Abstract
We investigate mapping properties of noncentered Hardy–Littlewood maximal operators related to the exponential measure \(d\mu (x) = \exp (x_1\cdots x_d)dx\) in \({\mathbb {R}}^d\). The mean values are taken over Euclidean balls or cubes (\(\ell ^{\infty }\) balls) or diamonds (\(\ell ^1\) balls). Assuming that \(d \ge 2\), in the cases of cubes and diamonds we prove the \(L^p\)boundedness for \(p>1\) and disprove the weak type (1, 1) estimate. The same is proved in the case of Euclidean balls, under the restriction \(d \le 4\) for the positive part.
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1 Introduction and statement of the results
Let \(d \ge 1\). Consider a metric measure space \(({\mathbb {R}}^d,\rho ,d\eta )\), with a Borel measure \(\eta \) which is nonnegative, nontrivial and locally finite. The associated noncentered Hardy–Littlewood maximal operator is defined by
where the supremum is taken over all open metric balls related to \(\rho \) that contain x and have strictly positive measure \(\eta \). Here f is any Borel measurable function on \({\mathbb {R}}^d\). The centered variant of \(M_{\eta }\), denoted by \(M_{\eta }^c\), arises by restricting the supremum to balls centered at x. Clearly, \(M_{\eta }^cf \le M_{\eta }f\). Furthermore, \(M_{\eta }\) is trivially bounded on \(L^{\infty }\).
When \(\eta \) is doubling, the two maximal operators are comparable and satisfy the weak type (1, 1) estimate with respect to \(\eta \). The latter follows from a Vitali type covering lemma, cf. [4, Chapter 2]. Then, by interpolation, \(M_{\eta }\) and \(M_{\eta }^c\) are bounded on \(L^p(d\eta )\) for \(p > 1\).
It is also well known, at least for the Euclidean distance \(\rho \), that (see e.g. [2, p. 44]) whatever the measure \(\eta \) is, \(M_{\eta }^c\) is always of weak type (1, 1) with respect to \(\eta \), thus also bounded on \(L^p(d\eta )\) for \(p>1\). The former is a consequence of the Besicovitch–Morse covering lemma. In dimension one the larger uncentered operator \(M_{\eta }\) behaves in the same way (see [2, p. 45]), that is, it is weak type (1, 1) and bounded on \(L^p(d\eta )\), \(p>1\), independently of the doubling property of \(\eta \). However, this is no longer true in general in higher dimensions.
One of the authors [15] proved that for \(d = 2\) (implicitly \(d \ge 2\)) and either the Euclidean or the \(\ell ^{\infty }\) distance \(\rho \), and the Gaussian measure \(\eta \), the weak type (1, 1) estimate for \(M_{\eta }\) fails. Nevertheless, as shown by Forzani et al. [3], the \(L^p\)boundedness for \(p > 1\) in this case still holds, though the convenient interpolation argument is inapplicable. Similar results for certain classes of rotationally invariant measures \(\eta \) were established in [6, 14, 16, 17], among others. It is interesting to point out that there are radial measures \(\eta \) for which \(M_{\eta }\) is not even weak type (p, p) for any \(p<\infty \), see [5, 6, 17].
It should be mentioned that so far noncentered Hardy–Littlewood maximal operators for nondoubling measures were studied in various settings and spaces also different from \({\mathbb {R}}^d\), for example in the framework of cusped manifolds [8, 9].
The main aim of this paper is to study the maximal operator \(M_{\eta }\) when the distance \(\rho \) is the Euclidean one and for the particular exponential measure \(\eta =\mu \),
Our motivation is to provide both methods and results in this model case where the measure is nondoubling and nonradial, since the literature seems to lack a basic example of this kind. Only recently Li, Wu and one of the authors [10] considered \(M_{\eta }\) essentially for \(d\eta (x)=e^{x_1}dx\) in \({\mathbb {R}}^d\). In this case the measure, in contrast with \(\mu \), is neither finite nor even in each variable. Moreover, it has a simple structure that makes the associated analysis relatively straightforward.
The measure \(\mu \) is not radial in the sense of the Euclidean distance, nevertheless it is radial with respect to the \(\ell ^1\) metric. Thus one might wonder whether, perhaps, the maximal operator behaves better when \(\rho \) is the seemingly better matching \(\ell ^1\) distance. This issue led us to study \(M_{\mu }\) also when \(\rho \) is the \(\ell ^1\) metric, as well as in the opposite extreme case where \(\rho \) is the \(\ell ^{\infty }\) metric.
Denote by \(M_{\mu }^{{\mathcal {B}}}\), \(M_{\mu }^{{\mathcal {Q}}}\), \(M_{\mu }^{{\mathcal {D}}}\) the maximal operators \(M_{\mu }\) with the underlying \(\ell ^2\) or \(\ell ^{\infty }\) or \(\ell ^1\) metric, respectively. Note that the metric balls in the first case are just the Euclidean balls \({\mathcal {B}}\), and in the second case the Euclidean cubes \({\mathcal {Q}}\) with sides parallel to the coordinate axes. The third case is geometrically somewhat more complicated, and we call the metric balls diamonds \({\mathcal {D}}\) in this situation. Notice that in dimension \(d=2\) the diamonds are simply rotated cubes (or actually squares), but there is no similar relation in higher dimensions.
Our main result is the following theorem. We strongly believe it will be an inspiration for considering \(M_{\eta }\) with more general nonradial and nondoubling \(\eta \), and for further research in the future.
Theorem 1
Let \(d \ge 2\).

(A)
None of the maximal operators \(M_{\mu }^{{\mathcal {B}}}\), \(M_{\mu }^{{\mathcal {Q}}}\), \(M_{\mu }^{{\mathcal {D}}}\) is weak type (1, 1).

(B)
The operators \(M_{\mu }^{{\mathcal {Q}}}\) and \(M_{\mu }^{{\mathcal {D}}}\) are bounded on \(L^p(d\mu )\) for \(p>1\). The same is true for \(M_{\mu }^{{\mathcal {B}}}\), provided that \(d \le 4\).
Remark 1.1
The restriction \(d \le 4\) in Theorem 1(B), the case of \(M_{\mu }^{{\mathcal {B}}}\), is caused by substantial technical difficulties of geometrical nature in proving the result in dimensions \(d=5\) and higher. Nevertheless, we strongly believe that the result is true for any \(d \ge 2\).
When \(d=1\), in view of what was said above, all the three maximal operators coincide and are weak type (1, 1) and bounded on \(L^p(d\mu )\), \(p>1\). Note that the latter readily implies Theorem 1(B) for \(M_{\mu }^{{\mathcal {Q}}}\). Indeed, due to the product structure of the cubes \(M_{\mu }^{{\mathcal {Q}}}\) can be controlled by a composition of the onedimensional operators.
Theorem 1 reveals that the \(L^p\) behavior of \(M_{\mu }^{{\mathcal {B}}}\) and \(M_{\mu }^{{\mathcal {Q}}}\) is exactly the same as in case of their counterparts for the Gaussian measure [3, 15]. In particular, we see that the local doubling property (see Sect. 2), satisfied by \(\mu \) but not by the Gaussian measure, does not lead here to any improvement.
An interesting but technically quite complicated problem is to generalize Theorem 1 to Laguerretype measures of the form
where \(\alpha = (\alpha _1,\ldots ,\alpha _d) \in (1,\infty )^d\) is a fixed multiparameter. Clearly, the special choice \(\alpha =(0,\ldots ,0)\) gives \(\mu \). The restriction of the measure space \(({\mathbb {R}}^d,d\mu _{\alpha })\) to \((0,\infty )^d\) forms a natural environment for analysis related to the classical Laguerre operator. Analysis of various objects in this context has already received considerable attention; see for instance [1, 11,12,13] and references given there. Thus any knowledge about the noncentered Hardy–Littlewood maximal operator \(M_{\mu _{\alpha }}\) or its variants would be potentially useful. For some negative results, see Remark 3.1 below, which says that \(M_{\mu _{\alpha }}\) is not of weak type (1, 1) when the underlying metric is either \(\ell ^2\) or \(\ell ^{\infty }\).
The remaining part of the paper is devoted to the proof of Theorem 1. The subsequent sections contain technical preliminaries, the proof of Theorem 1(A) and the proof of Theorem 1(B), respectively.
2 Technical preliminaries
Denote \({\mathbb {R}}^d_+ = (0,\infty )^d\), \(d \ge 1\). For brevity the restriction of \(\mu \) to \({\mathbb {R}}^d_+\) will be denoted by the same symbol. We write \(\cdot _{q}\) for the \(\ell ^q\), \(1 \le q \le \infty \), norm in \({\mathbb {R}}^d\),
Of course, this norm generates a metric \(\rho _q\) both in \({\mathbb {R}}^d\) and \({\mathbb {R}}^d_+\). For \(q=1,2,\infty \) we denote the families of open balls in the metric measure spaces \(({\mathbb {R}}^d_+,\rho _q,d\mu )\) by \({\mathcal {D}}_+\), \({\mathcal {B}}_+\), \({\mathcal {Q}}_+\), respectively. Notice that these are exactly diamonds, Euclidean balls and cubes, respectively, centered in and intersected with \({\mathbb {R}}^d_+\).
Bring in the noncentered Hardy–Littlewood maximal operator
and analogously \(M_{\mu }^{{\mathcal {Q}}_+}\) and \(M_{\mu }^{{\mathcal {D}}_+}\). The following elementary result shows that proving Theorem 1 can be reduced to (actually is equivalent to) a similar analysis for \(M_{\mu }^{{\mathcal {B}}_+}\), \(M_{\mu }^{{\mathcal {Q}}_+}\) and \(M_{\mu }^{{\mathcal {D}}_+}\).
Proposition 2.1
Let \(d \ge 1\) and \(p > 1\) be fixed. The operator \(M_{\mu }^{{\mathcal {B}}}\) is bounded on \(L^p({\mathbb {R}}^d,d\mu )\) (is weak type (1, 1) with respect to \(({\mathbb {R}}^d,d\mu )\)) if and only if \(M_{\mu }^{{\mathcal {B}}_+}\) is bounded on \(L^p({\mathbb {R}}^d_+,d\mu )\) (is weak type (1, 1) with respect to \(({\mathbb {R}}^d_+,d\mu )\)).
The same relations hold between \(M_{\mu }^{{\mathcal {Q}}}\) and \(M_{\mu }^{{\mathcal {Q}}_+}\), as well as between \(M_{\mu }^{{\mathcal {D}}}\) and \(M_{\mu }^{{\mathcal {D}}_+}\).
Proof
This is a consequence of the symmetries involved. Use either the even (more precisely even with respect to each coordinate axis) extension to \({\mathbb {R}}^d\) of \(f_+\) on \({\mathbb {R}}^d_+\) or, for the other implication, the decomposition of f on \({\mathbb {R}}^d\) into its symmetric components which are either even or odd with respect to each coordinate axis. \(\square \)
Thus, from now on, we focus on the restricted operators \(M_{\mu }^{{\mathcal {B}}_+}\), \(M_{\mu }^{{\mathcal {Q}}_+}\) and \(M_{\mu }^{{\mathcal {D}}_+}\). This is a crucial reduction from a technical point of view, since in \({\mathbb {R}}_+^d\) the measure \(\mu \) has a simpler analytic structure than in \({\mathbb {R}}^d\) (no absolute values involved). From now on \(\mu \) will denote the restriction of the measure with density \(\exp (x_1)\) to \({\mathbb {R}}^d_+\).
In what follows we shall write \(X \lesssim Y\) with \(Y>0\) to indicate that \(X \le C Y\) with a constant \(C>0\) depending only on the dimension and on p in the proofs of \(L^p\) estimates, and also on \(\alpha \) in Remarks 2.4 and 3.1. We write \(X\simeq Y\) when simultaneously \(X \lesssim Y\) and \(Y \lesssim X\).
We will occasionally refer to the strong maximal operator in Euclidean space with Lebesgue measure. It is defined as
where the supremum is taken over all rectangles with edges parallel with the coordinate axes and containing x. It is well known that \(M_{\textrm{str}}\) is bounded on \(L^p(dx)\) for \(1<p \le \infty \), as seen by iterating the onedimensional estimate.
The following notation will be used for \(\ell ^1\), \(\ell ^2\) and \(\ell ^{\infty }\) balls in \({\mathbb {R}}^d_+\). For \(x \in {\mathbb {R}}^d_+\) and \(r > 0\)
Euclidean balls in all of \({\mathbb {R}}^d\) will be written as
Further, we denote
The measure \(\mu \) is not doubling in \(({\mathbb {R}}^d_+,\rho _q,d\mu )\), \(q=1,2,\infty \); nevertheless it is locally doubling in the following sense.
Lemma 2.2
Let \(d \ge 1\). Given \(R>0\), there exists a constant \(C_R>0\) such that
The same holds if Q above is replaced either by B or by D.
Proof
This is elementary, since in any of the balls considered the density of \(\mu \) varies at most by a factor depending only on R. \(\square \)
Let \(d \ge 1\). We now give sharp estimates for the measure of large cubes, balls and diamonds provided that they are disjoint with the boundary of \({\mathbb {R}}^d_+\). Consider a ball in one of the three metrics \(\ell ^\infty ,\,\ell ^2,\, \ell ^1\) with center \(x \in {\mathbb {R}}^d_+\) and radius r satisfying \(1 \le r \le \min _{1 \le i \le d} x_i\). We select a point \(z_q = z_q(x,r)\) in the closure of this ball where \(\cdot _1\) is minimal, i.e., the density of \(\mu \) is maximal, as follows:
Notice that \(z_\infty \) and \(z_2\) are unique points with this minimizing property, but \(z_1\) is not.
Lemma 2.3
Let \(x \in {\mathbb {R}}^d_+\) and \( 1 \le r \le x_i,\, i = 1,\dots , d\). Then the balls Q(x, r), B(x, r) and D(x, r) are contained in \({\mathbb {R}}^d_+\) and
The implicit constants here depend only on d.
Proof
The inclusions follow, since if y is in one of the balls, then \(y_ix_i <r\) for each i, so that \(y_i>0\).
The estimate for cubes is straightforward. One has
To deal with the case of Euclidean balls, observe that any point in B(x, r) can be written as \(z_2 + \frac{s}{\sqrt{d}}\,{\textbf{1}} + y\), where \(s>0\) and \(y\perp {\textbf{1}}\). Using the expression for \(z_2\), we see that this point is in B(x, r) precisely when \((rs)^2 + y^2 < r^2\) or equivalently \(y< \sqrt{2rss^2}\) and \(0<s<2r\). We now integrate in y in a hyperplane orthogonal to \({\textbf{1}}\) and then in s, taking the density of \(\mu \) into account. For the upper estimate, we simply write
To obtain the lower estimate, we observe that \(2rss^2 > rs\) for \(0<s<r\) and argue similarly. Since \(r \ge 1\), we get
As for the diamonds, note that \(z_1_1 = x_1r\). For \(s>0\) the diameter of the intersection of D(x, r) with the hyperplane \(\{y: y_1 = x_1r +s\}\) is \({\mathcal {O}}(r)\). Integrating as before, we obtain the upper estimate.
On the other hand, consider the following set
The \(\ell ^1\) distance from x to a point in this set is
Thus D(x, r) contains the set, and the lower estimate follows by integration. \(\square \)
Remark 2.4
Lemmas 2.2 and 2.3 can be generalized to the space \(({\mathbb {R}}^d_+,\rho _q,d\mu _{\alpha })\), where \(q \in \{1,2,\infty \}\) and \(\mu _{\alpha }\) is the restriction of the measure defined in (1.1). This means that \(\mu _{\alpha }\) is locally doubling (but not doubling) in the context of this space. Moreover,
uniformly in \(x \in {\mathbb {R}}^d_+\) and \(1 \le r \le \min _{1\le i \le d} x_i\); here \(E_q(x,r)\) is the open ball in \(({\mathbb {R}}^d_+,\rho _q)\) centered at x and of radius r.
Proposition 2.1 can also be generalized in a similar spirit.
We now pass to the proof of Theorem 1. It is worth indicating that the radiality of \(\mu \) with respect to the \(\ell ^1\) norm will be heavily exploited, often implicitly, throughout our reasonings.
3 Proof of Theorem 1(A)
In this section we prove Theorem 1(A) working with the operators restricted to \({\mathbb {R}}^d_+\), see Proposition 2.1. The cases of \(M_{\mu }^{{\mathcal {Q}}_+}\) and \(M_{\mu }^{{\mathcal {B}}_+}\) will be treated together, since the argument is essentially the same. This argument has the advantage that it can be rather easily generalized to cover \(M_{\mu _{\alpha }}^{{\mathcal {Q}}_+}\) and \(M_{\mu _{\alpha }}^{{\mathcal {B}}_+}\) (analogues of \(M_{\mu }^{{\mathcal {Q}}_+}\) and \(M_{\mu }^{{\mathcal {B}}_+}\) for the measure \(\mu _{\alpha }\)), see Remark 3.1 below. Unfortunately, this argument does not apply to \(M_{\mu }^{{\mathcal {D}}_+}\) since it uses essentially the nonradiality of the measure with respect to the norm. Therefore, we give a different argument for \(M_{\mu }^{{\mathcal {D}}_+}\), but the question of its generalization to \(M_{\mu _{\alpha }}^{{\mathcal {D}}_+}\) seems to be technically difficult and remains open.
3.1 Proof of Theorem 1(A), the cases of \(\textbf{M}_{\mu }^{\mathcal {Q}}\) and \(\textbf{M}_{\mu }^{\mathcal {B}}\)
We first consider the case \(d=2\) and then indicate the changes needed for \(d\ge 3\). We begin with the operator \(M_{\mu }^{{\mathcal {Q}}_+}\). Let \(Q_s\), \(s\ge 2\), denote the square centered at (s, s) and of ‘radius’ \(s/2\). Further, let \(\widehat{Q_s}\) be the union of all squares obtained by moving \(Q_s\) (or rather its center) along the line segment \(\Delta _s\) which is the intersection of \(\frac{1}{2}Q_s\) with the line \(x+y=2s\), see Fig. .
Assuming a contrario that \(M_{\mu }^{{\mathcal {Q}}_+}\) is of weak type (1, 1), we claim that
To see this, take \((x_0,y_0)\in \widehat{Q_s}\) and find a square \(Q^0=Q((x',y'),s/2)\) with center on \(\Delta _s\) and of side length s, such that \((x_0,y_0)\in Q^0\). It is clear that \(\frac{1}{2}Q_s\subset Q^0\), and by Lemma 2.3\(\mu (Q^0) = \mu (Q_s) \simeq e^{s}\). Thus, for the \(L^1\)normalized function \({\widetilde{\chi }}=\frac{1}{\mu (\frac{1}{2}Q_s)}\chi _{\frac{1}{2}Q_s}\) one has
We conclude that
hence (3.1) follows. On the other hand, since \(\widehat{Q_s}\) contains the rectangle \(R_s\) with basis \(\Delta _s  (\frac{s}{2},\frac{s}{2})\) and height \(\sqrt{2}s\), we have
For large s this contradicts (3.1) since, as already noted, \(\mu (Q_s)\simeq e^{s}\).
We now continue with the operator \(M_{\mu }^{{\mathcal {B}}_+}\) in dimension \(d=2\). Let \(B_s\), \(s\ge 1\), denote the discs with center at (s, s) and radius s/2 (thus \(B_s\) is an ordinary Euclidean disc), and let \(\widehat{B_s}\) be the union of all balls obtained by moving \(B_s\) (or rather its center) along the line segment \(\widetilde{\Delta _s}\) which is the intersection of \(\frac{1}{2}B_s\) with the line \(x+y=2s\), see Fig. .
Again, assuming a contrario that \(M_{\mu }^{{\mathcal {B}}_+}\) is of weak type (1, 1), we claim that
The argument is similar to that for squares. Lemma 2.3 yields
Since \(\widehat{B_s}\) contains the rectangle \(\widetilde{R_s}\) with basis \(\widetilde{\Delta _s}(\frac{s}{2\sqrt{2}},\frac{s}{2\sqrt{2}})\) and height s (that contains \(\frac{1}{2} B_s\)), we have
which for large s contradicts (3.2).
We pass to explaining the changes necessary for \(d\ge 3\). Let \(Q_s\), \(s\ge 1\), denote the cube centered at \(s{\textbf{1}}\), of side length s, and let \(\widehat{Q_s}\) be the union of all cubes emerging from moving the center of \(Q_s\) along the hypersegment \(\Delta _s\) obtained by intersecting \(\frac{1}{2}Q_s\) with the hyperplane \(x_1+\cdots +x_d=ds\). With the present notation the justification of (3.1), assuming a contrario the weak type (1, 1) of \(M_{\mu }^{{\mathcal {Q}}_+}\), is analogous to that for the case \(d=2\) and involves the estimate (see Lemma 2.3)
Now (3.1) is contradicted for large s by
To justify the last estimate, observe that \(\widehat{Q_s}\) contains the hyperprism \(R_s\) with basis \(\Delta _s\frac{s}{2} {\textbf {1}}\) and height \(\sqrt{d}s\). Then
Similarly, let \(B_s\), \(s\ge 1\), denote the ball with center at \(s{\textbf {1}}\) and radius \(s/2\), and let \(\widehat{B_s}\) be the union of balls emerging from moving the center of \(B_s\) along the hypersegment \(\Delta _s\) obtained by intersecting \(\frac{1}{2}B_s\) with the hyperplane \(x_1+\cdots +x_d=ds\). Assuming again a contrario the weak type (1, 1) of \(M_{\mu }^{{\mathcal {B}}_+}\), we prove (3.2) in a way analogous to that for the case \(d=2\) with the estimate
included. Let \(\widetilde{R_s}\) be the cylinder with basis \(\widetilde{\Delta _s}\frac{s}{2\sqrt{d}}\) and height s that includes \(\frac{1}{2}B_s\). Since \(\widetilde{R_s}\subset \widehat{B_s}\), we have
For large s, this contradicts \(\mu (B_s)\simeq s^{\frac{d1}{2}}e^{s(d\frac{\sqrt{d}}{2})}\). This finishes the proof. \(\square \)
Remark 3.1
In view of Remark 2.4, the above proof extends in a straightforward manner to the context of the measure \(\mu _{\alpha }\) given in (1.1). Consequently, \(M_{\mu _{\alpha }}^{{\mathcal {Q}}_+}\) and \(M_{\mu _{\alpha }}^{{\mathcal {B}}_+}\) are not weak type (1, 1).
3.2 Alternative condensed version of the proof of Theorem 1(A), the cases of \(\textbf{M}_{\mu }^{\mathcal {Q}}\) and \(\textbf{M}_{\mu }^{\mathcal {B}}\)
Consider first \(\textbf{M}_{\mu }^{\mathcal {Q}}\). With \(s>1\), we choose \(0 \le f \in L^1(d\mu )\) so that the measure \(f d\mu \) is a close approximation of the Dirac measure \(\delta _{2s{\textbf{1}}}\). The cube \(Q(2s{\textbf{1}}+y,s)\) will contain the point \(2s{\textbf{1}}\) if \(y\perp {\textbf{1}}\) and \(y_\infty <s\), and this cube is contained in \({\mathbb {R}}^d_+\). Then any point \(x \in Q(2\,s{\textbf{1}}+y,s)\) will satisfy
where we applied Lemma 2.3, and \(z_\infty = z_\infty (2\,s{\textbf{1}}+y,s) = s{\textbf{1}}+y\). Notice that \(z_\infty _1 = ds\) does not depend on y. The union of these cubes taken over all admissible points y will contain the set
whose \(\mu \) measure is at least \(c\exp { \left( z_\infty _1\right) }\,s^{d1} \). Since (3.3) holds in this set, the weak type (1, 1) inequality is violated for large s.
In the case of \({M_{\mu }^{{\mathcal {B}}_+}}\), we proceed similarly, with the same f but with the balls \(B(2s{\textbf{1}}+y,s)\) instead of the cubes. In view of Lemma 2.3, the estimate (3.3) will now read \( M_{\mu }^{{\mathcal {B}}_+} f(x) \gtrsim \mu (B(2\,s{\textbf{1}}+y,s))^{1} \simeq \exp { \left( z_2_1\right) } \,s^{(1d)/2}\), where \(z_2 = z_2(2\,s{\textbf{1}}+y,s)\). The measure of the union of the balls will be at least constant times \(\exp { \left( z_2_1\right) }\,s^{d1}\). These two estimates together disprove the weak type inequality. \(\square \)
3.3 Proof of Theorem 1(A), the case of \(\textbf{M}_{\mu }^{\mathcal {D}}\)
Fixing a large \(N>0\), we now let \(f d\mu \) approximate \(\delta _{(0,\dots ,0,N)}\) (cf. the argument for \(M_{\mu }^{{\mathcal {Q}}_+}\)).
Let \(\xi \in {\mathbb {R}}^d_+\) with \(\xi _1 < N\), and write \(s = \xi _1\). To estimate \(M^{{\mathcal {D}}_+}_{\mu } f(\xi )\) from below, we introduce a (closed) diamond \(D=\{x\in {\mathbb {R}}^d_+:xc_1\le M\}\) with \(c_i=\xi _i\) for \(i<d\) and \(c_d=\xi _d+M\). Here \(M > N+s\). Then the points \(\xi \) and \((0,\dots ,0,N)\) are both in D, and \(x_1 \ge s\) if \(x \in D\). Since \(x_d\ge \xi _d\) for all points \(x\in D\), one has for \(h>0\)
and the \((d1)\)dimensional area of the last set here is \({\mathcal {O}}((s+h\xi _d)^{d1})\), as seen by projecting onto \({\mathbb {R}}^{d1}\). Thus
This implies that \(M^{{\mathcal {D}}_+}_{\mu } f(\xi )\gtrsim {e^s}/(1+s\xi _d)^{d1}\); observe that \(s\xi _d = \sum _1^{d1} \xi _i\).
Next we choose the level \(\lambda = {N^{1d}\,e^N}\) and examine when \(M^{{\mathcal {D}}_+}_{\mu } f(\xi )\gtrsim \lambda \). This occurs if \(1+(s\xi _d)^{d1} \lesssim N^{d1} e^{sN}\), in particular if
To find points \(\xi \in {\mathbb {R}}^d_+\) satisfying these two inequalities, we fix
We can then choose any \(\xi _i \in \left( 0,\, d^{1}\,N\,e^{(sN)/(d1)} \right) \), \(i= 1,\dots , d1\), and set \(\xi _d = s  \sum _1^{d1} \xi _i\). Indeed, for such points \(\xi \) the first inequality is clear, and the second one follows because
Here the last inequality assures that \(\xi _d\) is positive, and it holds since \(s<N\) implies \(e^s\,s^{1d} < e^N\,N^{1d}\) for large s and N.
Keeping still s fixed, we see that the \((d1)\)dimensional measure of the set of points \(\xi \) thus obtained is of order of magnitude \( N^{d1} e^{sN}\). Varying then s, we conclude that the \(\mu \)measure of the set of all points \(\xi \) obtained is greater than constant times
For large N, this contradicts the weaktype (1, 1) boundedness of \(M^{{\mathcal {D}}_+}_{\mu }\). \(\square \)
4 Proof of Theorem 1(B)
As remarked in Sect. 1, the case of \(M_{\mu }^{{\mathcal {Q}}}\) in Theorem 1(B) is an immediate consequence of the onedimensional result. The remaining two cases are much less straightforward and will be treated subsequently. We shall work with the operators restricted to \({\mathbb {R}}^d_+\), see Proposition 2.1. We make the following two preliminary reductions in proving the \(L^p\)boundedness of \(M_{\mu }^{{\mathcal {D}}_+}\) and \({M_{\mu }^{{\mathcal {B}}_+}}\).
Reduction 1
We may consider only diamonds (elements of \({\mathcal {D}}_+\)) or balls (elements of \({\mathcal {B}}_+\)) with radii bounded from below by any fixed positive constant, due to the local doubling property of \(\mu \), see Lemma 2.2.
Reduction 2
Among diamonds or balls remaining after Reduction 1, we may consider only those not intersecting \(t \Sigma _+^{d1} = \{x \in {\mathbb {R}}^d_+: x_{1} = t \}\) for \(0<t\le c\) with \(c>2\) arbitrary and fixed, since otherwise they have measures bounded from below (and above) by a positive constant.
We first consider the simpler case \(M^{{\mathcal {D}}_+}_{\mu }\). The reasoning in case of \(M^{{\mathcal {B}}_+}_{\mu }\) is more sophisticated, because of the geometry of the balls in \({\mathbb {R}}^d_{+}\), especially those touching the boundary of \({\mathbb {R}}^d_+\).
4.1 Proof of Theorem 1(B), the case of \(\textbf{M}_{\mu }^{\mathcal {D}}\)
Our aim is to prove that \(M_{\mu }^{{\mathcal {D}}_+}\) is bounded on \(L^p({\mathbb {R}}^d_+,d\mu )\) for \(1<p<\infty \). Recall that diamonds in \({\mathbb {R}}^d_+\) are denoted
Here \(r > 0\), and z will always be in \({\mathbb {R}}^d_+\).
For each \(x \in {\mathbb {R}}^d\) we denote \(x_0 = \sum _1^d x_j\). Then
is a hyperplane for each \(t \in {\mathbb {R}}\), and we write \(\lambda _t\) for the Lebesgue measure in \(\Pi _t\). Further, \(x_t\) will for \(t>0\) denote the orthogonal projection on \(\Pi _t\) of any point x.
Let f be a nonnegative function in \( L^1(d\mu )\), which we extend by 0 in \( {\mathbb {R}}^d \setminus {\mathbb {R}}^d_+\). We want to estimate \(M_{\mu }^{{\mathcal {D}}_+}f\) at a point \(\xi \in {\mathbb {R}}^d_+\). So we take a diamond \({D} = {D}(z,r)\) with \(z \in {\mathbb {R}}^d_+\) and such that \(\xi \in D\), and estimate the mean
Reductions 1 and 2 allow us to assume that the quantities r and \(z_0  r > 2\) are large. It will be convenient to write \(b = z_0r\), which indicates the “bottom” of the diamond.
Denoting slices of D as \(D_t = D \cap \Pi _t\), we can write this mean as
The inner integral here will be estimated in terms of a \((d1)\)dimensional maximal operator. We define V as the set consisting of the ddimensional vector \(v = (1,1,0,\dots ,0)\) and all the vectors obtained from v by permuting the coordinates.
Proposition 4.1
For each \(t \in (b,b+2r)\) there exists a \((d1)\)dimensional parallelepiped \(P_t \subset \Pi _t\) containing \(D_t\) and containing \(\xi _{t}\) such that
and whose edges are all parallel to vectors in V.
Before proving this proposition, we use it to finish the proof of the \(L^p(d\mu )\)boundedness of \(M_{\mu }^{{\mathcal {D}}_+}\). Here \(1<p<\infty \).
In the iterated integral in (4.1), we extend the inner integration to \(P_{t}\) and insert the factor
Thus (4.1) is controlled by
The mean over \(P_{t}\) here can be estimated in terms of the noncentered maximal operator \({\mathcal {M}}_t\) in \(\Pi _{t}\) associated with parallelepipeds having edges with directions from V, evaluated at \(\xi _t\). So the iterated integral is at most
We consider the exponent \(t+b\) here. Since \(\xi _0>b\) and \(t>b\), we have
with \(c = c(p) > 0\); in the last step we used the simple fact that \((tb) \vee (\xi _0b) \ge \xi _0t\). After inserting this estimate in the integral (4.2), we can delete the factors \(e^{c\,[(tb) \vee (\xi _0b)]}\,\big [1 + (tb) \vee (\xi _0b)\big ]^{d1}\), and thus estimate (4.2) by constant times
Now we apply Hölder’s inequality, with \(e^{ c \,\xi _0t/p'}\) as one factor. It follows that (4.1) is not larger than constant times
Since this quantity is independent of the choice of the diamond D, it gives an upper bound for \(M_{\mu }^{{\mathcal {D}}_+}f(\xi )\).
Integrating pth powers with respect to \(d\mu (\xi )\), one obtains
In the righthand side here, we integrate first in \(\xi _t\), using the fact^{Footnote 1} that the operator \({\mathcal {M}}_t\) is bounded on \(L^p(d\lambda _t)\) uniformly in t. Thus the triple integral is at most constant times
Integrating next in \(\xi _0\), we conclude that
and this proves the \(L^p(d\mu )\)boundedness of \(M_{\mu }^{{\mathcal {D}}_+}\).
Proof of Proposition 4.1
We fix \(\xi \in D\) and \(t \in (b, b+2r)\), and for convenience we also write \(t = b +h = z_0  r +h\) with \(0< h < 2r\). Further, we renumber the coordinates so that
Denote
Obviously \(D_t\subset G_{t,\xi }\) but also \(\xi _t\in G_{t,\xi }\). Indeed, \((\xi _t)_i>(\xi _t)_i\xi _i\ge \xi _t\xi _1=t\xi _0\) and \(z\xi _t_1\le z\xi _1+ \xi  \xi _{t}_1 < r+t  \xi _{0}\).
In order to include \( G_{t,\xi }\) in a parallelepiped in \(\Pi _t\), we let \(x\in G_{t,\xi }\). Since \(zx_1 < r + t  \xi _{0}\) and \((zx)_0=rh\), we then have for each \(i=1,\ldots ,d\)
Switching coordinates to \(y_i = z_ix_i+(h+t  \xi _0)/2\), we get
Further, \((zx)_0=rh\) implies, since \(y_d > 0\),
We need a simple lemma.
Lemma 4.2
Let \(m \ge 2\) and consider the set \(E \subset {\mathbb {R}}^m\) defined by
for some \(a_i,\, R > 0\). Then E is contained in the mdimensional rectangle
and the Lebesgue measures satisfy \(E \simeq \big {{\widetilde{E}}}\big  = \prod _1^m a_i\wedge R\).
(In expressions like the last product here, we always mean the product of the minima.)
To get the lower estimate for E in the lemma, one observes that \(E \supset \prod _1^m (0, (a_i\wedge R)/m)\), and the other parts are trivial.
Let the projection \(\tau _t:\Pi _{t} \rightarrow {\mathbb {R}}^{d1}\) be given by suppression of the last coordinate. The lemma, applied with \(m = d1\) and in the coordinates \((y_1,\dots ,y_{d1})\), implies that the projection \(\tau _t(G_{t,\xi })\) is contained in a rectangle \({{\widetilde{E}}}\) in \({\mathbb {R}}^{d1}\) with sides parallel to the y (or equivalently x) coordinate axes. Then \(G_{t,\xi }\) is contained in \(\tau _t^{1}\big ({{\widetilde{E}}}\big )\), which is seen to be a parallelepiped \(P_{t}\) fulfilling the conditions of Proposition 4.1, except that the estimate we get for its Lebesgue measure is
In addition to (4.5), we will deduce a similar estimate by writing first
Combining this estimate with (4.4) and applying Lemma 4.2 in the coordinates \(y_i = x_i + t  \xi _0\), we can argue as above. As a result, we find a parallelepiped \(P_{t}\) containing \(G_{t,\xi }\) and verifying
Next we derive two different lower estimates for \(\mu (D)\), whose validity will depend on the condition
We shall verify that
when (4.7) holds (upper), and when (4.7) is false (lower), respectively.
These two estimates will end the proof of Proposition 4.1 when combined with (4.5) and (4.6), respectively, since
and
recall here that \(z_0  r > 2\), see Reduction 2.
To verify (4.8), it is enough to show that for \(1<h<2\), under relevant assumptions,
because one can then integrate with respect to \(e^{bh}\,dh\) over the interval (1, 2).
Observe that the last coordinate of any point \(x \in \Pi _{b+h}\) is given by
Let \(h \in (1, 2)\). Aiming at the lower case in (4.9) and thus assuming (4.7) false, we define the set
We claim that the inverse projection, or lift, \(\tau _{b+h}^{1}(E)\) is contained in \(D_{b+h}\). Indeed, let \(x \in \tau _{b+h}^{1}(E)\) and consider the last coordinate \(x_d\) of x. From (4.10) we conclude
the last step since (4.7) is false. Thus \(x \in {\mathbb {R}}^d_+ \cap \Pi _{b+h}\). Further,
so that \(x \in D\). The claim follows.
For the measures, we then get \(\lambda _{b+h}(D_{b+h}) \ge \lambda _{b+h}(\tau _{b+h}^{1}(E)) \simeq E\), where \(\cdot \) denotes Lebesgue measure in \({\mathbb {R}}^{d1}\). Lemma 4.2 yields that \(E \simeq \prod _1^{d1} (z_i+1) \wedge (z_0  r)\). This proves (4.9) and thus (4.8), for the lower lines.
Next, we verify (4.9) (upper) under the assumption (4.7); recall that \(1< h <2\). We start with the case \(z_d \ge r\), and here we argue almost as above. Define now
Points in \(E'\) clearly satisfy
As before, we take a point \(x \in \tau _{b+h}^{1}(E')\) and verify that \(x \in D_{b+h}\). From (4.10) combined with \(z_d \ge r\), we now get
It follows that \(x \in {\mathbb {R}}^d_+ \cap \Pi _{b+h}\) and that (4.11) remains valid. This proves the inclusion \(\tau _{b+h}^{1}(E') \subset D_{b+h}\).
Thus \(\lambda _{b+h}(D_{b+h}) \gtrsim E'\), and \(E'\) can be estimated by means of Lemma 4.2 and the coordinates \(y_i = z_i  x_i + h/(2d),\; i=1,\dots , d1\). Since \(0< y_i < z_i+h/(2d)\simeq z_i + 1\) for each i and \(\sum _{1}^{d1} y_i < rh + (d1)h/(2d) \simeq r\), the result is \(E' \simeq \prod _1^{d1} (z_i+1) \wedge r\). This proves (4.9) (upper) when \(z_d \ge r\).
In the complementary case \(z_d<r\), we can suppress \(\wedge r\) in (4.9) (upper) because of (4.3). Define \(s,\sigma \in {\mathbb {R}}\) by
They satisfy \(0 \le s< \sigma < 1\), where the first inequality follows from (4.7), the second because \(h<2<z_d\) and the third from \(z_d < r\). Consider now the set
Clearly, any point \(x \in S\) satisfies
so for its last coordinate, (4.10) implies \(0<h<x_d<z_d\). Thus \(S \subset {\mathbb {R}}^d_+\), and (4.11) holds again, since for each \(i=1,\dots , d1\)
It follows that \(S \subset D_{b+h}\).
For the measures, we have
To finish the proof of (4.9) (upper), it is enough to verify that \(\sigma s \gtrsim 1\). But
the last inequality because of (4.3). Proposition 4.1 is proved. \(\square \)
4.2 Proof of Theorem 1(B), the case of \(\textbf{M}_{\mu }^{\mathcal {B}}\)
Our strategy of proving the \(L^p\)boundedness of \({M_{\mu }^{{\mathcal {B}}_+}}\) is heavily inspired by [10]. Thus we first rotate suitably the whole situation and then use a slicing argument together with \(L^p\)boundedness of certain standard maximal functions. The details are as follows.
Rotate simultaneously the cone \({\mathbb {R}}^d_+\) and all the objects considered (measure, truncated balls, etc.) so that the rotation of \(\Sigma _+^{d1}\) is orthogonal to the first coordinate axis and contained in the halfspace \(\{x \in {\mathbb {R}}^d: x_1 > 0\}\). Then denote by \(C_+\) the rotated open cone, in which the rotated measure is, up to a multiplicative constant and scaling,
Clearly, the above formula extends \(\nu \) from \(C_+\) to all of \({\mathbb {R}}^d\). We shall sometimes use this extension without explicit indication. Further, denote
Our aim is to prove that \(M_{\mu }^{{\mathcal {B}}_+}\) is bounded on \(L^p({\mathbb {R}}^d_+,d\mu )\) for \(1<p<\infty \). After rotation and scaling and keeping the same symbols, we consider \(M_{\mu }^{{\mathcal {B}}_+}\) as a maximal operator acting on functions living on \(C_+\), related to the family \(\widetilde{{\mathcal {B}}}_+\) of truncated Euclidean balls in \({\mathbb {R}}^d\) with centers in \(C_+\), the truncation being relative to \(C_+\). Then the \(L^p\)boundedness concerns \(L^p(C_+,d\nu )\).
Thus it is enough that we prove the \(L^p(d\nu )\)boundedness, \(1<p<\infty \), of the maximal operator
where the supremum is taken over all truncated balls
called simply balls henceforth, such that \(m \in C_+\) and \(x \in {\widetilde{B}}\). Further, we may assume that f is nonnegative and defined in all of \({\mathbb {R}}^d\) but supported in the closure of \(C_+\).
In what follows points in \({\mathbb {R}}^d\) will be written as \(x=(x_1,x')\). We shall always assume that the centers of balls \({\widetilde{B}}\) are in \(C_+\). Given \({\widetilde{B}}\), the minimum
(\({{\,\textrm{cl}\,}}\) meaning closure in \({\mathbb {R}}^d\)) is taken at a unique point \(a=a({\widetilde{B}})=(a_1,a') \in \partial {\widetilde{B}}\).
We now make some preliminary observations that lead to an essential reduction of the class of truncated balls over which the supremum in (4.12) is taken.
Observation 1
We may restrict to balls \({\widetilde{B}}(m,r)\) with radii uniformly bounded from below by a positive constant, see Reduction 1 above. In addition we may assume that \(a_1({\widetilde{B}}) > 2\), see Reduction 2.
Observation 2
We may further restrict to balls \({\widetilde{B}}(m,r)\) such that \(a=a({\widetilde{B}}) \in \partial C_+\). (In particular, we exclude untruncated balls \({\widetilde{B}} = {\textbf{B}}(m,r)\) entirely contained in \(C_+\).) Indeed, if \(a \notin \partial C_+\), i.e., a is in (the interior of) \(C_+\), then \(m_1 = a_1 + r\) and \(m'=a'\), and one considers the following two complementary cases.
If \(1 \lesssim \sqrt{r} < a_1\), then \(\nu ({\widetilde{B}}(m,r)) \simeq r^{(d1)/2}e^{a_1} \simeq \nu ({\textbf{B}}(m,r))\) (for the last relation, see the proof of Lemma 2.3) and the result is a simple consequence of [10, Theorem 3].
On the other hand, letting \(M_0\) be that part of the maximal operator M given by restricting the supremum in (4.12) to balls \({\widetilde{B}}(m,r)\) remaining after Observation 1 and such that \(a({\widetilde{B}}) \notin \partial C_+\) and \(a_1 \le \sqrt{r}\), we have the following.
Claim \(M_0\) is \(L^p(d\nu )\)bounded for \(1< p < \infty \).
To justify the Claim, notice that any \({\widetilde{B}}\) under consideration contains a cylinder parallel to the \(x_1\) axis, with one face contained in \(\pi _{a_1 + 1}\), of essentially unit width and radius comparable to \(a_1\), so \(\nu ({\widetilde{B}}) \gtrsim a_1^{d1}e^{a_1}\). Given that, consider the projections
of \(d\nu \) and f, respectively, on the \(x_1\) axis (here we omit multiplicative constants, which are irrelevant for the argument). Notice that \(\int f d\nu = \int F d\tau \). Thus we have
where \(I_{a_1} = (a_1,\infty )\). Now observe that the onedimensional maximal operator
(the supremum taken over all intervals \(I \subset {\mathbb {R}}_+\) such that \(s \in I\)) is of weak type (1, 1) with respect to the measure space \(({\mathbb {R}}_+,d\tau )\), and it controls \(M_0\). Therefore \(M_0\) is of weak type (1, 1) with respect to \((C_+,d\nu )\), and the \(L^p(d\nu )\)boundedness of \(M_0\) follows by interpolation with the \(L^{\infty }\)boundedness. This finishes proving the Claim and ends Observation 2.
Summing up, in the analysis of (4.12) we may assume that \({\widetilde{B}}={\widetilde{B}}(m,r)\) is a ball such that \(m \in C_+\) and
By convention, we define the supremum in (4.12) as zero if there is no admissible ball \({\widetilde{B}}\) containing x.
We shall first prove the result in the simplest situation when the dimension \(d=2\). This will give us some intuition needed for higher dimensions.
Dimension \(\varvec{d=2}\) When \(d=2\) we write points simply \(x=(x_1,x_2)\) rather than \(x=(x_1,x')\). Our rotated cone is \(C_+ = \{ x \in {\mathbb {R}}^2: x_2 < x_1\}\). We can assume that the balls \({\widetilde{B}}(m,r)\) under consideration are such that \(m_2 \ge 0\), by symmetry. Then \(a({\widetilde{B}}) = ({\mathfrak {a}},{\mathfrak {a}})\) with \({\mathfrak {a}}> 2\), and also \(r > \sqrt{2}\) and \(m_2 \ge {\mathfrak {a}}\); see (4.13). Notice that \(r/\sqrt{2} < m_1  {\mathfrak {a}}\le r\) and, of course,
See Fig. .
We shall now split into cases. In each case, we consider the maximal operator obtained by imposing some conditions on \({\widetilde{B}}\), in addition to (4.13).
Case 1 \({\widetilde{B}}\) contains the point \(({\mathfrak {a}}+1,0)\). Then \(\nu ({\widetilde{B}}) \gtrsim {\mathfrak {a}}e^{{\mathfrak {a}}}\), since \({\widetilde{B}}\) contains a rectangle of unit width and height \({\mathfrak {a}}\), with one of the vertical edges contained in \(\pi _{{\mathfrak {a}}+1}\). Thus the projection argument from Observation 2 gives the desired conclusion.
Case 2 \({\widetilde{B}}\) does not contain the point \(({\mathfrak {a}}+1,0)\). We first find the lower intersection of the line \(x_1 = {\mathfrak {a}}+ h\), \(0 < h \le r/\sqrt{2}\), with \(\partial {\textbf{B}}\), denoted \(({\mathfrak {a}}+h, {\mathfrak {a}}\xi )\); here \({\textbf{B}}\) is the untruncated prototype of \({\widetilde{B}}\) and \(\xi = \xi (h) > 0\). Notice that the condition defining Case 2 can be written as \(\xi (1) \le {\mathfrak {a}}\).
We have
Subtracting (4.14) from this equation leads to
Dividing by \(\xi ^2\), solving for \(1/\xi \) and taking into account that \(\xi > 0\), we get
Note that \(h < m_1  {\mathfrak {a}}\) (recall that \(r/\sqrt{2} \le m_1{\mathfrak {a}}< r\)). Then \(2(m_1{\mathfrak {a}})h  h^2 \simeq (m_1{\mathfrak {a}})h \simeq rh\). Consequently,
To estimate \(\nu ({\widetilde{B}})\) from below, observe that \({\widetilde{B}}\) contains the triangle T whose vertices are \(({\mathfrak {a}}, {\mathfrak {a}})\), \(({\mathfrak {a}}+1, {\mathfrak {a}}\xi (1))\) and \(({\mathfrak {a}}+1, {\mathfrak {a}})\), and \(\nu (T)\simeq \xi (1) e^{{\mathfrak {a}}}\). Thus (4.15) implies
Next, we consider all \(h > 0\) and estimate from above the measures of the intersections \(\pi _{{\mathfrak {a}}+h} \cap {{\,\textrm{shw}\,}}{\widetilde{B}}\), where
is the shadow of \({\widetilde{B}}\) in the positive \(x_1\) direction. By the geometry of the situation and (4.15), observing also that \(m_2  {\mathfrak {a}}< r/\sqrt{2}\), we have
Using this together with (4.16), by an elementary analysis of cases we see that
uniformly in \({\mathfrak {a}}\) and \({\widetilde{B}}\).
Now, let \(M_2\) be the part of the maximal operator (4.12) under consideration, i.e., with the supremum taken only over balls \({\widetilde{B}}\) considered in Case 2. We will apply the slicing argument from [10].
Similarly as in [10], consider the unit slices
In \(S_i\) one has \(e^{i1}dx \le d\nu (x) \le e^{i}dx\). Let
Since \(M_2 f \le \sum _{k \in {\mathbb {Z}}} M_2^kf\), it is enough to prove that \(\Vert M_2^kf\Vert _{L^p(d\nu )} \lesssim 2^{\delta k/p} \Vert f\Vert _{L^p(d\nu )}\) with some \(\delta > 0\), because then one can sum the estimates and get the conclusion. Thus we must show that
With \(i,j \ge 1\), we let \(x \in S_j\) and \({\widetilde{B}}\) be a ball containing x, and we will estimate first the mean
In our situation \(x \in {\widetilde{B}} \cap S_j\) and \(y \in {\widetilde{B}} \cap S_i\). Observing that the sets \(\{z_2 \in {\mathbb {R}}: \exists z_1 \; (z_1,z_2) \in \pi _{{\mathfrak {a}}+h} \cap {{\,\textrm{shw}\,}}{\widetilde{B}}\}\) form an increasing family of intervals with respect to \(h > 0\), we get
notice that here \(i,j \ge 1 \vee ({\mathfrak {a}}1) = {\mathfrak {a}}1\). Then, using (4.17), we obtain
where the implicit multiplicative constant is independent of \(i,j \ge {\mathfrak {a}}1\), the ball \({\widetilde{B}}\) and the point \(x \in {\widetilde{B}} \cap S_j\), and of f. Here \({\mathcal {M}}\) is the onedimensional noncentered Hardy–Littlewood maximal function acting on the second coordinate. Note that \({\mathcal {M}}\) is bounded on \(L^p({\mathbb {R}},dx_2)\) for \(p > 1\).
We now estimate the factor in front of the integral in (4.19). Write
where the last inequality follows from the bound \(i \wedge j \ge {\mathfrak {a}}1\). Thus
with \(\epsilon = \frac{1}{2}(\frac{1}{p} \wedge \frac{1}{p'})\), uniformly in \({\mathfrak {a}}> 2\) and \(i,j \ge {\mathfrak {a}}1\).
With the bound just obtained, taking the supremum of the lefthand side of (4.19) and using Hölder’s inequality on the righthand side there, we arrive at
Raising to power p and integrating this estimate in \(x=(x_1,x_2) \in S_j\) we further get
Finally, we use the \(L^p\)boundedness of \({\mathcal {M}}\) to write
and (4.18) with \(\delta = \epsilon p\) follows. This finishes the proof in the case of dimension \(d=2\).
Remark
Cases 1 and 2 considered above can be merged. Indeed, right after (4.14) one can estimate \(\xi (h)\), as it was done in Case 2, getting (4.15). Then it follows that
Further, we can estimate measures of the intersections \(\pi _{{\mathfrak {a}}+h} \cap {{\,\textrm{shw}\,}}{\widetilde{B}}\) as (observe that the expression \(2({\mathfrak {a}}+ h)\) appears as the measure of \(C_{+} \cap \pi _{{\mathfrak {a}}+h}\))
Using this together with an elementary analysis of cases we get the key bound
uniformly in \({\mathfrak {a}}\) and \({\widetilde{B}}\). From here the slicing argument goes as described in Case 2 above.
Dimension \(\varvec{d=3}\) From now on we will write points \(x=(x_1,x')\), with \(x_1 > 0\) and \(x' \in {\mathbb {R}}^2\). Our fixed rotated cone \(C_+\) is contained in \({\mathbb {R}}_+ \times {\mathbb {R}}^2\), its vertex is the origin of \({\mathbb {R}}^3\), and its central axis is the \(Ox_1\) axis. For any \(\xi >0\), the intersection \(C_+ \cap \pi _{\xi }\) is an open equilateral triangle of side \(\sqrt{6}\,\xi \).
In order to estimate Mf defined in (4.12), we let \({\widetilde{B}} = {\widetilde{B}}(m,r)\) be a truncated ball with \(m \in C_+\) verifying (4.13).
For any set \(E \subset {\mathbb {R}}^3\), we define its shadow in the direction of the \(x_1\) axis as
We claim that
where only the second inequality needs to be verified. For this we fix \(m_1\) and \(a_1\) and use Fig. .
Each part of this figure shows the triangles \(\pi _{a_1} \cap \partial C_+\) and \(\pi _{m_1} \cap \partial C_+\), and inside the latter the triangle \(\pi _{m_1} \cap {{\,\textrm{shw}\,}}(\pi _{a_1} \cap \partial C_+)\). Notice that the point m cannot be in the interior of this last triangle, since a is on the boundary of \(C_+\). Given the position of a, the figure illustrates the possible positions of m. To the left, a is on an open face of the cone \(C_+\), and then m is seen to be in the short, closed segment indicated. In the righthand part of the figure, a is on an edge of \(C_+\), and m has to belong to the closed quadrilateral marked in the figure. From this, we see that the minimal value of the quotient \((m_1  a_1)/r = (m_1  a_1)/m  a\) occurs when a and m are situated on the same edge of \(C_+\), and then the quotient equals \(1/\sqrt{3}\). We have verified the claim (4.20).
For \(0 \le h < r+(m_1a_1)\) we define
with \({\textbf{B}}= {\textbf{B}}(m,r)\). Observe that \(B_h\) would be empty if defined in this way for \(h \ge r+(m_1a_1)\).
Case I \(a({\widetilde{B}})\) lies on an edge of \(C_+\).
We intersect \(C_+\) and \({\textbf{B}}\) with \(\pi _{a_1}\), see Fig. .
Then \(C_0\) is an equilateral triangle with one vertex at \(a'\), and \(B_0\) is an open disc with center \(m'\) and radius R satisfying
The definition of a implies that \(a' \in \partial B_0 \cap \partial C_0\) and also that the tangent line, denoted t, to \(B_0\) through \(a'\) does not intersect \(C_0\). Thus \(R = a'm'\).
The point \(a'\) is the endpoint of two edges of \(C_0\), and we consider the angles at \(a'\) between t and these two edges. Let \(\beta \) denote the smallest such angle and let \(e_1\) be the corresponding edge. Then \(0 \le \beta \le \pi /3\), and the other edge \(e_2\) forms the angle of \(\beta + \pi /3\) with the same tangent.
We now consider the intersection of \({\textbf{B}}\) and \({{\,\textrm{shw}\,}}(C_+ \cap \pi _{a_1})\) with the plane \(\pi _{a_1+h}\), assuming that \(0 < h \le r/\sqrt{3}\); see Fig. . Then \(a'\) is an inner point of the disc \(B_h\).
From \(a'\) we move first along the edge \(e_1\) and then possibly continue beyond it in the same direction until we hit \(\partial B_h\), say at distance \(p_h\) from \(a'\). Then
Subtracting (4.21), we get
We rewrite this as a quadratic equation in \(1/p_h\) which we solve, getting
Since \(h \le r/\sqrt{3} < m_1a_1 \le r\), we see that
We next repeat the above, but moving in the direction of \(e_2\) until we leave \(B_h\), after covering a distance \(q_h\), say. The same argument applies, but instead of \(\beta \) we have \(\beta + \pi /3 \in [\pi /3,2\pi /3]\). The result is
We now estimate the measure of \({\widetilde{B}}\) from below. Consider for the time being only \(h \in (1/2,1)\). Then \(p_h \simeq p_1\) and \(q_h \simeq q_1\). Thus we can find one point on each edge \(e_1\) and \(e_2\) belonging to the closure of \(B_h \cap C_h\) whose distance from \(a'\) is comparable to \(a_1 \wedge p_1\) and \(a_1 \wedge q_1\), respectively (recall that \(e_1=e_2\simeq a_1\)). The triangle formed by these two points and \(a'\) is also contained in \(B_h \cap C_h\) by convexity, and its area is comparable to \((a_1 \wedge p_1)(a_1\wedge q_1)\). Integrating over \(1/2< h < 1\), we see that
Next, we consider all \(h \in (0, r+(m_1a_1))\). We shall need the following.
Proposition 4.3
There is an increasing family of parallelograms \(\{{\mathcal {P}}_h: 0< h < r+(m_1a_1)\}\) in \({\mathbb {R}}^2\) with sides parallel to \(e_1\) and \(e_2\) and side lengths controlled (up to multiplicative absolute constants) by \((a_1 + h) \wedge p_h\) and \((a_1 + h) \wedge q_h\), respectively, in case \(0 < h \le r/3\), and by r in case \(h > r/3\), such that \(B_h \cap C_h \subset {\mathcal {P}}_h\) for \(0< h < r+(m_1a_1)\).
Proof
Consider first \(h \le r/3\). The triangle \(C_h\) is a concentric scaling of \(C_0\), and all its points have a distance of at most \(\sqrt{2}\,h\) from \(C_0\). In particular, \(C_h\) has a vertex \(a_h'\), corresponding to \(a'\), which is at the distance \(\sqrt{2}\,h \sin (\beta +\pi /6)\) from t, and at the distance \(\sqrt{2}\,h \cos (\beta + \pi /6)\) from the line perpendicular to t and passing through \(a'\) (and \(m'\)); see Fig. .
Bring in the “vertical coordinate”
in the plane \(\pi _{a_1+h}\).
We take as \({\mathcal {P}}_h\) the smallest open parallelogram having one vertex at \(a'_h\), sides parallel to \(e_1\) and \(e_2\), and containing \(B_h \cap C_h\). Then we must show that the side lengths of \({\mathcal {P}}_h\) are controlled by \((a_1+h) \wedge p_h\) and \((a_1+h) \wedge q_h\). We shall separate the cases when \(a'_h\) lies below or above the level of \(m'\), see Fig. 7.
Assume first that \(\tau (a'_h) \le 0\), i.e., \(a'_h\) does not exceed the level of \(m'\). This means that \(\sqrt{2}\,h \sin (\beta +\pi /6) \ge a'  m' = R\), which implies \(h \ge R/\sqrt{2}\). In this situation, see (4.23) and (4.24), one has \(p_h \simeq \sqrt{rh} \simeq q_h\). On the other hand, the radius of \(B_h\) is also comparable with \(R+ q_h \simeq \sqrt{rh}\).
Now, observe that the edges of \({\mathcal {P}}_h\) have lengths controlled by the quantity
thus by \((a_1+h)\wedge \sqrt{rh} \simeq (a_1+h)\wedge p_h \simeq (a_1+h)\wedge q_h\), as desired.
Next, assume that \(\tau (a'_h) > 0\), i.e., \(a'_h\) is above the level of \(m'\). We shall construct a parallelogram \({\mathcal {P}}_h^*\) containing \({\mathcal {P}}_h\), having vertex at \(a'_h\) and sides parallel to \(e_1\) and \(e_2\), whose side lengths satisfy the desired estimates. Clearly, this will be enough for our purpose.
In the plane \(\pi _{a_1+h}\), let w be the line through \(a'_h\) parallel to t. Define u to be the line parallel to w given by \(u = \{x': \tau (x') = (\sup _{B_h}\tau ) \wedge (\sup _{C_h}\tau )\}\). Observe that two cases may occur (call them (a) and (b), respectively): u is tangent to \(B_h\) (if the last minimum is realized by \(\sup _{B_h}\tau \), see Fig. 7) or u passes through the vertex of \(C_h\) of maximal distance from w.
The intersection \(B_h \cap C_h\) is contained in the band between w and u. In case (a), the width of this band, see Fig. 7, is not larger than (actually comparable with) \(q_h + \sqrt{2}\,h \sin (\beta +\pi /6)\), and this quantity, in view of (4.24), is comparable to \(q_h\). In case (b), the width of this band is comparable with the side length of \(C_h\), i.e., with \(a_1 + h\).
Now consider the segment along the \(e_2\) direction with one endpoint at \(a_h'\), and whose other endpoint lies on u. Since \(e_2\) forms the angle \(\beta +\pi /3\) with u, which is separated from 0, the segment in question has length comparable with the width of the band, thus with \((a_1+h) \wedge q_h\). We take this segment as a side of our \({\mathcal {P}}_h^*\).
As the other side of \({\mathcal {P}}_h^*\) we shall take the segment along the \(e_1\) direction with one endpoint at \(a_h'\), and the other endpoint \(b_h'\) lies either on the boundary of \(B_h\), inside the band, or is the vertex of \(C_h\) in case \(B_h\) is so large that \(\partial B_h\) does not cross this (\(e_1\)directed) side of \(C_h\). See again Fig. 7. Denote by \({\mathfrak {p}}_h\) the length of this segment. Clearly, \({\mathfrak {p}}_h\) is comparable with \(a_1+h\) in case \(b_h'\) is the vertex. Assuming the other case \(b_h' \in \partial B_h\), we will show that \({\mathfrak {p}}_h\) is comparable to \(p_h\), a fact that is intuitively clear from the picture. Since \({\mathcal {P}}_h^*\) just defined contains^{Footnote 2}\(B_h \cap C_h\), this will finish the reasoning when \(h\le r/3\).
Observe that, cf. (4.22),
Subtracting (4.21) and solving for \(1/{\mathfrak {p}}_h\) (see the analysis leading from (4.22) to (4.23)), we get after some elementary computations and applications of basic trigonometric identities
Then, recalling that \(r/\sqrt{3} < m_1a_1 \le r\), \(R < r\) and \(h \le r/3\), we arrive at
Considering \(h > r /3\), take as \({\mathcal {P}}_h\) the smallest (open) parallelogram, with sides parallel to \(e_1\) and \(e_2\), containing both \({\mathcal {P}}_{r/3}\) and \(B_{m_1a_1}\). This parallelogram has side lengths comparable to r, by the geometry of the situation.
The fact that the family \(\{{\mathcal {P}}_h: h > 0\}\) is increasing is clear from the construction. Proposition 4.3 follows. \(\square \)
In view of (4.25), for \({\mathcal {P}}_h\) from Proposition 4.3 we have the bound
To estimate the righthand side here we use (4.23) and (4.24), and apply an elementary analysis of cases. Considering \(h \le r/3\), if \(a_1 \wedge p_1 = a_1\), then (recall that \(a_1 > 2\))
if \(a_1 \wedge p_1 = p_1\), then
For \(h > r/3\) we have
The factors involving \(q_h\) are treated similarly. Thus we arrive at the key bound
uniformly in \(a_1\) and \({\widetilde{B}}\).
We are now in a position to apply the slicing argument. Let M be the part of the maximal operator (4.12) under consideration. As in dimension 2, we define \(S_{i}\) for \(i \ge 1\) as \(\{x \in C_+: i < x_1 \le i+1\}\), and in \(S_i\), \(e^{i1}dx \le d\nu (x) < e^{i}dx\). It is enough to prove that for some constant \(\delta > 0\)
see (4.18) and the preceding comments.
To prove (4.28), let \(i,j \ge 1\). Let \(x \in {\widetilde{B}} \cap S_j\) and \(y \in {\widetilde{B}} \cap S_i\). Proposition 4.3 tells us that \(x'\) and \(y'\) are contained in a certain parallelogram \({\mathcal {P}}_h\), and both parallelograms are contained in the one given by Proposition 4.3 with \(h = (ia_1 + 1) \vee (ja_1+1)\); notice that here \(i,j \ge {\mathfrak {a}}1\). Define
for any locally integrable function g in \({\mathbb {R}}^2\), where the supremum is taken over all parallelograms \({\mathcal {P}}\) containing z and with sides parallel to two sides of the triangle \(C_0\), and \({\mathcal {P}}\) denotes the area of \({\mathcal {P}}\). Then we can write the estimate
Note that \({\mathcal {M}}'\) is bounded on \(L^p({\mathbb {R}}^2)\) for \(1< p < \infty \). Indeed, \({\mathcal {M}}'\) splits naturally into three components, each determined by two edges of \(C_0\). Then a linear transformation makes each component coincide with the strong maximal operator \(M_{\textrm{str}}\) in \({\mathbb {R}}^2\).
Combining (4.29) with (4.27) we obtain
This is an analogue of (4.19). From here one proceeds as before, arguing as done after (4.19), getting \(L^p(d\nu )\)boundedness of the considered part of our maximal operator.
Case II \(a({\widetilde{B}})\) lies on a face of \(C_+\).
Then \(a'\) is an inner point of a side of the triangle \(C_0\); see Fig. .
We split \(C_0\) into its intersections with three twodimensional cones, by introducing two rays from \(a'\) forming angles of \(\pi /3\) with the side of \(C_0\). Then we apply the arguments from Case I, using instead of \(C_0\) each of these three intersections, with \(\beta =0\) twice and with \(\beta = \pi /3\) once, as seen in Fig. 8. That intersection which has \(\beta = \pi /3\) is not a triangle but a parallelogram. But notice that the hexpansion of this parallelogram, analogous to \(C_h\) in Case I, will necessarily be contained in the analog of the parallelogram \({\mathcal {P}}_h\) constructed in Proposition 4.3. To get the lower estimate (4.25), it is enough to argue as in Case I for the larger of the two intersections with \(\beta =0\). In each of the three intersections, we can now follow the pattern of Case I for all upper estimates of integrals, and divide by \(\nu ({\widetilde{B}})\).
This ends the case of dimension 3.
Dimension \(\varvec{d=4}\) We largely follow the threedimensional argument. Recall that
The assumptions (4.13) remain in force. As in dimension three, we define for \(0 \le h < r + (m_1  a_1)\)
which is an open regular tetrahedron of edge \((a_1+h)\,\sqrt{8}\), and
Observe that (4.13) implies \(a' \in \partial C_0\). The radius of the ball \(B_h\) will be denoted by \(R_h\), and as before we write R for \(R_0\).
In \(\pi _{a_1}\), which we identify with \({\mathbb {R}}^3\), we now let T be the tangent plane of the ball \(B_0\) passing through \(a'\). Moreover, \(T_+\) will denote that closed halfspace in \(\pi _{a_1}\) whose boundary is T and which contains \(C_0\).
Instead of (4.20), we now have
The equality (4.21) remains valid and implies
When \(h<r\), we similarly get for \(R_h\) in view of (4.31)
We also have
the last step by (4.31).
The “vertical coordinate” \(\tau \) in \({\mathbb {R}}^3\) is defined by (4.26), as in the threedimensional case.
We will need some angles connected with a regular tetrahedron. The angle at a vertex between an edge and the axis of symmetry from that vertex is \(\gamma \), where \(\sin \gamma = 1/\sqrt{3}\), and the angle between two faces of the tetrahedron is \(2\gamma \). Further, the angle between a face and an edge not in that face is \(\kappa \), where \(\sin \kappa = \sqrt{2/3}\). Using this last angle, one finds that the ratio between the height and the edge of the tetrahedron is \( \sqrt{2/3} > 1/2\); the height is the distance between a vertex and the opposite face.
Case I \(a'\) is a vertex of \(C_0\).
In \({\mathbb {R}}^3\), the point \(a'\) is now an endpoint of three edges \(e_1\), \(e_2\) and \(e_3\) of the tetrahedron \(C_0\). Let \(\beta _i,\; i= 1,2,3\), denote the angle at \(a'\) between \(e_i\) and the plane T. Then \(0 \le \beta _i \le \pi /2\), and at most two of the \(\beta _i\) can be small.
Clearly \(a'\) is an inner point of the ball \(B_h\) when \( 0< h < r + (m_1  a_1)\). We consider for \(i= 1,2,3\) the intersection of \(B_h\) and the ray in the direction of \(e_i\) emanating from \(a'\). Let \(p_h^{i}\) be the length of this intersection. We can determine the \(p_h^{i}\) exactly like \(p_h\) in dimension three, and instead of (4.23) we get for \(0< h< r/2\)
The argument leading to (4.25) also carries over, so that
As before, \(a'_h\) denotes the vertex of \(C_h\) that corresponds to \(a'\); one finds that the distance from \(a'\) to \(a'_h\) is \(\sqrt{3}\,h\).
Let \({\mathcal {P}}_h \subset {\mathbb {R}}^3\) for \(0 < h \le r/2\) be the minimal parallelepiped containing \(B_h \cap C_h\) which has one vertex at \(a'_h\) and edges parallel to \(e_1\), \(e_2\) and \(e_3\). Then \({\mathcal {P}}_h\) increases with h.
Proposition 4.4
For \(0 < h \le r/2\), the edges of \({\mathcal {P}}_h\) are bounded by constant times \((a_1+h) \wedge p_h^{i}\), \(i= 1,2,3\).
To prove this, we fix \(h \in (0, r/2)\) and deal first with the simple case when \( h \ge c_0 R\), for some small constant \(c_0 > 0\) to be determined. Then the \(p_h^{i}\) are all of magnitude \(\sqrt{rh}\), and (4.34) implies
the last step since here \(R \lesssim h\). Thus \(R_h \lesssim \sqrt{rh}\).
Comparing the sides of \({\mathcal {P}}_h\) with the minimum of \(R_h\) and the edge of \(C_h\), we arrive at the conclusion of the proposition, when \(h \ge c_0 \,R\).
Consider now the remaining case \(0< h < c_0 R\), and observe that then \(a'_h \in B_h.\) Let \(i\in \{ 1,2,3\}\). We define \(\rho _i\) as the ray parallel with \(e_i\), with endpoint at \(a'_h\) and contained in the halfspace \( \{x': \tau (x') \ge \tau (a'_h) \} \). If \(\sin \beta _i \ge 1/32\), we denote by \(b'_i\) the point of intersection of \(\rho _i\) and \(\partial H\), where H is the halfspace
When \(\sin \beta _i < 1/32\), we define \(b'_i\) similarly, but now with the intersection point of \(\rho _i\) and \(\partial {B_h}\). Finally, let \(v_i\) be the vector \(b'_i  a'_h\), which is parallel with \(e_i\). See Fig. .
Define now
a parallelepiped with one vertex at \(a'_h\) and edge lengths \(v_i\). It is increasing in h.
We will need the following two lemmas, whose proofs are given after the end of the proof of Proposition 4.4.
Lemma 4.5
If \(0< h<c_0R\) with \(c_0\) small enough, then for \(i= 1,2,3\)
and if moreover \(\sin \beta _i < 1/32\), then
Lemma 4.6
If \(0< h<c_0R\) with \(c_0\) small enough, then
Given these lemmas, and still assuming \(0< h < c_0 R\), let \( {\mathcal {P}}''_h\) be the minimal parallelepiped with one vertex at \(a'_h\) that contains \(C_h\). Then the parallelepiped \({\mathcal {P}}^*_h = {\mathcal {P}}'_h \cap {\mathcal {P}}''_h\) will contain \(B_h \cap C_h\) because of Lemma 4.6. From Lemma 4.5 and the fact that the edges of \( {\mathcal {P}}''_h\) are of order of magnitude \(a_1+h\), it follows that the edges of \({\mathcal {P}}^*_h\) are as stated in Proposition 4.4. The minimality of \({\mathcal {P}}_h\) shows that \({\mathcal {P}}_h \subset {\mathcal {P}}^*_h\), and this concludes the proof of Proposition 4.4.
In the proofs of the two lemmas, we will denote by \(\omega \) the angle at \(a'\) between the central axis of \(C_0\) emanating from \(a'\) and the plane T. Notice that \(\omega \gtrsim 1\), since \(\omega \) is at least as large as the angle between the central axis and a face of \(C_0\).
Proof of Lemma 4.5
Consider first the case \(\sin \beta _i \ge 1/32\). The vertical distance \(\tau (b'_i)  \tau (a'_h)\) is \(R_h  R + \sqrt{3}\, h \sin \omega \), see Fig. 9. This gives an expression for \(v_i\), and then we use in turn (4.34), (4.33), (4.32) and then (4.35). As a result,
In the opposite case \(\sin \beta _i < 1/32\), the quantity \(v_i\) is the length of a segment from \(a'_h\) to a point on \(\partial B_h\). The segment forms an angle \(\beta _i\) with the plane \(W = \{x': \tau (x') = \tau (a'_h)\}\) (and is on the same side of W as the point \(a'\)), as seen in Fig. 9.
Project this segment and also the central axis of \(C_h\) starting at \(a'_h\) orthogonally onto the plane W. Let \(\theta \) denote the angle between these two projections at their common point \(a'_h\).
Since the endpoint of the segment is on \(\partial B_h\), the following equation will have the positive solution \(z = v_i\), and also a negative solution. We temporarily write \(\ell = R  \sqrt{3} \,h\sin \omega \), which is the vertical distance between \(m'\) and \(a'_h\). The equation is
or simplified
where \(K = \ell \sin \beta _i  \sqrt{3} \,h \cos \omega \cos \beta _i \cos \theta \) and \(L =\ell ^2 + 3\,h^2\cos ^2 \omega  R_h^2\). We consider this equation for all \(\theta \in [0, \pi ]\). Since the two roots of the equation have opposite signs, the constant term L is negative, which can also be seen geometrically. Let us now vary only \(\theta \), and write the positive solution as \(z = z(\theta )\). Differentiating the equation with respect to \(\theta \), we get
Since z is the positive solution of the equation, \(z+K\) equals the square root that appears in the wellknown formula for the solutions, so it is positive. Thus \(dz/d\theta < 0\) for \(0< \theta < \pi \). It follows that the minimal and maximal values of \(z(\theta )\) are \(z(\pi )\) and z(0), respectively, so that \(z(\pi ) \le v_i \le z(0)\). We now rewrite the equation with these two values of \(\theta \), and replace \(K,\,L\) and also \(\ell \) by their explicit expressions. Using some elementary trigonometry, one obtains the result
where the ± signs should be read as plus for \(z(\pi )\) and minus for z(0). We now solve this equation for 1/z, denoting
The positive solution z is given by
We estimate the numerator and the denominator in (4.37) from above and below, choosing \(c_0\) small enough whenever needed. Because of (4.34), we find
and
where we also used (4.20). Further, (4.38) implies that
From (4.39), we obtain
These four inequalities hold whether the ± signs are read as plus or minus.
Combining (4.38) and (4.41) with (4.37), we conclude that
because of (4.35). If \(\sin \beta _i < 1/32\), (4.39) and (4.40) similarly yield
The last two formulas end the proof of Lemma 4.5. \(\square \)
Proof of Lemma 4.6
Any point \(x \in C_h\) can be written \(x = a'_h + \sum _1^3 \alpha _j v_j\) with \(\alpha _j\ge 0\). Assume now that \(x \in B_h \cap C_h\). We will show that \(\alpha _j \le 1\) for each j, so that \(x \in {\mathcal {P}}'_h\). Thus we fix \(j\in \{ 1,2,3\}\). Observe that to prove the inequality \(\alpha _j \le 1\), we may assume that \(\alpha _j \ge 1\), since the opposite case is clear.
Since \(x= a'_h + \sum _1^3 \alpha _i v_i \in H\) and the function \(\alpha _i \mapsto \tau (a'_h + \sum _1^3 \alpha _i v_i)\) is nondecreasing for each i, the point \( a'_h +\alpha _j v_j\) is also in H. If \(\sin \beta _j \ge 1/32\), this implies \(\alpha _j \le 1\), by the definitions of \(v_j\) and \(b'_h\).
When instead \(\sin \beta _j < 1/32\), we will similarly show that \(\alpha _j \le 1\) by proving that \( a'_h +\alpha _j v_j \in B_h\). We know that \(a'_h + \sum _1^3 \alpha _j v_j\in B_h\), so it is enough to verify that the distance \(a'_h + \sum _1^3 \alpha _i v_im'\) is increasing in \(\alpha _i\) for \(i \ne j\). But
and here all the terms to the right except possibly the second one are nondecreasing in \(\alpha _i\). Further, \(\langle v_i, v_k \rangle = v_i v_k/2\). Consider for \(i \ne j\) the following two terms from the righthand side
Now
so (4.42) equals
It is enough to verify that the three terms in this parenthesis have a positive sum. The middle term is positive, since \(c_0\) is small. Recall that we assumed \(\alpha _j \ge 1\) and also \(\sin \beta _j < 1/32\) which implies \( v_j \ge 4h \) because of Lemma 4.5. Thus the first term in the above parenthesis is dominated by the third term, the parenthesis is positive and the expression in (4.42) is increasing in \(\alpha _i\), as desired. Lemma 4.6 is proved. \(\square \)
We can now continue Case I as in three dimensions, but using the three quantities \({p}_h^{i}\) instead of \(p_h\) and \(q_h\). In the estimate (4.27) the exponent of h will be 3 instead of 2. We extend the definition of \({\mathcal {P}}_h\) by setting it equal to the smallest parallelepiped containing \({\mathcal {P}}_{r/2} \cap B_{m_1a_1}\) for \(r/2< h < r+(m_1a_1)\); cf. the end of the proof of Proposition 4.3 in the threedimensional case. We leave the details finishing Case I to the reader.
Case II \(a'\) is an inner point of a face of \(C_0\).
This face of \(C_0\) is contained in the plane T, and we consider the three translates \(C_0^{j},\, j= 1,2,3\), of \(C_0\) along T which have a vertex at \(a'\) (see Fig. , where for clarity only that face of \(C_0\) contained in T is marked).
The angles at \(a'\) between T and the edges of each \(C_0^{j}\) are now \(0,\;0,\;\kappa \). The dilations \(C_h\) are given by (4.30), and we can define for \(0 \le h < r + (m_1  a_1)\) analogous dilations \(C_h^{j},\;j = 1,2, 3\), of the \(C_0^{j}\) by replacing in (4.30) \(C_+\) by the fourdimensional cone generated by \(C_0^{j} \times \{a_1\}\) and the origin. In analogy with the beginning of Case I, we consider for each j the intersection with \(B_h\) of the three rays emanating from \(a'\) and containing an edge of \(C_h^{j}\). As in Case I, we write \(p_h^{i},\, i= 1,2,3\), for the lengths of these intersections. The \(p_h^{i}\) will not depend on j, and from (4.35) we see that their orders of magnitude are
At least one of the intersections \(C_0\cap C_0^{j},\, j= 1,2,3\), is comparable in volume to \(C_0\). To estimate the measure of \({{\widetilde{B}}}\) from below, we can thus for one value of j argue as in Case I with \(C_0^{j}\) and \(C_h^{j}\). Hence we still have the lower estimate (4.36). The corresponding upper estimate will now be verified.
In addition to \(C_0^{j},\, j= 1,2,3\), we will consider a finite number of tetrahedra \(C_0^{j},\,j=4,\dots , N\), of the same size. They will all have a vertex at \(a'\) and be contained in \(T_+\). We select them so that the \(C_0^{j},\,j=1,\dots N\), together cover a neighborhood of \(a'\) in \(T_+\). Here N will be an absolute constant. Of the three angles at \(a'\) between the plane T and an edge of any \(C_0^{j},\;j = 4,\dots , N\), at least one must stay away from 0, since \(C_0^{j} \subset T_+\). (In fact, the largest of these three angles is at least \(\pi /4\).) This implies that the corresponding lengths \(p_h^{i,j}\) (which will now depend also on j) have orders of magnitude no larger than those in (4.43).
By \(C_0^{j,2}\) we denote the result of a scaling of \(C_0^{j}\) centered at \(a'\) by a factor of 2. Thus \(a'\) is a vertex also of \(C_0^{j,2}\). The \(C_0^{j,2},\;j = 1,\dots , N\), will together contain the intersection of \(T_+\) and the ball of center \(a'\) and radius equal to the height of \(C_0^{j,2}\). Since this height is larger than the diameter, i.e., the edge, of \(C_0^j\), we conclude that
The arguments from Case I will apply to each scaled tetrahedron \(C_h^{j,2}\). In particular, we choose as there minimal parallelepipeds \({\mathcal {P}}_h^j\) containing \(B_h \cap C_h^{j,2},\;\;j = 1,\dots , N\), which together cover \(B_h \cap C_h\). The proofs of Lemmas 4.5 and 4.6 and then also that of Proposition 4.4 will go through for each \({\mathcal {P}}_h^j\), and this allows us to conclude Case II like Case I.
Case III \(a'\) is an inner point of an edge of \(C_0\).
This edge of \(C_0\) will be called \(e_0\). It is contained in T, and it is the intersection of two faces of \(C_0\). We denote by \(\Pi '\) and \(\Pi ''\) the planes containing these faces. The angle between \(\Pi '\) and \(\Pi ''\) is \(2\gamma \).
Consider the translates \(C_0^{1}\) and \(C_0^{2}\) of \(C_0\) along \(e_0\) which have one vertex at \(a'\) (see Fig. ). Both \(C_0^{1}\) and \(C_0^{2}\) have three edges with endpoint \(a'\): one in \(\Pi ' \cap \Pi ''\), one in \(\Pi '\) and one in \(\Pi ''\). These edges form angles with the plane T which are \(\beta _1 = 0\), \(\beta _2 > 0\) and \(\beta _3 > 0\). For \(h \in (0, r+(m_1a_1))\) we have dilations \(C_h\), \(C_h^{1}\) and \(C_h^{2}\) of \(C_0\), \(C_0^{1}\) and \(C_0^{2}\), where the latter two dilations are constructed as in Case II.
Following Case II, we introduce rays emanating from \(a'\) along the three edges of \(C_0^{1}\) and \(C_0^{2}\) and segments of lengths \(p_h^i,\, i = 1,2,3\). These \(p_h^i\) will satisfy (4.35). At least one of the intersections \(C_0 \cap C_0^{1}\) and \(C_0 \cap C_0^{2}\) must have volume comparable to that of \(C_0\). The argument leading to (4.25) can be applied to the corresponding \(C_0^{j}\); cf. (4.36). This gives the necessary lower estimate for \(\nu ({{\tilde{B}}})\).
To get the corresponding upper estimate, we follow the pattern of Case II. We will cover \(C_0\) by a finite number of (doubled) tetrahedra having one vertex at \(a'\), among them \(C_0^{1}\) and \(C_0^{2}\) doubled. This is done as follows.
Consider the wedge defined as that component of \({\mathbb {R}}^3\setminus \left( \Pi ' \cup \Pi '' \right) \) which contains \(C_0\). There is then a halfplane that splits this wedge in two congruent wedges denoted \(V'\) and \(V''\); of these \(V'\) shall be the one with boundary along \(\Pi '\).
We will next rotate \(C_0^{1}\), using as rotation axis the normal through \(a'\) of the plane \(\Pi '\). The rotation angle will go from 0 to \(2\pi /3\); the angle \(2\pi /3\) will bring \(C_0^{1}\) to \(C_0^{2}\). During this rotation, the edge of \(C_0^{1}\) from \(a'\) in \(\Pi ' \cap \Pi ''\) and that in \(\Pi '\) will both stay in \(\Pi '\). The edge from \(a'\) which is in \(\Pi ''\) before the rotation will describe a conic surface, and its angle with \(\Pi ''\) will be positive and increase until it reaches a maximum at the rotation angle \(\pi /3\). Then it will decrease back to 0. This maximum is seen to be \(2\gamma  \kappa \), and one has \(0< 2\gamma  \kappa < \gamma \), the last inequality since \(\kappa > \gamma \).
This implies that the rotations of \(C_0^{1}\) considered will together cover the intersection of \(V'\) with a neighborhood of \(a'\). We can then select a finite number of these rotated tetrahedra, say \(C_0^{j},\, j= 1,\dots N\), which together also cover a neighborhood of \(a'\) in \(V'\). Notice that \(C_0^{1}\) and \(C_0^{2}\) are included here. As in Case II, we consider the doubled tetrahedra \(C_0^{j,2}\) with a vertex at \(a'\) and conclude that
To deal similarly with \(V''\), we repeat the rotation procedure, swapping \(\Pi '\) and \(\Pi ''\) as well as \(V'\) and \(V''\).
The result will be that we cover \(C_0\) by a finite number of tetrahedra, each having a vertex at \(a'\). The edges of these tetrahedra will have angles with T which are larger than or equal to \(\beta _1\), \(\beta _2\) and \(\beta _3 \), respectively. This makes it possible to argue as in Cases I and II, considering dilations \(C_h\) and \(C_h^{j,2}\) for \(h \in (0, r+(m_1a_1))\) and also minimal parallelepipeds.
This ends Case III and the argument in dimension four.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Notes
There are finitely many components of \({\mathcal {M}}_t\) defined by fixing the directions of the edges of the parallelepipeds, and each of them is made by a linear transformation into the strong maximal operator \(M_{\textrm{str}}\) in \({\mathbb {R}}^{d1}\), see (2.1).
This inclusion is seen from the geometry of the situation, see Fig. 7. Perhaps the least obvious point is to ensure that in the case when \(b_h' \in \partial B_h\) the edge of \({\mathcal {P}}_h^*\) starting at \(b_h'\) and parallel to \(e_2\) does not cross \(B_h\). Indeed, this is true since the outward normal of \(B_h\) at \(b'_h\) enters into \(C_h\). Thus the angle between this normal and \(e_2\) is less than \(\pi /3\), and the inclusion follows.
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Research of Adam Nowak and Krzysztof Stempak was supported by the National Science Centre of Poland within the project OPUS 2013/09/B/ST1/02057. Adam Nowak was additionally supported by the grant OPUS 2017/27/B/ST1/01623 from the same institution.
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Nowak, A., Sasso, E., Sjögren, P. et al. On noncentered maximal operators related to a nondoubling and nonradial exponential measure. Math. Ann. 388, 2887–2929 (2024). https://doi.org/10.1007/s0020802302595w
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DOI: https://doi.org/10.1007/s0020802302595w