1 Introduction
The object of this paper is the study of two-weight norm inequalities for a vector-valued operator from a weighted
$L^{p}(\unicode[STIX]{x1D70E})$
-space to mixed norm
$L_{l^{s}}^{q}(\unicode[STIX]{x1D707})$
spaces,
$1<p<\infty$
,
$0<q<p$
. More precisely, we study the following problem. Let
${\mathcal{D}}$
be the standard dyadic system in
$\mathbb{R}^{n}$
, given by
${\mathcal{D}}:=\{2^{-k}(\,[0,1)^{n}+j);k\in \mathbb{Z},j\in \mathbb{Z}^{n}\}$
, let
$(\unicode[STIX]{x1D706}_{Q})_{Q\in {\mathcal{D}}}$
be a sequence of non-negative real numbers and let
$T$
be the operator defined by
$$\begin{eqnarray}T(f\,d\unicode[STIX]{x1D70E})=\biggl(\unicode[STIX]{x1D706}_{Q}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)\unicode[STIX]{x1D712}_{Q}\biggr)_{Q\in {\mathcal{D}}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D712}_{Q}$
denotes the characteristic function of
$Q$
. We ask the following question. Given
$1\leqslant s\leqslant \infty$
,
$1<p<\infty$
,
$0<q<\infty$
, which are the pair of positive Borel measures
$\unicode[STIX]{x1D707}$
,
$\unicode[STIX]{x1D70E}$
on
$\mathbb{R}^{n}$
such that, for any non-negative function
$f$
,
$$\begin{eqnarray}\Vert T(f\,d\unicode[STIX]{x1D70E})\Vert _{L_{l^{s}}^{q}(\unicode[STIX]{x1D707})}=\biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}?\end{eqnarray}$$
As usual, the above estimate for the case
$s=\infty$
has to be understood by replacing the sum by the sup-norm.
The possible characterizations of (1.1) depend heavily on the size of the parameters
$s,p$
and
$q$
involved.
For the range
$1<p\leqslant q<\infty$
, the problem is rather well understood.
If
$s=1$
, the operator
$f\rightarrow (\sum _{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\int _{Q}f\,d\unicode[STIX]{x1D70E})^{s}\unicode[STIX]{x1D712}_{Q})^{1/s}$
is linear and its boundedness is characterized by the Sawyer testing condition (testing the boundedness of the operator and its adjoint against characteristic functions of cubes); see [Reference Hytönen12, Reference Lacey, Sawyer and Uriarte-Tuero13, Reference Nazarov, Treil and Volberg16, Reference Treil21].
If
$s=\infty$
, the boundedness is characterized by the direct Sawyer condition [Reference Sawyer17].
If
$1<s<\infty$
and
$s\geqslant p$
, it was characterized also by the direct Sawyer’s conditions in [Reference Cascante and Ortega1] and, for
$s<p$
, it was reduced to the case when
$s=1$
.
The case when
$1<s<\infty$
and
$q\geqslant p$
was considered by Scurry in [Reference Scurry18], extending Lacey, Sawyer and Uriarte-Tuero’s proof of the case when
$s=1$
. For
$p=q$
, a different approach was considered by Hänninen in [Reference Hänninen9], by adapting Hytönen’s proof for the case when
$s=1$
(see [Reference Hytönen12]), based on a parallel stopping cubes method. Namely, they proved that if
$q=p$
and
$s>1$
and if we define
$T_{P}(f\,d\unicode[STIX]{x1D70E}):=(\unicode[STIX]{x1D706}_{Q}(\int _{Q}f\,d\unicode[STIX]{x1D70E})\unicode[STIX]{x1D712}_{Q})_{\substack{ Q\subset P \\ Q\in {\mathcal{D}}}}$
and its formal adjoint
$T_{P}^{\ast }$
by
$T_{P}^{\ast }(gd\unicode[STIX]{x1D707}):=\sum _{\substack{ Q\subset P, \\ Q\in {\mathcal{D}}}}\unicode[STIX]{x1D706}_{Q}(\int _{Q}g_{Q}\,d\unicode[STIX]{x1D707})\unicode[STIX]{x1D712}_{Q}$
, then (1.1) holds if and only if both of the following conditions are satisfied.
-
(i) For all
$P\in {\mathcal{D}}$
,
$\Vert T_{P}(d\unicode[STIX]{x1D70E})\Vert _{L_{l^{s}}^{p}(\unicode[STIX]{x1D707})}\leqslant C\unicode[STIX]{x1D70E}(P)^{1/p}$
. -
(ii) For all
$g=(g_{Q}\unicode[STIX]{x1D712}_{Q})_{Q\in {\mathcal{D}}}$
,
$g_{Q}\geqslant 0$
,
$$\begin{eqnarray}\Vert T_{P}^{\ast }(gd\unicode[STIX]{x1D707})\Vert _{L^{p^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C\Vert g\Vert _{L_{l^{s^{\prime }}}^{\infty }(\unicode[STIX]{x1D707})}\unicode[STIX]{x1D707}(P)^{1/p^{\prime }}.\end{eqnarray}$$
The approach was further extended to an abstract Banach-valued setting in [Reference Hänninen8].
The upper triangle case
$q<p$
in the linear case
$s=1$
was considered in [Reference Cascante, Ortega and Verbitsky2], assuming some extra conditions on the operator and the measure
$\unicode[STIX]{x1D70E}$
; conditions that were removed by Tanaka in [Reference Tanaka20]. We also recall that, in [Reference Hänninen, Hytönen and Li11] (see also [Reference Vuorinen23]), a unified characterization was obtained that does not depend on the relative position of
$p$
and
$q$
. Namely, for the particular case when
$q<p$
, the characterization obtained by Hänninen et al in [Reference Hänninen, Hytönen and Li11] was the following. The operator
$T$
is bounded from
$L^{p}(\unicode[STIX]{x1D70E})$
to
$L^{q}(\unicode[STIX]{x1D707})$
if and only if there exists a constant
$C>0$
such that:
-
(i)
$$\begin{eqnarray}\sup _{{\mathcal{F}}}\biggl\|\biggl\{\frac{\Vert T_{F}(d\unicode[STIX]{x1D70E})\Vert _{L^{q}(\unicode[STIX]{x1D707})}}{\unicode[STIX]{x1D70E}(F)^{1/p}}\biggr\}_{F\in {\mathcal{F}}}\biggr\|_{l^{r}({\mathcal{F}})}\leqslant C;\end{eqnarray}$$
-
(ii)
$$\begin{eqnarray}\sup _{{\mathcal{G}}}\biggl\|\biggl\{\frac{\Vert T_{G}^{\ast }(d\unicode[STIX]{x1D707})\Vert _{L^{p^{\prime }}(\unicode[STIX]{x1D70E})}}{\unicode[STIX]{x1D707}(G)^{1/q^{\prime }}}\biggr\}_{G\in {\mathcal{G}}}\biggr\|_{l^{r}({\mathcal{G}})}\leqslant C,\end{eqnarray}$$
where the supremums are taken over all subcollections
${\mathcal{F}}$
and
${\mathcal{G}}$
of
${\mathcal{D}}$
that are sparse (in the sense of the definition given in §3) with respect to
$\unicode[STIX]{x1D70E}$
and
$\unicode[STIX]{x1D707}$
, respectively, and
$1/r=1/q-1/p$
. In the same paper, a characterization is obtained for the maximal dyadic function (
$s=\infty$
) and
$q<p$
(see also [Reference Hänninen10]).
We will see, in the next section, that the cases where
$s\leqslant q<p$
can be reduced to
$s=1$
and, in consequence, to the linear case considered in [Reference Cascante, Ortega and Verbitsky3, Reference Tanaka20] and in [Reference Hänninen, Hytönen and Li11]. Hence, we limit ourselves to the case where
$q<p$
and
$s>q$
.
In this paper, we obtain a characterization of (1.1) for the cases where
$q<p$
and
$s>q$
: namely, the following theorem.
Theorem 1.1. Assume that
$1<p<\infty$
,
$0<q<p<\infty$
and
$q\leqslant s\leqslant \infty$
. Let
$\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}$
be positive locally finite measures. Assume that, in addition, the measure
$\unicode[STIX]{x1D707}$
has no point masses. Let
$T(f\,d\unicode[STIX]{x1D70E})=(\unicode[STIX]{x1D706}_{Q}(\int _{Q}f\,d\unicode[STIX]{x1D70E})\unicode[STIX]{x1D712}_{Q})_{Q\in {\mathcal{D}}}$
. Then there exists
$C>0$
such that, for any
$f\in L^{p}(\unicode[STIX]{x1D70E})$
,
$$\begin{eqnarray}\Vert T(f\,d\unicode[STIX]{x1D70E})\Vert _{L_{l^{s}}^{q}(\unicode[STIX]{x1D707})}\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}\end{eqnarray}$$
if and only if there exists
$C_{1}>0$
such that, for any
$(E_{Q})_{Q\in {\mathcal{D}}}$
pairwise disjoint
$\unicode[STIX]{x1D707}$
-measurable sets such that
$E_{Q}\subset Q$
,
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C_{1},\end{eqnarray}$$
where
$\tilde{s}=s/q$
and
$\tilde{p}=p/q$
.
The proof of the theorem follows an approach that is different from the ones stated before for the other relative positions of the parameters
$p,q,s$
. One key argument is a technical lemma that shows that, for a positive measure
$\unicode[STIX]{x1D707}$
with no point masses, for any measurable set
$A$
and any intermediate value between zero and
$\unicode[STIX]{x1D707}(A)$
, there exists a measurable subset of
$A$
that attains this intermediate value. It also shows that we can use a “canonical” procedure to obtain this set. This key lemma is then used to obtain a direct proof of an equivalence between the so-called “
$\unicode[STIX]{x1D707}$
-sparse” and “
$\unicode[STIX]{x1D707}$
- Carleson” coefficients. If
$\unicode[STIX]{x1D707}$
is a locally finite positive measure on
$\mathbb{R}^{n}$
, we say that a set of non-negative real numbers
$(\unicode[STIX]{x1D706}_{Q})_{Q\in {\mathcal{D}}}$
, associated to a dyadic system
${\mathcal{D}}$
, satisfy a
$\unicode[STIX]{x1D707}$
-Carleson condition with constant
$C>0$
if, for every
$P\in {\mathcal{D}}$
,
$$\begin{eqnarray}\mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D706}_{Q}\leqslant C\unicode[STIX]{x1D707}(P).\end{eqnarray}$$
We say that the coefficients are
$\unicode[STIX]{x1D707}$
-sparse with constant
$C>0$
if there exist pairwise disjoint sets
$E_{Q}\subset Q$
such that, for any
$Q\in {\mathcal{D}}$
,
$\unicode[STIX]{x1D706}_{Q}\leqslant C\unicode[STIX]{x1D707}(E_{Q})$
. In §3, we give a direct proof of the equivalence of these two types of coefficient, for the case where
$\unicode[STIX]{x1D707}$
has no point masses. The other ingredients in the proof of the theorem are a reduction to the case when
$q=1$
and an argument based on duality.
In the last section, we apply Theorem 1.1 to the characterization of the pair of measures
$\unicode[STIX]{x1D707}$
and
$\unicode[STIX]{x1D70E}$
such that, for any
$f\in L^{p}(\unicode[STIX]{x1D70E})$
,
$$\begin{eqnarray}\Vert {\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E})^{1/s}\Vert _{L^{q}(\unicode[STIX]{x1D707})}\lesssim \Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})},\end{eqnarray}$$
where
${\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}$
is the dyadic Wolff’s potential, defined by
$$\begin{eqnarray}{\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E})(x)=\mathop{\sum }_{Q\in {\mathcal{D}}}\biggl(\frac{\int _{Q}f\,d\unicode[STIX]{x1D70E}(x)}{|Q|^{1-\unicode[STIX]{x1D6FC}/n}}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}(x),\end{eqnarray}$$
where
$|Q|$
denotes the Lebesgue measure.
Throughout the paper, we will adopt the usual convention of writing
$A\lesssim B$
if there exists an absolute constant
$C>0$
such that
$A\leqslant CB$
.
The paper is organized as follows. In §2, we rewrite the estimate (1.1) as a problem on discrete multipliers. In §3 we give the equivalence between “sparse” and “Carleson” coefficients. In §4, we give the proof of Theorem 1.1 and, in the last section, we give an application of this last result to the boundedness of Wolff-type potentials.
2 Some preliminaries
The following lemma is a consequence of the boundedness of the dyadic maximal function and it shows that (1.1) can be rewritten in terms of discrete multipliers (see, for example, [Reference Verbitsky22, Lemma 1] or [Reference Cascante and Ortega1, Lemma 2.1]). For completeness, we include a proof here.
Lemma 2.1. Let
$1<p\leqslant \infty$
and let
$0<q,s\leqslant \infty$
. Then estimate (1.1), that is,
$$\begin{eqnarray}\Vert T(f\,d\unicode[STIX]{x1D70E})\Vert _{L_{l^{s}}^{q}(\unicode[STIX]{x1D707})}=\biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})},\end{eqnarray}$$
holds if and only if there exists
$C_{1}>0$
such that, for any sequence
$(\unicode[STIX]{x1D70C}_{Q})_{Q}$
of non-negative numbers,
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s}\unicode[STIX]{x1D70C}_{Q}^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(d\unicode[STIX]{x1D707})}\leqslant C_{1}\Vert \sup _{Q\in {\mathcal{D}}}(\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
Proof. Assume that (1.1) holds and that
$s<\infty$
. Let
$f=\sup _{Q}(\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q})$
. Since
$$\begin{eqnarray}\int _{Q}f\,d\unicode[STIX]{x1D70E}=\int _{Q}\sup _{Q^{\prime }\in {\mathcal{D}}}(\unicode[STIX]{x1D70C}_{Q^{\prime }}\unicode[STIX]{x1D712}_{Q^{\prime }})\,d\unicode[STIX]{x1D70E}\geqslant \int _{Q}\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q}\,d\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D70E}(Q),\end{eqnarray}$$
(1.1) gives that
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s}\unicode[STIX]{x1D70C}_{Q}^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\leqslant C\Vert \sup _{Q\in {\mathcal{D}}}(\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{p}(\unicode[STIX]{x1D70E})},\end{eqnarray}$$
which is (2.1).
Conversely, assume that (2.1) holds and let
$\unicode[STIX]{x1D70C}_{Q}=(1/\unicode[STIX]{x1D70E}(Q))\int _{Q}f\,d\unicode[STIX]{x1D70E}$
,
$f\geqslant 0$
. Since
$p>1$
, the dyadic maximal operator with respect to
$\unicode[STIX]{x1D70E}$
,
$M_{\unicode[STIX]{x1D70E}}^{{\mathcal{D}}}$
, given by
$$\begin{eqnarray}M_{\unicode[STIX]{x1D70E}}^{{\mathcal{D}}}f(x)=\sup _{x\in Q}\frac{1}{\unicode[STIX]{x1D70E}(Q)}\int _{Q}f\,d\unicode[STIX]{x1D70E},\end{eqnarray}$$
is of strong type
$(p,p)$
with respect to
$\unicode[STIX]{x1D70E}$
. Hence the hypothesis (2.1) gives that
$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s}\biggl(\frac{1}{\unicode[STIX]{x1D70E}(Q)}\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant C_{1}\Vert \sup _{Q\in {\mathcal{D}}}(\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{p}(\unicode[STIX]{x1D70E})}=C_{1}\Vert M_{\unicode[STIX]{x1D70E}}^{{\mathcal{D}}}f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\nonumber\end{eqnarray}$$
The case when
$s=\infty$
is proved analogously. ◻
Since the proof of Lemma 2.1 is only based on the monotony and positivity of the expressions on the left-hand side of the inequalities involved and using the fact that the dyadic maximal function is of strong type, the same argument gives the following proposition that generalizes Lemma 2.1.
Proposition 2.2. Let
$1<p\leqslant \infty$
. Let
$\unicode[STIX]{x1D70E}$
be a locally finite Borel measure. Let
$F:[0,\infty ]^{{\mathcal{D}}}\rightarrow [0,\infty ]$
be a function that is increasing in the sense that, for any pair of sequences
$\{a_{Q}\}_{Q\in {\mathcal{D}}}$
and
$\{b_{Q}\}_{Q\in {\mathcal{D}}}$
of non-negative real numbers such that
$a_{Q}\leqslant b_{Q}$
for every
$Q\in {\mathcal{D}}$
,
$F(\{a_{Q}\}_{Q\in {\mathcal{D}}})\leqslant F(\{b_{Q}\}_{Q\in {\mathcal{D}}})$
. Then the following assertions are equivalent.
-
(i) There exists
$C>0$
such that, for any
$f\geqslant 0$
,
$$\begin{eqnarray}F\biggl(\biggl\{\frac{1}{\unicode[STIX]{x1D70E}(Q)}\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr\}_{Q\in {\mathcal{D}}}\biggr)\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
-
(ii) There exists
$C>0$
such that, for any sequence
$\{a_{Q}\}_{Q\in {\mathcal{D}}}$
of non-negative real numbers,
$$\begin{eqnarray}F(\{a_{Q}\}_{Q\in {\mathcal{D}}})\leqslant C\Vert \sup _{Q\in {\mathcal{D}}}a_{Q}\unicode[STIX]{x1D712}_{Q}\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
Before we prove our results, we will point out some observations concerning the cases where
$s\leqslant q$
and
$q<p$
. We observe that, when
$s\leqslant q$
, if we write
$t_{Q}=\unicode[STIX]{x1D70C}_{Q}^{s}$
and denote
$\tilde{p}=p/s$
and
$\tilde{q}=q/s$
, (2.1) can be rewritten as
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s}t_{Q}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{q}}(d\unicode[STIX]{x1D707})}\leqslant C_{1}^{s}\Vert \sup _{Q\in {\mathcal{D}}}(t_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{\tilde{p}}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
By Lemma 2.1, the above estimate is equivalent to the boundedness from
$L^{\tilde{p}}(\unicode[STIX]{x1D70E})$
to
$L^{\tilde{q}}(\unicode[STIX]{x1D707})$
of the linear operator
$$\begin{eqnarray}\displaystyle \tilde{T}(f\,d\unicode[STIX]{x1D70E})=\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s}\frac{\int _{Q}f\,d\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q},\end{eqnarray}$$
that is,
$$\begin{eqnarray}\Vert \tilde{T}(f\,d\unicode[STIX]{x1D70E})\Vert _{L^{\tilde{q}}(\unicode[STIX]{x1D707})}\lesssim \Vert f\Vert _{L^{\tilde{p}}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
As we have already said in the introduction, if we assume, in addition, that
$s<q$
, that is
$\tilde{q}>1$
, this estimate was characterized in [Reference Tanaka20] (see also [Reference Cascante, Ortega and Verbitsky3]) and with a different approach in [Reference Hänninen, Hytönen and Li11].
Next, if
$s=q<p$
,
$\tilde{q}=1$
and, consequently,
$$\begin{eqnarray}\displaystyle \Vert \tilde{T}(f\,d\unicode[STIX]{x1D70E})\Vert _{L^{1}(\unicode[STIX]{x1D707})} & = & \displaystyle \biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s-1}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{1}(\unicode[STIX]{x1D707})}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s-1}\int _{Q}f\,d\unicode[STIX]{x1D70E}\unicode[STIX]{x1D707}(Q)\nonumber\\ \displaystyle & = & \displaystyle \int _{\mathbb{R}^{n}}\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s-1}\unicode[STIX]{x1D707}(Q)f\unicode[STIX]{x1D712}_{Q}\,d\unicode[STIX]{x1D70E}.\nonumber\end{eqnarray}$$
Hence, duality now gives that (2.2) holds if and only if
$$\begin{eqnarray}\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}(\unicode[STIX]{x1D70E}(Q))^{s-1}\unicode[STIX]{x1D707}(Q)\unicode[STIX]{x1D712}_{Q}\in L^{(p/q)^{\prime }}(\unicode[STIX]{x1D70E}).\end{eqnarray}$$
We are left to deal with the case when
$1<p<\infty$
,
$0<q<p$
and
$s>q$
. In the rest of the paper, we will assume that this is the situation.
3 Equivalence between “sparse coefficients” and “Carleson coefficients”
In this section, we will prove an equivalence between “sparse coefficients” and “Carleson coefficients” associated to the dyadic system
${\mathcal{D}}$
and the measure
$\unicode[STIX]{x1D707}$
(see Theorem 3.4). An essential condition for this equivalence is that the measure
$\unicode[STIX]{x1D707}$
has no point masses.
We recall that a family of dyadic cubes
${\mathcal{F}}$
is
$\unicode[STIX]{x1D70E}$
-sparse (or sparse with respect to
$\unicode[STIX]{x1D70E}$
) if, for each
$F\in {\mathcal{F}}$
, there exists
$E_{F}\subset F$
such that
$\unicode[STIX]{x1D70E}(E_{F})\geqslant (1/2)\unicode[STIX]{x1D70E}(F)$
and the sets
$(E_{F})_{F\in {\mathcal{F}}}$
are pairwise disjoints. Of course, the constant
$1/2$
can be replaced by any other fixed constant
$\unicode[STIX]{x1D6FF}\in (0,1)$
.
It is important to recall that if a family
${\mathcal{F}}$
is
$\unicode[STIX]{x1D70E}$
-sparse, then
${\mathcal{F}}$
is
$\unicode[STIX]{x1D70E}$
-Carleson in the sense that, for any
$P\in {\mathcal{F}}$
,
$$\begin{eqnarray}\mathop{\sum }_{Q\in {\mathcal{F}},Q\subset P}\unicode[STIX]{x1D70E}(Q)\leqslant 2\unicode[STIX]{x1D70E}(P).\end{eqnarray}$$
Indeed, if
${\mathcal{F}}$
is
$\unicode[STIX]{x1D70E}$
-sparse, for each
$F\in {\mathcal{F}}$
, there exists
$E_{F}\subset F$
pairwise disjoint such that
$\unicode[STIX]{x1D70E}(E_{F})\geqslant (1/2)\unicode[STIX]{x1D70E}(F)$
. Then
$$\begin{eqnarray}\mathop{\sum }_{Q\in {\mathcal{F}},Q\subset P}\unicode[STIX]{x1D70E}(Q)\leqslant 2\mathop{\sum }_{Q\in {\mathcal{F}},Q\subset P}\unicode[STIX]{x1D70E}(E_{Q})\leqslant 2\unicode[STIX]{x1D70E}(P).\end{eqnarray}$$
Nevertheless, the reciprocal is not true in general. However, the converse holds for the Lebesgue measure, as proved in [Reference Lerner and Nazarov14]. The presented proof of the equivalence between sparse and Carleson coefficients in the case of any measure without point masses (Theorem 3.4) is a generalization of Lerner and Nazarov’s proof of the equivalence between sparse and Carleson collections in the case of the Lebesgue measure.
We recall that one of the main interests of
$\unicode[STIX]{x1D70E}$
-Carleson families lies in the well-known dyadic Carleson embedding which states that if
$1<p<\infty$
and
${\mathcal{F}}$
is a
$\unicode[STIX]{x1D70E}$
-Carleson family, then
$$\begin{eqnarray}\biggl(\mathop{\sum }_{F\in {\mathcal{F}}}(\langle f\rangle _{F}^{\unicode[STIX]{x1D70E}})^{p}\unicode[STIX]{x1D70E}(F)\biggr)^{1/p}\lesssim \Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})},\end{eqnarray}$$
where
$\langle f\rangle _{F}^{\unicode[STIX]{x1D70E}}=(\int _{F}f\,d\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70E}(F))$
.
Observe also that the concept of sequences of
$\unicode[STIX]{x1D70E}$
-sparse and
$\unicode[STIX]{x1D70E}$
-Carleson coefficients, given in the introduction, generalizes the concept of
$\unicode[STIX]{x1D70E}$
-sparse and
$\unicode[STIX]{x1D70E}$
-Carleson families, respectively (consider
$\unicode[STIX]{x1D706}_{Q}=\unicode[STIX]{x1D70E}(Q)$
if
$Q\in {\mathcal{F}}$
and
$\unicode[STIX]{x1D706}_{Q}=0$
if
$Q\notin {\mathcal{F}}$
).
The first theorem will give, for a measure
$\unicode[STIX]{x1D707}$
with no point masses, a “canonical” way of choosing sets with a prescribed mass. Here the word “canonical” does not have a sense of uniqueness, but it only means that it is prefixed so as to give a way of choosing the sets with the required properties.
The proof of the result will be based on an induction process. In this process we will consider cubes
$Q$
that contain some “faces”. In the first step, we fix an order on the family of the “faces” of
$Q$
that are the (
$n-1$
)-dimensional cubes in
$\unicode[STIX]{x2202}Q$
, which we may assume are pairwise disjoint if we subtract some of the “edges”. This procedure will be repeated in an analogous way in the successive generations to order the corresponding family of lower-dimensional “faces”.
Theorem 3.1. Let
$\unicode[STIX]{x1D707}$
be a positive locally finite measure on
$\mathbb{R}^{n}$
with no point masses. Let
$Q$
be an
$n$
-dimensional cube in
$\mathbb{R}^{n}$
which contains some of its “faces” and let
$0<m\leqslant \unicode[STIX]{x1D707}(Q)$
. Then there exists a set
$H_{Q}=H_{Q}(m)\subset Q$
such that
$\unicode[STIX]{x1D707}(H_{Q})=m$
. This set can be chosen “canonically” in such a way that if
$0<m_{1}\leqslant m_{2}\leqslant \unicode[STIX]{x1D707}(Q)$
, then the corresponding “canonical” sets
$H_{Q}(m_{1})$
and
$H_{Q}(m_{2})$
satisfy that
$H_{Q}(m_{1})\subset H_{Q}(m_{2})$
.
Proof. Let
$Q$
an
$n$
-dimensional cube in
$\mathbb{R}^{n}$
. Let
$x(Q)$
be the center of
$Q$
and let
$0<m\leqslant \unicode[STIX]{x1D707}(Q)$
. Let
$Q_{n}(t)$
be a
$t$
-homotetic cube in
$Q$
with the same center as
$Q$
, that is,
$Q_{n}(t)=x(Q)+t(Q-x(Q))$
,
$0\leqslant t\leqslant 1$
. Let
$f_{n}$
be the function defined on
$[0,1]$
by
$f_{n}(t):=\unicode[STIX]{x1D707}(Q_{n}(t))$
. The function
$f_{n}$
is a non-decreasing function with
$f(0)=0$
since
$x(Q)$
is not a point mass and
$f(1)=\unicode[STIX]{x1D707}(Q)$
.
Let
$t_{n}^{0}=\sup \{t\in [0,1];\unicode[STIX]{x1D707}(Q_{n}(t))<m\}$
. We first observe that the fact that the measure
$\unicode[STIX]{x1D707}$
has no point masses, gives that
$t_{n}^{0}>0$
.
There are two possibilities:
-
(i)
$f_{n}$
is continuous at
$t_{n}^{0}$
; -
(ii)
$f_{n}$
has a jump discontinuity at
$t_{n}^{0}$
.
Assume that (i) holds. If
$t_{n}^{0}<1$
,
$$\begin{eqnarray}m\geqslant \lim _{t\rightarrow (t_{n}^{0})^{-}}f_{n}(t)=\unicode[STIX]{x1D707}((Q_{n}(t_{n}^{0}))^{\circ })=\lim _{t\rightarrow (t_{n}^{0})^{+}}f_{n}(t)=\unicode[STIX]{x1D707}(\overline{Q_{n}(t_{n}^{0})})\geqslant m.\end{eqnarray}$$
Then we choose the set
$H_{Q}=(Q_{n}(t_{n}^{0}))^{\circ }$
and
$\unicode[STIX]{x1D707}(H_{Q})=m$
.
If
$t_{n}^{0}=1$
, then
$$\begin{eqnarray}\lim _{t\rightarrow 1^{-}}f_{n}(t)=\unicode[STIX]{x1D707}((Q)^{\circ })=\unicode[STIX]{x1D707}(Q)=m\end{eqnarray}$$
and we also take
$H_{Q}=(Q)^{\circ }$
.
Assume now that (ii) holds. If
$t_{n}^{0}<1$
,
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}:=\unicode[STIX]{x1D707}((Q_{n}(t_{n}^{0}))^{\circ })=\lim _{t\rightarrow (t_{n}^{0})^{-}}f_{n}(t)<\lim _{t\rightarrow (t_{n}^{0})^{+}}f_{n}(t)=\unicode[STIX]{x1D707}(\overline{Q_{n}(t_{n}^{0})}):=\unicode[STIX]{x1D6FD}.\end{eqnarray}$$
Hence
$\unicode[STIX]{x1D707}(\unicode[STIX]{x2202}Q_{n}(t_{n}^{0}))=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}$
.
If
$t_{n}^{0}=1$
, we replace
$\unicode[STIX]{x2202}Q_{n}(t_{n}^{0})$
by
$Q_{n}(1)\setminus (Q_{n}(1))^{\circ }=Q\setminus (Q)^{\circ }$
. From now on, we will assume that
$t_{n}^{0}<1$
, with the obvious changes for
$t_{n}^{0}=1$
.
We have that
$\unicode[STIX]{x1D6FC}\leqslant m\leqslant \unicode[STIX]{x1D6FD}$
. Then
$m-\unicode[STIX]{x1D6FC}\in [0,\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}]$
and we want to choose, in a “canonical” way, a measurable set in
$\unicode[STIX]{x2202}Q_{n}(t_{n}^{0})$
of measure
$m-\unicode[STIX]{x1D6FC}$
.
The set
$\unicode[STIX]{x2202}Q_{n}(t_{n}^{0})$
can be identified as a finite union of
$(n-1)$
-dimensional cubes that we have made pairwise disjoint by subtracting some edges. We order them in the fixed way,
$Q_{n-1}^{1},\ldots ,Q_{n-1}^{i_{n}}$
, as given earlier. We also consider the center of each of these cubes,
$x(Q_{n-1}^{i})$
,
$1\leqslant i\leqslant i_{n}$
.
Take
$i$
the lowest index such that
$$\begin{eqnarray}\unicode[STIX]{x1D707}(Q_{n-1}^{1})+\cdots +\unicode[STIX]{x1D707}(Q_{n-1}^{i-1})\leqslant m-\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D707}(Q_{n-1}^{1})+\cdots +\unicode[STIX]{x1D707}(Q_{n-1}^{i}).\end{eqnarray}$$
If
$\unicode[STIX]{x1D707}(Q_{n-1}^{1})+\cdots +\unicode[STIX]{x1D707}(Q_{n-1}^{i-1})=m-\unicode[STIX]{x1D6FC}$
, then we will choose
$H_{Q}=(Q_{n}(t_{n}^{0}))^{\circ }\cup (\bigcup _{j=1}^{i-1}Q_{n-1}^{j})$
as “canonical” set.
If not, we continue iterating. We have that
$m-\unicode[STIX]{x1D6FC}-(\unicode[STIX]{x1D707}(Q_{n-1}^{1})+\cdots +\unicode[STIX]{x1D707}(Q_{n-1}^{i-1}))\in (0,\unicode[STIX]{x1D707}(Q_{n-1}^{i}))$
. We consider the function
$f_{n-1}^{i}(t):=\unicode[STIX]{x1D707}(Q_{n-1}^{i}(t))$
, where
$Q_{n-1}^{i}(t)$
denotes the
$t$
-homotetic cube of
$Q_{n-1}^{i}$
with respect to the center
$x(Q_{n-1}^{i})$
. Let
$t_{n-1}^{i}=\sup \{t\in [0,1];\unicode[STIX]{x1D707}(Q_{n-1}^{i}(t))<m-\unicode[STIX]{x1D6FC}-(\unicode[STIX]{x1D707}(Q_{n-1}^{1})+\cdots +\unicode[STIX]{x1D707}(Q_{n-1}^{i-1}))\}$
. If the function
$f_{n-1}^{i}$
is continuous at
$t_{n-1}^{i}$
, then we consider
$H_{Q}=(Q_{n}(t_{n}^{0}))^{\circ }\cup (\bigcup _{j=1}^{i-1}Q_{n-1}^{j})\cup (Q_{n-1}^{i}(t_{n-1}^{i}))^{\circ }$
as “canonical” set and we have finished. If not, we can continue iterating. If this iteration stops at some stage, then we have finished.
We next show that the iteration has to stop at some stage. Assume that this is not the case. Then, finally, we can find a line segment
$Q_{1}$
and a function
$f_{1}(t):=\unicode[STIX]{x1D707}(Q_{1}(t))$
such that the function
$f_{1}$
is discontinuous at some
$t$
, which is equivalent to saying that there is a point
$x\in Q_{1}$
such that
$\unicode[STIX]{x1D707}(\{x\})>0$
. This contradicts the assumption that
$\unicode[STIX]{x1D707}$
has no point masses. Therefore, the iteration has to stop. Observe that
$H_{Q}$
is always a subset of
$Q$
and that it is a cube with part of its boundary. We will call such a set a “canonical” extended cube.
Finally, observe that the “canonical” method chosen to construct the sets
$H_{Q}$
gives that if
$0<m_{1}<m_{2}<\unicode[STIX]{x1D707}(Q)$
, then the corresponding “canonical” extended cubes
$H_{Q}(m_{1})$
and
$H_{Q}(m_{2})$
satisfy
$H_{Q}(m_{1})\subset H_{Q}(m_{2})$
. ◻
Remark 1. We remark that the set
$H_{Q}(m)$
constructed in Theorem 3.1 clearly depends on the measure
$\unicode[STIX]{x1D707}$
so, whenever we want to specify the measure, we will denote the set
$H_{Q}(m)$
corresponding to the measure
$\unicode[STIX]{x1D707}$
by
$H_{Q}^{\unicode[STIX]{x1D707}}(m)$
.
Let
$F$
be a measurable set and let
$\unicode[STIX]{x1D707}_{|F^{c}}$
be the restricted measure defined by
$\unicode[STIX]{x1D707}_{|F^{c}}(A):=\unicode[STIX]{x1D707}(A\cap F^{c})=\unicode[STIX]{x1D707}(A\setminus F)$
for any measurable set
$A$
. Observe that if
$F\subset Q$
and
$m\in [0,\unicode[STIX]{x1D707}(Q\setminus F)]$
, then the “canonical” cube associated to
$\unicode[STIX]{x1D707}_{|F^{c}}$
,
$H_{Q}^{\unicode[STIX]{x1D707}_{|F^{c}}}(m)\subset Q$
defines a set which we will denote by
$H_{Q,F}(m)$
such that
$\unicode[STIX]{x1D707}(H_{Q,F}(m)\setminus F)=m$
.
We also observe that if
$\unicode[STIX]{x1D707}\leqslant \unicode[STIX]{x1D708}$
, then, for any
$0<m\leqslant \unicode[STIX]{x1D707}(Q)$
,
$H_{Q}^{\unicode[STIX]{x1D708}}(m)\subset H_{Q}^{\unicode[STIX]{x1D707}}(m)$
. In particular, let
$F,G$
be measurable sets such that
$F\subset G$
. Since
$\unicode[STIX]{x1D707}_{|G^{c}}\leqslant \unicode[STIX]{x1D707}_{|F^{c}}$
, for all
$0<m\leqslant \unicode[STIX]{x1D707}_{|G^{c}}(Q)=\unicode[STIX]{x1D707}(Q\setminus G)$
, the corresponding “canonical” cubes
$H_{Q,F}(m)$
and
$H_{Q,G}(m)$
satisfy that
$$\begin{eqnarray}H_{Q,F}(m)\subset H_{Q,G}(m).\end{eqnarray}$$
Corollary 3.2. Let
$\unicode[STIX]{x1D707}$
be a positive locally finite Borel measure on
$\mathbb{R}^{n}$
. Then the following assertions are equivalent.
-
(i) The measure
$\unicode[STIX]{x1D707}$
has no point masses. -
(ii) For each measurable set
$A$
and for every
$m\in [0,\unicode[STIX]{x1D707}(A)]$
, there exists a measurable subset
$H\subset A$
such that
$\unicode[STIX]{x1D707}(H)=m$
.
Proof. It is clear that (ii) implies (i). Let’s prove that (i) implies (ii). Let
$A$
be a measurable set. By a limiting argument, we may assume that the set
$A$
is contained in a cube. Theorem 3.1 applied to the measure
$\unicode[STIX]{x1D707}_{|A}$
finishes the proof. ◻
Our next theorem (Theorem 3.4) was stated in [Reference Verbitsky22, Corollary 2] using a result of [Reference Dor5] under some implicit conditions on the involved measures. Here we will give a direct proof with the only assumption that the positive locally finite Borel measure has no point masses. We will follow the arguments used in [Reference Lerner and Nazarov14, Lemma 6.3], where an equivalence between sparse and Carleson families of dyadic sets is established with respect to the Lebesgue measure.
At some point in the proof, we will use some properties of non-monotonic sequences of sets which we recall before we state the theorem. We briefly summarize the precise definitions needed. Let
$\{A_{k}\}_{k=1}^{\infty }$
be a sequence of sets in
$\mathbb{R}^{n}$
. Then the limit inferior
$\liminf _{k\rightarrow \infty }A_{k}$
is defined as
$$\begin{eqnarray}\liminf _{k\rightarrow \infty }A_{k}:=\mathop{\bigcup }_{N=1}^{\infty }\mathop{\bigcap }_{k=N}^{\infty }A_{n}\end{eqnarray}$$
and the limit superior
$\limsup _{k\rightarrow \infty }A_{k}$
is defined as
$$\begin{eqnarray}\limsup _{k\rightarrow \infty }A_{k}:=\mathop{\bigcap }_{N=1}^{\infty }\mathop{\bigcup }_{k=N}^{\infty }A_{n}.\end{eqnarray}$$
The sequence
$\{A_{k}\}_{k=1}^{\infty }$
of sets in
$\mathbb{R}^{n}$
is said to have a limit if its limit inferior and limit superior are equal to each other. Then the limit is defined as the set
$$\begin{eqnarray}\lim _{k\rightarrow \infty }A_{k}:=\liminf _{k\rightarrow \infty }A_{k}=\limsup _{k\rightarrow \infty }A_{k}.\end{eqnarray}$$
In the following lemma, we summarize the main properties of set-theoretic limits that we need.
Lemma 3.3. Let
$\{A_{k}\}_{k=1}^{\infty }$
and
$\{B_{k}\}_{k=1}^{\infty }$
be sequences of sets in
$\mathbb{R}^{n}$
.
-
(i) If the sequences
$\{A_{k}\}_{k=1}^{\infty }$
and
$\{B_{k}\}_{k=1}^{\infty }$
each have a limit, then the sequence
$\{A_{k}\cap B_{k}\}_{k=1}^{\infty }$
has also a limit and
$$\begin{eqnarray}\lim _{k\rightarrow \infty }(A_{k}\cap B_{k})=\Bigl(\lim _{k\rightarrow \infty }A_{k}\Bigr)\cap \Bigl(\lim _{k\rightarrow \infty }B_{k}\Bigr).\end{eqnarray}$$
In particular, if the sequences
$\{A_{k}\}_{k=1}^{\infty }$
and
$\{B_{k}\}_{k=1}^{\infty }$
are non-decreasing, then
$$\begin{eqnarray}\lim _{k\rightarrow \infty }(A_{k}\setminus B_{k})=\biggl(\mathop{\bigcup }_{k=1}^{\infty }A_{k}\biggr)\bigg\backslash\biggl(\mathop{\bigcup }_{k=1}^{\infty }B_{k}\biggr).\end{eqnarray}$$
-
(ii) Let
$\unicode[STIX]{x1D707}$
be a positive locally finite measure on
$\mathbb{R}^{n}$
. Then
$\unicode[STIX]{x1D707}(\liminf _{k\rightarrow \infty }A_{k})\leqslant \liminf _{k\rightarrow \infty }\unicode[STIX]{x1D707}(A_{k})$
.If, in addition,
$\unicode[STIX]{x1D707}(\bigcup _{k=1}^{\infty }A_{k})<\infty$
, then
$$\begin{eqnarray}\unicode[STIX]{x1D707}\Bigl(\limsup _{k\rightarrow \infty }A_{k}\Bigr)\geqslant \limsup _{k\rightarrow \infty }\unicode[STIX]{x1D707}(A_{k}).\end{eqnarray}$$
In particular, if the sequence
$\{A_{k}\}_{k=1}^{\infty }$
has a limit and
$\unicode[STIX]{x1D707}$
satisfies that
$\unicode[STIX]{x1D707}(\bigcup _{k=1}^{\infty }A_{k})<\infty$
, then
$\unicode[STIX]{x1D707}(\lim _{k\rightarrow \infty }A_{k})=\lim _{k\rightarrow \infty }\unicode[STIX]{x1D707}(A_{k})$
.
Proof. For any pair of sequences of sets
$\{A_{k}\}_{k=1}^{\infty }$
and
$\{B_{k}\}_{k=1}^{\infty }$
, it is immediate to check that
$$\begin{eqnarray}\displaystyle & \displaystyle \liminf _{k\rightarrow \infty }(A_{k}\cap B_{k})=\Bigl(\liminf _{k\rightarrow \infty }A_{k}\Bigr)\cap \Bigl(\liminf _{k\rightarrow \infty }B_{k}\Bigr)\quad \text{ and} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \limsup _{k\rightarrow \infty }(A_{k}\cap B_{k})\subset \Bigl(\limsup _{k\rightarrow \infty }A_{k}\Bigr)\cap \Bigl(\limsup _{k\rightarrow \infty }B_{k}\Bigr). & \displaystyle \nonumber\end{eqnarray}$$
Then, if both sequences have a limit, then there exists
$\lim _{k\rightarrow \infty }(A_{k}\cap B_{k})$
and case (i) holds.
The proof of (ii) can be found, for instance, in [Reference Monroe15, Ch. II]. ◻
Theorem 3.4. Let
$\unicode[STIX]{x1D707}$
be a locally finite measure on
$\mathbb{R}^{n}$
without point masses. Let
$(\unicode[STIX]{x1D706}_{Q})_{Q}$
be non-negative real numbers and let
$C>0$
. Then the following assertions are equivalent.
-
(i) The coefficients
$(\unicode[STIX]{x1D706}_{Q})_{Q}$
satisfy the Carleson condition with constant
$C$
, that is, for every dyadic cube
$P\in {\mathcal{D}}$
,
$$\begin{eqnarray}\mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D706}_{Q}\leqslant C\unicode[STIX]{x1D707}(P).\end{eqnarray}$$
-
(ii) The coefficients
$(\unicode[STIX]{x1D706}_{Q})_{Q}$
satisfy the sparse condition with respect to
$C$
, that is, there exist pairwise disjoint sets
$E_{Q}\subset Q$
such that, for any
$Q\in {\mathcal{D}}$
,
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{Q}\leqslant C\unicode[STIX]{x1D707}(E_{Q}).\end{eqnarray}$$
Proof. It is clear that (ii) implies (i). Assume that (i) holds, that is,
$$\begin{eqnarray}\mathop{\sum }_{Q\subset P}\frac{\unicode[STIX]{x1D706}_{Q}}{C}\leqslant \unicode[STIX]{x1D707}(P).\end{eqnarray}$$
This gives, in particular, that
$$\begin{eqnarray}\frac{\unicode[STIX]{x1D706}_{Q}}{C}\leqslant \unicode[STIX]{x1D707}(Q)\end{eqnarray}$$
and
$$\begin{eqnarray}\frac{\unicode[STIX]{x1D706}_{P}}{C}\leqslant \unicode[STIX]{x1D707}(P)-\mathop{\sum }_{Q\subsetneq P}\frac{\unicode[STIX]{x1D706}_{Q}}{C}.\end{eqnarray}$$
One way of interpreting the meaning of the key observation (3.3) is to say that if from
$\unicode[STIX]{x1D707}(P)$
we take
$(1/C)\unicode[STIX]{x1D706}_{Q}$
of mass off every strict cube
$Q\subsetneq P$
, then there is still at least
$(1/C)\unicode[STIX]{x1D706}_{P}$
of mass left in
$P$
.
For each
$k\geqslant 1$
, denote by
${\mathcal{D}}_{k}$
the family of dyadic cubes of generation
$k$
, that is, cubes with side length
$1/2^{k}$
. If the cubes in
${\mathcal{D}}$
such that
$\unicode[STIX]{x1D706}_{Q}\neq 0$
are of size bounded from below by a positive number (that is, if there exists
$k_{0}$
such that for any
$k\geqslant k_{0}$
and for any
$Q\in {\mathcal{D}}_{k}$
,
$\unicode[STIX]{x1D706}_{Q}=0$
), we can proceed by an argument starting from the bottom up and consider only cubes in
$\bigcup _{i\leqslant k_{0}}{\mathcal{D}}_{i}$
. For each
$Q\in {\mathcal{D}}_{k_{0}}$
, by Lemma 3.2, choose any set
$E_{Q}\subset Q$
with the mass
$\unicode[STIX]{x1D707}(E_{Q})=\unicode[STIX]{x1D706}_{Q}/C$
. The chosen sets
$E_{Q}$
are, of course, pairwise disjoint. For the cubes in the next generation
${\mathcal{D}}_{k_{0}-1}$
, if
$P\in {\mathcal{D}}_{k_{0}-1}$
, by (3.3),
$\unicode[STIX]{x1D707}(P\setminus \bigcup _{Q\subsetneq P}E_{Q})\geqslant (1/C)\unicode[STIX]{x1D706}_{P}$
. Hence, Corollary 3.2 applied to the set
$P\setminus \bigcup _{Q\subsetneq P}E_{Q}$
gives that there exists a measurable set
$E_{P}$
that is disjoint with any
$E_{Q}$
with
$Q\subsetneq P$
and such that
$\unicode[STIX]{x1D707}(E_{P})=(1/C)\unicode[STIX]{x1D706}_{P}$
. Iterating this process, we obtain (ii) in this particular case. We observe that, in this situation, the “canonical” choices of the extended cubes
$E_{Q}$
are irrelevant.
For the general case, we will follow a limiting argument as in [Reference Lerner and Nazarov14] as well as using the “canonical” extended cubes with prescribed mass given in Theorem 3.1.
Let
$K\in \mathbb{Z}$
be fixed and let
$Q\in \bigcup _{k\leqslant K}{\mathcal{D}}_{k}$
,
$k\leqslant K$
. We define sets
$\hat{E}_{Q}^{K}$
“canonically”, by induction on
$k$
, satisfying the following properties.
-
(a)
$\hat{E}_{Q}^{K}\subset Q$
. -
(b) If we define
then
$$\begin{eqnarray}F_{Q}^{K}:=\mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in \mathop{\bigcup }_{k+1\leqslant i\leqslant K}{\mathcal{D}}_{i}}}\hat{E}_{R}^{K},\end{eqnarray}$$
$\unicode[STIX]{x1D707}(\hat{E}_{Q}^{K}\setminus F_{Q}^{K})=(1/C)\unicode[STIX]{x1D706}_{Q}$
.
Indeed, let
$Q\in {\mathcal{D}}_{K}$
. Since
$(1/C)\unicode[STIX]{x1D706}_{Q}\leqslant \unicode[STIX]{x1D707}(Q)$
, if
$\hat{E}_{Q}^{K}\in Q$
is the extended cube given in Theorem 3.1,
$$\begin{eqnarray}\unicode[STIX]{x1D707}(\hat{E}_{Q}^{K})=\frac{1}{C}\unicode[STIX]{x1D706}_{Q}.\end{eqnarray}$$
If we consider
$F_{Q}^{K}=\emptyset$
, we have the construction for the first generation.
Assume now that we have defined
$\hat{E}_{R}^{K}\subset R$
for
$R\in \bigcup _{k+1\leqslant i\leqslant K}{\mathcal{D}}_{i}$
satisfying (a) and (b). We first check that, for each
$Q\in {\mathcal{D}}_{k}$
,
$\unicode[STIX]{x1D707}(Q\setminus F_{Q}^{K})\geqslant (1/C)\unicode[STIX]{x1D706}_{Q}$
. We write
$$\begin{eqnarray}\displaystyle F_{Q}^{K} & = & \displaystyle \mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in \mathop{\bigcup }_{k+1\leqslant i\leqslant K}{\mathcal{D}}_{i}}}\hat{E}_{R}^{K}\nonumber\\ \displaystyle & = & \displaystyle \biggl(\mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\biggl(\hat{E}_{R}^{K}\bigg\backslash\biggl(\mathop{\bigcup }_{\substack{ S\subsetneq R \\ S\in \mathop{\bigcup }_{k+2\leqslant i\leqslant K}{\mathcal{D}}_{i}}}\hat{E}_{S}^{K}\biggr)\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle \cup \,\biggl(\mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\biggl(\mathop{\bigcup }_{\substack{ S\subsetneq R \\ S\in \mathop{\bigcup }_{k+2\leqslant i\leqslant K}{\mathcal{D}}_{i}}}\hat{E}_{S}^{K}\biggr)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}(\hat{E}_{R}^{K}\setminus F_{R}^{K})\cup \mathop{\bigcup }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}F_{R}^{K}.\nonumber\end{eqnarray}$$
Hence, using the induction hypothesis (b) (by which
$\unicode[STIX]{x1D707}(\hat{E}_{R}^{K}\setminus F_{R}^{K})=(1/C)\unicode[STIX]{x1D706}_{R}$
for every
$R\in \bigcup _{k+1\leqslant i\leqslant K}{\mathcal{D}}_{i}$
), we obtain
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}(F_{Q}^{K}) & {\leqslant} & \displaystyle \mathop{\sum }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\unicode[STIX]{x1D707}(\hat{E}_{R}^{K}\setminus F_{R}^{K})+\mathop{\sum }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\unicode[STIX]{x1D707}(F_{R}^{K})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\frac{1}{C}\unicode[STIX]{x1D706}_{R}+\mathop{\sum }_{\substack{ R\subsetneq Q \\ R\in {\mathcal{D}}_{k+1}}}\unicode[STIX]{x1D707}(F_{R}^{K}).\nonumber\end{eqnarray}$$
By iterating this argument and using the Carleson condition (i), we obtain
$$\begin{eqnarray}\unicode[STIX]{x1D707}(F_{Q}^{K})\leqslant \frac{1}{C}\mathop{\sum }_{\substack{ R\subsetneq Q \\ R\in \mathop{\bigcup }_{k+1\leqslant i\leqslant K}{\mathcal{D}}_{i}}}\unicode[STIX]{x1D706}_{R}\leqslant \frac{1}{C}\mathop{\sum }_{R\subsetneq Q}\unicode[STIX]{x1D706}_{R}\leqslant \unicode[STIX]{x1D707}(Q)-\frac{1}{C}\unicode[STIX]{x1D706}_{Q},\end{eqnarray}$$
which implies, since
$\unicode[STIX]{x1D707}$
is locally finite, that
$\unicode[STIX]{x1D707}(Q\setminus F_{Q}^{K})\geqslant (1/C)\unicode[STIX]{x1D706}_{Q}$
.
This estimate permits us to construct the “canonical” extended cube
$H_{Q,F_{Q}^{K}}^{K}\subset Q$
(see Remark 1) satisfying
$$\begin{eqnarray}\unicode[STIX]{x1D707}(H_{Q,F_{Q}^{K}}^{K}\setminus F_{Q}^{K})=\frac{1}{C}\unicode[STIX]{x1D706}_{Q}.\end{eqnarray}$$
We define
$$\begin{eqnarray}\hat{E}_{Q}^{K}:=F_{Q}^{K}\cup H_{Q,F_{Q}^{K}}^{K}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\unicode[STIX]{x1D707}(\hat{E}_{Q}^{K}\setminus F_{Q}^{K})=\unicode[STIX]{x1D707}(H_{Q,F_{Q}^{K}}^{K}\setminus F_{Q}^{K})=\frac{1}{C}\unicode[STIX]{x1D706}_{Q}\end{eqnarray}$$
and we have completed the induction.
Let us check that, with this construction, we have that for any
$Q\in {\mathcal{D}}_{k}$
,
$k\leqslant K$
,
$\hat{E}_{Q}^{K}\subset \hat{E}_{Q}^{K+1}$
.
We proceed inductively on
$k$
. Indeed, if
$Q\in {\mathcal{D}}_{K}$
, then
$\hat{E}_{Q}^{K}$
is the canonical set included in
$Q$
such that
$\unicode[STIX]{x1D707}(\hat{E}_{Q}^{K})=(1/C)\unicode[STIX]{x1D706}_{Q}$
and
$\hat{E}_{Q}^{K+1}=F_{Q}^{K+1}\cup H_{Q,F_{Q}^{K+1}}^{K+1}$
. In addition, Remark 1 gives that
$\hat{E}_{Q}^{K}\subset H_{Q,F_{Q}^{K+1}}^{K+1}$
. Then, for
$Q\in {\mathcal{D}}_{K}$
,
$\hat{E}_{Q}^{K}\subset \hat{E}_{Q}^{K+1}$
.
Assume now that
$\hat{E}_{Q_{1}}^{K}\subset \hat{E}_{Q_{1}}^{K+1}$
for
$Q_{1}\in \bigcup _{k<i\leqslant K}{\mathcal{D}}_{k}$
. Let
$Q\in {\mathcal{D}}_{k}$
. We want to prove that
$\hat{E}_{Q}^{K}\subset \hat{E}_{Q}^{K+1}$
. By hypothesis,
$F_{Q}^{K}\subset F_{Q}^{K+1}$
and, consequently, by Remark 1,
$H_{Q,F_{Q}^{K}}^{K}\subset H_{Q,F_{Q}^{K}}^{K+1}$
. Thus
$\hat{E}_{Q}^{K}\subset \hat{E}_{Q}^{K+1}$
.
Now let
$Q\in {\mathcal{D}}_{k}$
. We define
$$\begin{eqnarray}\hat{E}_{Q}:=\lim _{K}\hat{E}_{Q}^{K}=\mathop{\bigcup }_{K\geqslant k}\hat{E}_{Q}^{K}\subset Q.\end{eqnarray}$$
Since the sequences of sets
$(F_{Q}^{K})_{K}$
and
$(\hat{E}_{Q}^{K})_{K}$
are non-decreasing and
$F_{Q}^{K}\subset \hat{E}_{Q}^{K}$
, Lemma 3.3 gives that there exists
$$\begin{eqnarray}\lim _{K\rightarrow \infty }(\hat{E}_{Q}^{K}\setminus F_{Q}^{K})\end{eqnarray}$$
and this coincides with
$(\bigcup _{K\geqslant k}\hat{E}_{Q}^{K}\setminus \bigcup _{K\geqslant k}F_{Q}^{K})$
.
We choose the sets
$E_{Q}$
of (ii) as
$$\begin{eqnarray}E_{Q}:=\lim _{K\rightarrow \infty }(\hat{E}_{Q}^{K}\setminus F_{Q}^{K}).\end{eqnarray}$$
Since, for each
$K$
,
$\unicode[STIX]{x1D707}(\hat{E}_{Q}^{K}\setminus F_{Q}^{K})=(1/C)\unicode[STIX]{x1D706}_{Q}$
, by Lemma 3.3 (using again that
$\unicode[STIX]{x1D707}$
is locally finite),
$\unicode[STIX]{x1D707}(E_{Q})=(1/C)\unicode[STIX]{x1D706}_{Q}$
. Next
$$\begin{eqnarray}E_{Q}=\biggl(\mathop{\bigcup }_{K\geqslant k}\hat{E}_{Q}^{K}\bigg\backslash\mathop{\bigcup }_{K\geqslant k}F_{Q}^{K}\biggr)=\hat{E}_{Q}\bigg\backslash\biggl(\mathop{\bigcup }_{R\subsetneq Q}\hat{E}_{R}\biggr)\end{eqnarray}$$
and, consequently, the sets
$E_{Q}$
, which are subsets of
$\hat{E}_{Q}$
, are pairwise disjoint.◻
Corollary 3.5. Let
$\unicode[STIX]{x1D707}$
be a positive locally finite Borel measure on
$\mathbb{R}^{n}$
with no point masses. Let
$(\unicode[STIX]{x1D706}_{Q})_{Q}$
be non-negative real numbers. Then, if we define
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{1}:=\sup _{P}\frac{\mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D706}_{Q}}{\unicode[STIX]{x1D707}(P)}\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{2}:=\inf _{\substack{ E_{Q}\subset Q \\ \text{pairwise}~\text{disjoints}}}\sup _{Q}\frac{\unicode[STIX]{x1D706}_{Q}}{\unicode[STIX]{x1D707}(E_{Q})},\end{eqnarray}$$
$\unicode[STIX]{x1D6EC}_{1}=\unicode[STIX]{x1D6EC}_{2}$
.
Proof. Since, for each
$P$
,
$\sum _{Q\subset P}\unicode[STIX]{x1D706}_{Q}\leqslant \unicode[STIX]{x1D6EC}_{1}\unicode[STIX]{x1D707}(P)$
, the equivalence between (i) and (ii) of Theorem 3.4 shows that there exists
$E_{Q}\subset Q$
pairwise disjoint such that
$\unicode[STIX]{x1D706}_{Q}\leqslant \unicode[STIX]{x1D6EC}_{1}\unicode[STIX]{x1D707}(E_{Q})$
, and hence
$\unicode[STIX]{x1D6EC}_{2}\leqslant \unicode[STIX]{x1D6EC}_{1}$
. Reciprocally, if
$\unicode[STIX]{x1D6EC}_{2}<C$
, the equivalence between (i) and (ii) shows that, for each
$P$
,
$\sum _{Q\subset P}\unicode[STIX]{x1D706}_{Q}\leqslant C\unicode[STIX]{x1D707}(P)$
. Consequently,
$\unicode[STIX]{x1D6EC}_{1}\leqslant C$
and then
$\unicode[STIX]{x1D6EC}_{1}\leqslant \unicode[STIX]{x1D6EC}_{2}$
. ◻
Remark 2. If the measure
$\unicode[STIX]{x1D707}$
has point masses, the above Theorem 3.4 may fail. Indeed, take two nested dyadic cubes
$Q_{1}\subset Q_{2}$
, two non-zero coefficients,
$\unicode[STIX]{x1D706}_{Q_{1}}$
and
$\unicode[STIX]{x1D706}_{Q_{2}}$
, and a point mass
$\unicode[STIX]{x1D707}$
contained in both the cubes. Then, clearly, the Carleson condition (i) holds but the condition (ii) fails since one cannot divide the mass point.
4 Proof of Theorem 1.1
We begin the section with a result that will be used in the proof of Theorem 1.1. For the Lebesgue measure and the case
$1<s<\infty$
, this was proved in [Reference Frazier and Jawerth7, Corollary 5.12] and, in the general setting, in an unpublished work by Verbitsky, using a different method. For a sake of completeness, we include a sketch of the proof here.
Lemma 4.1. Let
$1<s<\infty$
and
$\unicode[STIX]{x1D707}$
a locally finite measure on
$\mathbb{R}^{n}$
. Let
$\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D706}_{Q})_{Q}\subset [0,\infty )$
be a sequence such that
$\unicode[STIX]{x1D706}_{Q}=0$
if
$\unicode[STIX]{x1D707}(Q)=0$
. Define
$$\begin{eqnarray}A_{1}(\unicode[STIX]{x1D6EC})=\biggl\|\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{1}(\unicode[STIX]{x1D707})}\end{eqnarray}$$
and
$$\begin{eqnarray}A_{2}(\unicode[STIX]{x1D6EC})=\sup _{(\unicode[STIX]{x1D6FC}_{Q})_{Q\in {\mathcal{D}}}}\frac{\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D6FC}_{Q}}{\sup _{\substack{ P\in {\mathcal{D}};\unicode[STIX]{x1D707}(P)\neq 0}}((1/\unicode[STIX]{x1D707}(P))\mathop{\sum }_{Q\subset P}(\unicode[STIX]{x1D6FC}_{Q}/\unicode[STIX]{x1D707}(Q))^{s^{\prime }}\unicode[STIX]{x1D707}(Q))^{1/s^{\prime }}},\end{eqnarray}$$
where the supremum is taken over all sequences
$(\unicode[STIX]{x1D6FC}_{Q})_{Q}\subset [0,\infty )$
such that
$\unicode[STIX]{x1D6FC}_{Q}=0$
if
$\unicode[STIX]{x1D707}(Q)=0$
. Then there exists
$C=C(s)$
such that
$$\begin{eqnarray}CA_{2}(\unicode[STIX]{x1D6EC})\leqslant A_{1}(\unicode[STIX]{x1D6EC})\leqslant A_{2}(\unicode[STIX]{x1D6EC}).\end{eqnarray}$$
Proof. The estimate
$CA_{2}(\unicode[STIX]{x1D6EC})\leqslant A_{1}(\unicode[STIX]{x1D6EC})$
was proved in [Reference Verbitsky22, Theorem 4.b] (consider
$p=r=1<q$
in that Theorem). For the reverse estimate,
$$\begin{eqnarray}\displaystyle A_{1}(\unicode[STIX]{x1D6EC}) & = & \displaystyle \int _{\mathbb{R}^{n}}\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\,d\unicode[STIX]{x1D707}\nonumber\\ \displaystyle & = & \displaystyle \int _{\mathbb{R}^{n}}\biggl(\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1-1/s^{\prime }}\,d\unicode[STIX]{x1D707}=\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{s}\int _{Q}\frac{d\unicode[STIX]{x1D707}}{(\mathop{\sum }_{R\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{R}^{s}\unicode[STIX]{x1D712}_{R})^{1/s^{\prime }}}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D6FC}_{Q},\nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{Q}=\unicode[STIX]{x1D706}_{Q}^{s-1}\int _{Q}\frac{d\unicode[STIX]{x1D707}}{(\mathop{\sum }_{R\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{R}^{s}\unicode[STIX]{x1D712}_{R})^{1/s^{\prime }}},\quad Q\in {\mathcal{D}}.\end{eqnarray}$$
Next, Hölder’s inequality gives that, for every
$P\in {\mathcal{D}}$
,
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{Q\subset P}\biggl(\frac{\unicode[STIX]{x1D6FC}_{Q}}{\unicode[STIX]{x1D707}(Q)}\biggr)^{s^{\prime }}\unicode[STIX]{x1D707}(Q) & = & \displaystyle \mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D6FC}_{Q}^{s^{\prime }}\unicode[STIX]{x1D707}(Q)^{1-s^{\prime }}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D706}_{Q}^{s}\unicode[STIX]{x1D707}(Q)^{1-s^{\prime }}\biggl(\int _{Q}\frac{d\unicode[STIX]{x1D707}}{(\mathop{\sum }_{R\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{R}^{s}\unicode[STIX]{x1D712}_{R})^{1/s^{\prime }}}\biggr)^{s^{\prime }}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D706}_{Q}^{s}\int _{Q}\frac{d\unicode[STIX]{x1D707}}{\mathop{\sum }_{R\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{R}^{s}\unicode[STIX]{x1D712}_{R}}\leqslant \int _{P}\,d\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}(P).\nonumber\end{eqnarray}$$
Consequently, for the set of chosen
$(\unicode[STIX]{x1D6FC}_{Q})_{Q}$
,
$$\begin{eqnarray}\displaystyle A_{1}(\unicode[STIX]{x1D6EC}) & = & \displaystyle \mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D6FC}_{Q}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{\mathop{\sum }_{Q\in {\mathcal{D}}}\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D6FC}_{Q}}{\sup _{P\in {\mathcal{D}}}((1/\unicode[STIX]{x1D707}(P))\mathop{\sum }_{Q\subset P}\unicode[STIX]{x1D6FC}_{Q}^{s^{\prime }}\unicode[STIX]{x1D707}(Q)^{-1/(s-1)})^{1/s^{\prime }}}\leqslant A_{2}(\unicode[STIX]{x1D6EC}).\square\nonumber\end{eqnarray}$$
Remark 3. This lemma is also true for
$s=1$
and
$s=\infty$
, but we do not include the proof, since it will not be necessary for our purposes.
As a consequence of Theorem 3.4 and Lemma 4.1, we have the following result.
Lemma 4.2. Let
$(b_{Q})_{Q}$
be a sequence of non-negative real numbers. Let
$0<q<\infty$
and
$q\leqslant s\leqslant \infty$
. Let
$\unicode[STIX]{x1D707}$
be a positive locally finite Borel measure with no point masses. Then the following assertions are equivalent.
-
(i)
$\Vert (\sum _{Q}b_{Q}^{s}\unicode[STIX]{x1D712}_{Q})^{1/s}\Vert _{L^{q}(\unicode[STIX]{x1D707})}\lesssim C$
. -
(ii) Define
$\tilde{s}:=s/q$
. Then, for every collection
$(E_{Q})_{Q}$
of pairwise disjoint sets with
$E_{Q}\subset Q$
, where here
$$\begin{eqnarray}\biggl(\mathop{\sum }_{Q}b_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\biggr)^{1/q}\lesssim C,\end{eqnarray}$$
${\lesssim}$
means that the constants involved may depend on
$q$
and
$s$
, but not on the sequence
$(b_{Q})_{Q}$
.
Proof. Notice that the endpoint cases are trivial: for
$s=q$
, the expressions coincide and, for
$s=\infty$
(in which case the summation is interpreted as the supremum), the assertion is clear by linearizing the supremum (we can write
$\sup _{Q}b_{Q}\unicode[STIX]{x1D712}_{Q}=\sum _{Q}b_{Q}\unicode[STIX]{x1D712}_{E_{Q}}$
for some pairwise disjoint sets
$E_{Q}\subset Q$
).
Assume now that
$q<s<\infty$
. Taking
$\tilde{s}:=s/q$
, (i) can be rewritten as
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q}(b_{Q}^{q})^{\tilde{s}}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/\tilde{s}}\biggr\|_{L^{1}(\unicode[STIX]{x1D707})}\lesssim C^{q}.\end{eqnarray}$$
By Lemma 4.1, (i) holds if and only if, for every sequence
$(\unicode[STIX]{x1D6FC}_{Q})_{Q}$
of non-negative real numbers such that
$\unicode[STIX]{x1D6FC}_{Q}=0$
if
$\unicode[STIX]{x1D707}(Q)=0$
,
$$\begin{eqnarray}\mathop{\sum }_{Q}b_{Q}^{q}\unicode[STIX]{x1D6FC}_{Q}\lesssim C^{q}\sup _{P}\biggl(\frac{1}{\unicode[STIX]{x1D707}(P)}\mathop{\sum }_{Q\subset P}\biggl(\frac{\unicode[STIX]{x1D6FC}_{Q}}{\unicode[STIX]{x1D707}(Q)}\biggr)^{\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)\biggr)^{1/\tilde{s^{\prime }}}.\end{eqnarray}$$
Next, Corollary 3.5 gives that the above estimate holds if and only if, for every family
$(E_{Q})_{Q}$
of pairwise disjoint sets
$E_{Q}\subset Q$
,
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{Q}b_{Q}^{q}\unicode[STIX]{x1D6FC}_{Q} & {\lesssim} & \displaystyle C^{q}\sup _{Q}\biggl(\frac{(\unicode[STIX]{x1D6FC}_{Q}/\unicode[STIX]{x1D707}(Q))^{\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)}{\unicode[STIX]{x1D707}(E_{Q})}\biggr)^{1/\tilde{s^{\prime }}}=C^{q}\sup _{Q}\frac{\unicode[STIX]{x1D6FC}_{Q}}{\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}}.\nonumber\end{eqnarray}$$
We define
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{Q}:=\frac{\unicode[STIX]{x1D6FC}_{Q}}{\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\mathop{\sum }_{Q}b_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D6FD}_{Q}\lesssim C^{q}\sup _{Q}\unicode[STIX]{x1D6FD}_{Q},\end{eqnarray}$$
which is equivalent to
$$\begin{eqnarray}\biggl(\mathop{\sum }_{Q}b_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\biggr)^{1/q}\lesssim C.\square\end{eqnarray}$$
Now we can finish the proof of Theorem 1.1.
Proof. The condition (1.2) says that
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{s}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/s}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\leqslant C\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
Since
$q\leqslant s\leqslant \infty$
, by Lemma 4.2, this estimate holds if and only if, for every collection
$(E_{Q})_{Q}$
of pairwise disjoint sets with
$E_{Q}\subset Q$
and where
$\tilde{s}=s/q$
,
$$\begin{eqnarray}\biggl(\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{q}\biggl(\int _{Q}f\,d\unicode[STIX]{x1D70E}\biggr)^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\biggr)^{1/q}\lesssim \Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
By Lemma 2.1, considering
$s:=q$
and
$p:=p$
in (1.1) and taking
$\unicode[STIX]{x1D706}_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}$
instead of
$\unicode[STIX]{x1D706}_{Q}^{s}$
, we obtain that the above is equivalent to
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D70E}(Q)^{q}\unicode[STIX]{x1D70C}_{Q}^{q}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/q}\biggr\|_{L^{q}(\unicode[STIX]{x1D707})}\lesssim \Vert \sup _{Q}(\unicode[STIX]{x1D70C}_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{p}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
Substituting
$s_{Q}^{1/q}$
for
$\unicode[STIX]{x1D70C}_{Q}$
, the previous estimate is equivalent to
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D70E}(Q)^{q}s_{Q}\unicode[STIX]{x1D712}_{Q}\biggr)^{1/q}\biggr\|_{L^{1}(\unicode[STIX]{x1D707})}\lesssim \Vert \sup _{Q}(a_{Q}\unicode[STIX]{x1D712}_{Q})\Vert _{L^{p/q}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
Since
$q<p$
, by Lemma 2.1, this estimate can be rewritten, taking
$\tilde{p}=p/q$
, as
$$\begin{eqnarray}\mathop{\sum }_{Q}\unicode[STIX]{x1D706}_{Q}^{q}\biggl(\frac{\int _{Q}f\,d\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70E}(Q)}\biggr)\unicode[STIX]{x1D70E}(Q)^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\lesssim \Vert f\Vert _{L^{\tilde{p}}(\unicode[STIX]{x1D70E})},\end{eqnarray}$$
that is,
$$\begin{eqnarray}\int _{\mathbb{R}^{n}}\biggl(\mathop{\sum }_{Q}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D712}_{Q}}{\unicode[STIX]{x1D70E}(Q)}\biggr)f\,d\unicode[STIX]{x1D70E}\lesssim \Vert f\Vert _{L^{\tilde{p}}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$
By duality, we finally have that
$$\begin{eqnarray}\biggl\|\biggl(\mathop{\sum }_{Q}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D712}_{Q}}{\unicode[STIX]{x1D70E}(Q)}\biggr)\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C_{1}.\end{eqnarray}$$
◻
Remark 4. Observe that if
$s=q$
(equivalently,
$\tilde{s}=1$
), (4.1) says simply that
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(Q)}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C,\end{eqnarray}$$
which is the trivial condition for
$\tilde{s}=1$
and
$\tilde{p}>1$
that can be obtained directly by duality.
In the other extreme case,
$s=\infty$
, which corresponds to the dyadic maximal function, the condition is that, for every subcollection of pairwise disjoint sets
$E_{Q}\subset Q$
,
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(E_{Q})}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C.\end{eqnarray}$$
This characterization for the extreme case
$s=\infty$
has also been proved in [Reference Hänninen10, Theorem 7.8] using a similar argument.
Remark 5. We observe that if
$T$
is such that there exists a subcollection of cubes
${\mathcal{S}}$
that are
$\unicode[STIX]{x1D707}$
-sparse and if
$T(f)=(\unicode[STIX]{x1D706}_{Q}(\int _{Q}f\,d\unicode[STIX]{x1D70E})\unicode[STIX]{x1D712}_{Q})_{Q\in {\mathcal{S}}}$
, then, for this operator, Theorem 1.1 gives that
$T$
is bounded from
$L^{p}(\unicode[STIX]{x1D70E})$
to
$L_{l^{s}}^{q}(\unicode[STIX]{x1D707})$
if and only if
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{S}}}\frac{(\unicode[STIX]{x1D706}_{Q}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(Q)}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C.\end{eqnarray}$$
Observe that this condition is independent of
$s$
.
5 Application to Wolff potentials
If
$\unicode[STIX]{x1D6FC},s>0$
,
$0<\unicode[STIX]{x1D6FC}<n$
, we consider the dyadic Wolff’s potential defined by
$$\begin{eqnarray}{\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E})(x)=\mathop{\sum }_{Q\in {\mathcal{D}}}\biggl(\frac{\int _{Q}f\,d\unicode[STIX]{x1D70E}(x)}{|Q|^{1-\unicode[STIX]{x1D6FC}/n}}\biggr)^{s}\unicode[STIX]{x1D712}_{Q}(x),\end{eqnarray}$$
where
$|Q|$
denotes the Lebesgue measure.
If
$1<q<p$
and
$q\leqslant s$
, the question that we want to consider is the following. Which are the measures
$\unicode[STIX]{x1D707},\unicode[STIX]{x1D70E}$
such that
$$\begin{eqnarray}\Vert {\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E})^{1/s}\Vert _{L^{q}(\unicode[STIX]{x1D707})}\lesssim \Vert f\Vert _{L^{p}(\unicode[STIX]{x1D70E})}?\end{eqnarray}$$
The relationship between Wolff’s potentials and the Riesz potentials and some applications are given in [Reference Cascante, Ortega and Verbitsky2].
Theorem 1.1 gives that, if
$\unicode[STIX]{x1D707}$
has no point masses, then (5.1) holds if and only if there exists
$C>0$
such that, for any
$(E_{Q})$
that are pairwise disjoint and such that
$E_{Q}\subset Q$
,
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{D}}}\frac{(|Q|^{\unicode[STIX]{x1D6FC}/n-1}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(Q)^{1/\tilde{s}}\unicode[STIX]{x1D707}(E_{Q})^{1/\tilde{s}^{\prime }}}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C,\end{eqnarray}$$
where
$\tilde{s}=s/q$
and
$\tilde{p}=p/q$
.
The following corollary gives a characterization of the estimate (5.1) for the particular case where
$\unicode[STIX]{x1D707}$
is an
$A_{\infty }$
weight. There exist several equivalent definitions of
$A_{\infty }$
weights. We recall here the one that is most convenient for our purposes (see [Reference Stein19, Ch. V]). A positive measure
$\unicode[STIX]{x1D707}$
is in
$A_{\infty }$
if, for each
$0<\unicode[STIX]{x1D6FC}<1$
, there exists
$0<\unicode[STIX]{x1D6FD}<1$
such that, for any
$E_{Q}\subset Q$
satisfying
$|E_{Q}|\geqslant \unicode[STIX]{x1D6FC}|Q|$
,
$\unicode[STIX]{x1D707}(E_{Q})\geqslant \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}(Q)$
.
For the general properties of weights in
$A_{\infty }$
, see, for instance, [Reference Duoandikoetxea6] or [Reference Stein19].
We then have the following corollary.
Corollary 5.1. Let
$\unicode[STIX]{x1D707}$
be an
$A_{\infty }$
-weight. Estimate (5.1) holds if and only if there exists a constant
$C>0$
such that, for any subcollection
${\mathcal{S}}$
of dyadic cubes that are
$\unicode[STIX]{x1D707}$
-sparse,
$$\begin{eqnarray}\biggl\|\mathop{\sum }_{Q\in {\mathcal{S}}}\frac{(|Q|^{\unicode[STIX]{x1D6FC}/n-1}\unicode[STIX]{x1D70E}(Q))^{q}\unicode[STIX]{x1D707}(Q)}{\unicode[STIX]{x1D70E}(Q)}\unicode[STIX]{x1D712}_{Q}\biggr\|_{L^{\tilde{p}^{\prime }}(\unicode[STIX]{x1D70E})}\leqslant C.\end{eqnarray}$$
Proof. We first observe that if
$\unicode[STIX]{x1D707}$
is an
$A_{\infty }$
weight, then any subcollection
${\mathcal{S}}$
sparse with respect to the Lebesgue measure is also
$\unicode[STIX]{x1D707}$
-sparse.
Then the proof of the corollary is an immediate consequence of Remark 5 and the following lemma.
Lemma 5.2. Let
${\mathcal{D}}$
be a dyadic grid in
$\mathbb{R}^{n}$
and a non-negative compactly supported integrable function
$f$
. There exists a subcollection
${\mathcal{S}}$
of dyadic cubes that are sparse with respect to the Lebesgue measure, such that
$$\begin{eqnarray}{\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E})\lesssim {\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{S}}}(f\,d\unicode[STIX]{x1D70E}).\end{eqnarray}$$
Of course, for every subcollection
${\mathcal{S}}_{1}$
of dyadic cubes that are sparse with respect to the Lebesgue measure, it is also true that
$$\begin{eqnarray}{\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{S}}_{1}}(f\,d\unicode[STIX]{x1D70E})\leqslant {\mathcal{W}}_{\unicode[STIX]{x1D6FC},s}^{{\mathcal{D}}}(f\,d\unicode[STIX]{x1D70E}).\end{eqnarray}$$
Proof. The proof follows closely the proof in [Reference Cruz-Uribe4, Proposition 3.9], where a similar pointwise estimate is obtained for a fractional operator, that is,
$s=1$
. It gives a pointwise estimate of the dyadic Wolff’s potential in terms of dyadic Wolff’s potentials defined on subcollections of dyadic cubes that are sparse with respect to the Lebesgue measure. ◻
Acknowledgement
The authors would like to thank the referee for his careful reading of the previous versions of this manuscript and, in particular, for his observations that cover some gaps and his suggestions on some aspects of the text that we hope have made the paper easier to read. We would like to point out that he suggested the direct proof of the equivalence between sparse and Carleson coefficients.
