Abstract
For the natural two-parameter filtration \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) \) on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on \(L^p(\Omega _0)\) for \(p \in (1, \infty )\). At the end, we consider \(L^p(\Omega _0)\) boundedness of martingale transforms. If the building is of \({\text {GL}}(3, \mathbb {Q}_p)\), then \(\Omega _0\) can be identified with p-adic Heisenberg group.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Avoid common mistakes on your manuscript.
1 Introduction
Let \((\Omega , \mathcal {F}, \pi )\) be a \(\sigma \)-finite measure space. A sequence of \(\sigma \)-algebras \((\mathcal {F}_n : n \in \mathbb {Z})\) is a filtration if \(\mathcal {F}_n \subset \mathcal {F}_{n+1}\). Given f a locally integrable function on \(\Omega \) by \(\mathbb {E}[f | \mathcal {F}_n]\), we denote its conditional expectation value with respect to \(\mathcal {F}_n\). Let \(M^*\) and S denote, respectively, the maximal function and the square function defined by
and
where \(d_n f = f_n - f_{n-1}\). The Hardy and Littlewood maximal estimate (see [8]) implies that
from where it is easy to deduce that for \(p \in (1, \infty ]\)
For the square function, if \(p \in (1, \infty )\), then there is \(C_p > 1\) such that
The inequality (1.2) goes back to Paley [12], and has been reproved in many ways, for example, [2,3,4, 7, 10]. Its main application is in proving the \(L^p\)-boundedness of martingale transforms (see [2]), that is, for operators of the form
where \((a_n : n \in \mathbb {Z})\) is a sequence of uniformly bounded functions such that \(a_{n+1}\) is \(\mathcal {F}_n\)-measurable.
In 1975, Cairoli and Walsh (see [5]) have started to generalize the theory of martingales to two-parameter cases. Let us recall that a sequence of \(\sigma \)-fields \((\mathcal {F}_{n, m} : n, m \in \mathbb {Z})\) is a two-parameter filtration if
Then \((f_{n, m} : n, m \in \mathbb {Z})\) is a two-parameter martingale if
Observe that conditions (1.3) and (1.4) impose a structure only for comparable indices. In that generality, it is hard, if not impossible, to build the Littlewood–Paley theory. This lead to the introduction of other (smaller) classes of martingales (see [19, 20]). In particular, in [5], Cairoli and Walsh introduced the following condition
where
Under (\(F_4\)), the result obtained by Jensen, Marcinkiewicz, and Zygmund in [9] implies that the maximal function
is bounded on \(L^p(\Omega )\) for \(p \in (1, \infty ]\). In this context, the square function is defined by
where \(d_{n, m}\) denote the double difference operator, i.e.
In [11], it was observed by Metraux that the boundedness of S on \(L^p(\Omega )\) for \(p \in (1, \infty )\) is implied by the one parameter Littlewood–Paley theory. Also the concept of a martingale transform has a natural generalization, that is,
where \((a_{n, m} : n,m \in \mathbb {Z})\) is a sequence of uniformly bounded functions such that \(a_{n+1, m+1}\) is \(\mathcal {F}_{n, m}\)-measurable.
In this article, we are interested in a case when the condition (\(F_4\)) is not satisfied. The simplest example may be obtained by considering the Heisenberg group together with the non-isotropic two parameter dilations
Since in this setup the dyadic cubes do not posses the same properties as the Euclidean cubes, it is more convenient to work on the p-adic version of the Heisenberg group. We observe that this group can be identified with \(\Omega _0\), a subset of a boundary of the building of \({\text {GL}}(3, \mathbb {Q}_p)\) consisting of the points opposite to a given \(\omega _0\). The set \(\Omega _0\) has a natural two-parameter filtration \((\mathcal {F}_{n, m} : n, m \in \mathbb {Z})\) (see Sect. 2 for details). The maximal function and the square function are defined by (1.5) and (1.6), respectively. The results we obtain are summarized in the following three theorems.
Theorem A
For each \(p \in (1, \infty ]\), there is \(C_p > 0\) such that for all \(f \in L^p\big (\Omega _0\big )\)
Theorem B
For each \(p \in (1, \infty )\), there is \(C_p > 1\) such that for all \(f \in L^p\big (\Omega _0\big )\)
Theorem C
If \((a_{n, m} : n, m \in \mathbb {Z})\) is a sequence of uniformly bounded functions such that \(a_{n+1, m+1}\) is \(\mathcal {F}_{n, m}\)-measurable, then the martingale transform
is bounded on \(L^p\big (\Omega _0\big )\), for all \(p \in (1, \infty )\).
Let us briefly describe methods we use. First, we observe that instead of (\(F_4\)) the stochastic basis satisfies the remarkable identity (2.2). Based on it, we show that the following pointwise estimate holds
proving the maximal theorem. Thanks to the two-parameter Khintchine’s inequality, to bound the square function S, it is enough to show Theorem C. To do so, we define a new square function \(\mathcal {S}\) which has a nature similar to the square function used in the presence of (\(F_4\)). Then, we adapt the technique developed by Duoandikoetxea and Rubio de Francia in [6] (see Theorem 3). This implies \(L^p\)-boundedness of S. Since S does not preserve the \(L^2\) norm, the lower bound requires an extra argument. Namely, we view the square function S as an operator with values in \(L^p(\ell ^2)\) and take its dual. As a consequence of Theorem 3 and the identity (4.7), the latter is bounded on \(L^p\).
Finally, let us comment on the behavior of the maximal function \(M^*\) close to \(L^1\). Based on the pointwise estimate (1.7), in view of [8], we conclude that \(M^*\) is of weak-type for functions in the Orlicz space \(L (\log L)^3\). To better understand the maximal function \(M^*\), we investigate exact behavior close to \(L^1\). This together with weighted estimates is the subject of the forthcoming paper. It is also interesting how to extend Theorems A, B and C to higher rank and other types of affine buildings.
1.1 Notation
For two quantities \(A>0\) and \(B>0\), we say that \(A \lesssim B\) (\(A \gtrsim B\)) if there exists an absolute constant \(C>0\) such that \(A\le CB\) (\(A\ge CB\)).
If \(\lambda \in P\) we set \({|{\lambda } |} = \max \left\{ {|{\lambda _1} |}, {|{\lambda _2} |}\right\} \).
2 Triangle Buildings
2.1 Coxeter Complex
We recall basic facts about the \(A_2\) root system and the \(\tilde{A}_2\) Coxeter group. A general reference is [1]. Let \(\mathfrak {a}\) be the hyperplane in \(\mathbb {R}^3\) defined as
We denote by \(\{e_1, e_2, e_3\}\) the canonical orthonormal basis of \(\mathbb {R}^3\) with respect to the standard scalar product \({\langle \cdot , \cdot \rangle }\). We set \(\alpha _1 = e_2 - e_1\), \(\alpha _2 = e_3 - e_2\), \(\alpha _0 = e_3 - e_1\) and \(I = \{0, 1, 2 \}\). The \(A_2\) root system is defined by
We choose the base \(\{\alpha _1, \alpha _2\}\) of \(\Phi \). The corresponding positive roots are \(\Phi ^+ = \{\alpha _0, \alpha _1, \alpha _2\}\). Denote by \(\{\lambda _1, \lambda _2\}\) the basis dual to \(\{\alpha _1, \alpha _2\}\); its elements are called the fundamental co-weights. Their integer combinations, form the co-weight lattice P. As
in Fig. 1, we always draw \(\lambda _1\) pointing up and to the left and \(\lambda _2\) up and to the right. Likewise \(\lambda _1-\lambda _2\) is drawn pointing directly left, while \(\lambda _2-\lambda _1\) points directly right. Because \({\langle \lambda _1, \alpha _0\rangle }={\langle \lambda _2, \alpha _0\rangle }=1\), we see that for any \(\lambda \in P\) the expression \({\langle \lambda , \alpha _0\rangle }\) represents the vertical level of \(\lambda \). For \(\lambda =i\lambda _1+j\lambda _2\), that level is \(i+j\).
Let \(\mathcal {H}\) be the family of affine hyperplanes, called walls,
where \(j \in I\), \(k \in \mathbb {Z}\). To each wall \(H_{j; k}\), we associate \(r_{j;k}\) the orthogonal reflection in \(\mathfrak {a}\), i.e.
Set \(r_1 = r_{1; 0}\), \(r_2 = r_{2; 0}\) and \(r_0 = r_{0; 1}\). The finite Weyl group \(W_0\) is the subgroup of \({\text {GL}}(\mathfrak {a})\) generated by \(r_1\) and \(r_2\). The affine Weyl group W is the subgroup of \({\text {Aff}}(\mathfrak {a})\) generated by \(r_0\), \(r_1\) and \(r_2\).
Let \(\mathcal {C}\) be the family of open connected components of \(\mathfrak {a} \setminus \bigcup _{H \in \mathcal {H}} H\). The elements of \(\mathcal {C}\) are called chambers. By \(C_0\), we denote the fundamental chamber, i.e.
The group W acts simply transitively on \(\mathcal {C}\). Moreover, \(\overline{C_0}\) is a fundamental domain for the action of W on \(\mathfrak {a}\) (see e.g. [1, VI, §1-3]). The vertices of \(C_0\) are \(\{0, \lambda _1, \lambda _2\}\). The set of all vertices of all \(C\in \mathcal {C}\) is denoted by \(V(\Sigma )\). Under the action of W, \(V(\Sigma )\) is made up of three orbits, W(0), \(W(\lambda _1)\), and \(W(\lambda _2)\). Vertices in the same orbit are said to have the same type. Any chamber \(C\in \mathcal {C}\) has one vertex in each orbit or in other words one vertex of each of the three types.
The family \(\mathcal {C}\) may be regarded as a simplicial complex \(\Sigma \) by taking as the simplexes all non-empty subsets of vertices of C, for all \(C \in \mathcal {C}\). Two chambers C and \(C^{\prime }\) are i-adjacent for \(i \in I\) if \(C = C'\) or if there is \(w \in W\) such that \(C=wC_0\) and \(C^{\prime }=wr_iC_0\). Since \(r_i^2=1\) this defines an equivalence relation.
The fundamental sector is defined by
Given \(\lambda \in P\) and \(w \in W_0\) the set \(\lambda + w \mathcal {S}_0\) is called a sector in \(\Sigma \) with base vertex \(\lambda \). The angle spanned by a sector at its base vertex is \(\pi /3\).
2.2 The Definition of Triangle Buildings
For the theory of affine buildings, we refer the reader to [13]. See also the first author’s expository paper [14], for an elementary introduction to the p-adics, and to precisely the sort of the buildings which this paper deals with.
A simplicial complex \(\mathscr {X}\) is an \(\tilde{A}_2\) building, or as we like to call it, a triangle building, if each of its vertices is assigned one of the three types, and if it contains a family of subcomplexes called apartments such that
-
1.
Each apartment is type-isomorphic to \(\Sigma \),
-
2.
Any two simplexes of \(\mathscr {X}\) lie in a common apartment,
-
3.
For any two apartments, \(\mathscr {A}\) and \(\mathscr {A}^{\prime }\), having a chamber in common, there is a type-preserving isomorphism \(\psi : \mathscr {A} \rightarrow \mathscr {A}^{\prime }\) fixing \(\mathscr {A} \cap \mathscr {A}^{\prime }\) pointwise.
We assume also that the system of apartments is complete, meaning that any subcomplex of \(\mathscr {X}\) type-isomorphic to \(\Sigma \) is an apartment. A simplex C is a chamber in \(\mathscr {X}\) if it is a chamber for some apartment. Two chambers of \(\mathscr {X}\) are i-adjacent if they are i-adjacent in some apartment. For \(i\in I\) and for a chamber C of \(\mathscr {X}\), let \(q_i(C)\) be equal to
It may be proved that \(q_i(C)\) is independent of C and of i. Denote the common value by q, and assume local finiteness: \(q<\infty \). Any edge of \(\mathscr {X}\), i.e., any 1-simplex, is contained in precisely \(q+1\) chambers.
It follows from the axioms that the ball of radius one about any vertex x of \(\mathscr {X}\) is made up of x itself, which is of one type, \(q^2+q+1\) vertices of a second type, and a further \(q^2+q+1\) vertices of the third type. Moreover, adjacency between vertices of the second and third types makes them into, respectively, the points and the lines of a finite projective plane.
A subcomplex \(\mathscr {S}\) is called a sector of \(\mathscr {X}\) if it is a sector in some apartment. Two sectors are called equivalent if they contain a common subsector. Let \(\Omega \) denote the set of equivalence classes of sectors. If x is a vertex of \(\mathscr {X}\) and \(\omega \in \Omega \), there is a unique sector denoted \([x,\omega ]\) which has base vertex x and represents \(\omega \).
Given any two points \(\omega \) and \(\omega ^{\prime } \in \Omega \), one can find two sectors representing them which lie in a common apartment. If that apartment is unique, we say that \(\omega \) and \(\omega ^{\prime }\) are opposite, and denote the unique apartment by \([\omega ,\omega ^{\prime }]\). In fact, \(\omega \) and \(\omega '\) are opposite precisely when the two sectors in the common apartment point in opposite directions in the Euclidean sense.
2.3 Filtrations
We fix once and for all an origin vertex \(O \in \mathscr {X}\) and a point \(\omega _0 \in \Omega \). Choose O so that it has the same type as the origin of \(\Sigma \). Let \(\mathscr {S}_0=[O,\omega _0]\) be the sector representing \(\omega _0\) with base vertex O. By \(\Omega _0\), we denote the subset of \(\Omega \) consisting of \(\omega \)’s opposite to \(\omega _0\). For purposes of motivation only, we recall that if \(\mathscr {X}\) is the building of \({\text {GL}}(3,\mathbb {Q}_p)\), then \(\Omega _0\) can be identified with the p-adic Heisenberg group (see Appendix 1 for details).
Let \(\mathscr {A}_0\) be any apartment containing \(\mathscr {S}_0\). By \(\psi \), we denote the type-preserving isomorphism between \(\mathscr {A}_0\) and \(\Sigma \) such that \(\psi (\mathscr {S}_0) = -S_0\). We set \(\rho = \psi \circ \rho _0\) where \(\rho _0\) is the retraction from \(\mathscr {X}\) to \(\mathscr {A}_0\). With these definitions, \(\rho :\mathscr {X}\rightarrow \Sigma \) is a type-preserving simplicial map, and for any \(\omega \in \Omega _0\) the apartment \([\omega ,\omega _0]\) maps bijectively to \(\Sigma \) with \(\omega _0\) mapping to the bottom (of Fig. 1) and \(\omega \) mapping to the top.
For any vertex x of \(\mathscr {X}\), define the subset \(E_x \subset \Omega _0\) to consist of all \(\omega \)’s such that x belongs to \([\omega , \omega _0]\); an equivalent condition is that \([x,\omega _0]\subseteq [\omega ,\omega _0]\). Fix \(\lambda \in P\). By \(\mathcal {F}_\lambda \), we denote the \(\sigma \)-field generated by sets \(E_x\) for \(x \in \mathscr {X}\) with \(\rho (x) = \lambda \). There are countably many such x, and the corresponding sets \(E_x\) are mutually disjoint, and hence, \(\mathcal {F}_\lambda \) is a countably generated atomic \(\sigma \)-field.
Let \(\preceq \) denote the partial order on P where \(\lambda \preceq \mu \) if and only if \({\langle \lambda - \mu , \alpha _1\rangle } \le 0\) and \({\langle \lambda - \mu , \alpha _2\rangle } \le 0\). If we draw and orient \(\Sigma \) as in Fig. 1, then \(\lambda \preceq \mu \) exactly when \(\mu \) lies in the sector pointing upward from \(\lambda \).
Proposition 2.1
If \(\lambda \preceq \mu \), then \(\mathcal {F}_\lambda \subset \mathcal {F}_\mu \).
Proof
Choose any vertex x so that \(\rho (x)=\mu \). Because \(\lambda \preceq \mu \), there is a unique vertex y in the sector \([x,\omega _0]\) so that \(\rho (y)=\lambda \). For any \(\omega \in E_x\), the apartment \([\omega ,\omega _0]\) contains x, and hence, it contains \([x,\omega _0]\), which hence contains y. This establishes that \(E_x\subseteq E_y\). In other words, each atom of \(\mathcal {F}_\mu \) is a subset of some atom of \(\mathcal {F}_\lambda \). Hence, each atom of \(\mathcal {F}_\lambda \) is a disjoint union of atoms of \(\mathcal {F}_\mu \). \(\square \)
In fact, Proposition 2.1 says that \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) =\left( {\mathcal {F}_{i\lambda _1+j\lambda _2}}: {i,j\in \mathbb {Z}}\right) \) is a two parameter filtration. Let
Let \(\pi \) denote the unique \(\sigma \)-additive measure on \((\Omega _0, \mathcal {F})\) such that for \(E_x \in \mathcal {F}_\lambda \)
All \(\sigma \)-fields in this paper should be extended so as to include \(\pi \)-null sets.
A function \(f(\omega )\) on \(\Omega _0\) is \(\mathcal {F}_\lambda \)-measurable if it depends only on that part of the apartment \([\omega ,\omega _0]\) which retracts under \(\rho \) to the sector pointing downward from \(\lambda \). For \(i, j \in \mathbb {Z}\) set
A function \(f(\omega )\) on \(\Omega _0\) is \(\mathcal {F}_{i,\infty }\)-measurable (respectively \(\mathcal {F}_{\infty ,j}\)-measurable) if it depends only on that part of the apartment which retracts to a certain “lower” half-plane with boundary parallel to \(\lambda _2\) (respectively \(\lambda _1\)).
If \(\mathcal {F}^{\prime }\) is \(\sigma \)-subfield of \(\mathcal {F}\), we denote by \(\mathbb {E}[f | \mathcal {F}^{\prime }]\) the Radon–Nikodym derivative with respect to \(\mathcal {F}^{\prime }\). If \(\mathcal {F}^{\prime \prime }\) is another \(\sigma \)-subfield of \(\mathcal {F}\), we write
The \(\sigma \)-field generated by \(\mathcal {F}^{\prime } \cup \mathcal {F}^{\prime \prime }\) is denoted by \(\mathcal {F}^{\prime } \vee \mathcal {F}^{\prime \prime }\). We write \(f_\lambda = \mathbb {E}_\lambda f = \mathbb {E}[f | \mathcal {F}_\lambda ]\) for \(\lambda \in P\). If \(\lambda \preceq \mu \), then it follows from Proposition 2.1 that \(\mathbb {E}_\mu \mathbb {E}_\lambda =\mathbb {E}_\lambda \mathbb {E}_\mu =\mathbb {E}_\lambda \).
We note that the Cairoli–Walsh condition (\(F_4\)) introduced in [5] is not satisfied, i.e.
Instead of (\(F_4\)), we have
Lemma 2.2
For a locally integrable function f on \(\Omega _0\)
and likewise if we exchange \(\lambda _1\) and \(\lambda _2\).
Proof
For the proof of (2.1) it is enough to consider \(f = {\mathbf {1}_{{E_{p_1}}}}\) where \(p_1\) is a vertex in \(\mathscr {X}\) such that \(\rho (p_1) = \lambda + \lambda _1\). Let \(\mathscr {S}\) be the sector \([p_1,\omega _0]\) and let x be the unique vertex of \(\mathscr {S}\) with \(\rho (x) = \lambda \). The ball in \(\mathscr {X}\) of radius 1 around x has the structure of a finite projective plane.
In Fig. 2, the spot marked x is for vertices of \(\mathscr {X}\) which retract via \(\rho \) to \(\lambda \). Recall that \(E_x\) is an atom of the \(\sigma \)-field \(\mathcal {F}_\lambda \). The spot marked \(p_1\) is for vertices retracting to \(\lambda +\lambda _1\); the spot marked l is for vertices retracting to \(\lambda +\lambda _2\); the spot marked \(l_1\) is for vertices retracting to \(\lambda +\lambda _1-\lambda _2\); etc. In the ball of radius 1 around x, only x itself retracts to the spot marked x. The line-type vertex known as \(l_0\) is the only vertex in the ball retracting to its spot; q line-type vertices retract to the same spot as \(l_1\); the remaining \(q^2\) line-type vertices retract to the spot marked l. Likewise, \(p_0\) is the unique point-type vertex of the ball retracting to its spot; q point-type vertices retract to the spot marked p; \(q^2\) retract to the same spot as \(p_1\). It follows that
and
where \(p'\) runs through the point-type vertices of the ball, l runs through the line-type vertices of the ball, and \(\sim \) stands for the incidence relation. We have
Therefore, we obtain
which finishes the proof of (2.1). Applying one more average to the next to the last expression of (2.4), we get
For any line \(l \not \sim p_0\), there are q points \(p'\) such that \(p' \sim l\) and \(p^{\prime } \not \sim l_0\) and among them there is exactly one incident to \(l_1\). Hence, in the last sum, each line \(l \not \sim p_0\) appears \(q-1\) times. Thus, we can write
proving (2.2). \(\square \)
The following lemma describes the composition of projections on the same level.
Lemma 2.3
If \(k, j \in \mathbb {Z}\) are such that \(k \ge j \ge 0\) or \(k \le j \le 0\) then
Proof
We carry out the proof for \(k\ge j\ge 0\). For any \(\omega \in \Omega _0\), there is a connected chain of vertices \(\left( {x_i}: {0\le i\le k}\right) \subseteq [\omega ,\omega _0]\) with \(\rho (x_i)=\lambda +k(\lambda _2-\lambda _1)\). Suppose, conversely, that \(\left( {x_i}: {0\le i\le k}\right) \) is a connected chain of vertices and that \(\rho (x_i)= \lambda +k(\lambda _2-\lambda _1)\). Construct a subcomplex \(\mathscr {B}\subset \mathscr {X}\) by putting together \(\left( {[x_i,\omega _0]}: {0\le i\le k}\right) \), the edges between the \(x_i\)’s and the triangles pointing downward from those edges to \(\omega _0\). Referring to Fig. 3, the extra triangle pointing downward from the first edge has vertices \(x_0\), \(x_1\), and \(y_0\). Note that \([x_0,\omega _0]\cap [x_1,\omega _0]=[y_0,\omega _0]\). Proceeding one step at a time, one may verify that the restriction of \(\rho \) to \(\mathscr {B}\) is an injection and that \(\mathscr {B}\) and \(\rho (\mathscr {B})\) are isomorphic complexes.
By basic properties of affine buildings, one knows it is possible to extend \(\mathscr {B}\) to an apartment. Any such apartment will retract bijectively to \(\Sigma \), and will be of the form form \([\omega ,\omega _0]\) where \(\omega \) is the equivalence class represented by the upward pointing sectors of the apartment. Moreover, using the definition of \(\pi \) one may calculate that
The important point is that the measure of the set depends only on the level of \(\lambda \) and the length of the chain.
Basic properties of affine buildings imply that any apartment containing \(x_0\) and \(x_k\) contains the entire chain. Hence,
Fix \(x_0\). Proceeding one step at a time, one sees there are \(q^k\) connected chains \(\left( {x_i}: {0\le i\le k}\right) \) with \(\rho (x_i)=\lambda +k(\lambda _2-\lambda _1)\). Consequently
Likewise
which is the same thing. \(\square \)
Consider \(\mathbb {E}_\lambda \mathbb {E}_\mu \). If \(\lambda \preceq \mu \) then the product is equal to \(\mathbb {E}_\lambda \); similarly if \(\mu \preceq \lambda \). If \(\lambda \) and \(\mu \) are incomparable, the following lemma allows us to reduce to the case where \(\lambda \) and \(\mu \) are on the same level.
Lemma 2.4
Suppose \(\lambda \in P\) and
for \(i, k \in \mathbb {N}\). Then for any locally integrable function f on \(\Omega _0\)
and likewise if we exchange \(\lambda _1\) and \(\lambda _2\).
Proof
We first prove (2.6) for \(i=1\) and \(k=1\). Because \(\mathbb {E}[f|\mathcal {F}_{\lambda ^{\prime }}] =\mathbb {E}[f|\mathcal {F}_\lambda |\mathcal {F}_{\lambda ^{\prime }}]\), it is sufficient to consider \(f={\mathbf {1}_{{E_{p_1}}}}\) where \(\rho (p_1)=\lambda \). Use Fig. 2 to fix the notation, and note that if \(p_1\) retracts to \(\lambda \), then x retracts to \(\lambda '\) and p to \(\mu \). One calculates:
Next consider the case \(i=1\), \(k>1\). Set \(\mu ^{\prime }=\mu +\lambda _1\), \(\nu = \mu + \lambda _1 - \lambda _2\) and \(\nu ' = \nu + \lambda _1\) (see Fig. 4). Since \(\mathcal {F}_\mu \) is a subfield of \(\mathcal {F}_{\mu ^{\prime }}\), we have
Thus, applying Lemma 2.3, we obtain
where in the last step we have used the case \(k = 1\). Now apply induction on k and Lemma 2.3 again to get
To extend to the case \(i>1\), use induction on i and observe that
The proof of (2.8) is analogous, starting with the case \(i=1\), \(k=0\). Identity that (2.6) can be read as \(\mathbb {E}_\mu \mathbb {E}_\lambda = \mathbb {E}_\mu \mathbb {E}_{\lambda '}\). The expectation operators are orthogonal projections with respect to the usual inner product, and taking adjoints gives \(\mathbb {E}_\lambda \mathbb {E}_\mu = \mathbb {E}_{\lambda '} \mathbb {E}_\mu \) which is (2.7). To be more precise, one takes the inner product of either side of (2.7) with some nice test function, applies self-adjointness, and reduces to (2.6). Likewise, (2.9) follows from (2.8). \(\square \)
Lemma 2.5
Suppose \(\lambda = i \lambda _1 + j \lambda _2\), \(\mu = \lambda + k(\lambda _1 - \lambda _2)\). Then for any locally integrable function f on \(\Omega _0\)
Proof
Suppose \(k \ge 0\). By Lemma 2.4 for any \(j' \ge 0\), we have
So if g is \(\mathcal {F}_{\lambda + j^{\prime } \lambda _2}\)-measurable and compactly supported, then
The test functions g which we use are sufficient to distinguish between one \(\mathcal {F}_{i,\infty }\)-measurable function and another. Since \(\mathbb {E}_{i,\infty }\mathbb {E}_\mu f\) and \(\mathbb {E}_\lambda \mathbb {E}_\mu f\) are both \(\mathcal {F}_{i,\infty }\)-measurable, the proof is done.
3 Littlewood-Paley Theory
3.1 Maximal Functions
The natural maximal function \(M^*\) for a locally integrable function f on \(\Omega _0\) is defined by
In addition, we define two auxiliary single-parameter maximal functions
Lemma 3.1
Let \(\lambda \in P\) and \(k \in \mathbb {N}\). For any non-negative locally integrable function f on \(\Omega _0\)
Proof
We may assume \(\lambda = 0\). Let us define (see Fig. 5)
We show
Let \(g = \mathbb {E}[f| \mathcal {F}_{\mu }]\). By two applications of Lemma 2.3, we can write
and by Lemma 2.2
Hence,
By repeated application of Lemma 2.4, we have
and
which finishes the proof of (3.1). By iteration of (3.1), we obtain
which together with Lemma 2.2 implies
\(\square \)
Theorem 1
For each \(p \in (1, \infty ]\) there is \(C_p > 0\) such that
Proof
Inequalities (3.2) are two instances of Doob’s well-known maximal inequality for single parameter martingales (see e.g. [15]). To show (3.3), consider a non-negative \(f \in L^p(\Omega _0, \mathcal {F}_\mu )\). Fix \(\lambda \in P\). Since \(f \in L^p(\Omega _0, \mathcal {F}_{\mu '})\) for any \(\mu ' \succeq \mu \) we may assume \(\mu \succeq \lambda \). Let
By Lemma 3.1,
If \(\lambda = i \lambda _1 + j \lambda _2\), then repeated application of Lemma 2.5 gives
By taking the supremum over \(\lambda \in P\), we get
Hence, by (3.2), we obtain (3.3) for \(f \in L^p(\Omega _0, \mathcal {F}_\mu )\). Finally, a standard Fatou’s lemma argument establishes the theorem for arbitrary \(f \in L^p(\Omega _0)\).
3.2 Square Function
Let f be a locally integrable function on \(\Omega _0\). Given \(i, j \in \mathbb {Z}\), we define projections
Note that \(L_i\) (respectively \(R_j\)) is the martingale difference operator for the filtration \(\left( {\mathcal {F}_{i, \infty }}: {i \in \mathbb {Z}}\right) \) (respectively \(\left( {\mathcal {F}_{\infty , j}}: {j \in \mathbb {Z}}\right) \)). For \(\lambda = i \lambda _1 + j \lambda _2\), we set
The following development is inspired by that of Stein and Street in [17]. We start by defining the corresponding square function.
We will also need its dual counterpart
Theorem 2
For every \(p \in (1, \infty )\) there is \(C_p > 1\) such that
Moreover, on \(L^2(\Omega _0)\) square functions \(\mathcal {S}\) and \(\mathcal {S}^\star \) preserve the norm.
Proof
Since
preserve the norm on \(L^2(\Omega _0)\), we have
Hence, \(\mathcal {S}\) preserves the norm.
For \(p \ne 2\), we use the two-parameter Khintchine inequality (see [12]) and bounds on single parameter martingale transforms (see [2, 15, 18]). Let \(\left( {\epsilon _i}: {i \in \mathbb {Z}}\right) \) and \((\epsilon ^{\prime }_j : j \in \mathbb {Z})\) be sequences of real numbers, with absolute values bounded above by 1. For \(N \in \mathbb {N}\), we consider the operator
which may be written as a composition \(\mathcal {L}_N \mathcal {R}_N\) where
Since by Burkholder’s inequality (see [2, 15]) the operators \(\mathcal {R}_N\) and \(\mathcal {L}_N\) are bounded on \(L^p(\Omega _0)\) with bounds uniform in N, we have
Setting \(r_k\) to be the Rademacher function, by Khintchine’s inequality, we get
which is bounded by \({\left||f \right||}_{L^p}^p\). Finally, let N approach infinity and use the monotone convergence theorem to get
For the opposite inequality, we take \(f \in L^p(\Omega _0) \cap L^2(\Omega _0)\) and \(g \in L^{p^{\prime }}(\Omega _0) \cap L^2(\Omega _0)\) where \(1/p'+1/p = 1\). By polarization of (3.4) and the Cauchy–Schwarz and Hölder inequalities, we obtain
\(\square \)
Given a set \(\{v_\lambda : \lambda \in P\}\) of vectors in a Banach space, we say that \(\sum _{\lambda \in P} v_\lambda \) converges unconditionally if, whenever we choose a bijection \(\phi : \mathbb {N}\rightarrow P\),
Equivalently, we may ask that for any increasing, exhaustive sequence \(\left( {F_N}: {N \in \mathbb {N}}\right) \) of finite subsets of P, the limit
The following proposition provides a Calderón reproducing formula.
Proposition 3.2
For each \(p \in (1, \infty )\) and any \(f \in L^p(\Omega _0)\),
where the sum converges in \(L^p(\Omega _0)\) unconditionally.
Proof
Fix an increasing and exhaustive sequence \(\left( {F_N}: {N \in \mathbb {N}}\right) \) of finite subsets of P. Let
For \(f \in L^p(\Omega _0)\) and \(g \in L^{p^{\prime }}(\Omega _0)\), where \(1/p + 1/{p^{\prime }} = 1\), we have
In particular,
whence \({\left||I_N(f) \right||}_{L^p} \lesssim {\left||f \right||}_{L^p}\) uniformly in N. Consequently, it is enough to prove convergence for \(f \in L^p(\Omega _0) \cap L^2(\Omega _0)\). From (3.5) and the bounded convergence theorem, it follows that for any positive \(\epsilon \), \({\left||I_N(f) - I_M(f) \right||}_{L^p} \le \epsilon \) whenever M and N are large enough. This shows that the limit exists. Finally, for \(g \in L^{p^{\prime }}(\Omega _0)\cap L^2(\Omega _0)\), the polarized version of (3.4) gives
\(\square \)
Theorem 3
Let \(\left( {T_\lambda }: {\lambda \in P}\right) \) be a family of operators such that for some \(\delta > 0\) and \(p_0 \in (1, 2)\)
Then for any \(p \in (p_0, 2]\) the sum \(\sum _{\lambda \in P} T_\lambda \) converges unconditionally in the strong operator topology for operators on \(L^p(\Omega _0)\).
Proof
First, recall that the Cotlar–Stein Lemma (see e.g. [16]) states that (3.7) implies the unconditional convergence of \(\sum _{\lambda \in P} T_\lambda \) in the strong operator topology on \(L^2(\Omega _0)\). Let \(\left( {F_N}: {N \in \mathbb {N}}\right) \) be an arbitrary increasing and exhaustive sequence of finite subsets of P. For \(N > 0\), we set
By (3.6), (3.7) and interpolation, each \(T_\mu \) is bounded on \(L^p\) for \(p\in [1,2]\) and the same holds for the finite sum \(V_N\). We consider \(f \in L^p(\Omega _0)\) for \(p \in (p_0, 2)\). By Proposition 3.2 and Theorem 2, we
Finally, by change of variables, we get
Assuming there is \(\delta _p > 0\) such that
we can estimate
Theorem 2, Proposition 3.2 and (3.11) imply that the \(V_M\) are uniformly bounded on \(L^p\).
For the proof of (3.10), we consider an operator \(\mathcal {T}\) defined for \(\mathbf {f} \in L^p\big (\pi , \ell ^2(P)\big )\) by
Since \({{\left||{D_\lambda } \right||}_{L^{1} \rightarrow L^{1}}} \lesssim 1\) and \({{\left||{T_\mu } \right||}_{L^{1} \rightarrow L^{1}}} \lesssim 1\), we have
Also, by (3.8), we can estimate
Therefore, using interpolation between \(L^{1}\left( \pi , \ell ^{1}(P) \right) \) and \(L^{2}\left( \pi , \ell ^{2}(P) \right) \) we obtain that there is \(\delta ^{\prime } > 0\) such that
Because \({|{D_\lambda g} |} \lesssim L^* R^*( {|{g} |} )\), and because Theorem 1 says that \(L^*\) and \(R^*\) are bounded on \(L^{p_0}\), we know that \(\left( {D_\lambda }: {\lambda \in P}\right) \) is bounded on \(L^{p_0}(\pi , \ell ^\infty (P))\). Of course the same holds for \(\left( {D_{\lambda +\gamma +\gamma '}}: {\lambda \in P}\right) \). Hence, by (3.9), we get
Next, interpolating between \(L^{p_0}\left( \pi , \ell ^{p_0}(P) \right) \) and \(L^{p_0}\left( \pi , \ell ^{\infty }(P) \right) \) gives a \(\delta ^{\prime \prime } > 0\) such that
Finally, interpolating between \(L^{p_0}\left( \pi , \ell ^{2}(P) \right) \) and \(L^{2}\left( \pi , \ell ^{2}(P) \right) \), we obtain (3.10).
To complete the proof, we are going to show that \(\left( {V_N f}: {N \in \mathbb {N}}\right) \) is a Cauchy sequence in \(L^p(\Omega _0)\). Let us consider \(g \in L^p(\Omega _0) \cap L^2(\Omega _0)\). Setting
and using the log-convexity of the \(L^q\)-norms, we get
Since \(\left( {V_N g}: {N \in \mathbb {N}}\right) \) converges in \(L^2(\Omega _0)\) and is uniformly bounded on \(L^{\tilde{p}}(\Omega _0)\) it is a Cauchy sequence in \(L^p(\Omega _0)\). For an arbitrary \(f \in L^p(\Omega _0)\) use the density of g’s as above. We have
Thus, \(\left( {V_N f}: {N\in \mathbb {N}}\right) \) also converges, and this finishes the proof of the theorem. \(\square \)
4 Double Differences
The martingale transforms are expressed in terms of double differences defined for a martingale \(f = \left( {f_\lambda }: {\lambda \in P}\right) \) as
4.1 Martingale Transforms
The following proposition is our key tool.
Proposition 4.1
Let \(f \in L^2(\Omega _0)\) and \(\lambda \in P\). If \(f_{\lambda - j \lambda _1} = 0\) for \(j \in \mathbb {N}\) then for each \(k \ge j\)
Analogously, for \(\lambda _1\) and \(\lambda _2\) exchanged.
Proof
Suppose \(j = 1\). We are going to show that if \(f_{\lambda - \lambda _1} = 0\) then for all \(k \ge 1\)
Indeed, if \(k = 1\) then by (2.1) of Lemma 2.2
If \(k > 1\), we use Lemma 2.3 to write
Since, by Lemma 2.4,
we can use induction to obtain
Let us consider \(j > 1\). For each \(i = 0, 1, \ldots , j-1\), we set
By Lemma 2.4 and (4.1), we have
Hence,
which finishes the proof since
\(\square \)
We have the following
Proposition 4.2
For any \(\lambda , \lambda ^{\prime }, \mu \in P\) and \(m \ge 1\)
Proof
We observe that for \(f \in L^2(\Omega _0)\), \(d_\mu f \in L^2(\pi , \mathcal {F}_\mu )\) and
whenever \({\langle \nu , \alpha _0\rangle } \le {\langle \mu , \alpha _0\rangle } - 2\). For the proof it is enough to analyze the case \(\nu = \mu - 2 \lambda _2\). By Lemma 2.4, we can write
Suppose \(\lambda = i \lambda _1 + j \lambda _2\). Let us consider \(R_j d_\mu \). If \(j \ge {\langle \mu , \alpha _2\rangle } + 1\) then \(R_j d_\mu f = 0\). For \(j \le {\langle \mu , \alpha _2\rangle } - 2\), in view of (4.2) we can use Proposition 4.1 to estimate
Next, if \({\langle \lambda , \alpha _0\rangle } \ge {\langle \mu , \alpha _0\rangle } + 2\) then \(D_\lambda d_\mu f = 0\), because \(d_\mu f\) is \(\mathcal {F}_\mu \)-measurable. For \({\langle \lambda , \alpha _0\rangle } \le {\langle \mu , \alpha _0\rangle } - 4\) and \({\langle \lambda , \alpha _2\rangle } \le {\langle \mu , \alpha _2\rangle }\), by Lemma 2.5, we can write \(D_\lambda d_\mu f = L_i g\) where
and \(\nu = ({\langle \mu , \alpha _0\rangle } - j)\lambda _1 + j \lambda _2\). By Lemma 2.5, we have
We notice that by Lemma 2.4 and (4.2)
Similarly, one can show
Therefore, \(\mathbb {E}[g | \mathcal {F}_{\nu - 2\lambda _1}] = 0\). Now, by Proposition 4.1, we obtain
Combining (4.4) with (4.3), we get
since \({\langle \nu , \alpha _0\rangle } = {\langle \mu , \alpha _0\rangle }\). By analogous reasoning one can show the corresponding norm estimates for \(D_{\lambda '}^\star d_\mu \). Hence, taking adjoint
Finally, (4.5) and (4.6) allow us to conclude the proof of the first inequality.
For the second, we may assume \(0 \le {\langle \mu - \lambda , \alpha _0\rangle } \le 1\). Suppose \({\langle \mu - \lambda , \alpha _0\rangle } = 0\) and \({\langle \mu - \lambda , \alpha _2\rangle } \ge 2\). Since \(d_\mu f \in L^2(\pi , \mathcal {F}_\mu )\), by (4.2) and Proposition 4.1
Similarly, we deal with the case \({\langle \mu - \lambda , \alpha _0\rangle } = 1\). We can assume \({\langle \mu - \lambda , \alpha _2\rangle } \ge 1\). By Lemma 2.4, we have
Hence, by Proposition 4.1,
\(\square \)
Let \(\left( {a_\lambda }: {\lambda \in P}\right) \) be an uniformly bounded predictable family of functions, i.e. each function \(a_\lambda \) is measurable with respect to \(\mathcal {F}_{\lambda -\lambda _1-\lambda _2}\) and
Predictability is the condition needed to ensure that \(d_\lambda \big (a_\lambda f\big ) = a_\lambda d_\lambda f\). By Theorems 1 and 3, Proposition 4.2 and duality when \(p > 2\), we get
Theorem 4
For each \(p \in (1, \infty )\) and \(m \in \mathbb {N}\) the series
converges unconditionally in the strong operator topology for the operators on \(L^p(\Omega _0)\), and defines the operator with norm bounded by a constant multiply of
4.2 Martingale Square Function
For a martingale \(f = \left( {f_\lambda }: {\lambda \in P}\right) \) there is the natural square function defined by
Although S does not preserve \(L^2\) norm, we have
Theorem 5
For every \(p \in (1, \infty )\) there is \(C_p > 0\) such that
Proof
We start from proving the identity
Let us notice that
Therefore, consecutively we have
Hence, by Lemma 2.2,
which together with (4.8) implies (4.7).
Next, we consider an operator \(\mathcal {T}\) defined for a function \(f \in L^p(\Omega _0)\) by
We also need an operator \(\widetilde{\mathcal {T}}\) acting on \(g \in L^{p'}(\Omega _0)\) as
We observe that by two-parameter Khinchine’s inequality and Theorem 4 we have
The dual operator \(\mathcal {T}^\star : L^{p^{\prime }}\big (\pi , \ell ^2(\mathbb {Z}^2)\big ) \rightarrow L^{p^{\prime }}(\Omega _0)\) is given by
Since \(\widetilde{\mathcal {T}} g \in L^{p^{\prime }}\big (\pi , \ell ^2(\mathbb {Z}^2)\big )\), by (4.7) and Theorem 4,
Therefore, by Cauchy–Schwarz and Hölder inequalities
and since \({\left||\mathcal {T}f \right||}_{L^p(\ell ^2)} = {\left||S f \right||}_{L^p}\) the proof is finished.
Finally, the method of the proof of Theorem 3, together with Theorems 4 and 5 shows the following
Theorem 6
Let \(\left( {T_\lambda }: {\lambda \in P}\right) \) be a family of operators such that for some \(\delta > 0\) and \(p_0 \in (1, 2)\)
Then for any \(p \in (p_0, 2]\) the sum \(\sum _{\lambda \in P} T_\lambda \) converges unconditionally in the strong operator topology for the operators on \(L^p(\Omega _0)\).
Notes
A topological field is an algebraic field with a topology making addition, multiplication and multiplicative inverse a continuous mappings.
References
Bourbaki, N.: Groupes et algèbres de Lie. Hermann, Paris (1968)
Burkholder, D.: Martingale transforms. Ann. Stat. 37, 1494–1504 (1966)
Burkholder, D.: Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973)
Burkholder, D.: The best constant in the Davis inequality for the expectation of the martingale square function. Trans. Am. Math. Soc. 354(1), 91–105 (2001)
Cairoli, R., Walsh, J.: Stochastic integrals in the plane. Acta Math. 134(1), 111–183 (1975)
Duoandikoetxea, J., Rubio de Francia, J.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84(3), 541–561 (1986)
Gundy, R.: A decomposition of \(L^1\) bounded martingales. Ann. Math. Stat. 39, 134–138 (1968)
Hardy, G., Littlewood, J.: A maximal theorem with function-theoretic applications. Acta Math. 54, 81–161 (1930)
Jensen, J., Marcinkiewicz, J., Zygmund, A.: Note on the differentiability of multiple integrals. Fundam. Math. 25, 217–234 (1935)
Marcinkiewicz, J.: Quelques théorèmes sur les séris orthogonales. Ann. Soc. Polon. Math. 16, 84–96 (1937)
Metraux, C.: Quelques inégalités pour martingales à parameter bidimensional. In: Dellacherie, C., Meyer, P.A., Weil, M. (eds.) éminaire de Probabilités XII, vol. 649, pp. 170–179. Springer, New York (1978)
Paley, R.: A remarkable series series of orthogonal functions (I). Proc. Lond. Math. Soc. 34, 241–279 (1932)
Ronan, M.: Lectures on Buildings. Perspectives in Mathematics. Academic Press, Cambridge (1989)
Steger, T.: Local fields and buildings. In: Koranyi, A. (ed.) Harmonic Functions on Trees and Buildings, vol. 206, pp. 79–107. American Mathematical Society, Providence (1997)
Stein, E.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies. Princeton University Press, Princeton (1970)
Stein, E., Murphy, T.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. Princeton University Press, Princeton (1993)
Stein, E., Street, B.: Multi-parameter singular Radon transforms II: the \(L^p\) theory. Adv. Math. 248, 736–783 (2013)
Weisz, F.: Martingale Hardy spaces and their applications in Fourier analysis, vol. 1568. Springer, New York (1996)
Wong, E., Zakai, M.: Martingales and stochastic integrals for processes with a multidimensional parameter. Z. Wahrscheinlichkeitstheorie verw. Gebiete 29, 109–122 (1974)
Zakai, M.: Some classes of two-parameter martingales. Ann. Prob. 9(2), 255–265 (1981)
Author information
Authors and Affiliations
Corresponding author
Appendix: About \(\Omega _0\) and Heisenberg Group
Appendix: About \(\Omega _0\) and Heisenberg Group
In some cases \(\Omega _0\) can be identified with a Heisenberg group over a nonarchimedean local field. Let us recall, that F is a nonarchimedean local field if it is a topological fieldFootnote 1 that is locally compact, second countable, non-discrete and totally disconnected. Since F together with the additive structure is a locally compact topological group it has a Haar measure \(\mu \) that is unique up to multiplicative constant. Observe that for each \(x \in F\), the measure \(\mu _x(B) = \mu (x B)\) is also a Haar measure. We set
where B is any measurable set with finite and positive measure. By \(\mathcal {O}= \{x \in F : {|{x} |} \le 1\}\), we denote the ring of integers in F. We fix \(\pi \in \mathfrak {p} - \mathfrak {p}^2\), where
We are going to sketch the construction of a building associated to \({\text {GL}}(3, F)\). For more details, we refer to [14]. A lattice is a subset \(L \subset F^3\) of the form
where \(\{v_1, v_2, v_3\}\) is a basis of \(F^3\). We say that two lattices \(L_1\) and \(L_2\) are equivalent if and only if \(L_1 = a L_2\) for some nonzero \(a \in F\). Then \(\mathscr {X}\), the building of \({\text {GL}}(3, F)\), is the set of equivalence classes of lattices in \(F^3\). For \(x, y \in \mathscr {X}\) there are a basis \(\{v_1, v_2, v_3\}\) of \(F^3\) and integers \(j_1 \le j_2 \le j_3\) such that (see [14, Proposition 3.1])
We say that x and y are joined by an edge if and only if \(0 = j_1 \le j_2 \le j_3 = 1\). The subset
is called an apartment. A sector in \(\mathscr {A}\) is a subset of the form
where \(\sigma \) is a permutation of \(\{1, 2, 3\}\) and \(x \in \mathscr {A}\). Thus, a subsector of \(\mathcal {S}\) is
for some \(0 \le k_{\sigma (1)} \le k_{\sigma (2)} \le k_{\sigma (3)}\). Finally, two sectors
and
are opposite if \(\sigma ^{\prime } \circ \sigma ^{-1} = (3\, 2\, 1)\).
A sector in \(\mathscr {X}\) is a sector in one of its apartments. Two sectors in \(\mathscr {X}\) are equivalent if and only if its intersection contains a sector. By \(\Omega \), we denote the equivalence classes of sectors in \(\mathscr {X}\). Let \(\omega _0\) and \(\omega _0^{\prime }\) be the equivalence class of
and
respectively. Two sectors \(\mathscr {S}\) and \(\mathscr {S}^{\prime }\) are opposite in \(\mathscr {X}\) if there are subsectors of \(\mathscr {S}\) and \(\mathscr {S}^{\prime }\) opposite in a common apartment. By \(\Omega _0\), we denote the equivalence classes of sectors opposite to \(\mathscr {S}_0\).
Suppose that \(\omega ^{\prime } \in \Omega _0\). Let \(\{v_1, v_2, v_3\}\) be a basis of \(F^3\), and \(k_1 \le k_2 \le k_3\) and \(k_1^{\prime } \ge k_2^{\prime } \ge k_3^{\prime }\) be integers such that
and
belong to \(\omega _0\) and \(\omega ^{\prime }\), respectively. Since the sector (4.9) belongs to \(\omega _0\), we have
for some \(b_{ij} \in F\) such that \(b_{11}, b_{22}, b_{33} \ne 0\). Hence, the matrix
satisfies \(g e_j = v_j\). In particular, \(g\omega ^{\prime }_0 = \omega ^{\prime }\). Therefore, the group of upper triangular matrices acts transitively on \(\Omega _0\). Observe also that the stabilizer of \(\omega _0'\) in \({\text {GL}}(3, F)\) is the group of lower triangular matrices. Thus, the group
acts simply transitively on \(\Omega _0\).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Steger, T., Trojan, B. Littlewood–Paley Theory for Triangle Buildings. J Geom Anal 28, 1122–1150 (2018). https://doi.org/10.1007/s12220-017-9856-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-017-9856-6
Keywords
- Affine building
- Littlewood–Paley theory
- Square function
- Maximal function
- Multi-index filtration
- Heisenberg group
- p-adic numbers