Abstract
We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on \(L^1\). This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on \(L^1\) of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on \(L^p\), \(p>1\), which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The work presented in this paper has two parts. The first part is a one-parameter pointwise ergodic theorem on \(L^1\), which contributes another counterexample for a celebrated conjecture of Rosenblatt and Wierdl, and the second part is a multi-parameter pointwise result in the spirit of Dunford and Zygmund, see for example 1.11. Before precisely stating our results, some historical remarks are in order.
1.1 Brief Historical Remarks
In 1991 Rosenblatt and Wierdl [25, Conjecture 4.1] formulated a famous conjecture asserting that for any arithmetical set A with zero Banach density and for any \((X,{\mathcal {B}},\mu ,T)\) aperiodic probability dynamical system, there exists a function \(f\in L^1_{ \mu }(X)\), such that
i.e. the set of \(x\in X\) such that \(\lim _{N\rightarrow \infty }M_{A,N}f(x)\) does not exist has positive measure. This was disproven in 2006 by Buczolich [5], where he provided a counterexample by constructing inductively an appropriate set A of zero Banach density for which one gets the pointwise convergence of the ergodic averages \(M_{A,N}f\) for all \(f\in L^1\).
Calderón’s transference principle suggests that such questions are closely related to the study of weak type (1, 1) estimates for the maximal function corresponding to those averages over the integer shift system, namely for the operator
A year later it was shown [26] that for \(A=\big \{\lfloor n^c\rfloor :\,n\in {\mathbb {N}}\big \}\) with \(c\in \big (1,\frac{1001}{1000}\big )\), the operator \({\mathcal {M}}_{A}\) is of weak type (1, 1), and as a corollary the authors proved pointwise convergence on \(L^1\) for the corresponding ergodic averages along the set A, providing a counterexample of the aforementioned conjecture given by a concrete formula. This class of examples was extended in [20] where the author established the weak type (1,1) bounds for \({\mathcal {M}}_{A}\) and the corresponding pointwise ergodic theorem on \(L^1\) for sets of the form \(\big \{\lfloor n^{c}\ell (n) \rfloor :\,n\in {\mathbb {N}}\big \}\), where c is close to 1 and \(\ell \) is a certain kind of slowly varying function, for example any iterate of \(\log \), see Definitions 1.1,1.2 below. Finally, another class of examples appeared in [7], where for any \(r\in {\mathbb {Q}}\cap (1,\infty )\), certain sequences \(\{a_n\}_{n\in {\mathbb {N}}}\) of algebraic nature were constructed such that \(a_n\simeq n^r\) and such that the associated maximal operator is of weak type (1, 1). Comparing this with the failure of the estimate for \(A=\{n^k:\,n\in {\mathbb {N}}\}\) for any \(k\ge 2\), see [6] and [18], one sees that determining whether the weak type (1, 1) bound holds for \({\mathcal {M}}_A\) is a delicate problem which is rather sensitive to perturbations of the sequence. For example, even the case correspoding to \(A=\{\lfloor n\log n\rfloor :\,n\in {\mathbb {N}}\}\) remains open.
One of the main results of the present work is a natural extension of the result from [20] and in order to formulate it, we must introduce two families of functions that one may think of as slowly varying and regularly varying functions respectively.
Definition 1.1
Fix \(x_0\ge 1\) and let \({\mathcal {L}}\) denote the set of all functions \(\ell :[x_0,\infty )\rightarrow [1,\infty )\) such that
where \(\vartheta \in {\mathcal {C}}^2([x_0,\infty ))\) is a real-valued function satisfying
Definition 1.2
Fix \(x_0\ge 1\) and let \({\mathcal {L}}_0\) denote the set of all functions \(\ell :[x_0,\infty )\rightarrow [1,\infty )\) such that
where \(\vartheta \in {\mathcal {C}}^2([x_0,\infty ))\) is a positive and decreasing function satisfying
and such that for all \(\varepsilon >0\) we have \(\vartheta (x)\gtrsim _{\varepsilon }x^{-\varepsilon }\) and \(\lim _{x\rightarrow \infty }\ell (x)=\infty \).
We note that \({\mathcal {L}}_0\subseteq {\mathcal {L}}\) and we give some examples of these slowly varying functions. For \(a>0\), \(b\in (0,1)\), \(k\in {\mathbb {N}}\) and \(c\in {\mathbb {R}}\), we have
We proceed by defining a family of regularly varying functions.
Definition 1.3
Fix \(x_0\ge 1\), \(c\in (1,\infty )\) and let \({\mathcal {R}}_c\) be the set of all functions \(h:[x_0,\infty )\rightarrow [1,\infty )\) such that h is strictly increasing, convex and of the form \(h(x)=x^c\ell (x)\) for some \(\ell \in {\mathcal {L}}\). We define \({\mathcal {R}}_1\) analogously with the extra assumption that \(\ell \in {\mathcal {L}}_0\).
We are now ready to give the definitions of the arithmetic sets we are interested in. Let \(c_1,c_2\in [1,2)\) and let us fix \(h_1\) and \(h_2\) in \({\mathcal {R}}_{c_1}\) and \({\mathcal {R}}_{c_2}\) respectively. Let \(\varphi _1\) and \(\varphi _2\) be the compositional inverses of \(h_1\) and \(h_2\) and for convenience, let \(\gamma _1=1/c_1\) and \(\gamma _2=1/c_2\). Fix a function \(\psi :[1,\infty )\rightarrow (0,1/2]\), \(\psi \in {\mathcal {C}}^2([1,\infty ))\) such that
We can now define \(B_+=\{\,n\in {\mathbb {N}}:\{\varphi _1(n)\}<\psi (n)\,\}\) and \(B_-=\{\,n\in {\mathbb {N}}:\{-\varphi _1(n)\}<\psi (n)\,\}\), where \(\{x\}=x-\lfloor x \rfloor \).
Those sets have been introduced and studied in [17], where the authors proved that the Hardy-Littlewood majorant property holds for them, as a corollary of a restriction theorem. Recently, the author [9] proved an analogous result for \({\mathbb {P}}\cap B_{\pm }\), see Theorem 1.8 in [9], as well as Roth’s theorem in these sets, namely, it was shown that any subset of the primes of the form \(B_{\pm }\) with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions.
Following [9] we repeat and extend some comments on the sophisticated nature of the sets \(B_{\pm }\). To motivate the definition of \(B_{\pm }\), we note that
For \(\gamma \in (0,1)\), \(h_1(x)=h_2(x)=x^{1/\gamma }\) and \(\psi (x)=\varphi _1(x+1)-\varphi _1(x)\), note that the last condition becomes \(m^{1/\gamma }\in [n,n+1)\) or \(n=\lfloor m^{1/\gamma }\rfloor \), thus \(B_-=\{\,\lfloor m^{1/\gamma } \rfloor :m\in {\mathbb {N}}\,\}\). It is not difficult to see that any set
can also be brought in the form \(B_-\) by similar appropriate choices. Thus the family of sets we consider includes the fractional powers with exponent close to 1 and even the more general sets considered in [20].
1.2 One-Parameter Ergodic Theorem on \(L^1\)
We state one of the main results of our paper. Due to some technical complications, we demand further that \(\varphi _1\simeq \varphi _2\), and note that this implies that \(c_1=c_2\).
Theorem 1.5
(Weak-type (1,1) inequality for \({\mathcal {M}}_{B_{\pm }}\)) Assume \(c_1\in (1,30/29)\) and \(\varphi _1\simeq \varphi _2\). Then the maximal function
is of weak-type (1,1), i.e.:
By interpolation, this implies that for all \(p\in (1,\infty ]\), there exists a constant \(C_p>0\) such that
We use this, together with 2-oscillation estimates, see Theorem 1.12, to obtain the following pointwise convergence result.
Theorem 1.6
(Pointwise ergodic theorem) Assume \(c\in (1,30/29)\), \(\varphi _1\simeq \varphi _2\) and let \((X,{\mathcal {B}},\mu ,T)\) be an invertible \(\sigma \)-finite measure preserving system. For any \(p\in [1,\infty )\) and any \(f\in L_{\mu }^p(X)\), we have that
Before discussing the strategy of our proofs, we would like to further examine the sets \(B_{\pm }\) and give some concrete examples. Note that
Now assume that \(n\in B_+\), and \(m\in {\mathbb {N}}\) is such that \(m\in (\varphi _1(n)-\psi (n),\varphi _1(n)]\) and assume that \(n_0\) is the smallest integer such that \(m\in (\varphi _1(n_0)-\psi (n_0),\varphi _1(n_0)]\). Even in simple examples, we should expect that \(B_+\) will contain a lot of consecutive integers after \(n_0\). For example, for any \(\varphi _1\) inverse of a function in \({\mathcal {R}}_c\), let \(\varphi _2=C100\varphi _1\), where C is the doubling constant of \(\varphi _1'\), namely, \(\varphi _1'(x)\le C \varphi '(2x)\), and let \(\psi =\varphi _2'\). Since \(\varphi _1\) is increasing and \(\psi =\varphi _2'\) is decreasing, we get that \(\varphi _1-\psi \) is increasing and thus since \(m\notin (\varphi _1(n_0-1)-\psi (n_0-1),\varphi _1(n_0-1)]\), we get that \(m>\varphi _1(n_0-1)\). We claim that for all \(l\in \{0,\cdots ,99\}\), we get that \(m\in (\varphi _1(n_0+l)-\psi (n_0+l),\varphi _1(n_0+l)]\), which implies that \(B_+\) contains 99 consecutive integers after \(n_0\). Clearly, \(m\le \varphi _1(n_0+l)\) for all \(l\in {\mathbb {N}}_0\). If we assume for the sake of a contradiction that for some \(l\in \{0,\dotsc ,99\}\) we have that \(m<\varphi _1(n_0+l)-\psi _1(n_0+l)\), then
and by the Mean Value Theorem, there exists a \(\xi _{n_0,l}\in (n_0-1,n_0+l)\) such that
Thus
This shows that the set \(B_+\) considered here contains infinitely many full blocks of 100 consecutive integers. Such a set \(B_+\) stands in sharp contrast to the sets of the form \({\mathbb {N}}_{h}=\{\,\lfloor h(m) \rfloor :m\in {\mathbb {N}}\,\}\), \(h\in {\mathcal {R}}_c\), as the gaps between members of such sets tend to infinity.
In general, the constant \(\sup _{x\in [1,\infty )}\frac{\varphi _2'(x)}{\varphi _1'(x)}\) determines an important qualitative aspect of the form of the sets \(B_{\pm }\), see Lemma 2.4. Loosely speaking, for big intervals of integers where the ratio is bigger than L, we expect that \(B_{\pm }\) will contain blocks of length at least L/C, where C is the doubling constant of \(\varphi _1'\). The technical issues that arose when trying to handle the case where \(B_{\pm }\) contains arbitrarily long intervals of integers (specifically in the counting Lemma 5.1) forced the author to impose the restriction \(\varphi _1'\simeq \varphi _2'\), which is equivalent to \(\varphi _1\simeq \varphi _2\). We note that \(\frac{\varphi _2'(x)}{\varphi _1'(x)}\) could oscillate and thus \(B_{\pm }\) could contain blocks of various oscillating lengths.
We give certain explicit examples to enhance and extend the heuristics described above as well as to illustrate the reach of the main theorems of the present work. We begin by considering the sets \({\mathbb {N}}_c=\{\lfloor m^c\rfloor :\,m\in {\mathbb {N}}\}\), and by observing that
which means that \(n\in {\mathbb {N}}_c\) if and only if the interval \([n,n+1)\) “contains a c-power”. Introducing the sets \(B_{\pm }\) allows us to consider intervals of various lengths.
Intervals of fixed length Let \(c>1\), \(\gamma =1/c\). With the choices
we get
or equivalently \({B_-}={\mathbb {N}}\cap \Big \{\frac{\lfloor 100m^{c} \rfloor }{100}:\,m\in {\mathbb {N}} \Big \}\).
The previous example is readily generalizable. Let \(c>1\), \(\gamma =1/c\) and fix \(A>0\), \(B\ge 0\) and \(C\in {\mathbb {N}}\), and let
Then
and thus
Similarly, appropriate choices of parameters give \(B_-\) the following form
For \(\alpha _2-\alpha _1\ge 2\), we have that \(B_-\) is a union of intervals of length at least \(\alpha _2-\alpha _1-1\) “around c-powers”, see also the proof of Lemma 2.4. This set highlights the aforementioned qualitative difference of \(B_{\pm }\) with any \({\mathbb {N}}_h\) for \(h\in {\mathcal {R}}_c\) considered in [20], namely, the sets \(B_{\pm }\) can contain consecutive integers.
Intervals of oscillating lengths For certain choices of parameters one produces a set \(B_-\) which is comprised of intervals of oscillating lengths around c-powers. Let \(c>1\), \(\gamma =1/c\) and
Then \(n\in B_-\iff n\le m^c < n+99\sin (\log \log n)+100\), and thus
We believe that the examples illustrate the flexibility in the choices of parameters defining \(B_{\pm }\) and consequently the reach of Theorem 1.6.
Let us remark that the examples given are readily generalizable, for example, \(m^c\) can be replaced by \(h_1(m)\) with \(h_1\in {\mathcal {R}}_{c_1}\). Routine but lengthy calculations show that all the choices we made for our parameters are compatible with the conditions in the definition of the set \(B_{\pm }\), as well as the technical assumption \(\varphi _1\simeq \varphi _2\), except possibly for small values of x, which is insignificant since only the tail of the set \(B_{\pm }\) plays a role in our results.
Let us restrict our attention to the sets \(B_+\), which we call B from now on, as the results for \(B_-\) are of equal difficulty. We make the following remarks.
Remark 1.9
(Smooth dyadic maximal operator) To establish Theorem 1.5, it is convenient to work with a smooth dyadic variant of the maximal operator. More precisely, let us fix \(\eta \in {\mathcal {C}}^{\infty }({\mathbb {R}})\) such that \(0\le \eta (x)\le 1\) for all \(x\in {\mathbb {R}}\), \({{\,\textrm{supp}\,}}(\eta )\subseteq (1/2,4)\) and \(\eta (x)=1\) for all \(x\in [1,2]\). We define
and note that
The first inequality is straightforward, and for the second one, note that Lemma 2.1 implies that
Thus it suffices to establish the weak type (1,1) bound for \({\mathcal {M}}^{(sd)}_B\) which we denote by \({\mathcal {M}}\), and let
so that \({\mathcal {M}}f(x)=\sup _{k\in {\mathbb {N}}_0} \{K_{2^k}*|f|(x)\}\).
We give a brief overview of the main ideas of the proof of Theorem 1.5. We use a subtle variation of the Calderón-Zygmund decomposition that was introduced by Fefferman [11], see also [8], in a similar manner to [26] and [20]. More specifically, after approximating \(K_n*{\widetilde{K}}_n\) by suitable well-behaving functions, see Lemma 5.2, we employ a refined Calderón-Zygmund decomposition which allows us to use \(\ell ^2\)-estimates for the “very bad” part of the decomposition, see subsection 5.2. The aforementioned approximation is analogous to the one presented in section 5 of [20] and similar techniques are used here. The novelty lies in the sophisticated nature of the sets B which complicates the situation substancially. For example, bounding \(|K_n*{\widetilde{K}}_n(x)|\) for small values of x, see Lemma 5.1, is precisely what forced the author to impose the extra assumption \(\varphi _1\simeq \varphi _2\). To carry out the approximation one needs to estimate certain exponential sums and the main tool is Van der Corput’s inequality. Some of the necessary exponential sum estimates can be readily found in section 3 of [20] and suitable extensions are already established by the author in [9]. Finally, we formulate an abstract result, see Theorem 5.5, which is a generalization of Theorem 6.1 of [20], adapted to our approximation for \(K_n*{\widetilde{K}}_n\). We give the full proof of Theorem 1.5 in Sect. 5 and the reader is encouraged to compare it with section 3 of [26], and section 5 and 6 of [20].
Combining Theorem 1.5 with the trivial estimate \(\Vert {\mathcal {M}}_B\Vert _{\ell ^{\infty }({\mathbb {Z}})\rightarrow \ell ^{\infty }({\mathbb {Z}})}\lesssim 1\), we obtain by interpolation that \(\Vert {\mathcal {M}}_{B}\Vert _{\ell ^{p}({\mathbb {Z}})\rightarrow \ell ^{p}({\mathbb {Z}})}\lesssim _p 1\), for all \(p\in (1,\infty ]\). Calderón’s transference principle implies that for any invertible \(\sigma \)-finite measure preserving system \((X,{\mathcal {B}},\mu ,T)\) we have
and a standard argument shows that \({\mathcal {L}}^p_{B}=\big \{f\in L^p_{\mu }(X):\,\lim _{N\rightarrow \infty }M_{B,N}f \text{ exists } \mu \text{-a.e. }\big \}\) is closed in \(L_{\mu }^p(X)\), for all \(p\in [1,\infty )\), and thus to establish Theorem 1.6 it suffices to exhibit an \(L_{\mu }^p\)-dense class of functions \({\mathcal {D}}_p\) contained in \({\mathcal {L}}^p_{B}\). The exponential sum estimates of Lemma 2.1 together with a straightforward adaptation of the argument presented in section 3 of [20], which uses ideas from [4], shows that one may take \({\mathcal {D}}_p=L^p_{\mu }(X)\cap L^2_{\mu }(X)\) and conclude. Instead of this, we derive the pointwise convergence immediately by the much stronger uniform 2-oscillation \(L^p_{\mu }\)-estimates of Theorem 1.12, which are also exploited in the sequel.
1.3 Multi-Parameter Ergodic Theorem
The second main result of our paper is a multi-parameter variant of Theorem 1.6. Here we discard the assumption \(\varphi _1\simeq \varphi _2\) and the acceptable range of \(c_1\) and \(c_2\) is considerably larger. In contrast to the one-parameter situation, weak-type (1, 1) estimates do not hold here, which is comparable with the failure of Lebesgue’s differentiation theorem for rectangles with sides parallel to the axes on \(L^1\).
To make the exposition slightly cleaner, let us fix \(k\in {\mathbb {N}}\) and \(B_1,\dotsc ,B_k\) as in the introduction, where \((h_{1,i},h_{2,i})\in {\mathcal {R}}_{c_{1,i}}\times {\mathcal {R}}_{c_{2,i}} \) are as in the definition of \(B_i\), and we assume that \(c_{1,i}\in [1,2)\) and \(c_{2,i}\in [1,6/5)\) for all \(i\in [k]\). We are ready to state the second main result.
Theorem 1.10
Assume \((X,{\mathcal {B}},\mu )\) is a \(\sigma \)-finite measure space and \(\{S_i:\,i\in [k]\}\) is a family of invertible \(\mu \)-invariant commuting transformations. Then for any \(p\in (1,\infty )\) and any \(f\in L^p_{\mu }(X)\) we have that
where \(S^{(l_1,\dotsc ,l_k)}=S^{l_1}_1\circ \dots \circ S^{l_k}_k\).
Apart from the larger range of exponents, the absence of the condition \(\varphi _1\simeq \varphi _2\) allows us to consider a variety of additional sets here. We provide some examples.
Shrinking intervals The fact that we can choose \(c_1< c_2\) allows us to consider shrinking intervals. Let \(1<c_1<c_2\), \(\gamma _1=1/c_1\) and \(\gamma _2=1/c_2\), and let
Then
which gives \(B_-\) the following form,
for \(c>1\), \(\theta >0\) and close to 1 and 0 respectively, which in turn, means that we only consider integers that “approach c-powers” rapidly.
Expanding intervals Analogously, with appropriate choices with \(c_1>c_2\), we get
for \(c>1\), \(\theta >0\) and close to 1 and 0 respectively, which is a union of intervals of expanding lengths, and again, such sets are not of the form \({\mathbb {N}}_h\). For an example of expanding intervals when \(c_1=c_2\), let \(c>1\), \(\gamma =1/c\) and let
Note that then
We make some brief historical remarks. In 1951 Dunford [10] and Zygmund [28] independently showed that given a \(\sigma \)-finite measure space \((X,{\mathcal {B}},\mu )\) and a family of \(\mu \)-invariant transformations \(\{T_i:\,i\in [k]\}\), for any \(p\in (1,\infty )\) and any \(f\in L^p_{\mu }(X)\), we have
Notably, for \(k\ge 2\), pointwise convergence fails on \(L^1\). Motivated by that observation and after his seminal work on pointwise ergodic theory [2,3,4], Bourgain showed that for any \(p\in (1,\infty )\) and any \(f\in L^p_{\mu }(X)\), we have
where \(P_1,\dotsc , P_k\in {\mathbb {Z}}[x]\), \(P_1(0)=\dots =P_k(0)=0\) and \(\{T_i:\,i\in [k]\}\) is a family of commuting and invertible \(\mu \)-invariant transformations. In contrast to Dunford and Zygmund’s result, the commutativity assumption turns out to be indispensable for the polynomial case. For a more thorough exposition on the matter we refer the reader to Section 1.2 in [23] as well as the introduction from [1], see page 3. In the spirit of the above, Theorem 1.10 establishes the multi-parameter result for orbits along sets of the form B. For example, for appropriate choices of parameters, Theorem 1.10 implies that for any \(p\in (1,\infty )\) and any \(f\in L^p_{\mu }(X)\), we have
where \(c_1,\dotsc ,c_k\in (1,6/5)\) and \(\{T_i:\,i\in [k]\}\) is a family of commuting and invertible \(\mu \)-invariant transformations. To the best of the author’s knowledge, this is the first time multi-parameter pointwise convergence is established for orbits along fractional powers in the spirit of Dunford and Zygmund.
Using an abstract multi-parameter oscillation result from [23], we reduce the task of proving the above theorem to establishing some useful quantitative uniform estimates, which may be of independent interest.
Before precisely formulating these estimates, we note that for any \(Y\subseteq X\subseteq {\mathbb {R}}\), with \(|X|>2\), we have that
For the basic properties of oscillations as well as the definition of multi-parameter oscillations we refer the reader to section 2 from [23].
Theorem 1.12
(Uniform 2-oscillation and vector-valued maximal estimates) Assume \(c_1\in [1,2)\), \(c_2\in [1,6/5)\) and B as in the introduction. Assume \((X,{\mathcal {B}},\mu )\) is a \(\sigma \)-finite measure space and T is an invertible \(\mu \)-invariant transformation. Let
Then for any \(p\in (1,\infty )\), there exists a constant \(C_p\) such that
and such that
We now comment on the proof of Theorem 1.12. Again, Calderón’s transference principle suggests that it suffices to establish these estimates for the integer shift system. Ultimately, those estimates are derived from the analogous ones for the standard discrete Hardy–Littlewood averaging operator. For the vector-valued maximal inequality we use the exponential sum estimates of Lemma 2.1 together with the fact that \(\psi \) behaves “like a constant” in dyadic blocks in order to eventually be able to use the corresponding estimates for the Hardy–Littlewood averaging operator (for example see Theorem 1 in [12] or Theorem C in [21]). The situation is much more complicated for the oscillations. We follow the strategy from [22] and [27], and we break our analysis into short and long oscillations, and instead of opting to handle as our “long oscillations” the rather natural choice \(\{2^n:\,n\in {\mathbb {N}}_0\}\), we choose a much denser set, namely \(\{\lfloor 2^{n^{\tau }}\rfloor :\,n\in {\mathbb {N}}_0\}\), for \(\tau \) small. This affords us to bound the short oscillations straightforwardly. Loosely speaking, the long oscillations are treated in similar manner to the vector-valued maximal inequality, but the fact that the 2-oscillations are not a positive operator makes the use of the fact that \(\psi \) behaves nicely in dyadic blocks difficult. Here, we adapt the argument from section 5 in [24] to our oscillation setting in order to compare averages with different weights. Again, we use the uniform oscillation estimates for the discrete Hardy–Littlewood averaging operator to conclude (which one may find for example in [15] or [19]). Finally, we mention that the exponential sum estimates help us understand some error terms on \(\ell ^2\), and Riesz–Thorin interpolation together with trivial bounds coming from the fact that we deal with averaging operators help us establish the corresponding \(\ell ^p\) bounds.
1.4 Notation
We denote by C a positive constant that may change from occurrence to occurrence. If A, B are two non-negative quantities, we write \(A\lesssim B\) or \(B \gtrsim A\) to denote that there exists a positive constant C such that \(A\le C B\). Whenever \(A\lesssim B\) and \(A\gtrsim B\) we write \(A\simeq B\). For two complex-valued functions f, g we write \(f\sim g\) to denote that \(\lim _{x\rightarrow \infty }\frac{f(x)}{g(x)}=1\). For any topological space X, we denote by \({\mathcal {C}}(X)\) the set of all complex-valued continuous functions. We denote the average value of a function \(f:{\mathbb {Z}}\rightarrow {\mathbb {C}}\) over a finite set \(Q\subseteq {\mathbb {Z}}\) by \([f]_Q=\frac{1}{|Q|}\sum _{x\in Q}f(x)\). For any natural number N, we let \([N]=\{1,2,\dotsc ,N\}\).
2 Basic Properties of the Sets B
In this section we collect some useful properties of the sets B. We begin by stating an exponential sum estimate proven in [17]. Here, and for the following two sections, we fix two constants \(c_1,c_2\) such that \(c_1\in [1,2)\) and \(c_2\in [1,6/5)\), as well as \(h_1\), \(h_2\in {\mathcal {R}}_{c_1}\) and \({\mathcal {R}}_{c_2}\) respectively and \(\psi \), \(\gamma _1\), \(\gamma _2\), B are as in the introduction and all the implied constants may depend on them. We note that we use the basic properties of those functions as described in Lemma 2.6 and Lemma 2.14 from [20] without further mention.
Lemma 2.1
Assume \(\chi >0\) is such that \((1-\gamma _1)+3(1-\gamma _2)+6\chi <1\). Then there exists a real number \(\chi '>0\) such that
as well as
where the implied constant does not depend on \(\xi \) or N.
Proof
We note that one is derived from the other using summation by parts. The proof can be found in page 6 as well as Lemma 3.2 in [17]. \(\square \)
Lemma 2.4
If \(\varphi _2\lesssim \varphi _1\), then B does not contain arbitrarily long intervals in \({\mathbb {Z}}\).
Proof
Assume that \(\varphi _2\lesssim \varphi _1\) and let \(\{n,n+1,\dotsc ,n+l-1\}\subseteq B\), we wish to bound l. First of all let us notice that B may be partitioned as follows
where \(B_m=\{\,n\in {\mathbb {N}}\,:\,0\le \varphi _1(n)-m<\psi (n)\,\}\), note that \(B_m\cap B_k=\emptyset \) for \(m\ne k\). For sufficiently large m, k with \(m<k\), we have that \(dist(B_m,B_k)\ge 2\) since if we assume for the sake of a contradiction that \(dist(B_m,B_k)=1\) then there exists \(n\in B_m\) such that and \(n+1\in B_k\). But then
and thus
which implies
By the Mean Value Theorem there exists \(\xi _n\in (n,n+1)\) such that \(\varphi _1'(\xi _n)>1/2\), which will be a contradiction for sufficiently large n, since \(\varphi _1'(x)\rightarrow 0\), as \(x\rightarrow \infty \). Thus, ignoring some first few terms of B, the fact that \(\{n,n+1,\dotsc ,n+l-1\}\subseteq B\), together with our previous observation, imply that there exists an \(m\in {\mathbb {N}}\) such that \(\{n,n+1,\dotsc ,n+l-1\}\subseteq B_m\). Then by combining \(n,n+l-1\in B_m\), we obtain
which by the Mean Value Theorem becomes
and thus \((l-1)\lesssim \frac{\varphi _2'(n+l)}{\varphi _1'(n+l)}\lesssim \frac{\varphi _2(n+l)}{\varphi _1(n+l)}\lesssim 1\), since \(\varphi _2\lesssim \varphi _1\). Thus B does not contain arbitrarily long intervals. \(\square \)
3 Uniform 2-oscillation Estimates
We prove the first half of Proposition 1.12, namely we establish the estimate (1.13). By the Calderón Transference Principle, in order to establish that estimate for any \(\sigma \)-finite measure preserving system, it suffices to establish it for the integer shift system, namely for \(({\mathbb {Z}},{\mathcal {P}}({\mathbb {Z}}),|\cdot |,S)\) where S is the shift map \(S(x)=x-1\) and the \(|\cdot |\) is the counting measure. To simplify the notation, we let \(B_t=B\cap [1,t]\) and write
We therefore wish to show that for any \(p\in (1,\infty )\), there exists a contant \(C_p\) such that
To obtain this result, we break the 2-oscillations into short and long ones, and we will need to carefully choose some parameters first. Let \(p_0\in (1,\infty )\) be such that \(p\in (p_0,p_0')\), let \(\tau \in (0,\min \big \{\frac{p_0-1}{2},\frac{1}{2}\big \})\) and let \({\mathbb {D}}_{\tau }=\{2^{n^{\tau }}:\,n\in {\mathbb {N}}_0\}\). It is not difficult to see that
where
see [23] page 17. One may adapt the argument appearing in Lemma 1.3 in [16] to establish this. We deal with the second term of (3.2). Note that
where we have used the fact that \(\Vert \cdot \Vert _{\ell ^2}\le \Vert \cdot \Vert _{\ell ^1}\), together with the fact that \(M_tf(x)=M_sf(x)\) whenever \(B_t=B_s\). For any \(n \in {\mathbb {N}}\), let \(\big \{\beta _{m}^{(n)}:\,m\in \{0,\dotsc ,l_n\}\big \}\) be an increasing enumeration of \(\big [2^{(n-1)^{\tau }},2^{(n+2)^{\tau }}\big )\cap B\). Then we use the triangle inequality to bound the last expression by
Let \(K_t(x)=\frac{1}{|B_t|}\sum _{n\in B_t}\delta _n(x)\), where \(\delta _n(x)=1_{\{n\}}(x)\), and note that \(M_tf(x)=K_t*f(x)\). Thus we rewrite the expression above as
We firstly consider the case \(p\in (2,\infty )\). We get
where we have used Minkowski’s inequality for \(p/2>1\) and \(p>1\), and then Young’s convolution inequality. In the case where \(p\in (1,2]\), we note
where we have used Minkowski’s inequality and Young’s convolution inequality. Combining the two cases gives
where \(q=\min \{2,p\}\). We focus on \(\big \Vert K_{\beta _{m}^{(n)}}-K_{\beta _{m-1}^{(n)}}\big \Vert _{\ell ^1({\mathbb {Z}})}\), note that
thus
We know that there exists \(\varepsilon >0\) such that \(|B_t|=\varphi _2(t)(1+O(t^{-\varepsilon }))\), (see [17], page 5), thus we get
Note that \(\frac{\varphi _2(2^{(n+2)^{\tau }})2^{-\varepsilon (n-1)^{\tau }}}{\varphi _2(2^{(n-1)^{\tau }})}\le \frac{\varphi _2(2^{(n-1)^{\tau }+3^{\tau }})2^{-\varepsilon (n-1)^{\tau }}}{\varphi _2(2^{(n-1)^{\tau }})}\lesssim 2^{-\varepsilon (n-1)^{\tau }}\), since \(0<\tau \le 1/2\), \(\varphi _2\) is increasing and \(\varphi _2(4x)\lesssim \varphi _2(x)\). We also note that by the Mean Value Theorem for \(f(x)=\varphi _2(2^{x^{\tau }})\) on the interval \([n-1,n+2]\) we get that there exists \(x_n\in (n-1,n+2)\) such that
Thus
which implies
Note that if \(p>2\) then \(q=2\) and then \(q(1-\tau )>2(1-1/2)=1\) thus the first sum converges. Similarly, if \(p\le 2\), then \(q=p\). Note that \(\tau<(p_0-1)/2<(p_0-1)/p_0=1/p_0'\), but then \(q(1-\tau )>p(1-1/p_0')=p/p_0>1\), as desired. In either case, the first series is summable. The second series is also summable, since for example \(2^{-q\varepsilon n^{\tau }}\lesssim _{p,\tau } n^{-2}\), and thus (3.3) becomes
To establish the estimate (3.2), it remains to bound the first term. Note that
and we now establish the following estimate.
Claim
There exists a constant \(C_{p,\tau }>0\) such that
Proof
We introduce some auxiliary averaging operators. Let
where \(L_t(x)=\frac{1}{|B_t|}\sum _{1\le s\le t}\psi (s)\delta _{s}(x)\). We may compare \(M_t\) with \(A_t\) as follows
The first term of the expression (3.4) will be bounded by using Lemma 2.1 and interpolation. More specifically, we start with \(p=2\), and we note
and for each \(n\in {\mathbb {N}}_0\) we have
and note that there exists \(\chi >0\) such that for any \(\xi \in {\mathbb {T}}\) we get
and thus
which gives
For the case of \(p\ne 2\), firstly, let us assume that \(p\in (2,\infty )\). Note that there exists a positive constant C such that
since
where we have used the trigonometric estimates [17], page 6, for \(\xi =0\). We may choose \(\theta \in (0,1)\) such that \(\frac{1}{p}=\frac{\theta }{2}+\frac{1-\theta }{p_0'}\) and use Riesz–Thorin interpolation theorem. Since for any \(n\in {\mathbb {N}}_0\) we have
we interpolate to obtain
Note that since \(p/2>1\), we have
The case of \(p\in (1,2)\) is similar; we choose \(\theta \in (0,1)\) such that \(\frac{1}{p}=\frac{\theta }{2}+\frac{1-\theta }{p_0}\). Riesz-Thorin interpolation theorem yields the same estimate as before
and we note that
We have appropriately bounded the first term of equation (3.4). We now focus on the second term. We will reduce the 2-oscillation \(\ell ^p\) estimates for \(A_{2^{n^{\tau }}}\) to the corresponding ones for the standard \(H_{2^{n^{\tau }}}\). Firstly, the analysis of the 2-oscillations will be made easier if we adjust \(A_t\) to the following very similar operator
Note that
Thus
One may use a similar argument to the one presented earlier to compare \(A_{2^{n^{\tau }}}\) with \(D_{2^{n^{\tau }}}\) and obtain
Using the estimate \(\Vert A_{2^{n^{\tau }}}f-D_{2^{n^{\tau }}}f\Vert _{\ell ^p({\mathbb {Z}})}\lesssim 2^{-\chi n^{\tau }} \Vert f\Vert _{\ell ^p({\mathbb {Z}})}\), we can bound the first term of equation (3.5) by \(C_{p,\tau }\Vert f\Vert _{\ell ^p({\mathbb {Z}})}\) and our task has been reduced to estimating the 2-oscillations of \(D_{2^{n^{\tau }}}\). In fact, we will be able to estimate the 2-oscillations of \(D_{n}\) by comparing it with \(H_n\). For convenience, let \(\Psi (k)=\sum _{1\le s\le k}\psi (s)\). We perform summation by parts
where
We have shown \(D_kf(x)=\sum _{s=1}^{\infty }\lambda _{s}^kH_sf(x)\). Without loss of generality, \(\psi (s)\) is decreasing, and thus \((\lambda _s^k)_{s,k\in {\mathbb {N}}}\) is a family of non-negative real numbers (in the spirit of Lemma 2 from [24]) and we note that for any \(k\in {\mathbb {N}}\) we have that
and we also have that for any fixed \(N\in {\mathbb {N}}\) the sequence \(\sum _{s=1}^N\lambda _s^k\) is decreasing in k, since
and for any \(1\le N\le k-1\) we have
For any \(k\in {\mathbb {N}}\) we introduce the function \(N_k:[0,1)\rightarrow {\mathbb {N}}\) such that \(N_k(t)=\min \{N\in {\mathbb {N}}:\,\sum _{i=1}^N\lambda ^k_i>t\}\) and we also introduce \(I_s^k=N_k^{-1}(\{s\})=\{t\in [0,1):\,N_k(t)=s\}\). We note that for all \(k\in {\mathbb {N}}\), \(N_k\) is increasing in t. Also, since for any fixed \(N\in {\mathbb {N}}\) the sequence \(\sum _{s=1}^N\lambda _s^k\) is decreasing in k, we have that for any fixed \(t\in [0,1)\), the sequence \(N_k(t)\) is increasing in k. Note that \(|I_s^k|=\lambda ^k_s\). Then
Finally, for any \(J\in {\mathbb {N}}_0\) and any \(I=\{I_0,\dotsc ,I_J\}\in \mathfrak {S}_J({\mathbb {N}}_0)\), we have that
and we finish the argument by noting that
where we have used the uniform 2-oscillation \(\ell ^p\)-estimates for the standard averaging operator for \(p\in (1,\infty )\), see [15] or [19], or more precisely
We note that
This establishes the estimate (1.13). \(\square \)
4 Vector-Valued Maximal Estimates and Concluding the Proof of Theorem 1.10
In this section we establish the vector-valued estimates for the maximal function corresponding to \(M_t\), namely the estimate (1.14). By the Calderón Transference Principle, it suffices to prove the following.
Proposition 4.1
For any \(p\in (1,\infty )\), there exists a constant \(C_p\) such that for any \((f_j)_{j\in {\mathbb {Z}}}\in \ell ^p\big ({\mathbb {Z}};\ell ^2({\mathbb {Z}})\big )\) we have
Proof
Firstly, we note that \(\sup _{t\in [1,\infty )}M_t|f|(x)\lesssim \sup _{n\in {\mathbb {N}}_0}M_{2^n}|f|(x)\) and thus
We focus on the first term, for \(p=2\) we note that
By Plancherel theorem combined with Lemma 2.1 we obtain the following
and thus
For the case of \(p\ne 2\), we proceed in a manner identical to the one of the previous section. Firstly, let us assume that \(p\in (2,\infty )\). We fix a \(p_0>p\) and we note that there exists a positive constant C such that
since
We may choose \(\theta \in (0,1)\) such that \(\frac{1}{p}=\frac{\theta }{2}+\frac{1-\theta }{p_0}\) and use Riesz-Thorin interpolation theorem. Since for any \(n\in {\mathbb {N}}_0\) we have
we interpolate to obtain
Thus if we let \(T_n:\ell ^{p}({\mathbb {Z}})\rightarrow \ell ^p({\mathbb {Z}})\) such that \(T_nf=M_{2^n}f-A_{2^n}f\), then we know that \(T_n\) is a bounded linear operator with \(\Vert T_n\Vert _{\ell ^p({\mathbb {Z}})\rightarrow \ell ^p({\mathbb {Z}})}\le C(2^{\chi \theta })^{-n}\). Thus, we know that \(T_n\) has an \(\ell ^2\)-valued extension (see for example [13], page 386) with the same norm, that is
Finally, since \(p/2>1\), we get
For \(p\in (1,2)\) the situation is similar, we choose \(p_0\in (1,p)\), and \(\theta \in (0,1)\) such that \(\frac{1}{p}=\frac{\theta }{2}+\frac{1-\theta }{p_0}\) and Riesz-Thorin interpolation theorem yields the estimate of (4.3), which in turn implies the estimate (4.4). Since \(p<2\), we have
We have bounded appropriately the first term of (4.2) and all is left is to bound the second term. We firstly observe that for any \(n\in {\mathbb {N}}_0\) we get
since
Since \(n\in {\mathbb {N}}_0\) was arbitrary, we have shown that \(\sup _{n\in {\mathbb {N}}_0}A_{2^n}|f|(x)\lesssim \sup _{n\in {\mathbb {N}}_0}H_{2^n}|f|(x)\), and thus the second term of 4.2 may dominated by
This completes the proof. \(\square \)
The work of Sects. 3 and 4 proves Proposition 1.12. We describe how Proposition 1.12 implies Theorem 1.10.
Proof of Theorem 1.10 assuming Proposition 1.12
We simply apply Proposition 4.1 from [23] to establish a multi-parameter uniform 2-oscillation estimate, which according to Remark 2.4 together with Proposition 2.8, page 15, from the same paper yield the desired result. \(\square \)
5 Proof of the Weak-Type (1,1) Inequality
Throughout this section we have fixed a set B with \(\varphi _1\simeq \varphi _2\) and \(c_1=c_2=c\in (1,30/29)\). All constants may depend on \(\varphi _1,\varphi _2\) and \(\psi \) but on nothing else, unless stated otherwise. We remind the reader that it suffices to establish the weak type (1,1) bound for the smooth dyadic maximal function
The next two lemmas are devoted to studying the properties of \(K_N*{\widetilde{K}}_N\) and they will be key ingredients for establishing the weak-type (1,1) bound.
Lemma 5.1
There exists a positive constant C such that for all \(N\in {\mathbb {N}}\) and all \(x\in {\mathbb {Z}}\) with \(C\le |x|\le \varphi _1(N)\), we have that \(|K_N*{\tilde{K}}_N(x)|\le C N^{-1}\).
Proof
We have assumed that \(\varphi _2\simeq \varphi _1\) and thus, by Lemma 2.4, there exists a uniform bound \(C_B\) for the length of intervals contained in B. For any \(x\in {\mathbb {Z}}\) such that \(|x|\ge C_B+1\), we get
and with a change of variables we see that \(K_N*{\widetilde{K}}_N(x)=K_N*{\widetilde{K}}_N(-x)\), and thus, without loss of generality, let us assume that \(x \ge C_B+1\). Since \({{\,\textrm{supp}\,}}(\eta )\subseteq (1/2,4)\) and \(0\le \eta (x)\le 1\), we have
and all is left to do is to estimate the cardinality of that set. Let \({\mathcal {A}}_N^x=\{\,n\in {\mathbb {Z}}\,:\,n,n+x\in B\cap (N/2,4N)\,\}\), and notice that for any \(n\in {\mathcal {A}}_N^x\), we have that there exists a unique \(s,m\in {\mathbb {N}}_0\) such that
Notice that since \(x>C_B\), we have that \(s\ge 1\), since n and \(n+x\) cannot correspond to the same m. By combining the previous set of inequalities we obtain
For constants depending only on \(\varphi _1,\varphi _2\) and \(\psi \), we have
and similarly
Thus \(s\simeq \frac{x\varphi _1(N)}{N}\). We have that \(m\le \varphi _1(4N)\lesssim \varphi _1(N)\) and \(m>\varphi _1(n)-\psi (n)\ge \varphi _1(n)-C\varphi _1'(n)\ge \varphi _1(n)\big (1-C/n)\ge C\varphi _1(N)\), when \(n\in (N/2,4N)\) thus \(m\simeq \varphi _1(N)\). We also note that
and thus
Note that for all l we get that \(l-h_1(\varphi _1(l)-\psi (l))=h_1(\varphi _1(l))-h_1(\varphi _1(l)-\psi (l))=\psi (l)h_1'(\xi _l)\), for some \(\xi _l\in (\varphi _1(l)-\psi (l),\varphi _1(l))\), and thus \(l-h_1(\varphi _1(l)-\psi (l))\lesssim \varphi _2'(l)h_1'(\varphi _1(l))=\frac{\varphi _2'(l)}{\varphi _1'(l)}\). Since \(\varphi _2\simeq \varphi _1\), we get that there exists an absolute constant T, such that \(l-h_1(\varphi _1(l)-\psi (l))\le T\), and thus
Consider the set \({\mathcal {B}}_N^x=\{(s,m)\in {\mathbb {N}}\times {\mathbb {N}}\,:\,s\simeq \frac{x\varphi _1(N)}{N},\,m\simeq \varphi _1(N),\,h_1(m+s)-h_1(m)\in B(x,T)\}\) and note that for any \((s,m)\in {\mathcal {B}}^x_N\) there are at most \(C_B\) number of n’s in \({\mathcal {A}}^x_N\) corresponding to m. Therefore \(|{\mathcal {A}}^x_N|\lesssim |{\mathcal {B}}^x_N|\), and everything reduces to estimating \(|{\mathcal {B}}^x_N|\). For every \(s\ge 1 \) such that \(s\simeq \frac{x\varphi _1(N)}{N}\), we wish to estimate the number of m’s such that \((s,m)\in {\mathcal {B}}^x_N\). Notice that if we define \(g(m)=h_1(m+s)-h_1(m)\) then by the Mean Value Theorem we get
for some \(\xi _1,\xi _2\in (0,1)\) and \(\xi _3\in (m,m+s+1)\). Thus
Since \(x-T<h_1(m+s)-h_1(m)<x+T\), according to the previous calculation, for any fixed \(s\simeq \frac{x\varphi _1(N)}{N}\), we have that there are at most \(1+C \frac{T\varphi _1(N)}{x}\lesssim \frac{\varphi _1(N)}{x}\), where in the last estimate we used that \(x\le \varphi _1(N)\). The number of s’s in \([1,C\frac{x\varphi _1(N)}{N}]\), where C is the implied fixed constant appearing in \(s\simeq \frac{x\varphi _1(N)}{N}\), are bounded by \(C\frac{x\varphi _1(N)}{N}\), and therefore
This implies that
and the proof is complete. \(\square \)
Lemma 5.2
There exists a real number \(\chi >0\) such that \(K_N*{\widetilde{K}}_N(x)=G_N(x)+E_N(x)\) for all \(|x|>\varphi _1(N)\) where
We also have that \(G_N(x)\lesssim N^{-1}\), \(|G_N(x+h)-G_N(x)|\lesssim N^{-2}|h|\) and \(|E_N(x)|\lesssim N^{-1-\chi }\).
Proof
We note that for all \(n\in {\mathbb {N}}\) we have that \(1_B(n)=\lfloor \varphi _1(n) \rfloor -\lfloor \varphi _1(n)-\psi (n)\rfloor \), see [17, Lemma 2.2]. We can therefore split our kernel to several manageable pieces.
We will exploit a famous truncated Fourier Series. More precisely, we know that if \(\Phi (x)=\{x\}-1/2\) then for all \(M\in {\mathbb {N}}\) we get
Importantly, we also have
and
Finally, we can rewrite
Let us use the truncated Fourier Series and define
and
and thus
Thus
Returning back to the splitting
\(=I_1(x)+I_2(x)+\cdots +I_9(x)\). Let \(G_N(x)=I_1(x)\) and \(E_N(x)=\sum _{i=2}^9I_i(x)\). Let us firstly estimate \(I_1\), we have
and for any \(h\in {\mathbb {Z}}\) we have
It suffices to consider x, h such that \(n+x,n+x+h\in [N/2,4N]\) since the integrand is zero outside that interval. We get
This shows that \(G_N\) satisfies the desired properties. Now we bound \(E_N\). Let us start with \(I_2\). We can rewrite \(I_2\) as
According to Corollary 3.12 in [20], if we let \(F_m^x(t)=\psi (t)\eta \Big (\frac{t}{N}\Big )\eta \Big (\frac{t+x}{N}\Big )\big (e^{2\pi i m \psi (t+x)}-1\big )\), we have that for all \(m\in {\mathbb {Z}}{\setminus } \{0\}\)
Let us follow the notation of [20] and write \(N_{1,x}=\max \{N/2,N/2-x\}\) and \(N_{2,x}=\min \{4N,4N-x\}\). For all \(n\in (N_{1,x},N_{2,x}]\) we get
where we used that for all real numbers x we have \(|e^{ix}-1|\le |x|\). Similarly, after applying the mean value theorem to obtain \(|F_m^x(n+1)-F_m^x(n)|\le \sup _{t\in [n,n+1]}\Big |\frac{d(F_m^x(t))}{dt}\Big | \), we can use the following bound
Therefore we can bound \(I_2\)
Since \(\gamma \in (29/30,1)\) and \(\varphi _1\simeq \varphi _2\), we get that for \(M=N^{1+2\chi +\varepsilon }\varphi _2(N)^{-1}\), \(\chi =1-\gamma \) and \(\varepsilon <\chi /10\)
We use that for all \(\varepsilon _1>0\) we have \(\sigma _1(x)\gtrsim _{\varepsilon _1} x^{-\varepsilon _1}\) and \(\varphi _2(x)\gtrsim _{\varepsilon _1} x^{\gamma -\varepsilon _1}\) to get
For a fixed \(\varepsilon _1=\varepsilon \in (0,\chi /10)\) we get
since
which is true since \(\gamma >29/30\). Therefore we have shown that \(|I_2(x)|\lesssim N^{-1-\chi }\), as desired. The term \(I_4\) is treated similarly
Using Corollary 3.12 in [20], for \(G_m^x(n)=\psi (n+x)\eta \Big (\frac{n}{N}\Big )\eta \Big (\frac{n+x}{N}\Big )\big (e^{2\pi i m \psi (n)}-1\big )\), we obtain in an almost identical fashion the bound \(|I_4(x)|\lesssim N^{-1-\chi }\). To bound the remaining terms, we use Lemma 3.18 [20], which we state here.
Lemma 5.3
Let \(N\ge 1\), \(p,q\in \{0,1\}\), \(x\in {\mathbb {Z}}\) and \(M\ge 1\). Then
We will also use an appropriate extension of the lemma.
Lemma 5.4
Let \(N\ge 1\), \(p\in \{0,1\}\), \(x\in {\mathbb {Z}}\) and \(M\ge 1\). Then
Proof
We have that
and
Thus
where we used Lemma 4.1 from [9] to obtain the estimate for \(\big |\sum _{n+px\in (N/2,4N]}e^{2\pi i m (\varphi _1(n+px)-\psi (n+px))}\big |\). \(\square \)
Using the two lemmas above, together with the trivial estimates \(\psi (x),|\Delta _M(x)|,|\Pi _M(x)|\lesssim 1\) (since \(1_B(n)=\psi (n)+\Delta _M(n)+\Pi _M(n)\)), we may estimate
since \(\chi =1-\gamma \) and \(5/2+2\chi -3\gamma +3\varepsilon<0 \iff 10(1-\gamma )+6\varepsilon <1\) and since \(\gamma \in (29/30,1)\), one may choose such an appropriately small \(\varepsilon >0\). Finally, for \(I_6\) we get
where \(F^x_{m_1,m_2}(n)= \big (e^{2\pi i m_1 \psi (n)}-1\big )\big (e^{2\pi i m_2 \psi (n)}-1\big )\eta \Big (\frac{n}{N}\Big )\eta \Big (\frac{n+x}{N}\Big )\). Therefore we have that
For all \(n\in (N_{1,x},N_{2,x}]\) we have \(|F^x_{m_1,m_2}(n)|\lesssim |m_1m_2|\psi (N)^2\lesssim |m_1m_2|\frac{\varphi _2(N)^2}{N^2}\) and also by the mean value theorem together with the following calculation
we get \(\sup _{N_{1,x}<n\le N_{2,x}} \{|F_{m_1,m_2}^x(n+1)-F_{m_1,m_2}^x(n)|\}\lesssim |m_1m_2|\frac{\varphi _2(N)^2}{N^3} \). We let \(m=\max \{m_1,m_2\}\) and we use Corollary 3.12 from [20] for \(\alpha =0\) and \(\kappa =1\) to obtain
We can now finish our estimates for \(I_6\)
We wish to have that \(8/3+16/3\chi +8/3\varepsilon -10/3\gamma +2\varepsilon _1-2/3\le -1-\chi \), but we have that
and we can choose \(\varepsilon _1>0\) to make this true. \(\square \)
We use Lemma 5.1 and Lemma 5.2 to prove the weak type (1,1) estimates of Theorem 1.5. We state and prove a general Theorem that allows us to conclude. It is a natural extension of Theorem 6.1 in [20], and the novelty lies in our handling of the problematic initial part of \(K_N*{\widetilde{K}}_N\).
Theorem 5.5
Let \({\mathcal {M}}(f)(x)=\sup _{n\in {\mathbb {N}}}|K_n*f(x)|\) be the maximal function corresponding to a family of nonnegative kernels \(\big (K_n\big )_{n\in {\mathbb {N}}}\subseteq \ell ^1({\mathbb {Z}})\) such that \(||{\mathcal {M}}(f)||_{\ell ^{\infty }({\mathbb {Z}})}\lesssim ||f||_{\ell ^{\infty }({\mathbb {Z}})}\) for all \(f\in \ell ^{\infty }({\mathbb {Z}})\) and let \(\big (F_n\big )_{n\in {\mathbb {Z}}}\) be a family of nonnegative functions. Assume that there are sequences \((d_n)_{n\in {\mathbb {N}}}\), \((D_n)_{n\in {\mathbb {N}}}\subseteq [1,\infty )\) such that \(|supp(K_n)|=d_n\), \(supp(K_n)\subseteq [0,D_n]\), \(supp(F_n)\subseteq [-D_n,D_n]\), \(d_n\le D_n^{\varepsilon _0}\) for some \(\varepsilon _0\in (0,1)\) and assume there exists a finite constant \(M>1\) such that \(Md_n\le d_{n+1}\) and \(MD_n\le D_{n+1}\le 2^{n+1}\) for all \(n\in {\mathbb {N}}\). Also, assume that exists a real number \(\varepsilon _1>0\) such that for all \(n\in {\mathbb {N}}\) and \(x\in {\mathbb {Z}}\) we have
and assume that there exists a constant \(A>0\) such that
Finally, assume that there exists an \(\varepsilon _2\in (0,1]\) such that
Then we have that there exists a constant \(C>0\) such that
Before proving the theorem let us briefly show how it implies the weak-type (1,1) bound.
Proof of Theorem 1.5
By letting \(K_n(x)=\frac{1}{\varphi _2(2^n)}\sum _{k\in B}\delta _k(x)\eta \Big (\frac{k}{2^n}\Big )\), \(d_n\simeq \varphi _1(2^n)\), \(D_n\simeq 2^n\) we can apply the theorem for
Lemma 5.2 guarantees the existence of a real number \(\varepsilon _1>0\) such that \(|K_n*{\widetilde{K}}_n(x)-F_n(x)|\lesssim D_n^{-1-\varepsilon _1}\) for \(|x|> \varphi _1(2^n)\) and for smaller values of x the estimate is trivially established from the definition of \(F_n\). We also have
and thus if C is the constant appearing in Lemma 5.1, for all integers x such that \(|x|\le C\), we get that
and for all x such that \(C\le |x|\le \varphi _1(2^n)\) we get
We can conclude by letting \(\varepsilon _2=1\), and using the estimates from Lemmas 5.1 and 5.2. This completes the proof. \(\square \)
Proof of Theorem 5.5
Let \(f\in \ell ^1({\mathbb {Z}})\) such that \(f\ge 0\) and let \(\alpha >0\). We will perform a subtle variation of the Calderón-Zygmund decomposition. There exists a family of disjoint dyadic cubes \((Q_{s,j})_{(s,j)\in {\mathcal {B}}}\), where \({\mathcal {B}}\subseteq {\mathbb {N}}_0\times {\mathbb {Z}}\) and \(Q_{s,j}=[j2^s,(j+1)2^s)\cap {\mathbb {Z}}\) and functions g, b such that
-
\(f=g+b\),
-
\(\Vert g\Vert _{\ell ^1({\mathbb {Z}})}\le \Vert f\Vert _{\ell ^1({\mathbb {Z}})}\text { and }\Vert g\Vert _{\ell ^{\infty }({\mathbb {Z}})}\le 2\alpha \),
-
\(b=\sum _{(s,j)\in {\mathcal {B}}}b_{s,j}\text { where }b_{s,j} \text { is supported on }Q_{s,j}\),
-
\(\sum _{x\in Q_{s,j}}b_{s,j}(x)=0\),
-
\(\Vert b_{s,j}\Vert _{\ell ^1({\mathbb {Z}})}\le 4\alpha |Q_{s,j}|\),
-
\(\sum _{(s,j)\in {\mathcal {B}}}|Q_{s,j}|\le \alpha ^{-1}\Vert f\Vert _{\ell ^1({\mathbb {Z}})}\).
For every \(s\ge 0\) we let
and for every \(n\in {\mathbb {N}}_0\) we decompose further
-
\(b_s^n(x)=b_s(x)1_{\{y\in {\mathbb {Z}}:\,|b_s(y)|>\alpha d_n\}}(x)\),
-
\(h_s^n(x)=b_s^n(x)-b_s^n(x)=b_s(x)1_{\{y\in {\mathbb {Z}}:\,|b_s(y)|\le \alpha d_n\}}(x)\),
-
\( g_s^n(x)=\sum _{\begin{array}{c} j\in {\mathbb {Z}}:\\ (s,j)\in {\mathcal {B}} \end{array}} [h^n_s]_{Q_{s,j}}1_{Q_{s,j}}\text {,}\)
-
\(B_s^n(x)=h_s^n(x)-g_s^n(x)=\sum _{\begin{array}{c} j\in {\mathbb {Z}}:\\ (s,j)\in {\mathcal {B}} \end{array}} \big (h^n_s-[h^n_s]_{Q_{s,j}}\big )1_{Q_{s,j}} \text {.}\)
Let \(s(n)=\min \{s\in {\mathbb {N}}_0:\,2^s\ge D_n\}\) and decompose f as \(g+\sum _{s\ge 0}b_s=g+\sum _{s\ge 0}(b_s^n+g_s^n+B_s^n)=\bigg (g+\sum _{s\ge 0}g^n_{s}\bigg )+\sum _{s\ge 0}b^n_s+\sum _{s=0 }^{s(n)-1}B^n_s+\sum _{s=s(n) }^{\infty }B^n_s\). We have that
and our treatment will be different for each summand. The following subsections are devoted to this task, and the most difficult part will be to bound the final one, where we will exploit the cancellation properties of \(B^n_s\) together with the properties of \(F_n\).
5.1 Estimates for the First Three Summands
For the good part we will use \(\ell ^{\infty }\)-bounds together with the fact that \(\Vert {\mathcal {M}}\Vert _{\ell ^{\infty }\rightarrow \ell ^{\infty }}=T<\infty \). Note that
and \(g(x)\le 2\alpha \) for all \(x\in {\mathbb {Z}}\), and thus \(\Vert g+\sum _{s\ge 0}g^n_s\Vert _{\ell ^{\infty }({\mathbb {Z}})}\le 6\alpha \) and thus \(\big |K_n*\big (g+\sum _{s\ge 0}g^n_s\big )(x)\big |\le T6\alpha \) for all \(x\in {\mathbb {Z}}\) and \(n\in {\mathbb {N}}\). Thus for any \(C>24T\) we get
since it is the empty set.
For the second summand we use the lacunary nature of \((d_n)_{n\in {\mathbb {N}}}\) as well as the bounds for the cardinality of the support of \(K_n\). Specifically, we have
For the third summand we simply use the fact for any \(n\in {\mathbb {N}}\) and any \(s\ge s(n)\) we get that \(2^s\ge D_n\) and thus
where 3Q denotes the interval with the same center as Q and three times its radius. Therefore
5.2 Estimates for the Fourth Summand
The fourth summand is the most difficult to estimate and here we use the regularity of \(K_n*{\widetilde{K}}_n\). We have
We need the following result to conclude.
Claim
There exists \(0<\lambda <1\) such that for all \(n\in {\mathbb {N}}\) and \(0\le s_1\le s_2\le s(n)-1\) we get
Assuming that (5.8) holds, let us see how we can deduce the desired estimate. We have
For the first and the third term note that
where we have used the estimates from the Calderón-Zygmund decomposition in the beginning of the proof.
The fourth term is bounded as follows.
Fix \(|j|\le A\) and \(x\in {\mathbb {Z}}\) such that \(x-j\in {{\,\textrm{supp}\,}}(b_{s_0})\) for some integer \(s_0\). Since the supports of \(b_{s}\)’s are disjoint we have that there can be at most one integer \(s_0'\) such that \(x\in {{\,\textrm{supp}\,}}(b_{s_0'})\). Note also that \([|h^n_s|]_{Q_{s,j}}\lesssim \alpha \), as we observed earlier. We have
where we have used the existence of a finite constant \(M>1\) such that \(Md_n\le d_{n+1}\). We get that
A similar argument may be used to bound the second term. For the sake of completeness we note that
Fix \(|j|\le A\) and \(x\in {\mathbb {Z}}\) such that \(x-j\in {{\,\textrm{supp}\,}}(b_{s_0})\) for some integer \(s_0\). Since the supports of \(b_{s}\)’s are disjoint we have that there can be at most one integer \(s_0'\) such that \(x\in {{\,\textrm{supp}\,}}(b_{s_0'})\). We have
By comparing the inequality above with (5.9), we see that an argument identical to the one used previously may be used here. The proof will be completed once we have established the estimate (5.8) of the claim. We do this in the following subsection.
5.3 Proof of the Estimate (5.8)
Let \(n\in {\mathbb {N}}\) and \(0\le s_1\le s_2\le s(n)-1\) and let us note that
Now decompose \(F_n(x)=F_n(x)1_{|x|\le A}+F_n(x)1_{|x|> A}=\sum _{-A \le j\le A}F_n(j)\delta _j(x)+F_n(x)1_{|x|> A}\) and let \(G_n(x)=F_n(x)1_{|x|>A}\). We obtain
In the right hand side of our inequality one of the terms of the desired estimate already appeared and thus we can now focus on the other two summands. Let \(Z_{m,n}=\big [m2^{s(n)},(m+1)2^{s(n)}\big )\cap {\mathbb {Z}}\), \({\widetilde{Z}}_{m,n}=\big [(m-1)2^{s(n)},(m+2)2^{s(n)}\big )\cap {\mathbb {Z}}\) and \(E_n=K_n*{\widetilde{K}}_n-F_n\). Note that \({{\,\textrm{supp}\,}}(E_n)\subseteq [-D_n,D_n]\subseteq [-2^{s(n)},2^{s(n)}]\). We have that
Now we note that for any \(m\in {\mathbb {Z}}\) we have
On the one hand
and on the other hand \( |\{k\in {\mathbb {Z}}:\,(s_1,k)\in {\mathcal {B}}\, \& \,Q_{s_1,k}\cap {\widetilde{Z}}_{m,n}\ne \emptyset \}|\lesssim 2^{s(n)-s_1}\) since \({\widetilde{Z}}_{m,n}\) can be partitioned into 3 dyadic intervals, \(Q_{s_1,k}\) is a dyadic interval and \(s_1\le s(n)\). Therefore
and finally
since \(D_n\ge 2^{s(n)-1}\ge 2^{s(n)-s_1-1}\), we get
as desired.
Now we focus on the last term \(\big |\langle G_n*B^n_{s_1},B^n_{s_2} \rangle _{\ell ^2({\mathbb {Z}})}\big |\). Note that \({{\,\textrm{supp}\,}}(G_n)\subseteq [-2^{s(n)},2^{s(n)}]\). We have
Let us define \(B^n_{s,j}=B^n_{s}1_{Q_{s,j}}\), and note that for any \(m\in {\mathbb {Z}}\), \(x\in Z_{m,n}\), we have
We also note that \(\sum _{x\in {\mathbb {Z}}}B^n_{s_1,j}(x)1_{{\widetilde{Z}}_{m,n}}(x)=0\). To see this note that if \({{\,\textrm{supp}\,}}(B^n_{s_1,j})\cap {\widetilde{Z}}_{m,n}=\emptyset \), then it is trivial, and if they intersect, we must have that \({{\,\textrm{supp}\,}}(B^n_{s_1,j})\subseteq {\widetilde{Z}}_{m,n}\), since \({{\,\textrm{supp}\,}}(B^n_{s_1,j})\subseteq Q_{s_1,j}\) which is a dyadic interval of length \(2^{s_1}\) and \({\widetilde{Z}}_{m,n}\) is the union of three dyadic intervals of larger length. In the second case we get
from the definition of \(B^n_{s_1,j}(x)\). Fix \(m\in {\mathbb {Z}}\) and \(j\in {\mathbb {Z}}\) such that \((s_1,j)\in {\mathcal {B}}\) and let \(x_{s_1,j}\) be the center of the cube \(Q_{s_{1},j}\). Assume \(x\in Z_{m,n}\) is such that \(|x-x_{s_1,j}|\ge C d_n^{\varepsilon _2}+C2^{s_1}\). Using the cancellation property we have established together with the regularity assumptions for \(F_n\), we get
Here we have used the fact that \(|x-y|\ge |x-x_{s_1,j}|-|x_{s_1,j}-y|\ge C d_n^{\varepsilon _2}+C2^{s_1}-2^{s_1}\gtrsim d_n^{\varepsilon _2}\) and thus we may use (5.7). Taking into account (5.6) and the definition of \(G_n\), we get that for any \(x\in {\mathbb {Z}}\)
Now we may estimate as follows
For the first summand note that
The calculations from (5.10), (5.12) show that \(\sup _{m\in {\mathbb {Z}}}\Vert B^n_{s_1}1_{{\widetilde{Z}}_{m,n}}\Vert _{\ell ^1({\mathbb {Z}})}\lesssim \alpha D_n\) and thus the first summand is bounded by a constant multiple of
For the second summand we consider two cases. In the first case we assume that \(2^{s_1}\le d_n^{\varepsilon _2}\). Then, any interval of radius \(\lesssim d_n^{\varepsilon _2}\) contains at most \(\lesssim 2^{-s_1}d_n^{\varepsilon _2}\) sets of the form \(Q_{s_1,j}\). Thus we have
where we used the estimate (5.11), the fact that \(d_n\le D_n^{\varepsilon _0}\) and the fact that \(2^{s(n)-1}<D_n\le 2^{s(n)}\). We have established the appropriate bound for the first case.
In the second case, we assume that \(2^{s_1}> d_n^{\varepsilon _2}\). In that case, any interval of radius \(\lesssim 2^{s_1}\) contains at most \(\lesssim 2^{s_1}2^{-s_1}=1\) sets of the form \(Q_{s_1,j}\). Thus we have
This concludes the second case.
Combining everything we get
since \(2^{-1}\lesssim 2^{\varepsilon _0-1}\). For \(\lambda =\min \{2^{-\varepsilon _1},2^{\varepsilon _0-1}\}\in (0,1)\), we obtain the estimate (5.8) and the proof of Theorem 5.5 is complete. \(\square \)
References
Bourgain, J., Mirek, M., Stein, E.M., Wright, J.: On a multi-parameter variant of the Bellow-Furstenberg problem. Available at arXiv:2209.07358
Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61, 39–72 (1988)
Bourgain, J.: On the pointwise ergodic theorem on \(L^p\) for arithmetic sets. Israel J. Math. 61, 73–84 (1988)
Bourgain, J.: Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson, and D. S. Ornstein. Inst. Hautes Etudes Sci. Publ. Math. 69, 5–45 (1989)
Buczolich, Z.: Universally \(L^1\) good sequences with gaps tending to infinity. Acta Math. Hungar. 117(1–2), 91–14 (2007)
Buczolich, Z., Mauldin, R.D.: Divergent square averages. Ann. Math. 171(3), 1479–1530 (2010)
Christ, M.: A weak type (1,1) inequality for maximal averages over certain sparse sequences. Available at arXiv:1108.5664
Christ, M.: Weak type (1, 1) bounds for rough operators. Ann. Math. (2) 128(1), 19–42 (1988)
Daskalakis, L.: Roth’s theorem and the Hardy-Littlewood majorant problem for thin subsets of Primes. J. Funct. Anal. 286(5), 110291 (2024)
Dunford, N.: An individual ergodic theorem for non-commutative transformations. Acta Sci. Math. Szeged 14, 1–4 (1951)
Fefferman, C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)
Fefferman, C., Stein, E.M.: Some Maximal Inequalities. Am. J. Math. 93(1), 107–115 (1971)
Grafakos, L.: Classical Fourier Analysis. Vol. 249 of Graduate Texts in Mathematics. 3rd ed. Springer
Heath-Brown, D.R.: The Pjateckii-Sapiro prime number theorem. J. Num. Theo. 16, 242–266 (1983)
Jones, R.L., Kaufman, R., Rosenblatt, J.M., Wierdl, M.: Oscillation in ergodic theory. Ergodic Theo. Dynam. Syst. 18(4), 889–935 (1998)
Jones, R.L., Seeger, A., Wright, J.: Strong variational and jump inequalities in harmonic analysis. Trans. Amer. Math. Soc. 360(12), 6711–6742 (2008)
Krause, B., Mirek, M., Trojan, B.: On the Hardy-Littlewood majorant problem for arithmetic sets. J. Funct. Anal. 271(1), 164–181 (2016)
LaVictoire, P.: Universally \(L^1\) Bad Arithmetic Sequences. J. Anal. Math. 113(1), 241–263 (2011)
Mirek, M., Słomian, W., Szarek, T.Z.: Some remarks on oscillation inequalities. Ergodic Theo. Dynam. Syst. 1–30. https://doi.org/10.1017/etds.2022.77
Mirek, M.: Weak type (1, 1) inequalities for discrete rough maximal functions. J. Anal. Mat. 127, 303–337 (2015)
Mirek, M., Stein, E.M., Trojan, B.: \(\ell ^p({\mathbb{Z} }^d)\)-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates. Invent. Math. 209, 665–748 (2017)
Mirek, M., Stein, E.M., Zorin-Kranich, P.: Jump inequalities for translation-invariant operators of Radon type on \(\mathbb{Z} ^d\). Adv. Math. 365, 57 (2020)
Mirek, M., Szarek, T.Z., Wright, J.: Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives. Rev. Mat. Iberoam. 38(7), 2249–2284 (2022)
Mirek, M., Trojan, B., Zorin-Kranich, P.: Variational estimates for averages and truncated singular integrals along the prime numbers. Trans. Am. Math. Soc. 369(8), 5403–5423 (2017)
Rosenblatt, J.M., Wierdl, M.: Pointwise ergodic theorems via harmonic analysis. In: Petersen, K.E., Salama, I. (eds.). Ergodic Theory and Harmonic Analysis: Proceedings of the 1993 Alexandria Conference, Cambridge University Press, pp. 3–152 (1995)
Urban, R., Zienkiewicz, J.: Weak Type (1, 1) Estimates for a Class of Discrete Rough Maximal Functions. Math. Res. Lett. 14(2), 227–237 (2007)
Zorin-Kranich, P.: Variation estimates for averages along primes and polynomials. J. Funct. Anal. 268(1), 210–238 (2015)
Zygmund, A.: An individual ergodic theorem for non-commutative transformations. Acta Sci. Math. Szeged 14, 103–110 (1951)
Funding
This paper is based upon work supported by the National Science Foundation under Grant DMS-2154712.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Seeger.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Daskalakis, L. Weak-Type (1,1) Inequality for Discrete Maximal Functions and Pointwise Ergodic Theorems Along Thin Arithmetic Sets. J Fourier Anal Appl 30, 37 (2024). https://doi.org/10.1007/s00041-024-10093-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-024-10093-z