.3) and (2 . | W s || B s | ≤ | Q || B s | = (3 m ) n , for all s ∈ Γ, thus, | g t ( x ) | ≤ | g ( x ) | + (3 m ) n T g ( x ) , for all t ∈ Γ and x ∈ U t . Next, using the continuity of T stated in Lemma 2.6 andsome straightforward calculations we conclude (cid:88) t ∈ Γ (cid:107) g t (cid:107) qL q ( U t ) ≤ q − N (cid:18) m ) nq q qNq − (cid:19) (cid:107) g (cid:107) qL q ( Q ) ≤ q +2 n qq − m ) nq (cid:107) g (cid:107) qL q ( Q ) ≤ q +2 nq n qq − (cid:16) √ n + 3 (cid:17) nq τ − nq (cid:107) g (cid:107) qL q ( Q ) . Hence, we have a finite C -decomposition of any function in W subordinate to { U t } t ∈ Γ with the constant in the estimate equal to C = (cid:18) q +2 nq n qq − (cid:19) /q (cid:16) √ n + 3 (cid:17) n τ − n . Now, from Proposition 4.1 and using that m > √ n +3 τ , we can conclude that inequality(3.6) is valid on each U t with an uniform constant C = ( n + 3) n/ p ( τ L ) s . Thus, using Lemma 3.3 we can claim that (cid:107) u − u Q (cid:107) L p ( Q ) ≤ C C (cid:32)(cid:88) t ∈ Γ (cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p | x − y | n + sp d y d x (cid:33) /p . Finally, diam( U t ) ≤ √ n + 3 Lm ≤ τ L , thus U t ⊂ B ( x, τ L ) for any x ∈ U t , thus, usingthe control on the overlapping of the tree covering given by N = 2 n , it follows that (cid:107) u − u Q (cid:107) L p ( Q ) ≤ C n,p τ − n ( τ L ) s (cid:18)(cid:90) Q (cid:90) Q ∩ B ( x,τL ) | u ( x ) − u ( y ) | p | x − y | n + sp d y d x (cid:19) /p , where C n,p = 2 (cid:18) q +2 nq n qq − (cid:19) /q (cid:16) √ n + 3 (cid:17) n ( n + 3) n/ p (2 n ) /p . (4.1) (cid:3) On fractional Poincar´e inequalities on John domains
In this section, we apply the results obtained in the previous sections on an arbitrarybounded John domain Ω. Its definition is recalled below. The weight ω ( x ) is definedas d F ( x ) β , where d F ( x ) denotes the distance from x to an arbitrary compact set F in ∂ Ω and β ≥
0. In the particular case where F = ∂ Ω, d ∂ Ω ( x ) is simply denoted as d ( x ).Notice that ω p belongs to L (Ω) for being Ω bounded and β nonnegative.A Whitney decomposition of Ω is a collection { Q t } t ∈ Γ of closed pairwise disjointdyadic cubes, which verifies(1) Ω = (cid:83) t ∈ Γ Q t .(2) diam( Q t ) ≤ dist( Q t , ∂ Ω) ≤ Q t ).(3) diam( Q s ) ≤ diam( Q t ) ≤ Q s ), if Q s ∩ Q t (cid:54) = ∅ . Here, dist( Q t , ∂ Ω) is the Euclidean distance between Q t and the boundary of Ω, denotedby ∂ Ω. The diameter of the cube Q t is denoted by diam( Q t ) and the side length iswritten as (cid:96) ( Q t ).Two different cubes Q s and Q t with Q s ∩ Q t (cid:54) = ∅ are called neighbors . This kind ofcovering exists for any proper open set in R n (refer to [13, VI 1] for details). Moreover,each cube Q t has less than or equal to 12 n neighbors. And, if we fix 0 < (cid:15) < anddefine (1 + (cid:15) ) Q t as the cube with the same center as Q t and side length (1 + (cid:15) ) timesthe side length of Q t , then (1 + (cid:15) ) Q t touches (1 + (cid:15) ) Q s if and only if Q t and Q s areneighbors.Given a Whitney decomposition { Q t } t ∈ Γ of Ω we refer by an expanded Whitneydecomposition of Ω to the collection of open cubes { Q ∗ t } t ∈ Γ defined by Q ∗ t := 98 Q ◦ t . Observe that this collection of cubes satisfies that χ Ω ( x ) ≤ n (cid:88) t ∈ Γ χ Q ∗ t ( x ) ≤ (12 n ) χ Ω ( x )for all x ∈ R n . We recall the definition of a bounded John domain . A bounded domain Ω in R n is aJohn domain with constants a and b , 0 < a ≤ b < ∞ , if there is a point x in Ω suchthat for each point x in Ω there exists a rectifiable curve γ x in Ω, parametrized by itsarc length written as length( γ x ), such thatdist( γ x ( t ) , ∂ Ω) ≥ a length( γ x ) t for all t ∈ [0 , length( γ x )]and length( γ x ) ≤ b. Examples of John domains are convex domains, uniform domains, and also domainswith slits, for example B (0 , \ [0 , Definition 5.1.
A bounded domain Ω in R n is a John domain if for any Whitneydecomposition { Q t } t ∈ Γ , there exists a constant K > a , that satisfies Q s ⊆ KQ t , (5.1)for any s, t ∈ Γ with s (cid:23) t . In other words, the shadow of Q t written as W t is containedin KQ t ; refer to (2.2). Moreover, the intersection of the cubes associated to adjacentindices, Q t and Q t p , is an n − { Q t } t ∈ Γ of a bounded John domain Ω in R n ,with constant K in the sense of (5.1), we define the tree covering { U t } t ∈ Γ of expandedWhitney cubes such that U t := Q ∗ t . (5.2) The overlapping is bounded by N = 12 n . Now, each open cube B t in the collection { B t } t (cid:54) = a shares the center with the n − Q t ∩ Q t p and has side length l t , where l t is the side length of Q t . It follows from the third condition in the Whitneydecomposition, and some calculations, that this collection is pairwise disjoint and B t ⊂ Q ∗ t ∩ Q ∗ t p = U t ∩ U t p . Moreover, it can be seen that | W t || B t | ≤ ( K l t ) n ( l t ) n = 72 n K n , (5.3)for all t ∈ Γ, with t (cid:54) = a . Lemma 5.2.
Let Ω in R n be a John domain with the constant K in the sense of (5.1), F in ∂ Ω a compact set and d F ( x ) the distance from x to F . Then, sup y ∈ W t d F ( y ) ≤ K √ n inf x ∈ B t d F ( x ) , for all t ∈ Γ . A similar inequality is also valid if we consider the weight d βF ( x ) with a nonnegativepower of the distance to F . Thus, this lemma implies, via Lemma 2.7, the continuityof the operator T from L q (Ω , d − qβF ) to itself with an estimation of its constant. Then,there exists a C -decomposition with a weighted estimate for a certain weight. Proof.
Given t ∈ Γ, with t (cid:54) = a , x ∈ B t and y ∈ W t := ∪ s (cid:23) t U s , we have to prove that d F ( y ) ≤ Kd F ( x ). Notice that d ( x ) ≤ d F ( x ) for all x ∈ Ω. Moreover, Q s ⊆ KQ t forall s (cid:23) t , then W t ⊆ KU t . In addition, d F ( y ) ≤ | y − x | + d F ( x ) ≤ diam( W t ) + d F ( x ) ≤ K diam( U t ) + d F ( x )= K diam( Q t ) + d F ( x ) . Finally, using the second property stated in the Whitney decomposition it follows that3 Q t ⊂ Ω. Then, as dist( Q ∗ t , ∂ Ω) ≥ dist( Q ∗ t , (3 Q t ) c ) ≥ l t , doing some calculations we can assert thatdiam( Q t ) ≤ √ n dist( Q ∗ t , ∂ Ω) ≤ √ n dist( Q ∗ t , ∂ Ω) . Thus, d F ( y ) ≤ K √ n dist( Q ∗ t , ∂ Ω) + d F ( x ) ≤ K √ n d ( x ) + d F ( x ) ≤ K √ n d F ( x ) + d F ( x ) . (cid:3) Now we are able to prove Theorem 1.1 and also to give the dependence of the constant C on the given value of τ and the constant K from (5.1). Proof of Theorem 1.1.
This result follows from Lemma 3.3 with the tree covering { U t } t ∈ Γ of Ω defined in (5.2), ω ( x ) := d βF ( x ) and µ ( x, y ) := d ps ( x ) d pβF ( x ) χ B ( x,τd ( x )) ( y ) | x − y | n + sp . (5.4)Notice that ω p belongs to L (Ω), the condition assumed at the beginning of Section 3.The validity of (3.6) on a cube U t , with a uniform constant C , follows from Proposition4.2. Indeed, by using the fact that U t is an expanded Whitney cube by a factor 9 / F ⊆ ∂ Ω, it follows that sup x ∈ U t d βF ( x ) ≤ β inf x ∈ U t d βF ( x ) . Thus, we haveinf c ∈ R (cid:107) u ( x ) − c (cid:107) L p ( U t ,d pβF ) ≤ C n,p τ s − n L st β (cid:18)(cid:90) U t (cid:90) U t | u ( x ) − u ( y ) | p | x − y | n + sp d pβF ( x ) χ B ( x,τL t ) ( y ) d y d x (cid:19) /p , where L t is the side length of U t and C n,p is the constant in (4.1). Now, observe that L t ≤ d ( x ) for all x ∈ U t . Indeed, if x ∈ Q t then L t = l t < √ n l t = diam( Q t ) ≤ dist( Q t , ∂ Ω) ≤ d ( x ) , where l t is the side length of Q t . Now, if x ∈ U t \ Q t then √ n l t ≤ dist( Q t , ∂ Ω) ≤ dist( U t , ∂ Ω) + √ n l t , hence, √ n l t ≤ dist( U t , ∂ Ω) and L t = l t < √ n l t ≤ dist( U t , ∂ Ω) ≤ d ( x ) . Then, the validity of L t ≤ d ( x ) for all x ∈ U t implies (3.6) for all U t , where µ ( x, y )is the function defined in (5.4), and the uniform constant C = C n,p τ s − n β . (5.5)Next, by Theorem 2.4, there is a finite C -decomposition of functions { g t } t ∈ Γ subor-dinate to { U t } t ∈ Γ of any function g in W which satisfies (2.3) and (2 . | g t ( x ) | ≤ | g ( x ) | + (72 K ) n T g ( x ) , for all t ∈ Γ and x ∈ U t .Now, ω ( x ) := d βF ( x ) fulfills the hypothesis of Lemma 2.7 where the constant in(2.5) is C = (3 K √ n ) β . The last assertion uses Lemma 5.2. Thus, the operator T iscontinuous from L := L q (Ω , d − qβF ) to itself with the norm (cid:107) T (cid:107) L → L ≤ (cid:18) qNq − (cid:19) /q (3 K √ n ) β . Hence, (cid:88) t ∈ Γ (cid:107) g t (cid:107) qL q ( U t ,d − qβF ) ≤ q − (cid:40)(cid:32)(cid:88) t ∈ Γ (cid:90) U t | g ( x ) | q d − qβF ( x ) (cid:33) + (72 K ) qn (cid:32)(cid:88) t ∈ Γ (cid:90) U t | T g ( x ) | q d − qβF ( x ) (cid:33)(cid:41) ≤ q − N (cid:26)(cid:90) Ω | g ( x ) | q d − qβF ( x ) d x + (72 K ) qn (cid:90) Ω | T g ( x ) | q d − qβF ( x ) d x (cid:27) ≤ q − N (cid:26) K ) qn q (cid:18) qNq − (cid:19) (3 K √ n ) qβ (cid:27) (cid:107) g (cid:107) qL q (Ω ,d − qβF ) ≤ q N (72 K ) qn (cid:18) qq − (cid:19) (3 K √ n ) qβ (cid:107) g (cid:107) qL q (Ω ,d − qβF ) = 4 q n qn (3 √ n ) qβ (cid:18) qq − (cid:19) K q ( n + β ) (cid:107) g (cid:107) qL q (Ω ,d − qβF ) . Therefore, we have a C -decomposition subordinate to { U t } t ∈ Γ with constant C = 4(12) n/q (72) n (3 √ n ) β (cid:18) qq − (cid:19) /q K n + β . (5.6)Finally, inequality (3.10) and the control on the overlapping of the tree covering by N = 12 n implies (1.2). (cid:3) Remark 5.3.
Notice that the proof of Theorem 1.1 provides an explicit constant C = 2 C C for inequality (1.2), where C and C are described respectively in (5.6)and (5.5).The next result, similar to Proposition 4.1, follows from the H¨older inequality (equiv-alently, from Minkowski’s integral inequality). Proposition 5.4.
Let ρ : R n \ { } → R be a positive radial Lebesgue measurablefunction which is increasing with respect to the radius. Then, the fractional Poincar´etype inequality (cid:107) u ( x ) − u U (cid:107) L p ( U ) ≤ diam ( U ) n/p ρ (diam( U )) | U | /p (cid:18)(cid:90) U (cid:90) U | u ( y ) − u ( x ) | p | y − x | n ( ρ | y − x | ) p d y d x (cid:19) /p (5.7) holds for any bounded domain U in R n and < p < ∞ , where u U := | U | (cid:82) U u ( y )d y .Proof. (cid:90) U | u ( x ) − u U | p d x = (cid:90) U (cid:12)(cid:12)(cid:12)(cid:12) | U | (cid:90) U u ( x ) − u ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) p d x ≤ | U | (cid:90) U (cid:90) U | u ( x ) − u ( y ) | p d y d x ≤ diam( U ) n { ρ (diam( U )) } p | U | (cid:90) U (cid:90) U | u ( x ) − u ( y ) | p | x − y | n ( ρ | x − y | ) p d y d x. (cid:3) Remark 5.5. If ρ ( x ) = | x | s , with s ∈ (0 , We generalize the fractional Poincar´e inequality stated in Theorem 1.1 by replacingthe fractional derivatives given by the power functions | x | s , with 0 < s <
1, by generalincreasing and positive radial functions ρ | x | . Theorem 5.6.
Let Ω in R n be a bounded John domain and < p < ∞ . Given anarbitrary compact set F in ∂ Ω , a parameter β ≥ and a positive radial Lebesguemeasurable function ρ : R n \ { } → R increasing with respect to the radius, there existsa constant C such that (cid:18)(cid:90) Ω | u ( x ) − u Ω ,ω | p d pβF ( x )d x (cid:19) /p ≤ C (cid:18)(cid:90) Ω (cid:90) Ω ∩ B ( x,d ( x )) | u ( x ) − u ( y ) | p | x − y | n ( ρ | x − y | ) p [ ρ (2 d ( x ))] p d pβF ( x ) d y d x (cid:19) /p (5.8) for all function u ∈ L p (Ω , d ( x ) pβ ) . We denote by d ( x ) and d F ( x ) the distance from x to ∂ Ω and F respectively, and by u Ω ,ω the weighted average d pβF (Ω) (cid:82) Ω u ( z ) d pβF ( z )d z .In addition, the constant C in (5.8) can be written as C = C n,p,β K n + β , where K is the geometric constant introduced in (5.1).Proof. This proof mimics the one of Theorem 1.1 with Proposition 5.4 instead of Propo-sition 4.2. Indeed, we will use again the tree covering { U t } t ∈ Γ of Ω defined in (5.2) andthe weight ω ( x ) = d βF ( x ), however, in this case µ ( x, y ) is defined as µ ( x, y ) := [ ρ (2 d ( x ))] p d pβF ( x ) | x − y | n ( ρ | x − y | ) p . We only have to show that (3.6) is verified on U t , for all t , with uniform constant. Thisfact follows from (5.7) by using the inequality diam( U t ) ≤ d ( x ) for all x ∈ U t . (cid:3) Acknowledgements
This research was initiated when the second author visited the University of Helsinkiduring the Summer of 2016. This author gratefully acknowledges Professor MichelLapidus from University of California Riverside for his financial support for the ex-penses of the trip.
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