Abstract
We study the boundedness of Hardy–Littlewood maximal function on the spaces defined in terms of Choquet integrals associated with weighted Bessel and Riesz capacities. As a consequence, we obtain a class of weighted Sobolev inequalities.
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1 Introduction and statements of main results
Let \(\alpha >0\) and \(s>1\). The Bessel potential spaces \(H^{\alpha ,s}({\mathbb {R}}^{n})\), \(n\in {\mathbb {N}}\), are defined to be the completion of \({\mathscr {D}}({\mathbb {R}}^{n})\) with respect to the norm:
where \({\mathscr {D}}({\mathbb {R}}^{n})\) is the space of compactly supported smooth functions on \({\mathbb {R}}^{n}\), and \({\mathcal {F}}\) is the distributional Fourier transform. Note that a function u belongs to \(H^{\alpha ,s}({\mathbb {R}}^{n})\) if and only if
for some \(f\in L^{s}({\mathbb {R}}^{n})\) and \(\Vert u\Vert _{H^{\alpha ,s}({\mathbb {R}}^{n})}=\Vert f\Vert _{L^{s}({\mathbb {R}}^{n})}\), where \(G_{\alpha }(\cdot )\) is the Bessel kernel defined by \(G_{\alpha }(x)={\mathcal {F}}^{-1}\left( (1+|\xi |^{2})^{-\frac{\alpha }{2}}\right) (x)\), \(x\in {\mathbb {R}}^{n}\).
Recall that the Bessel capacities \(\text {Cap}_{\alpha ,s}(\cdot )\) associated with \(H^{\alpha ,s}({\mathbb {R}}^{n})\) are defined to be
Subsequently, a function \(f:{\mathbb {R}}^{n}\rightarrow [-\infty ,\infty ]\) is said to be defined quasi-everywhere (q.e.) with respect to \(\text {Cap}_{\alpha ,s}(\cdot )\) if it is defined everywhere on \({\mathbb {R}}^{n}\) except for a set of zero capacity \(\text {Cap}_{\alpha ,s}(\cdot )\) (the notion of q.e. with respect to a set function \({\mathcal {C}}(\cdot )\) is defined in the same fashion). In which case, we define the Choquet integral of f associated with Bessel capacity by
Besides that, we define the local Hardy–Littlewood maximal function \({\textbf{M}}^{\textrm{loc}}\) by
where \(f\in L_{\textrm{loc}}^{1}({\mathbb {R}}^{n})\). If the supremum above is taken over \(r\in (0,\infty )\), then we obtain the standard Hardy–Littlewood maximal function \({\textbf{M}}\).
One of the main results in [5] concerns the weak-type estimate that
where \(\alpha s<n\). As noted in [5, Remark 2.6], the exponent \((n-\alpha s)/n\) is sharp. For the strong-type estimate, we have
for \(p>(n-\alpha s)/n\). The present paper is to obtain the weighted version of (1.1) and (1.2), respectively. To begin with, let us introduce the weighted capacities. A non-negative function \(\omega \) on \({\mathbb {R}}^{n}\) is said to be a weight if \(\omega \in L_{\textrm{loc}}^{1}({\mathbb {R}}^{n})\) and \(\omega (x)>0\) a.e.. Subsequently, the weight \(\omega \) is called an \(A_{p}\) weight for \(1\le p<\infty \) if there exists a constant \(A>0\), such that for every ball B of \({\mathbb {R}}^{n}\)
The infimum of all such constants A is denoted by \([\omega ]_{A_{p}}\). A necessary and sufficient condition for \(\omega \) to be an \(A_{1}\) weight is given by
We refer the readers [9, Chapter 1] for a quick review of basic properties of \(A_{p}\) weights. For any weight \(\omega \), we define the weighted local Riesz capacities \(R_{\alpha ,s;\rho }^{\omega }(\cdot )\) by
where \(\alpha s<n\), \(0<\rho <\infty \), \({\mathcal {I}}_{\alpha ,\rho }f=I_{\alpha ,\rho }*f\),
and
Similarly, one defines the weighted Riesz capacities \(R_{\alpha ,s}^{\omega }(\cdot )\) by
where \(\alpha s<n\), \({\mathcal {I}}_{\alpha }f=I_{\alpha }*f\), and \(I_{\alpha }(x)=|x|^{-(n-\alpha )}\) is the usual Riesz kernel. Under the assumption that \(\omega \in A_{s}\), the weighted local Riesz capacities \(R_{\alpha ,s;\rho }^{\omega }(\cdot )\) are equivalent to the weighted Bessel capacities \(B_{\alpha ,s}^{\omega }(\cdot )\) defined by
More precisely, we have
(see [9, Theorem 3.3.7 and Lemma 3.3.8]). For technical reason, we prefer to work with the weighted local Riesz capacities in the sequel. Our first result is the weighted weak-type estimate, to wit:
Theorem 1.1
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), and \(\omega \in A_{1}\). For any Lebesgue measurable \(R_{\alpha ,s;1}^{\omega }(\cdot )\)-q.e. defined function f on \({\mathbb {R}}^{n}\), it holds that
The weighted strong-type estimate is given by the following theorem.
Theorem 1.2
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), and \(\omega \in A_{1}\). For any Lebesgue measurable \(R_{\alpha ,s;1}^{\omega }(\cdot )\)-q.e. defined function f on \({\mathbb {R}}^{n}\), it holds that
for \(p>(n-\alpha s)/n\).
For the results regarding to weighted Riesz capacities \(R_{\alpha ,s}^{\omega }(\cdot )\), one replaces the local Hardy–Littlewood maximal function \({\textbf{M}}^{\textrm{loc}}\) by the standard Hardy–Littlewood maximal function \({\textbf{M}}\) and the local weighted Riesz capacities \(R_{\alpha ,s;1}^{\omega }(\cdot )\) by \(R_{\alpha ,s}^{\omega }(\cdot )\) in Theorems 1.1 and 1.2 (see Sect. 4 for details).
Our results reminisce of the celebrated classical weighted norm estimates given by Muckenhoupt [4]. It is standard that a necessary and sufficient condition for \(\omega \) to be an \(A_{p}\) weight can given by
Although Theorems 1.1 and 1.2 require stronger assumption that \(\omega \) being \(A_{1}\), our weighted estimates are actually valid for certain exponents \(p<1\). On the other hand, the authors in [10] and [8] concern the weighted estimates in terms of Hausdorff content. The major techniques in their proofs use covering lemmas and certain properties of dyadic cubes decomposition, which appear also in the work of [6]. Note that the Hausdorff content is defined in terms of covering. It is difficult to see how the techniques in their works can be adapted to our setting since capacities are not defined in terms of covering. Our method of proof to Theorems 1.1 and 1.2 mainly uses the nonlinear potential theory.
Sophisticated readers may realize that Theorem 1.2 can be obtained by Theorem 1.1 through the Marcinkiewicz’s interpolation technique. Indeed, if \(\omega \) is a weight, such that the Lebesgue measure is absolutely continuous with respect to \(R_{\alpha ,s;1}^{\omega }(\cdot )\):
then \(|f(x)|\le C\) q.e. with respect to \(R_{\alpha ,s;1}^{\omega }(\cdot )\) implies \({\textbf{M}}^{\textrm{loc}}f(x)\le C\) everywhere and hence
where
Note that if \(\omega \in A_{s}\), then the absolute continuity given in (1.4) holds (see [9, Lemma 4.4.3]). Combining (1.5) and the weak-type estimate in Theorem 1.1, one can easily modify the proof of Marcinkiewicz’s interpolation theorem given in [2, Theorem 1.3.2] to obtain the strong-type estimate in Theorem 1.2. Nevertheless, we will give an independent proof of Theorem 1.2 without appealing to Theorem 1.1 and the interpolation technique. In fact, the crucial step in proving the strong-type estimate in Theorem 1.2 is given by
For the exponent \(p>1\), one observes that
and the strong-type estimate in Theorem 1.2 for \(p>1\) then reduces to the case where \(p=1\).
In the sequel, for q.e. everywhere defined (with respect to \(R_{\alpha ,s;1}^{\omega }(\cdot )\)) function u on \({\mathbb {R}}^{n}\) and \(0<q<\infty \), we write
Denote by \(L^{q}(R_{\alpha ,s;1}^{\omega })\) and \(L^{q,\infty }(R_{\alpha ,s;1}^{\omega })\) the spaces consisting of all functions u with finite quantities \(\Vert u\Vert _{L^{q}(R_{\alpha ,s;1}^{\omega })}\) and \(\Vert u\Vert _{L^{q,\infty }(R_{\alpha ,s;1}^{\omega })}\), respectively. An obvious notational modification will do for \(L^{q}(R_{\alpha ,s}^{\omega })\) and \(L^{q,\infty }(R_{\alpha ,s}^{\omega })\) with the corresponding \(\left\| \cdot \right\| _{L^{q}(R_{\alpha ,s}^{\omega })}\) and \(\left\| \cdot \right\| _{L^{q,\infty }(R_{\alpha ,s}^{\omega })}\). To illustrate an application of Theorems 1.1 and 1.2, let us first address a type of homogeneous weighted Sobolev inequalities.
Theorem 1.3
Let \(s>1\), \(0<\alpha <n/s\), \(q\ge (n-\alpha s)/n\), \(0<\beta <(n-\alpha s)/q\), and
For any Lebesgue measurable \(R_{\alpha ,s}^{\omega }(\cdot )\)-q.e. defined function \(f\in L^{q}\left( R_{\alpha ,s}^{\omega ^{(n-\alpha s)/n}}\right) \), \(\omega \in A_{1}\), it holds that
provided that \(q=(n-\alpha s)/n\), and
provided that \(q>(n-\alpha s)/n\).
To obtain the corresponding inhomogeneous weighted Sobolev inequalities, we impose an extra condition on the weight \(\omega \) that
Theorem 1.4
Let \(s>1\), \(0<\alpha <n/s\), \(\alpha \in {\mathbb {N}}\), \(q\ge (n-\alpha s)/n\), \(0<\beta <(n-\alpha s)/q\), and
For any Lebesgue measurable \(R_{\alpha ,s;1}^{\omega }(\cdot )\)-q.e. defined function \(f\in L^{q}\left( R_{\alpha ,s;1}^{\omega ^{(n-\alpha s)/n}}\right) \), \(\omega \in A_{1}\) which satisfies (1.9), it holds that
provided that \(q=(n-\alpha s)/n\), and
provided that \(q>(n-\alpha s)/n\).
Remark 1.5
Consider the power weights \(\omega (x)=|x|^{-\eta }\) for \(0\le \eta <n\). Certainly, \(\omega \in A_{1}\) and satisfies (1.9). Nevertheless, there are \(A_{1}\) weights \(\omega \) which fail to satisfy (1.9). For instance
where \(\omega _{i}(x)=\max \{i|x|^{-1/2},1\}\), \(i\in {\mathbb {N}}\), and \(x\in {\mathbb {R}}\).
In what follows, we assume that \(\alpha >0\), \(s>1\), \(n\in {\mathbb {N}}\), and \(\alpha s<n\). We denote by \({\mathscr {D}}({\mathbb {R}}^{n})\) the space of compactly supported infinitely differentiable functions, while \({\mathscr {S}}'({\mathbb {R}}^{n})\) refers to the class of tempered distributions. The characteristic function of a set \(E\subseteq {\mathbb {R}}^{n}\) is denoted by \(\chi _{E}\). While \(B_{r}(x)\) denotes the open ball with center \(x\in {\mathbb {R}}^{n}\) and radius \(r>0\).
2 Preliminaries on the weighted capacities
Let \(\omega \) be a weight on \({\mathbb {R}}^{n}\). We say that \(\omega \) satisfies the strong doubling property if there exist some constants \(C>0\) and \(\varepsilon >0\), such that for every ball B of \({\mathbb {R}}^{n}\) and Lebesgue measurable subset E of B, it holds that
It is a standard fact that if \(\omega \in A_{p}\), \(1\le p<\infty \), then one may take \(C=[\omega ]_{A_{p}}\) and \(\varepsilon =p\) in the above inequality (see [9, Proposition 1.2.7]).
As noted in the first section, the weighted local Riesz \(R_{\alpha ,s;\rho }^{\omega }(\cdot )\) and weighted Bessel capacities \(B_{\alpha ,s}^{\omega }(\cdot )\) are equivalent for \(\omega \in A_{s}\). Let us introduce a variant weighted local Riesz capacities \({\mathcal {R}}_{\alpha ,s;\rho }^{\omega }(\cdot )\) by
where \(E\subseteq {\mathbb {R}}^{n}\) and \(0<\rho <\infty \) (see [9, Section 3.6.1]). Under the same assumption that \(\omega \in A_{s}\), we have the equivalence that
(see [9, Corollary 3.6.7]). Combining with (1.3), we have
where \(0<\rho _{1},\rho _{2}<\infty \) are arbitrary positive numbers. Note again that the following absolute continuity holds for \(\omega \in A_{s}\) that
(see [9, Lemma 4.4.3]). In view of (2.2), one may replace \(R_{\alpha ,s;\rho }^{\omega }(\cdot )\) by \({\mathcal {R}}_{\alpha ,s;\rho }^{\omega }(\cdot )\) in (2.4). On the other hand, we define the nonlinear potential \({\mathcal {V}}_{\omega ;\rho }^{\mu }\) of a positive measure \(\mu \) on \({\mathbb {R}}^{n}\) by
As a consequence, following [9, Proposition 3.6.4], if E is an arbitrary subset of \({\mathbb {R}}^{n}\) and \(\omega \in A_{s}\), then there exists a positive measure \(\mu ^{E}\) supported in \({\overline{E}}\), such that
and
We may compare the nonlinear potential \({\mathcal {V}}_{\omega ;\rho }^{\mu }\) with the Wolff potential \(W_{\omega ;\rho }^{\mu }\) defined by
To this end, up to a constant depending on n and s, we prefer to express the Wolff potential by
We observe that \(B_{t}(y)\subseteq B_{2t}(x)\) for \(|x-y|<t\), then \(\omega \in A_{s}\) yields
Besides that, by writing \(1=\omega ^{1/s}\cdot \omega ^{-1/s}\), Hölder’s inequality gives
Since \(B_{t}(x)\subseteq B_{2t}(y)\) for \(|x-y|<t\), we have
Therefore, we obtain the estimate that
The Wolff potential possesses the bounded maximum principle.
Lemma 2.1
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\rho >0\), and \(\mu \) be a positive measure on \({\mathbb {R}}^{n}\). If \(\omega \) is a weight satisfying (2.1), then
where the constant \(C>0\) is as in (2.1).
Proof
Let \(x\notin \textrm{supp}(\mu )\) and \(x_{0}\in K=\textrm{supp}(\mu )\) be the point that minimizes the distance from x to \(\textrm{supp}(\mu )\). If \(B_{t}(x)\cap K\ne \emptyset \), then \(t>|x-x_{0}|\), which in turn implies that \(B_{t}(x)\subseteq B_{2t}(x_{0})\). Consequently, (2.1) implies that
and the bounded maximum principle follows. \(\square \)
Now, we show that the Wolff potential satisfies the weak-type estimate.
Proposition 2.2
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\rho >0\), \(\omega \in A_{s}\), and \(\mu \) be a positive measure on \({\mathbb {R}}^{n}\). Then
Proof
Let \(\gamma \) be the measure associated with a compact subset K of \(\{W_{\omega ;\rho }^{\mu }>t\}\), such that \(\gamma (K)={\mathcal {R}}_{\alpha ,s;2\rho }^{\omega }(K)\) as in (2.6). Suppose that \(x\in \text {supp}(\gamma )\) and we let
Then
since \(W_{\omega ;\rho }^{\gamma }(x)\le C_{n,\alpha ,s,\omega }\cdot {\mathcal {V}}_{\omega ;2\rho }^{\gamma }(x)\le C_{n,\alpha ,s,\omega }\) on \(\text {supp}(\gamma )\) by (2.5) and (2.7). Thus,
By Besicovitch covering theorem, there are \(c_{n}\) collections of balls \(A_{i}=\{B_{n_{i}}\}\), \(i=1,\ldots ,c_{n}\), such that \(A_{i}\) is disjoint and
As a consequence,
The result follows by noting that \({\mathcal {R}}_{\alpha ,s;\rho }^{\omega }(\cdot )\le C_{n,\alpha ,s,\rho ,\omega }\cdot {\mathcal {R}}_{\alpha ,s;2\rho }^{\omega }(\cdot )\) as in (2.3). \(\square \)
Remark 2.3
As noted in the remark following [1, Proposition 6.3.12], the weak-type estimate in Proposition 2.2 is false for the Wolff potential replaced by the nonlinear Bessel potential that \(V_{\alpha ,s}^{\mu }=G_{\alpha }*(G_{\alpha }*\mu )^{\frac{1}{s-1}}\), \(1<s\le 2-\alpha /n\).
The following is known to be the capacitary strong-type estimate, which will be used in justifying the normability of the space \(L^{1}(R_{\alpha ,s;1}^{\omega })\).
Proposition 2.4
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), and \(\omega \in A_{s}\). Then
for all \(f\ge 0\).
Proof
In view of (2.2), we will show that
Denote by \(s'=s/(s-1)\). Assume for the moment that \(f\in C_{0}({\mathbb {R}}^{n})\) is a compactly supported continuous function on \({\mathbb {R}}^{n}\). Let
Suppose that \(\mu _{t}\) is a measure associated with the compact set \(\{{\mathcal {I}}_{\alpha ,1}f\ge t\}\) which satisfies (2.6) with respect to \({\mathcal {R}}_{\alpha ,s;32}^{\omega }(\cdot )\), that is,
Using (2.7) and Lemma 2.1, one obtains the boundedness of Wolff potential \(W_{\omega ;8}^{\mu _{t}}\) that
We have
where
To conclude the proof, we will show that \(L\le C_{n,\alpha ,s,\omega }\cdot J\). Assume for the moment that \(s\ge 2\). Let
We have
where we have used [9, Theorem 3.1.2] in (2.9). On the other hand, integration by parts gives
Express \(\mu _{u}(B_{r}(y))=\mu _{u}(B_{r}(y))^{(2-s')s'}\cdot \mu _{u}(B_{r}(y))^{(s'-1)^{2}}\). Using Hölder’s inequality with respect to the exponents that \(1/(2-s')^{-1}+1/(s'-1)^{-1}=1\), we obtain \(L\le C_{n,\alpha ,s}\cdot L_{1}^{2-s'}\cdot L_{2}^{s'-1}\), where
and
We have
where we have used (2.7), (2.8), and (2.3) in (2.10), (2.11), and (2.12), respectively. Similarly
Now, we consider the case that \(1<s<2\). We write
and integration by parts gives
Thus, a similar estimate as before yields
where
Note that
which yields
and the result holds for the case where \(0\le f\in C_{0}({\mathbb {R}}^{n})\). For general \(f\in L^{s}(\omega )\), we approximate f by a sequence \(\{f_{j}\}\) of functions in \(C_{0}({\mathbb {R}}^{n})\). Indeed, if \(f_{j}\rightarrow f\) in \(L^{s}(\omega )\) and \(f_{j}(x)\rightarrow f(x)\) a.e., then
which finishes the proof. \(\square \)
Let us introduce an auxiliary functional that
Proposition 2.5
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), and \(\omega \in A_{s}\). The functional \(\gamma (\cdot )\) is sublinear. Furthermore,
In other words, the space \(L^{1}(R_{\alpha ,s;1}^{\omega })\) is normable.
Proof
We first show the sublinearity of \(\gamma \). To this end, let \({\mathcal {H}}\) be the set of all non-negative functions f on \({\mathbb {R}}^{n}\), such that \({\mathcal {I}}_{\alpha ,1}\varphi (x)\ge f(x)^{1/s}\) q.e. with respect to \(R_{\alpha ,s;1}^{\omega }(\cdot )\) for some non-negative function \(\varphi \in L^{s}(\omega )\) with \(\Vert \varphi \Vert _{L^{s}(\omega )}\le 1\). We claim that \({\mathcal {H}}\) is convex. Suppose that \(f_{1},f_{2}\in {\mathcal {H}}\) and \(0<c<1\). Then there are non-negative \(\varphi _{1},\varphi _{2}\in L^{s}(\omega )\), such that \({\mathcal {I}}_{\alpha ,1}\varphi _{i}(x)\ge f_{i}(x)^{1/s}\) q.e. with respect to \(R_{\alpha ,s;1}^{\omega }(\cdot )\) and \(\Vert \varphi _{i}\Vert _{L^{s}(\omega )}\le 1\) for \(i=1,2\). It follows by the reverse Minkowski’s inequality that
Subsequently, it also holds that
The convexity of \({\mathcal {H}}\) follows. Now, it is routine to check that
from which the sublinearity of \(\gamma (\cdot )\) follows.
Now, we show for (2.13). Proposition 2.4 gives the \(\le \) direction of (2.13). On the other hand, since
one has \(R_{\alpha ,s;1}^{\omega }(\{|f|=\infty \})=0\). Then we have by the sublinearity of \(\gamma (\cdot )\) that
However
which yields
and hence the \(\ge \) direction of (2.13). \(\square \)
We claim that the spaces \(L^{p}(R_{\alpha ,s;1}^{\omega })\) and \(L^{p,\infty }(R_{\alpha ,s;1}^{\omega })\) satisfy the p-convexity for \(0<p<1\).
Corollary 2.6
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(0<p\le 1\), and \(\omega \in A_{s}\). Then
Proof
where we have used Proposition 2.5 for the normability of \(L^{1}(R_{\alpha ,s;1}^{\omega })\) in (2.15). \(\square \)
Corollary 2.7
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(0<p<1\), and \(\omega \in A_{s}\). Then
Proof
As in [2, Exercise 1.1.14], the countable subadditivity of \(R_{\alpha ,s;1}^{\omega }(\cdot )\) implies
It is a standard fact that
for any increasing sequence \(\{E_{N}\}_{N=1}^{\infty }\) of subsets of \({\mathbb {R}}^{n}\). As a consequence, for any \(t>0\), one obtains
Let
It follows by (2.16) that
Now, we estimate I. Note that for any \(t>0\), we have
where we have used see Proposition 2.5 for the normability of \(L^{1}(R_{\alpha ,s;1}^{\omega })\) in (2.18). As a consequence
The proof is complete by combining (2.17) and (2.19). \(\square \)
3 Proofs of main results
We begin by introducing a technical lemma.
Lemma 3.1
For any \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\omega \in A_{1}\), and positive measure \(\mu \) on \({\mathbb {R}}^{n}\), it holds that
where \(\eta =(s-1)n/(n-\alpha s)\), \(0<r\le 1\), and the \(L^{\eta ,\infty }(B_{r}(x_{0}))\) indicates the weak \(L^{\eta }\) space over the ball \(B_{r}(x_{0})\).
Proof
Let
and
To bound \(P_{1}\), we may assume that \(\mu \) is supported in \(B_{2r}(x_{0})\). First note that if \(x\in B_{r}(x_{0})\) and \(0<t<r\), then \(B_{t}(x)\subseteq B_{4r}(x_{0})\). The strong doubling property (2.1) of \(\omega \in A_{1}\) gives
As a consequence,
Now, we claim that
where
Let \(0<\delta \le r\) to be determined later. We write
We have
On the other hand, we also have
In proving (3.1), the left-sided of (3.1) allows us to assume also that \(\mu \) is supported in \(B_{r}(x)\). As a consequence,
By letting
then \(0<\delta \le r\), and (3.1) follows by routine simplification of \(I_{1}\) and \(I_{2}\).
Subsequently, the weak-type (1, 1) boundedness of \({\textbf{M}}\) implies that
Now, we bound for \(P_{2}\). Observe that if \(x\in B_{r}(x_{0})\) and \(t\ge r\), then \(B_{t}(x)\subseteq B_{2t}(x_{0})\) and again the strong doubling property (2.1) of \(\omega \in A_{1}\) yields
which implies that
and the lemma now follows by combining the estimates \(P_{1}\) and \(P_{2}\). \(\square \)
Now, we prove Theorem 1.1 for the case where f is the characteristic function of a Lebesgue measurable subset E of \({\mathbb {R}}^{n}\).
Proposition 3.2
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\omega \in A_{1}\), and E be a measurable subset of \({\mathbb {R}}^{n}\). Then
Proof
By choosing the nonlinear potential \({\mathcal {V}}_{\omega ;2}^{\mu }\) as in (2.5), we have
Since \(\omega \in A_{1}\subseteq A_{s}\), the absolute continuity (2.4) entails
Thus, for any \(x_{0}\in {\mathbb {R}}^{n}\) and \(0<r\le 1\), we have by (2.7) and Lemma 3.1 that
which yields
As a consequence, for any \(t>0\), we have by Proposition 2.2 that
and the proof is complete by appealing to the equivalence of the capacities that (2.2) and (2.3). \(\square \)
Proof of Theorem 1.1
First of all, Corollary 2.7 entails
with \(p=(n-\alpha s)/n\in (0,1)\). Suppose that \(f\in L^{p}(R_{\alpha ,s;1}^{\omega })\). Then
gives \(R_{\alpha ,s;1}^{\omega }(\{|f|=\infty \})=0\). As a result, we can write
and hence
The absolutely continuity (2.4) implies
We obtain by (3.2) and Proposition 3.2 that
which yields
as expected. \(\square \)
For any positive measure \(\mu \) on \({\mathbb {R}}^{n}\) and \(\rho >0\), we denote by
Lemma 3.3
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\rho >0\), \(\omega \in A_{1}\), \(x\in {\mathbb {R}}^{n}\), and \(\mu \) be a compactly supported positive measure on \({\mathbb {R}}^{n}\). Then
Proof
Let \(0<\delta \le \rho \) to be determined later. We write
Note that the strong doubling property (2.1) of \(\omega \in A_{1}\) gives
We have
On the other hand,
Note that we may assume that \(\mu \) is supported in \(B_{\rho }(x)\), this gives
If we choose
then \(0<\delta \le \rho \) and
the proof is complete by routine simplification of (3.3) and (3.4). \(\square \)
Proposition 3.4
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\rho >0\), \(0<\delta <n(s-1)/(n-\alpha s)\), \(\omega \in A_{1}\), and \(x\in {\mathbb {R}}^{n}\). Then
Proof
Let \(0<r\le 1\) be given. We need to estimate that
where
with the convention that \(I_{2}=0\) if \(\rho \le r\). Assume without loss of generality that \(\rho >r\). For the integral \(I_{1}\), we may assume that \(\mu \) is supported in \(B_{2r}(x)\). In view of Lemma 3.3, it suffices to estimate
Let \(p=\delta (n-\alpha s)/(n(s-1))<1\). We appeal to the estimate that
for any measurable set \(E\subseteq {\mathbb {R}}^{n}\) with \(|E|<\infty \) and \(F\in L^{1,\infty }(E)\), here \(L^{1,\infty }(E)\) is the weak Lebesgue space on E (see [2, Exercise 1.1.11]). As a consequence,
where we have used and the weak-type (1, 1) boundedness of \({\textbf{M}}(\cdot )\) in (3.6) that
with \({\textbf{M}}_{r}(\cdot )\le {\textbf{M}}(\cdot )\) for all \(r>0\), and the strong doubling property (2.1) of \(\omega \in A_{1}\) in (3.7). For the integral \(I_{2}\), observe that \(B_{t}(y)\subseteq B_{2t}(x)\) for \(y\in B_{r}(x)\) and \(t>r\). Using the strong doubling property (2.1) of \(\omega \in A_{1}\) again, one obtains
The proof is complete by combining (3.5) and (3.8). \(\square \)
Remark 3.5
Assume that the weighted Riesz capacities \(R_{\alpha ,s}^{\omega }(\cdot )\) are taken into account. Following the similar argument given in the proof of Proposition 3.4, one may show that the Wolff potential \(W_{\omega }^{\mu }\) defined by
satisfies that
for \(x\in {\mathbb {R}}^{n}\) and \(0<\delta <n(s-1)/(n-\alpha s)\). In other words, \((W_{\omega }^{\mu })^{\delta }\) is an \(A_{1}\) weight for those admissible exponents \(\delta \).
Next we prove Theorem 1.2 for the case where f is the characteristic function of a Lebesgue measurable subset E of \({\mathbb {R}}^{n}\).
Proposition 3.6
Let \(\alpha >0\), \(s>1\), \(\alpha s<n\), \(\omega \in A_{1}\), and E be a measurable subset of \({\mathbb {R}}^{n}\). For any \(p>(n-\alpha s)/n\), it holds that
Proof
For any \(p>(n-\alpha s)/n\), choose an \(\varepsilon =\varepsilon _{p}>0\), such that
Let \(\delta =(s-1)n/(n-\alpha s+\varepsilon )\). By choosing the nonlinear potential \({\mathcal {V}}_{\omega ;1}^{\mu }\) as in (2.5) and using (2.7), one has
Since \(\omega \in A_{1}\subseteq A_{s}\), the absolute continuity (2.4) and (2.7) entail
then Proposition 3.4 yields
Note that \(\textbf{M}^{\textrm{loc}}(\chi _{E})\le 1\) and \((s-1)/(p\delta )<1\). Using the equivalence of capacities that (2.2) and (2.3), we obtain by Proposition 2.2 that
which completes the proof. \(\square \)
Proof of Theorem 1.2
It has been noted in (1.6) that we only need to prove the estimate for the exponent that \((n-\alpha s)/n<p\le 1\). By using Corollary 2.6 and Proposition 3.6, one obtains the result by repeating the argument given in the proof of Theorem 1.1. \(\square \)
Proof of Theorem 1.3
We appeal to the weighted Sobolev embedding that
where \(\omega \in A_{q}\), \(q=s^{*}/s'+1\), \(1/s^{*}=1/s-\alpha /n\) (see [9, Theorem 2.2.1]). The assumption that \(\omega \in A_{1}\) entails \(\omega \in A_{q}\) and \(\omega ^{\frac{n-\alpha s}{n}}\in A_{1}\subseteq A_{s}\). By recalling the definition of \(R_{\alpha ,s}^{\omega ^{(n-\alpha s)/n}}(\cdot )\), one obtains immediately that
and hence
As a consequence, we have
where \(L^{\frac{nq}{n-\alpha s},q}(\omega )\) is the Lorentz space.
Let \(\delta >0\) to be determined later. For any \(1\le p<n/\beta \), we compute that
Consequently, Hölder’s inequality gives
By choosing
we obtain
Let \(p=nq/(n-\alpha s)\). Note that \(q^{*}(1-\beta q/(n-\alpha s))=q\). If \(q=(n-\alpha s)/n\), then Theorem 1.1 (in terms of weighted Riesz capacities \(R_{\alpha ,s}^{\omega }(\cdot )\)) implies that
and the estimate (1.7) follows. For \(q>(n-\alpha s)/n\), the same argument will prove for the estimate (1.8) by using Theorem 1.2. The proof is now complete. \(\square \)
Let us address a localization principle. Assume that \(\gamma \in {\mathscr {D}}({\mathbb {R}}^{n})\) satisfies
Then [7, Theorem 2.21] shows that
where \(F_{s,q}^{\alpha ,\omega }\) are the weighted Triebel-Lizorkin spaces (see [7, Definition 2.4] for precise definition). In particular, whenever \(\alpha \in {\mathbb {N}}\), [7, Theorem 2.20] gives
If we let \(q=2\) in (3.12) and (3.13), then one can use [7, Theorem 1.10] and [9, Theorem 3.3.4] to deduce the quasi-additivity of \(R_{\alpha ,s;1}^{\omega }(\cdot )\) that
for any subset E of \({\mathbb {R}}^{n}\).
Proof of Theorem 1.4
By replacing \(I_{\alpha }(\cdot )\) with \(I_{\alpha ,1}(\cdot )\) in (3.9), one has
We deduce similarly as in (3.10) that
Now, we recall the pointwise behavior of the Bessel kernel \(G_{\beta }(\cdot )\) that
(see [1, Section 1.2.4]). For \(1\le p<n/\beta \), we have
Arguing as in (3.11), one has
In the sequel, denote by \(p=nq/(n-\alpha s)\). By Theorems 1.1 and 1.2, we deduce that for \(q=(n-\alpha s)/n\),
and \(L^{q^{*},\infty }\left( R_{\alpha ,s;1}^{\omega ^{(n-\alpha s)/n}}\right) \) is replaced by \(L^{q^{*}}\left( R_{\alpha ,s;1}^{\omega ^{(n-\alpha s)/n}}\right) \) whenever \(q>(n-\alpha s)/n\). To bound \({\mathcal {J}}(\cdot )\), we observe by Hölder’s inequality that
Consider the partition \(\{Q_{z}\}_{z\in {\mathbb {Z}}^{n}}\) of \({\mathbb {R}}^{n}\) defined by \(Q_{z}=z+[0,1/2)^{n}\). Then
Note that \(q/p=(n-\alpha s)/n<1\). Using Corollary 2.6, one obtains
Using the exponential decay of \(e^{-\textrm{dist}(\cdot ,Q_{z})/2}\), it is not hard to compute that
Recall that \(\omega \in A_{1}\) and hence \(\omega ^{(n-\alpha s)/n}\in A_{1}\subseteq A_{s}\). We use the size estimate that
(see [9, Lemma 3.3.12]), then assumption (1.9) yields
Now, we use (3.15) and the quasi-additivity (3.14) to deduce that
Similarly, for any \(r>p\), we obtain
where we have used the normability of \(L^{r}\left( R_{\alpha ,s;1}^{\omega ^{(n-\alpha s)/n}}\right) \) for \(r>1\) in (3.18) (see [3, Theorem 1.2]), and (3.15) in (3.19). Note that \(q^{*}>q\) and \(p>q\). Using (3.16), (3.17), and interpolation theorem, one has
The proof is now complete. \(\square \)
4 A note on Theorems 1.1 and 1.2 for weighted Riesz capacities
In this section, we show briefly that Theorems 1.1 and 1.2 hold for weighted Riesz capacities \(R_{\alpha ,s}^{\omega }(\cdot )\) and the usual Hardy–Littlewood maximal function \(\textbf{M}\) in place of \(R_{\alpha ,s;1}^{\omega }(\cdot )\) and \(\textbf{M}^{\textrm{loc}}\), respectively. More precisely, we have
We introduce a variant weighted Riesz capacities \({\mathcal {R}}_{\alpha ,s}^{\omega }(\cdot )\) defined by
The corresponding nonlinear potential \({\mathcal {V}}_{\omega }^{\mu }\) and Wolff potential \(W_{\omega }^{\mu }\) of a positive measure \(\mu \) on \({\mathbb {R}}^{n}\) are defined by
If \(\omega \in A_{s}\), then we have
The Wolff potential \(W_{\omega }^{\mu }\) satisfies the bounded maximum principle
for all weights \(\omega \) satisfying (2.1). Besides that, we have the following weak-type estimate of Wolff potential that
The capacitary strong-type estimate for \(R_{\alpha ,s}^{\omega }(\cdot )\) reads as
By introducing the auxiliary functional
one proves the normability of \(L^{1}(R_{\alpha ,s}^{\omega })\) by noting that
As a result, the following p-convexity is obtained.
We have similarly as in Lemma 3.1 that
where \(\eta =(s-1)n/(n-\alpha s)\). Consequently,
As a result, the estimate (4.1) is proved by using the p-convexity. To obtain the estimate (4.2), we have by Remark 3.5 that
Consequently, for \(p>(n-\alpha s)/n\), we have
Finally, the estimate (4.2) is obtained by using the p-convexity again.
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Ooi, K.H. Boundedness of maximal function for weighted Choquet integrals. Banach J. Math. Anal. 18, 8 (2024). https://doi.org/10.1007/s43037-023-00317-7
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DOI: https://doi.org/10.1007/s43037-023-00317-7