1 Introduction
1.1 Background and motivation
Let L stand for the standard Laplacian
$-\Delta $
or Hermite operator (also known as quantum harmonic oscillator)
$H=-\Delta +|x|^2$
on
$\mathbb R^n.$
We consider the heat equation of the following form:
$$ \begin{align} \begin{cases} \frac{\partial u}{\partial t}(x,t) = -L\, u(x,t) \\ u(x,0) = f(x) \end{cases} \quad (x,t) \in \mathbb{R}^n \times \mathbb{R}_+. \end{align} $$
The aim of this article is to study under what condition on initial data
$f,$
the solution
$u(x,t)$
to heat equation (1.1) – which is the heat semigroup as in (1.2) and (1.5) corresponding to Laplacian and Hermite operator, respectively, converges to f pointwise? Specifically, we wish to characterize the class of data for which the following limit
$$\begin{align*}\lim_{t\to 0} e^{tL}f(x)= f(x) \quad a.e.\end{align*}$$
holds true. The solution of (1.1) with
$L = -\Delta $
can be written as follows:
$$ \begin{align} u(x,t)= e^{-t\Delta}f(x)= h_t \ast f(x), \end{align} $$
where
$\ast $
denotes convolution on
$\mathbb {R}^n$
and the heat kernel
$$ \begin{align} h_{t}(x) = \frac{1}{(4 \pi t)^{\frac{n}{2}}} e^{-\frac{| x |^{2}}{4t}}. \end{align} $$
The spectral decomposition of the Hermite operator gives
$$ \begin{align} H = \sum_{k=0}^\infty (2k + n) P_k, \quad P_k f= \sum_{|\alpha|=k} \langle f, \Phi_\alpha \rangle\, \Phi_\alpha, \end{align} $$
where
$\Phi _\alpha $
are the normalized Hermite functions and
$P_k$
denotes the orthogonal projection onto the eigenspace corresponding to the eigenvalue
$2k + n,$
with the multi-index notation
$|\alpha | = \alpha _1 + \dots + \alpha _n$
. The Hermite heat semigroup is given by
$$ \begin{align} e^{-t H} f(x) =& \sum_{k=0}^\infty e^{-t(2k+n)} P_k f(x) = \int_{\mathbb{R}^n} h^H_{t}(x,y) f(y) \, dy, \end{align} $$
where
$h^H_t(x, y)$
denotes the corresponding heat kernel associated with H. See Section 6 for details. The pointwise convergence problem is closely connected with the boundedness of Hardy–Littlewood maximal operator
$\mathcal {M}f$
, which is defined by
$$ \begin{align} \mathcal{M}f(x) = \sup_{r> 0} \frac{1}{|B(x, r)|} \int_{B(x, r)} |f(y)|dy, \quad f\in L^1_{loc}(\mathbb R^n). \end{align} $$
Here,
$ B(x, r) $
denotes the ball of radius
$ r $
centered at
$ x $
, and
$ |B(x, r)| $
its Lebesgue measure. It is well-known [Reference Stein40, Chapter V] that
$\mathcal {M}$
is bounded on the weighted space
$ L^p_w(\mathbb {R}^n)$
(
$1<p< \infty $
) if and only if
$ w \in A_p $
, the class of Muckenhoupt weights.Footnote
1
Here, the weighted Lebesgue space norm is given by
$$\begin{align*}\|f\|^p_{L^p_w}=\int_{\mathbb R^n} |f|^p(x)w(x) dx. \end{align*}$$
As a consequence, we have
$$ \begin{align} \lim_{t \to 0} e^{t\Delta} f(x) = f(x) \quad for \ a.e. \ x \in \mathbb{R}^n, \ \forall f \in L^p_w(\mathbb{R}^n). \end{align} $$
Thus, for such classes of weights, the pointwise convergence holds for the heat semigroup (see [Reference Stein38, Theorem 2]). However, this result gives only a sufficient condition and does not yield a full characterization. On the other hand, the abstract Nikishin theory asserts that almost everywhere pointwise convergence implies the weak-type boundedness of the maximal operator
$\mathcal {M}$
from
$L^p_w(\mathbb {R}^n)$
into
$L^{p, \infty }_u(\mathbb {R}^n)$
for some weight function u (see [Reference García-Cuerva and de Francia23, Chapter VI]). However, establishing strong-type boundedness typically requires a vector-valued framework, as developed in [Reference Carleson and Jones16, Reference Mittag-Leffler33, Reference Rubio de Francia37]. In fact, Hartzstein et al. [Reference Hartzstein, Torrea and Viviani28] successfully adapted this approach to characterize the weight class
$$ \begin{align} D^h_p(\mathbb{R}^{n}) := \left\{ v: \mathbb R^{n}\to (0, \infty) \mid \exists\, t_0> 0 \ \text{ such that } \ h_{t_0} \in L^{p'}_{v^{-1}}(\mathbb{R}^n) \right\} \end{align} $$
for which (1.7) holds. Specifically, they established the following result.
Theorem 1.1 [Reference Hartzstein, Torrea and Viviani28]
Let v be a weight in
$\mathbb {R}^n, 1 \leq p<\infty $
, and heat kernel
$\left \{h_t\right \}_t$
be as in (1.3). Define
$$ \begin{align*}\mathcal{W}_R^* f(x)=\sup _{t<R}\left|h_t * f(x)\right|, \end{align*} $$
for some
$R,~ 0<R<\infty $
. The following statements are equivalent:
-
(1) There exists
$0<R<\infty $
and a weight u such that the operator
$f \mapsto \mathcal {W}_R^* f $
maps
$L^p_v$
into
$L^p_u$
for
$p>1$
. In the case
$p=1$
, it maps
$L^1_v$
into weak
$L^1_u$
. -
(2) There exists
$0<R<\infty $
and a weight u such that the operator
$ f \rightarrow \mathcal {W}_R^* f $
maps
$L^p_v$
into weak
$L^p_u$
. -
(3) There exists
$0<R<\infty $
such that
$h_R * f(x)<\infty $
a.e. x and the limit
$ \lim _{t \rightarrow 0} h_t * f(x) $
exists a.e. x for all
$f \in L^p_v$
. -
(4) There exists
$0<R<\infty $
such that
$ \mathcal {W}_R^* f(x)<\infty $
, a.e. x, for all
$f \in L^p_v$
. -
(5) The weight
$ v \in D_p^h. $
A similar characterization was also obtained for the Poisson semigroup in [Reference Hartzstein, Torrea and Viviani28]. Since then, many authors have studied this a.e. convergence problem in various settings:
-
- Heat-diffusion problems associated with the harmonic oscillator and the Ornstein–Uhlenbeck operator, by Abu-Falahah et al. [Reference Abu-Falahah, Stinga and Torrea1].
-
- Laguerre-type operators, by Garrigós et al. [Reference Garrigós, Hartzstein, Signes and Viviani24, Reference Viola and Viviani45].
-
- The Hermite operator
$ H = -\Delta + |x|^2 $
and the corresponding Poisson semigroup, by Garrigós et al. [Reference Garrigós, Hartzstein, Signes, Torrea and Viviani25]. -
- The Bessel operator, by Cardoso [Reference Cardoso14].
-
- More recently, the problem has been investigated in non-Euclidean settings: Cardoso [Reference Cardoso15], Bruno and Papageorgiou [Reference Bruno and Papageorgiou12], and Álvarez-Romero et al. [Reference Alvarez-Romero, Barrios and Betancor3] studied it on the Heisenberg group, symmetric spaces, and homogeneous trees, respectively. Moreover, Bhimani and Dalai [Reference Bhimani and Dalai8] have considered the corresponding problem on the torus and waveguide manifolds.
-
- The Laplace operator perturbed by a gradient, the fractional Laplacian, mixed local–nonlocal operators, the Laplacian with inverse square potential, the Laplacian on Riemannian manifolds, and the Dunkl Laplacian are studied by Bhimani–Biswas–Dalai in [Reference Bhimani, Biswas and Dalai7] and have provided general framework in the setting of metric measure spaces.
We conclude this section by highlighting the following remark. In all the aforementioned results, the characterized class of weights and initial data is based on
$L^p$
spaces. On the other hand, since the heat semigroup has been extensively studied on several other function spaces, it is natural to study the a.e. convergence problem there. The aim of this article is to study this problem in the realm of modulation spaces.
1.2 The weight-class in the framework of modulation spaces
The concept of modulation spaces was introduced in the pioneering work of Feichtinger [Reference Feichtinger, Krishna, Radha and Thangavelu21] in the early 1980s and has since become an integral tool in both pure and applied mathematics (see, e.g., [Reference Bényi and Okoudjou4, Reference Bhimani and Ratnakumar11, Reference Cardona, Chatzakou, Delgado, Kumar and Ruzhansky13, Reference Cordero19, Reference Cordero and Nicola20, Reference Feichtinger22, Reference Gröchenig26, Reference Nicola34, Reference Nicola and Trapasso35, Reference Toft43, 44, Reference Wang and Hudzik46, Reference Wang, Huo, Hao and Guo47]). In order to define these spaces, the idea is to impose integrability conditions on the short-time Fourier transform (STFT) with respect to a test function from the Schwartz space
$\mathcal {S}$
. Specifically, let
$0\neq \phi \in \mathcal {S}$
and
$f \in \mathcal {S}^{\prime }$
(tempered distributions), then the STFT of f with respect to
$\phi $
is defined as
$$ \begin{align} V_\phi f(x, \xi)=\int_{\mathbb{R}^n} f(y) \overline{\phi(y-x)} e^{-2 \pi i x \cdot \xi} d y \quad x,\xi\in\mathbb{R}^n, \end{align} $$
whenever the integral exists. Throughout this article, we assume that the weight function
$v:\mathbb R^{2n}\to (0, \infty )$
is sub-multiplicative, that is, a measurable function
$v : \mathbb {R}^{2n} \to (0,\infty )$
satisfies
$$ \begin{align*}\quad v(z_1+z_2) \leq v(z_1)\, v(z_2) \quad \text{for all} \quad z_1,z_2 \in \mathbb{R}^{2n}. \end{align*} $$
The weighted modulation space
$M_v^{p, q}=M_v^{p, q}(\mathbb {R}^n) \ \ (1\leq p, q \leq \infty )$
is the set of all
$f \in \mathcal {S}^{\prime } $
for which the following norm
$$ \begin{align} \|f\|_{M_v^{p, q}}= \|V_\phi f\|_{L_v^{p, q}}=\left(\int_{\mathbb{R}^n}\left(\int_{\mathbb{R}^n} \left|V_\phi f(x, \xi) \right|{}^p v(x,\xi)\, d x\right)^{\frac{q}{p}} d \xi\right)^{\frac{1}{q}} \end{align} $$
is finite, and with the natural modifications if
$p=\infty $
or
$q=\infty .$
Remark 1.2 The structure of the article requires several comments:
-
(1) The definition of the modulation space norm (1.10) is independent of the choice of the particular window function (see, e.g., [Reference Feichtinger, Krishna, Radha and Thangavelu21] and [Reference Gröchenig26, Proposition 11.3.2(c)]). Thus, regardless of the chosen test function
$\phi \in \mathcal {S}$
, the space
$M_v^{p, q}$
remains constant. In case of
$v\equiv 1,$
we simply denote
$M_1^{p, q}=M^{p, q}.$
See [Reference Bényi and Okoudjou4, Reference Cordero and Nicola20, Reference Gröchenig26, Reference Nicola and Trapasso35, Reference Wang and Hudzik46, Reference Wang, Huo, Hao and Guo47] for a comprehensive introduction to these spaces. -
(2) We have not considered the power of weight v in the modulation space norm (1.10), unlike in the existing literature, due to technical reasons. This will be clarified in the below pages (see also Theorem 1.1).
-
(3) The concept of a weight function arises frequently in harmonic analysis, which serves to quantify growth, decay, or smoothness. It turns out that the correct class of weight function to be considered is the “sub multiplicative weight” to study the weighted modulation spaces. This contrasts with the weights appearing in the classical setting, for example, the Muckenhoupt classes. The main drawback with the Muckenhoupt classes is their lack of invariance under time–frequency shifts in weighted modulation spaces. We refer to the excellent survey article by Gröchenig [Reference Gröchenig27] for the most relevant classes of weights appearing in the time–frequency analysis.
-
(4) The main objective in this article is to study pointwise convergence, and so it is essential that the initial data admit well-defined pointwise values. However, modulation spaces in general consist of tempered distributions, for which pointwise evaluation may not be meaningful. Moreover, the definition of the maximal function requires the initial data to be locally integrable. Thus, throughout the article, we shall assume that the initial data are also in the space of locally integrable functions
$L^1_{loc}(\mathbb R^n)$
. -
(5) The STFT (1.9) is also closely related to Fourier-Wigner and Bargmann transform (see [Reference Gröchenig26]). There is also an equivalent definition of modulation spaces via uniform frequency decomposition. This is quite similar in the spirit of Besov spaces, where we have dyadic decomposition (see [Reference Feichtinger, Krishna, Radha and Thangavelu21] and [Reference Wang and Hudzik46, Proposition 2.1]). See also [Reference Cleanthous and Georgiadis18] for more unified approach to these spaces (so-called
$\alpha $
-modulation spaces).
Denote
$p'$
the Hölder conjugate of
$p,$
that is,
$ 1/p + 1/{p'} = 1 $
. In order to state our main result, we define the following weight classes associated with heat kernel (1.3):
$$\begin{align*}D^h_{p,q}(\mathbb{R}^{2n})= \left\{ v: \mathbb R^{2n}\to \mathbb (0, \infty) \mid\exists\, t_0>0 \ \text{such that } h_{t_0} \in M^{p', q'}_{v^{-1}} (\mathbb R^n) \right\}.\end{align*}$$
Remark 1.3 We compare the weight classes that appear in this article:
-
(1) The class of Muckenhoupt weights
$A_p$
is a proper subset of the weight class
$D_p^h$
appeared in Theorem 1.1. -
(2) We have imposed sub-multiplicative assumption for the member of weight class
$D^h_{p,q}$
. As this gives flexibility to use desired window functions in modulation spaces (see Remark 1.2(1) and (3)). On the other hand, we do not require this assumption for the class
$D^h_{p}$
. -
(3) By fixing a frequency variable
$v_{\xi }=v(x, \xi )$
, we can embed
$D^{h}_{p,q}(\mathbb R^{2n})$
into
$D^{h}_{p}(\mathbb R^n)$
(see Lemma 5.2). -
(4) Our formulation of weight class
$D^h_{p,q}$
(defined using modulation spaces) is inspired from the weight class
$D^h_{p}$
(defined using Lebesgue spaces, see Theorem 1.1). This provides the natural way to introduce weighted structures in modulation spaces and turned out very effective in our analysis (see Remark 1.2(2)). -
(5) The Hermite heat kernel (1.5) can be estimated by the standard heat kernel; for the upper bound, see (6.4), and for the lower bound, see (6.5). Thus, the same weight class will work in Theorem 2.1 for both the Laplacian and Hermite operator.
2 Main results
We are now ready to state our main result.
Theorem 2.1 Let
$ v $
be a strictly positive sub-multiplicative weight on
$\mathbb {R}^{2n}, 1 \leq p, q < \infty $
and
$H=-\Delta + |x|^2.$
Suppose that
$f \in M_v^{p,q}(\mathbb R^n)\cap L^1_{loc}(\mathbb R^n)$
is nonnegative. Then the weight
$$ \begin{align*}v \in D^h_{p,q}(\mathbb{R}^{2n})\end{align*} $$
if and only if
$$\begin{align*}\lim_{t \rightarrow 0} h_t \ast f(x) = f(x) \quad for \ a.e. \ x\in\mathbb{R}^n \ \end{align*}$$
if and only if
$$\begin{align*}\lim_{t \to 0} e^{-tH} f(x)= f(x) \quad for \ a.e. \ x\in\mathbb{R}^n. \end{align*}$$
The boundedness of the local maximal function is not quite clear and seems challenging in modulation spaces. And so the approach described in Section 1.1 is not applicable directly to prove Theorem 2.1.
Remark 2.2 The hypothesis of Theorem 2.1 deserves several comments:
-
(1) For the sufficiency part, the proof relies on maximal function techniques, where the comparison between the semigroup and the Hardy–Littlewood maximal operator requires
$f \ge 0$
in order to apply pointwise domination arguments. See Remark 2.7 and the proof of Lemmas 4.1 and 4.2. -
(2) For the necessity part, the argument relies on the bounds of the heat kernel (and of the Hermite heat kernel) to derive the required lower estimates for the characterization of the weight class. In this direction, the non-negativity of f is not required.
-
(3) Due to certain additional algebraic properties of some modulation spaces, specifically when
$1 \leq p < \infty $
and
$1 \leq q \leq 2$
, the non-negativity assumption can be relaxed (see Remark 2.9 and Theorem 4.6). In general, our method does not allow us to remove the non-negativity hypothesis. Whether the equivalence in Theorem 2.1 remains valid for general (possibly sign-changing) functions in
$M^{p,q}_v$
is an interesting open question (see Section 7).
Remark 2.3 (Examples)
Theorem 2.1 is applicable to certain initial data that have not been studied before in the literature [Reference Abu-Falahah, Stinga and Torrea1, Reference Bhimani, Biswas and Dalai7, Reference Garrigós, Hartzstein, Signes, Torrea and Viviani25, Reference Hartzstein, Torrea and Viviani28]:
-
(1) Noticing the following strict embedding:
we deduce the existence of certain nonnegative functions
$$\begin{align*}M^{p, q_1} \subset L^p \subset M^{p, q_2}, \quad q_1 \leq \min \left\{p, p^{\prime}\right\}, \quad q_2 \geq \max \left\{p, p^{\prime}\right\}, \end{align*}$$
$ f \in M^{p, q} $
such that
$ f \notin L^p $
for
$ p \geq 2 $
.
-
(2) It is known (see, e.g., [Reference Bhimani, Manna, Nicola, Thangavelu and Trapasso9, Example 2.1]) that, for
$0< \alpha <n,$
we have While
$$\begin{align*}f_\alpha(x) = |x|^{-\alpha} \in M^{p,q} \quad \text{for} \quad p> n / \alpha, \ q > n / (n - \alpha). \end{align*}$$
$f_{\alpha }$
does not belong to any Lebesgue spaces
$L^p.$
Remark 2.4 The Ornstein–Uhlenbeck operator
$\mathcal {O} = -\Delta + 2x \cdot \nabla $
on
$\mathbb {R}^n$
is closely related to a small perturbation of the Hermite operator on
$\mathbb {R}^n$
, given by
$$\begin{align*}L = -\Delta + |x|^2 - n. \end{align*}$$
Indeed, by defining
$\tilde {u}(x) = e^{-|x|^2 / 2} u(x)$
, it follows directly that
$\mathcal {O} u(x) = e^{|x|^2 / 2}(L \tilde {u})(x)$
. Consequently, the heat semigroups associated with
$\mathcal {O}$
can be expressed as
$$\begin{align*}e^{-t \mathcal{O}} f(x) = e^{\frac{|x|^2}{2}} e^{-t L} \tilde{f}(x), \end{align*}$$
where
$\tilde {f}(x) = e^{-|x|^2 / 2} f(x)$
. This connection implies that the convergence properties of the semigroups associated with
$\mathcal {O}$
can be derived from those of L via the mapping
$f \mapsto \tilde {f}$
(see, e.g., [Reference Abu-Falahah, Stinga and Torrea1, Equation (3.1)] and [Reference Garrigós, Hartzstein, Signes, Torrea and Viviani25]). As a result, Theorem 2.1 for the operator
$\mathcal {O}$
is obtained directly from the corresponding results for L.
The study of nonlinear evolution equations with Cauchy data in modulation spaces has gained a lot of interest in recent years (see, e.g., [Reference Bhimani and Ratnakumar11, Reference Cordero and Nicola20, Reference Nicola34, Reference Wang and Hudzik46, Reference Wang, Huo, Hao and Guo47]). We briefly mention well-posedness theory for nonlinear heat equation
$$\begin{align*}u_t +\Delta u = u^{k}\end{align*}$$
with Cauchy data in certain modulation spaces. Iwabuchi [Reference Iwabuchi30] proved local and global well-posedness for small data in some weighted modulation spaces
$M^{p,q}_s$
. Later Chen–Deng–Ding–Fan [Reference Chen, Deng, Ding and Fan17] have obtained some space–time estimates for fractional heat semigroup
$e^{-t (-\Delta )^{\beta /2}}$
in modulation spaces. Further, they [Reference Chen, Deng, Ding and Fan17, Theorems 1.5 and 1.6] proved local and global well-posedness (for small data) in some weighted modulation spaces (see also [Reference Ru36, Theorem 3]). In [Reference Huang, Fan and Chen29, Reference Zheng, Qiang and Rui48] authors have found some critical exponent in modulation spaces and provide some local well-posedness and ill-posedness results. On the other hand, Bhimani [Reference Bhimani6] established some finite time blow-up. While Cordero [Reference Cordero19] and Bhimani et al. in [Reference Bhimani, Manna, Nicola, Thangavelu and Trapasso9, Reference Bhimani, Manna, Nicola, Thangavelu and Trapasso10] have studied well-posedness theory for the heat equation in
$M^{p,q}$
spaces associated with the Hermite operator. Later, Cardona et al. in [Reference Cardona, Chatzakou, Delgado, Kumar and Ruzhansky13] have extended these results for the heat equation associated with anharmonic oscillator. While Trapasso [44] has studied heat equation associated with the twisted Laplacian in modulation spaces. In light of these considerations and Remark 2.3, we would like to mention that Theorem 2.1 is new and nicely complements the ongoing research activity.
We shall now briefly comment on the proof of Theorem 2.1. These problems were motivated by the seminal work of Torrea et al. in [Reference Garrigós, Hartzstein, Signes, Torrea and Viviani25], where they addressed similar issues in the setting of weighted Lebesgue spaces on
$\mathbb {R}^n$
. We observe that pointwise convergence results and the boundedness of the maximal operator are closely related. However, to the best of authors’ knowledge, it is important to note that the boundedness of the maximal operator on modulation spaces remains an open problem (Section 2.1), requiring the development of novel techniques, which may be of independent interest. Specifically, to prove Theorem 2.1, we employ various properties of modulation spaces to reduce the problem to a stage where we can apply known results [Reference Abu-Falahah, Stinga and Torrea1, Reference Garrigós, Hartzstein, Signes, Torrea and Viviani25, Reference Hartzstein, Torrea and Viviani28] on weighted Lebesgue spaces. We refer to Remark 5.1 for a detailed proof strategy.
2.1 Maximal operator
The Hardy–Littlewood maximal operator is a fundamental tool in harmonic analysis. The renowned theorem of Hardy, Littlewood, and Wiener asserts that the maximal operator, defined by (1.6), is bounded on
$ L^p $
for
$ 1 < p \leq \infty $
, and weak-
$ L^1 $
bounded when
$ p = 1 $
(see [Reference Stein38, Reference Stein39]). This result is a cornerstone in harmonic analysis due to its profound applications, particularly in potential theory.
Our next theorem shows that the class of nonnegative functions in the modulation space
$ M^{p, \infty } \ (1<p\leq \infty ) $
is invariant under the action of the Hardy–Littlewood maximal operator. Specifically, we have the following theorem.
Theorem 2.5 Let
$1 < p \leq \infty $
and suppose
$f \in M^{p,\infty }(\mathbb R^n)\cap L^1_{loc}(\mathbb R^n)$
is nonnegative. Then
$\mathcal {M}f \in M^{p,\infty }(\mathbb R^n)$
.
Kinnunen in his seminal paper [Reference Kinnunen31] proved that
$\mathcal M$
maps the Sobolev spaces
$W^{1,p} \ (1 < p<\infty )$
into themselves, using functional analytic techniques. Since
$\mathcal M$
fails to be bounded in
$L^1,$
a vital question was whether the boundedness property holds for
$f\in W^{1,1}.$
Tanaka [Reference Tanaka41] provided a positive answer to this in the case of the uncentered maximal function (see definitions therein), which was further improved by [Reference Aldaz and Pérez Lázaro2] whenever
$f\in BV$
(set of functions whose total variation is finite). In the centered case (which coincides with (1.6)), Kurka [Reference Kurka32] showed the endpoint question to be true, that is,
$\mathcal {V}(\mathcal Mf)\leq C\mathcal {V}(f)$
with some constant
$C,$
where
$\mathcal {V}$
denotes the total variation of a function. However, to the best of the author’s knowledge, Theorem 2.5 is the first result of the maximal operator (1.6) established under the modulation space norm.
Remark 2.6 Theorem 2.5 encompasses new classes of functions that the boundedness of the maximal operator has not been previously examined. For instance, as discussed in Remark 2.3, the function
$f_\alpha (x) = |x|^{-\alpha } \in M^{p,q}$
for
$0< \alpha <n, \ p>n / \alpha $
, and
$q> n / (n - \alpha ),$
which does not belong to any
$L^p$
space but lies in
$M^{p,q} \subset M^{p,\infty }$
, satisfies
$\mathcal {M}f \in M^{p,\infty }$
.
Remark 2.7 Theorem 2.5 remains unknown for arbitrary functions (not necessarily nonnegative) within the modulation spaces. This discrepancy arises from the fact that the definition of the maximal operator involves
$ |f| $
, while certain modulation spaces do not necessarily remain closed under the modulus operation (see, e.g., [Reference Bhimani5, Corollary 1.3] and [Reference Bhimani and Ratnakumar11, Corollary 3.2]).
Remark 2.8 To establish Theorem 2.5, we first show that for any
$ f \in M^{p,\infty } $
, there exists a Schwartz class function
$ \phi $
such that
$ \mathcal {M}f * |\phi | \in L^{p} $
(see Lemma 4.1). Utilizing this result, we can derive the inequality
$\left |\mathcal {M}f * M_{\xi } \phi (x)\right | \leq \mathcal {M}(f * |\phi |)(x),$
for almost every
$ x $
, as shown in Lemma 4.2. These results facilitate the proof of the required boundedness. Since we are employing the
$ L^p $
boundedness of the maximal operator result, it is required to consider all nonnegative functions within this framework.
Remark 2.9 It is worth noting that, by exploiting certain algebraic properties inherent to modulation spaces, Theorem 2.5 remains valid for arbitrary functions (not necessarily nonnegative) belonging to specific modulation spaces, when
$f \in M^{p,q}$
with
$1 \leq p < \infty $
and
$1 \leq q \leq 2$
. Consequently, the pointwise convergence results stated in Theorem 2.1 also hold for such modulation spaces (see Theorem 4.6).
The article is organized as follows. In Section 3, we revisit the boundedness of the Hardy–Littlewood maximal operator and examine related results within the framework of Lebesgue spaces. Section 4 presents the proof of Theorem 2.5, along with the identification of additional modulation spaces where pointwise convergence is valid for general functions. In Section 5, we determine the relevant weight classes for modulation spaces in the Laplacian setting. In Section 6, we carry out the analysis for the Hermite setting and establish our main result (Theorem 2.1).
3 Preliminaries
A measurable function
$ f $
is said to be in the weak
$ L^p $
space if there exists a constant
$ C> 0 $
such that for all
$ \lambda> 0, $
$$\begin{align*}\left|\{x \in \mathbb{R}^n : |f(x)|> \lambda\}\right| \leq \frac{C^p}{\lambda^p}, \end{align*}$$
where
$|A|$
denotes the Lebesgue measure of the set A.
3.1 Maximal operator on Lebesgue spaces
Theorem 3.1 [Reference Stein40, Theorem 1] (Boundedness of maximal operator)
-
(1) If
$f \in L^p \ (1 \leq p \leq \infty )$
, then the function
$\mathcal {M}f$
is finite almost everywhere. -
(2) If
$f \in L^1$
, then
$ \lambda \cdot \left |\{x:\mathcal Mf(x)>\lambda \}\right | \lesssim _{n} \|f\|_{L^1}$
for every
$\lambda>0.$
-
(3) If
$f \in L^p \ (1<p \leq \infty )$
, then
$||\mathcal Mf||_{L^p}\lesssim _{p,n}||f||_{L^p}.$
Theorem 3.2 [Reference Stein38, Theorem 2]
Let
$\varphi $
be an integrable function on
$\mathbb {R}^n$
, and set
$\varphi _{t}(x)=t^{-n} \varphi (x / t)$
for
$ t>0.$
Suppose the least decreasing radial majorant of
$\varphi $
is integrable, that is,
$\psi (x) =\sup _{|y|\geq |x|} |\varphi (y)|$
and
$\int _{\mathbb {R}^n} \psi (x) d x=A<\infty $
. Then
-
(1)
$\sup _{t>0} \left |\varphi _{t} * f(x)\right | \leq A\, \mathcal Mf(x) \quad \mbox {a.e. for all } f \in L^p \ (1 \leq p \leq \infty )$
. -
(2) If
$\int _{\mathbb {R}^n} \varphi (x) d x=1$
, then
$\lim _{t \rightarrow 0}\left (\varphi _{t} * f\right )(x)=f(x)$
a.e. for all
$f\in L^p \ (1 \leq p \leq \infty ).$
-
(3) If
$1\leq p<\infty $
, then
$\left \|\varphi _{t} * f-f\right \|_{L^p} \rightarrow 0$
as
$t \rightarrow 0$
.
Remark 3.3 [Reference Stein40, Chapter V, Section 2.1, p. 198]
The following formulations provide equivalent representations of the Hardy–Littlewood maximal function.
-
(1) Let
$\varphi (x) = \frac {1}{|B_1|} \chi _{B_1}(x),\,B_{1} = \{x : |x| < 1\}$
. Then the Hardy–Littlewood maximal operator (1.6) can be represented as (3.1)
$$ \begin{align} \sup_{t>0} (\varphi_{t} *|f|)(x) =\, \mathcal Mf(x). \end{align} $$
-
(2) Denote
If
$$\begin{align*}\mathcal{X}=\left\{ \vartheta>0 : \vartheta \text{ is radial and decreasing such that } \int_{\mathbb{R}^n} \vartheta(x) \, dx = 1 \right\}. \end{align*}$$
$\vartheta \in \mathcal {X}$
, then
$\vartheta _{t}(\cdot )= t^{-n} \vartheta (\cdot /t) \in \mathcal {X}$
. We may rewrite (3.2)
$$ \begin{align} \mathcal{M}f(x) = \sup_{\vartheta \in \mathcal{X}} (\vartheta * |f|)(x). \end{align} $$
This shows that the maximal function defined in (3.2) coincides with the classical Hardy–Littlewood maximal function (3.1). Indeed, if we denote by
$B_t = \{x \in \mathbb {R}^n : |x| < t\}$
, then the normalized characteristic function
$|B_t|^{-1} \chi _{B_t}$
belongs to the class
$\mathcal {X}$
, and taking the supremum over such elements recovers the standard definition of
$\mathcal {M}f$
. Conversely, any element of
$\mathcal {X}$
can be approximated by limits of such normalized characteristic functions, thus establishing the equivalence of the two definitions.
Remark 3.4 Theorem 3.2 holds for heat and Poisson kernels corresponding to the standard Laplacian, that is, we may take
$\varphi _t=h_t$
or
$\varphi _t=p_t$
(among others).
3.2 Short-time Fourier transform
In the next lemma, we recall several useful facets of the STFT. To this end, for
$x,y, \xi \in \mathbb R^n, \epsilon>0,$
denote
-
•
$f^{*}(x)=\overline {f(-x)}$
(involution); -
•
$T_yf(x)=f(x-y)$
(translation/time shift by y); -
•
$D_\epsilon f(x)=\epsilon ^{-n}f(\epsilon ^{-1}x)$
(
$L^1$
-normalized
$\epsilon $
-dilation); -
•
$M_\xi f(x)=e^{2\pi i \xi \cdot x} f(x)$
(modulation/frequency shift by
$\xi ).$
Lemma 3.5 (Basic properties)
Let
$p_i, q_i \in [1, \infty ]$
for
$i=0,1,2.$
-
(1) [Reference Gröchenig26, Lemma 3.1.1] If
$f, \phi \in L^2,$
then
$V_g f$
is uniformly continuous on
$\mathbb {R}^{2 n}$
, and where
$$ \begin{align*} \begin{aligned} V_\phi f(x, \xi) & =\left(f \cdot T_x \bar{\phi}\right)^{\wedge}(\xi) =\left\langle f, M_\xi T_x \phi\right\rangle =\left\langle\hat{f}, T_\xi M_{-x} \hat{\phi}\right\rangle =e^{-2 \pi i x \cdot \xi}\left(\hat{f} \cdot T_\xi \overline{\hat{\phi}}\right)^{\wedge}(-x) \\ & =e^{-2 \pi i x \cdot \xi} V_{\hat{\phi}} \hat{f}(\xi,-x) =e^{-2 \pi i x \cdot \xi}\left(f * M_\xi \phi^*\right)(x) =\left(\hat{f} * M_{-x} \hat{\phi}^*\right)(\xi), \end{aligned} \end{align*} $$
$\hat {\cdot }$
denotes the Fourier transform.
-
(2) [Reference Gröchenig26, Theorem 3.2.1] (Moyal identity/orthogonality for STFT) If
$f_i, \phi _i \in L^2 $
for
$(i=1,2),$
then
$V_{\phi _i}f_i\in L^2(\mathbb R^{2n})$
and
$$ \begin{align*} \langle V_{\phi_1}f_1, V_{\phi_2}f_2 \rangle = \langle f_1, f_2 \rangle\, \overline{\langle \phi_1, \phi_2 \rangle}. \end{align*} $$
-
(3) [Reference Bényi and Okoudjou4, Corollary 4.21] If
$\frac {1}{p_{1}}+\frac {1}{p_{2}}=\frac {1}{p_{0}}$
and
$\frac {1}{q_{1}}+\frac {1}{q_{2}}=1+\frac {1}{q_{0}},$
then
$$ \begin{align*}\|f g\|_ {M^{p_{0},q_{0}}} \lesssim \|f\|_ {M^{p_{1},q_{1}}} \|g\|_ {M^{p_{2},q_{2}}}.\end{align*} $$
4 Hardy–Littlewood maximal operator on modulation spaces
In this section, we examine the Hardy–Littlewood maximal operator on modulation spaces. Before proceeding to the proof of Theorem 2.5, we first establish several preliminary results that play a fundamental role in the subsequent analysis. These auxiliary results provide the necessary groundwork and will be addressed in the following section.
Lemma 4.1 Let
$1 < p< \infty ,1 \leq q < \infty ,$
and
$f \in M^{p, q}\cap L^1_{loc}$
be nonnegative. Then there exists some
$\phi \in \mathcal S$
such that
$\mathcal Mf * |\phi |\in L^{p}.$
In the case
$p=1, \mathcal Mf * |\phi |$
is in weak-
$L^{1}.$
Proof In the definition of modulation spaces, since the choice of
$\phi \in \mathcal {S} \setminus \{0\}$
is arbitrary, we may select
$\phi $
such that
$\phi (y) \leq 0$
for a.e.
$y \in \mathbb {R}^n$
and ensure that
$f * |\phi | \in L^p$
by utilizing the following chain of embeddings (see [Reference Toft43]):
$$ \begin{align} M^{p,q} * \mathcal{S} \hookrightarrow M^{p,q} * M^{1, q'} \hookrightarrow M^{p,1} \hookrightarrow L^p. \end{align} $$
By (3.2), we have
$$ \begin{align} \mathcal Mf * \phi(x)=\left(\sup _{\vartheta \in \mathcal{X}} (\vartheta * f)\right)\ast\phi(x)=\int_{\mathbb R^n} \sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y) \,\phi(x-y) d y. \end{align} $$
It is clear that
$\vartheta * f$
is a nonnegative function since convolution of two nonnegative function is nonnegative. Hence,
$(\vartheta * f) (y) \phi (x-y)\leq 0$
for a.e.
$y\in \mathbb R^n.$
Recall
$\sup A =-\inf (-A)$
for
$A\subset \mathbb R.$
Now, we can write
$$ \begin{align*}\sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y) \phi(x-y)= -\inf _{\vartheta \in \mathcal{X}}(\vartheta * f)(y) |\phi|(x-y).\end{align*} $$
By putting the above equation in (4.2) and then using (3.2), we get
$$ \begin{align*}\begin{aligned} \mathcal Mf * |\phi|(x)&=-\left(\mathcal Mf * \phi\right)(x)\\ &=\int_{\mathbb R^n}-\sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y)\, \phi(x-y)dy\\ &= \int_{\mathbb R^n}\inf_{\vartheta \in \mathcal{X}} \left(\vartheta * f) (y) \,|\phi|(x-y)\right) d y\\ &\leq \int_{\mathbb R^n}\liminf_{n \to \infty} \left(\vartheta_n * f) (y) \,|\phi|(x-y)\right) d y, \end{aligned}\end{align*} $$
where
$\{\vartheta _n\} \subset \mathcal {X}$
be some sequence. Then, by Fatou’s lemma, we obtain
$$ \begin{align*}\begin{aligned} \mathcal Mf * |\phi|(x)&\leq\liminf_{n \to \infty}\int_{\mathbb R^n}(\vartheta_n * f) (y) |\phi|(x-y) d y \\ &\leq\sup_{\vartheta \in \mathcal{X}}\int_{\mathbb R^n}(\vartheta * f) (y) |\phi|(x-y) d y \\ & =\sup _{\vartheta \in \mathcal{X}} \left(\vartheta *(f * |\phi|)\right)(x) \\ & =\mathcal M(f * |\phi|)(x). \end{aligned}\end{align*} $$
Using Theorem 3.1, the
$L^p$
-boundedness of maximal operator
$\mathcal M(f * |\phi |)$
will be in
$L^{p}$
for
$1<p\leq \infty ,$
since
$f * |\phi |$
is in
$L^{p}.$
For the case
$p=1,$
we can also conclude our claim because
$\mathcal M(f * |\phi |)$
is weak-
$L^1$
whenever
$f * |\phi |$
is in
$L^1.$
Lemma 4.2 Let
$ f \in M^{p, q}\cap L^1_{loc},\, 1 \leq p, q < \infty , $
be nonnegative. Then, for almost every
$ \xi \in \mathbb {R}^n $
, the following inequality holds for some
$\phi \in \mathcal {S:}$
$$\begin{align*}\left|\mathcal{M}f * M_{\xi} \phi(x)\right| \leq \mathcal{M}(f * |\phi|)(x) \quad \text{for almost every } x \in \mathbb{R}^n. \end{align*}$$
Proof By (3.2), we have
$$ \begin{align} |\mathcal Mf * M_{\xi} \phi(x)| &\leq\int_{\mathbb R^n} \left|\sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y)\right|\left|M_{\xi} \phi(x-y) \right|d y\nonumber\\ &=\int_{\mathbb R^n}\sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y) |\phi|(x-y)dy. \end{align} $$
It is clear that for each
$\vartheta \in \mathcal {X},$
we have
$$ \begin{align} |(\vartheta * f)(y) |\phi|(x-y)|=(\vartheta * f)(y) |\phi|(x-y)\leq \mathcal Mf(y) |\phi|(x-y). \end{align} $$
The
$L^1$
norm of right-hand side of (4.4) is
$$ \begin{align*}\int_{\mathbb R^n}\mathcal Mf(y) |\phi|(x-y)dy=\mathcal Mf*|\phi|(x)<\infty\end{align*} $$
for almost all
$x\in \mathbb R^n,$
by Lemma 4.1. Hence, using dominated convergence theorem (DCT), (4.3) can be rewritten as
$$ \begin{align*}\begin{aligned} |\mathcal Mf * M_{\xi} \phi(x)|&\leq\int_{\mathbb R^n}\sup _{\vartheta \in \mathcal{X}}(\vartheta * f)(y) |\phi|(x-y)dy\\ &=\sup _{\vartheta \in \mathcal{X}}\int_{\mathbb R^n}(\vartheta * f)(y) |\phi|(x-y)dy\\ & =\sup _{\vartheta \in \mathcal{X}} \left(\vartheta *(f * |\phi|)\right)(x) \\ & =\mathcal M(f * |\phi|)(x) \end{aligned}\end{align*} $$
for almost every
$x\in \mathbb R^n.$
Hence, we conclude the result.
Now, we are prepared to prove the boundedness result within the modulation space.
Proof of Theorem 2.5
Given that
$f \in M^{p, \infty }$
for
$1 < p \leq \infty $
, showing that
$\mathcal Mf \in M^{p, \infty }$
is equivalent to showing (see Lemma 3.5(1))
$$\begin{align*}\operatorname*{\mbox{ess sup}}_{\xi\in \mathbb{R}^n}\left( \int_{\mathbb{R}^n} \left|\mathcal Mf * M_{\xi} {\phi^*}(x)\right|{}^p \, dx \right)^{\frac{1}{p}} < \infty.\end{align*}$$
We can choose the
$ {\phi } $
in such a way that
$ {\phi ^*} $
satisfies the required conditions of Lemma 4.2. By applying Lemma 4.2, it follows that
$$ \begin{align*}\left|\mathcal Mf * M_{\xi} {\phi^*}(x)\right| \leq \mathcal M(f * |{\phi^*}|)(x).\end{align*} $$
Furthermore, as per embeddings (4.1), we have
$f * |{\phi ^*}| \in L^{p}.$
So we can derive the following by combining these facts with the
$L^p$
-boundedness of the maximal operator:
$$ \begin{align*}\begin{aligned} & \operatorname*{\mbox{ess sup}}_{\xi\in \mathbb{R}^n}\left( \int_{\mathbb{R}^n} \left|\mathcal Mf * M_{\xi} {\phi^*}(x)\right|{}^p \, dx \right)^{\frac{1}{p}}\\ \leq&\operatorname*{\mbox{ess sup}}_{\xi\in \mathbb{R}^n}\left( \int_{\mathbb{R}^n} \left|\mathcal M \left(f * |{\phi^*}|\right)(x)\right|{}^{p} \, dx \right)^{\frac{1}{p}}\\ \lesssim&\operatorname*{\mbox{ess sup}}_{\xi\in \mathbb{R}^n}\left( \int_{\mathbb{R}^n} \left|f * |{\phi^*}|(x)\right|{}^{p} \, dx \right)^{\frac{1}{p}}<\infty. \end{aligned}\end{align*} $$
This establishes that
$\mathcal Mf \in M^{p, \infty }$
whenever
$f \in M^{p, \infty }$
for
$1<p \leq \infty .$
Theorem 4.3 Suppose that the least decreasing radial majorant of
$\varphi $
is integrable, that is,
$\psi (x)=\sup _{|y|\geq |x|} |\varphi (y)|$
and
$\int _{\mathbb {R}^n} \psi (x) d x=A<\infty $
. Let
$1 \leq p, q < \infty ,$
and
$f \in M_v^{p, q}\cap L^1_{loc}$
be nonnegative. Then
$$ \begin{align*}\sup_{t>0} \left|\varphi_{t} * f(x)\right| \leq A\, \mathcal Mf(x) ~\mbox{ almost everywhere}.\end{align*} $$
Proof The proof of this is similar to the
$L^{p}$
case as in Theorem 3.2(1). When
$f \in M_v^{p, q},$
since
$\left |\varphi _{t} * f(x)\right |\leq \psi _{t} * f(x),$
it is sufficient to show that
$$ \begin{align} \psi_{t} * f(x) \leq A\,\mathcal{M}f(x) \end{align} $$
holds for every
$t>0.$
Hence, showing (4.5) is the same as showing
$$ \begin{align} \psi_{t} * (T_xf)(0) \leq A\,\mathcal{M}(T_xf)(0), \end{align} $$
where
$T_xf(y)=f(y-x).$
In order to establish (4.6), consider
$$ \begin{align*}\begin{aligned} \psi_{t} * (T_xf)(0)&=\int_{\mathbb{R}^{n}} (T_xf)(y) \psi_t(y) d y\\ & =\int_{0}^{\infty} \int_{S^{n-1}} (T_xf)(r \theta) \psi_t(r) r^{n-1} d \sigma dr \\ & =\int_{0}^{\infty} \lambda(r) \psi_t(r) r^{n-1} d r, \end{aligned} \end{align*} $$
where
$\lambda (r)=\int _{S^{n-1}} (T_xf)(r \theta ) d \sigma $
and
$\sigma $
is a surface measure on
$S^{n-1}=\{x\in \mathbb {R}^n: |x|=1\}.$
Then, using limiting case and applying integration by parts, we can write
$$ \begin{align} \begin{aligned} \psi_{t} * (T_xf)(0)&=\lim _{\substack{\varepsilon \rightarrow 0 \\ N \rightarrow \infty}} \int_{\varepsilon}^{N} \lambda(r) \psi_t(r) r^{n-1} d r \\ & =\lim _{\substack{\varepsilon \rightarrow 0 \\ N \rightarrow \infty}} \left(\left. \Lambda(r) \psi_t(r) \right|{}_{\varepsilon} ^{N}-\int_{\varepsilon}^{N} \Lambda(r) d\psi_t(r)\right), \end{aligned} \end{align} $$
where
$\Lambda (r)=\int _{0}^{r} \lambda (t) t^{n-1} d t.$
Notice that
$$ \begin{align} 0\leq cr^n\psi(r)= \psi(r)\int_{\frac{r}{2}<|x|<r}dx\leq\int_{\frac{r}{2}<|x|<r}\psi(x)dx, \end{align} $$
where
$c=\Omega (1-\frac {1}{2^n})$
and
$\Omega $
is the volume of the unit ball. From the fact that
$\psi _t$
is in
$ L^{1}$
and decreasing, the right-hand side of (4.8) vanishes as
$r \rightarrow 0$
or
$r \rightarrow \infty ,$
so is
$r^{n} \psi _t(r) \rightarrow 0.$
Hence, using this observation and the following inequality,
$$ \begin{align*}\Lambda(r)=\int_{|y|<r} (T_xf)(y) d y \leq \Omega r^{n} \mathcal{M}(T_xf)(0),\end{align*} $$
we can show that the error term of (4.7),
$\Lambda (N)\psi _t(N)-\Lambda (\varepsilon )\psi _t(\varepsilon ),$
tends to zero as
$\varepsilon \rightarrow 0$
and
$N \rightarrow \infty .$
Thus, we have
$$ \begin{align*}\psi_{t} * (T_xf)(0)=\int_{0}^{N} \Lambda(r) d(-\psi_t(r)) \leq \Omega \mathcal{M}(T_xf)(0) \int_{0}^{\infty} r^{n} d(-\psi_t(r)).\\[-34pt] \end{align*} $$
Next, we will establish the pointwise convergence result for specific modulation spaces.
Proposition 4.4 Suppose
$\varphi $
is as defined in Theorem 4.3. Let
$1 \leq p, q < \infty ,$
and
$f \in M^{p, q}\cap L^1_{loc}$
be nonnegative. Then
$$ \begin{align} \lim _{t \rightarrow 0}(\varphi_{t} * f)(x)=f(x) \mbox{ almost everywhere}. \end{align} $$
Proof We begin by considering the following expression:
$$ \begin{align} \begin{aligned} \lim _{t \rightarrow 0} \left(\varphi_{t} * (f * M_{\xi} {\phi^*})\right)(x) &=\lim _{t\rightarrow 0}\left((\varphi_{t} * f) * M_{\xi} {\phi^*}\right)(x)\\ &=\lim _{t\rightarrow 0}\int_{\mathbb R^n} (\varphi_{t} * f)(y) M_{\xi} {\phi^*}(x-y) d y. \end{aligned} \end{align} $$
We aim to use the DCT on the right-hand side of equation (4.10). To do this, we define a sequence of functions as follows:
$$ \begin{align*}F_t(y):=(\varphi_{t} * f)(y) M_{\xi} {\phi^*}(x-y).\end{align*} $$
Now, applying Theorem 4.3, we obtain the inequality
$$ \begin{align} |F_t(y)|\leq A\,|\mathcal Mf(y) M_{\xi} {\phi^*}(x-y)|\leq A\,\mathcal Mf(y)|{\phi^*}|(x-y), \end{align} $$
for all
$t>0.$
In order to utilize DCT, we need the right-hand side of equation (4.11) to be an integrable function. Employing Lemma 4.1, it can be readily observed that
$$ \begin{align*}\int_{\mathbb R^n} \mathcal Mf(y)|{\phi^*}|(x-y)dy=\mathcal Mf * |{\phi^*}|(x)<\infty\end{align*} $$
for almost every
$x.$
Hence, by DCT, equation (4.10) yields
$$ \begin{align} \lim _{t \rightarrow 0} \left(\varphi_{t} * (f * M_{\xi} {\phi^*})\right)(x) &=\int_{\mathbb R^n}\lim _{t\rightarrow 0} \left(\varphi_{t} * f\right)(y) M_{\xi} {\phi^*}(x-y) d y\nonumber\\ &=\left(\lim _{t \rightarrow 0} (\varphi_{t} * f)\right) * M_{\xi} {\phi^*}(x). \end{align} $$
On the other hand, Theorem 3.2(2) implies
$$ \begin{align} \lim _{t \rightarrow 0} \left(\varphi_{t} *\left(f * M_{\xi} {\phi^*}\right)\right)(x)=(f * M_{\xi} {\phi^*})(x), \end{align} $$
since
$f * M_{\xi } {\phi ^*} \in L^{p}.$
Now, comparing equations (4.12) and (4.13), we obtain
$$ \begin{align*}\left(\lim _{t \rightarrow 0}\left(\varphi_{t} * f\right)-f\right) * M_{\xi} {\phi^*}(x)=0\end{align*} $$
for almost every
$x\in \mathbb R^n.$
Therefore, we can conclude that
$\lim _{t \rightarrow 0} (\varphi _{t} * f)(x)=f(x)$
almost everywhere.
Remark 4.5 In Proposition 4.4, we established a result analogous to Theorem 3.1 (2), although the boundedness of maximal function is not available in our setting. In this case, the result holds only in one direction, and no weight function has been considered. The main idea of the proof relies on the
$L^p$
-boundedness of the Hardy–Littlewood maximal operator applied to the convolution
$f * |\phi ^*|$
. In general, for an arbitrary function belonging to a modulation space, the
$L^p$
-boundedness of
$\mathcal {M}(f * |\phi ^*|)$
requires that
$|f| * |\phi ^*|$
lies in some
$L^p$
space. However, modulation spaces are not necessarily closed under the modulus operation, which makes it difficult to extend Proposition 4.4 to all functions in modulation spaces.
Moreover, by employing certain algebraic properties of modulation spaces, we can extend the validity of Proposition 4.4 to arbitrary functions (not necessarily restrict nonnegative) belonging to specific modulation spaces. We will discuss such spaces in the following result.
Theorem 4.6 Let
$1 \leq p< \infty $
and
$1 \leq q \leq 2.$
Then (4.9) holds for all
$f\in M^{p, q}\cap L^1_{loc}.$
Proof For
$f \in M^{p,q}$
, we have
$$ \begin{align} (|f| * |\phi^*|)^2 \leq f^2 * |\phi^*|^2. \end{align} $$
If it can be shown that the square of any function in
$M^{p,q}$
(
$1 \leq p < \infty ,\ 1 \leq q \leq 2$
) lies in some other modulation space, then the right-hand side of (4.14) will belong to some
$L^p$
space. This would imply
$|f| * |\phi ^*| \in L^{2p}$
, from which the desired result follows by an argument analogous to Proposition 4.4.
The final step is to determine the modulation space containing the square
$f^2$
of a given function f. By invoking the algebra property in Lemma 3.5(3), we conclude that
$f^2$
belongs to an appropriate modulation space, thereby completing the proof of Theorem 4.6.
5 Heat equation with Laplacian
In this section, we recall the definition of the weight classes
$D^h_{p,q}(\mathbb {R}^{2n})$
as introduced in the introduction. We then proceed to establish Theorem 2.1 (Laplacian part) by first proving a series of auxiliary lemmas that are essential to the overall argument.
Remark 5.1 (Proof strategy)
Let us briefly outline the key ideas used to characterize the weighted modulation spaces in which almost everywhere convergence holds (i.e., the strategy for proving Theorem 2.1):
-
- By proving Proposition 5.4, we can infer the appropriate weight classes in the context of modulation spaces.
-
- By Lemma 5.2, the weight class on
$\mathbb {R}^{2n}$
, as considered in Theorem 2.1, can be suitably related to the previously studied weight class on
$\mathbb {R}^n$
(see (1.8)). This correspondence ensures that the weight class on
$\mathbb {R}^{2n}$
satisfies the necessary structural conditions required to apply existing results from the theory of weighted Lebesgue spaces. -
- Then, by Lemma 5.3, we deduce that if
$f \in M_v^{p, q}$
is nonnegative, there exists a test function
$\phi \in \mathcal {S}$
such that
$f * |\phi | \in L_v^{p}$
for some suitable v on
$\mathbb {R}^n$
. This estimate allows us to apply known boundedness results within the framework of weighted Lebesgue spaces. -
- Conversely, pointwise convergence ensures that
$ f $
belongs to the required weighted space
$ M_v^{p,q} $
(as shown in the proof of Proposition 5.4). -
- However, in the case of the Hermite operator, additional effort is required to address this situation effectively (see Section 6 for details).
Lemma 5.2 Let v be a strictly positive weight in
$\mathbb {R}^{2n}.$
Denote
$v_\xi (x)=v(x,\xi )$
with
$\xi $
fixed. If
$v\in D^h_{p,q}(\mathbb {R}^{2n}),$
then
$v_\xi \in D^h_p(\mathbb {R}^{n})$
for almost every
$ \xi \in \mathbb {R}^n $
.
Proof Firstly, assume that
$v \in D_{p, q}^{h}$
. Then, by definition, there exist
$t_{0}$
such that
$h_{t_{0}} \in M_{v^{-1}}^{p, q}$
. That is,
$V_{\phi } h_{t_{0}} \in L_{v^{-1}}^{p, q},$
where
$V_{\phi } h_{t_{0}}$
is the STFT of
$h_{t_{0}}$
with respect to test function
$\phi \in \mathcal {S}\backslash \{0\}.$
Without loss of generality, we can choose
$\phi =h_{t_{0}}.$
We can write
$h_{t_{0}}(x)=D_{\sqrt {4 \pi {t_0}}}h(x),$
where the notation
$D_{t} h(x)=t^{-n} h\left (t^{-1} x\right )$
with
$h(x)=e^{-\pi |x|^2}$
. Now, consider
$V_{h_{t_{0}}} h_{t_{0}}$
:
$$ \begin{align} V_{h_{t_{0}}} h_{t_{0}}(x, \xi) & =\int_{\mathbb R^{n}} D_{\sqrt{4 \pi t_0}} h(y) \,D_{\sqrt{4 \pi t_0}} h(y-x)\, e^{-2 \pi i y \cdot \xi} d y \nonumber\\ & =\int_{\mathbb R^{n}} D_{\sqrt{8 \pi t_0}} h(x) \,D_{\sqrt{2 \pi t_0}} h\left(y-\frac{x}{2}\right)\, e^{-2 \pi i y \cdot \xi} d y \nonumber\\ & =D_{\sqrt{8 \pi t_0}} h(x) \,\mathcal{F}\left({T_{x / 2} D_{\sqrt{2 \pi t_0}}}h\right)(\xi) \nonumber\\ & =D_{\sqrt{8 \pi t_0}} h(x)\, M_{x / 2}\left(\mathcal{F}(h)\right)(\sqrt{2 \pi t_0} \xi) \nonumber\\ & = e^{-2 \pi^{2} t_0 |\xi|^{2}}\,e^{-\pi i x \cdot \xi}\, h_{\frac{t_0}{2}}(x). \end{align} $$
From equation (5.1), we deduce that
$ h_{\frac {t_0}{2}} \in L^p_{v_\xi } $
for almost every
$ \xi \in \mathbb {R}^n $
, since
$ V_{h_{t_{0}}} h_{t_{0}} \in L_{v^{-1}}^{p, q} $
. Hence, we can conclude that
$ v_\xi \in D^h_p $
for almost every
$ \xi \in \mathbb {R}^n .$
This completes the proof.
Lemma 5.3 Let
$1 \leq p,q< \infty ,$
and
$v \in D^h_{p,q}(\mathbb {R}^{2n}).$
Suppose
$f \in M_v^{p, q}\cap L^1_{loc}$
be nonnegative. Then there exists some
$\phi \in \mathcal S$
such that
$f * |\phi |\in L_v^{p}$
for some
$v\in D^h_p(\mathbb {R}^n).$
Moreover,
$\mathcal Mf * |\phi |\in L_u^{p}$
for some weight u on
$\mathbb {R}^{n}.$
Proof Let
$\rho \in L^{p'}(\mathbb {R}^n)$
, where
$p'$
is the Hölder conjugate of p. Denote
$\eta (x)=e^{- |x|^2}.$
Consider
$$ \begin{align*}\langle \eta(f * |\phi|),\,\rho \rangle &= \int_{\mathbb{R}^n} \eta(x)\,(f * |\phi|)(x)\, \overline{\rho(x)} \, dx\\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \eta(x)\,f(y) \,|\phi(x - y)| \,\overline{\rho(x)} \, dy \, dx. \end{align*} $$
Let’s choose
$\eta (x)=|\phi |(x)=e^{- |x|^2}.$
By Fubini’s theorem and Hölder’s inequality, we obtain
$$ \begin{align} |\langle \eta(f * |\phi|),\, \rho \rangle| &\leq \int_{\mathbb{R}^n} f(y) \left( \int_{\mathbb{R}^n} e^{- |x|^2}e^{- |x-y|^2} |\rho(x)| \, dx \right) dy \nonumber\\ &= \int_{\mathbb{R}^n} f(y)\, e^{- \frac{|y|^2}{2}}\left( \int_{\mathbb{R}^n} e^{- |x-\frac{y}{2}|^2} |\rho(x)| \, dx \right) dy \nonumber\\ &= \int_{\mathbb{R}^n} f(y)\,e^{- \frac{|y|^2}{2}} \left( \int_{\mathbb{R}^n} |T_{\frac{y}{2}} \phi(x)| \, |\rho(x)| \, dx \right) dy \nonumber\\ &\leq \int_{\mathbb{R}^n} f(y)\, e^{- \frac{|y|^2}{2}}\|T_{\frac{y}{2}} \phi\|_{L^p} \,\|\rho\|_{L^{p'}} \, dy \nonumber\\ &= \|T_{\frac{y}{2}} \phi\|_{L^p}\,\|\rho\|_{L^{p'}} \int_{\mathbb{R}^n} f(y) \, e^{- \frac{|y|^2}{2}} \, dy. \end{align} $$
Let
$f \in M_v^{p,q}$
with
$v \in D^h_{p,q}(\mathbb {R}^{2n})$
. We can choose
$t_0 \leq \frac {1}{2}$
such that
$\|h_{t_0}\|_{M_{v^{-1}}^{p', q'}} < \infty $
, where
$h_t$
denotes the heat kernel defined in (1.3). Then, we have
$$ \begin{align*}|\langle \eta(f * |\phi|), \rho \rangle| \lesssim \|T_{\frac{y}{2}} \phi\|_{L^p} \, \|\rho\|_{L^{p'}} \int_{\mathbb{R}^n} f(y) \, h_{t_0}(y) \, dy. \end{align*} $$
Using the duality between
$M_v^{p,q}$
and
$M_{v^{-1}}^{p', q'}$
, this yields
$$ \begin{align*}|\langle \eta(f * |\phi|), \rho \rangle| \lesssim \|T_{\frac{y}{2}} \phi\|_{L^p} \, \|\rho\|_{L^{p'}} \, \|f\|_{M_v^{p,q}} \, \|h_{t_0}\|_{M_{v^{-1}}^{p', q'}}. \end{align*} $$
Taking the supremum over all
$\rho \in L^{p'}$
with
$\|\rho \|_{L^{p'}} \leq 1$
, we obtain
$$ \begin{align*}\|\eta(f * |\phi|)\|_{L^p} \lesssim (2\pi)^{\frac{n}{2}} \, \|T_{\frac{y}{2}} \phi\|_{L^p} \, \|f\|_{M_v^{p,q}} \, \|h_{t_0}\|_{M_{v^{-1}}^{p', q'}}. \end{align*} $$
Therefore, we conclude that
$f * |\phi | \in L^p_{\eta ^p}$
, where
$\eta ^p \in D^h_p$
.
Proposition 5.4 (Characterization of
$D^h_{p,q}$
)
Suppose
$ 1 \leq p ,q< \infty $
. The weight
$ v \in D^h_{p,q}(\mathbb {R}^{2n}) $
if and only if there exist
$ t_0> 0 $
and a weight
$ u $
on
$\mathbb {R}^{2n}$
such that the operator
$ f \mapsto h_{t_0} * f $
maps
$ M_v^{p,q} $
into
$ M_u^{p,q} $
with the norm inequality
$$\begin{align*}\| h_{t_0} * f \|_{M_u^{p,q}} \lesssim \| f \|_{M_v^{p,q}}. \end{align*}$$
We need the following lemma to prove Proposition 5.4.
Lemma 5.5 Suppose
$ 1 \leq p, q < \infty $
. Define
$$\begin{align*}g_{t}(x) = \left\|h_{t}(x - \cdot)\right\|_{M_{v^{-1}}^{p', q'}} \quad\text{for }t>0.\end{align*}$$
If
$ v \in D^h_{p,q}(\mathbb {R}^{2n}) $
, then there exist
$ t_0>0 $
and a weight
$ u $
on
$\mathbb {R}^{2n}$
such that
$\left \|g_{t_0}\right \|_{M_u^{p, q}} < \infty .$
Proof Consider the function
$g_t(x)$
, which can be expressed as follows:
$$\begin{align*}g_t(x) = \left\| h_t(x - \cdot) \right\|_{M_{v^{-1}}^{p', q'}} \lesssim \left\| V_{h_t} h_t(x - \cdot, \cdot) \right\|_{L_{v^{-1}}^{p', q'}}. \end{align*}$$
Using equation (5.1), we can write
$ g_{t_0}(x) $
as
$$ \begin{align} g_{t_0}(x) = \left( \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} \left( e^{-2 \pi^2 {t_0} |\xi|^2} \, h_{\frac{t_0}{2}}(x - y) \right)^p v(y, \xi) \, dy \right)^{\frac{q}{p}} d\xi \right)^{\frac{1}{q}} \end{align} $$
for some
$t_0>0$
(from Lemma 5.2). Next, by using the conditions,
$$\begin{align*}\begin{cases} |x - y| \leq |x| & \Rightarrow |y| \leq |x - y| + |x| \leq 2|x| \\ |x - y|> |x| & \Rightarrow |y| \leq |x - y| + |x| \leq 2|x - y|, \end{cases} \end{align*}$$
we can deduce that
$ h_{t_0}(x - y) \lesssim h_{\frac {t_0}{4}}(y) $
. Substituting this estimate into equation (5.3), we obtain
$$\begin{align*}g_{t_0}(x) \lesssim \left\| V_{h_{\frac{t_0}{8}}} h_{\frac{t_0}{8}} \right\|_{L_{v^{-1}}^{p', q'}}. \end{align*}$$
Hence, it follows that
$ g_{t_0}(x) < \infty $
a.e. for
$ x $
, since we are given that
$$\begin{align*}\left\| h_{\frac{t_0}{8}} \right\|_{M_{v^{-1}}^{p', q'}} < \infty. \end{align*}$$
Now, we will choose a weight
$ u \in L_{\text {loc}}^1 $
such that
$ g_{t_0} \in M_u^{p, q} $
, which is equivalent to showing
$ V_\phi g_{t_0} \in L_u^{p, q} $
for some
$\phi \in \mathcal {S}.$
By choosing
$ u \in L_{\text {loc}}^1$
as
$$\begin{align*}u(x, \xi) = \begin{cases} 1 & \text{if } |V_\phi g_{t_0}(x, \xi)| \leq u'(x, \xi) \\ \frac{u'(x, \xi)}{|V_\phi g_{t_0}(x, \xi)|} & \text{if } |V_\phi g_{t_0}(x, \xi)|> u'(x, \xi), \end{cases} \end{align*}$$
where
$u'\in L^{p,q},$
we get that
$ g_{t_0} \in L_u^{p, q} $
. This means that we are choosing
$ u $
in such a way that
$ |V_\phi g_{t_0}| \cdot u$
is dominated by a function in
$ L^{p, q} $
.
Proof of Proposition 5.4
By using the Moyal identity (Lemma 3.5(2)) and Hölder’s inequality, for any
$ t> 0 $
, we have
$$ \begin{align*} \left\|h_t * f\right\|_{M_u^{p, q}}\nonumber & = \left\|\int h_t(\cdot - y) f(y) \, dy\right\|_{M_u^{p, q}} \\\nonumber & \lesssim \left\|\int V_\phi h_t(\cdot - y) V_\phi f(y) \, dy\right\|_{M_u^{p, q}} \\\nonumber & \leq \left\| \left\|f\right\|_{M_v^{p, q}}g_{t}(.)\right\|_{M_u^{p, q}}=\left\|f\right\|_{M_v^{p, q}} \left\|g_{t}\right\|_{M_u^{p, q}}, \end{align*} $$
where
$g_t$
is as defined in Lemma 5.5. Hence, we have
$ \left \|h_{t_0} * f\right \|_{M_u^{p, q}} \lesssim \left \| f \right \|_{M_v^{p, q}} $
for some
$ t_0>0$
, since Lemma 5.5 guarantees that there exist
$ t_0 $
and a weight
$ u $
such that
$\left \|g_{t_0}\right \|_{M_u^{p, q}} < \infty .$
Conversely, let
$$\begin{align*}\left\|h_{t_0} * f\right\|_{M_u^{p, q}} \lesssim \left\| f \right\|_{M_v^{p, q}} \end{align*}$$
for all
$ f \in M_v^{p, q} $
. Thus,
$ |h_{t_0} * f(x)| < \infty $
almost everywhere for all
$ f \in M_v^{p, q} $
. Fix
$ x_0 $
such that
$ h_{t_0} * f(x_0) < \infty $
. Then we will show that
$ h_{\frac {t_0}{4}} * f(x) < \infty $
for all
$ x $
. Let assume
$x \neq x_{0}$
. Note that
$$ \begin{align} \begin{cases} |x-y| \leq |x-x_{0}| \implies\left|y-x_{0}\right| \leq 2\left|x-x_{0}\right|\\ |x-y| \geq |x-x_{0}| \implies \left|x_{0}-y\right| \leq 2|x-y|. \end{cases} \end{align} $$
By (5.4), we obtain
$$ \begin{align*}h_{\frac{t_{0}}{4}}(x-y) \lesssim \begin{cases} \frac{h_{t_{0}}\left(x_{0}-y\right)}{h_{t_{0}}\left(2\left(x-x_{0}\right)\right)} \lesssim h_{t_{0}}\left(x_{0}-y\right) & if \ |x-y| \leq |x-x_{0}| \\ h_{\frac{t_{0}}{4}}\left(\frac{x_{0}-y}{2}\right) \lesssim h_{t_{0}}\left(x_{0}-y\right) & if \ |x-y| \geq |x-x_{0}|. \end{cases} \end{align*} $$
Thus, we have
$$ \begin{align*} h_{\frac{t_0}{4}} \ast f(x) & \lesssim \left(\int_{|x-y|<|x-x_{0}|} + \int_{|x-y| \geq |x-x_{0}|}\right) h_{t_{0}}\left(x_{0}-y\right) f(y) d y\nonumber\\ & \lesssim h_{t_0}\ast f (x_0) < \infty \end{align*} $$
for all
$x \in \mathbb {R}^{n} \backslash \{x_{0}\}$
. Note that the later inequality in (5.4) also holds for
$x=x_0.$
That is,
$$\begin{align*}0 \leq \int_{\mathbb{R}^{n}} h_{\frac{t_{0}}{4}}\left(x_{0}-y\right) f(y) d y \lesssim \int_{\mathbb{R}^{n}} h_{t_{0}}\left(x_{0}-y\right) f(y) d y < \infty.\end{align*}$$
Thus, we have
$$\begin{align*}\int_{\mathbb{R}^{n}} h_{\frac{t_{0}}{4}}(x-y) f(y) d y< \infty \quad for\ all\ \ x\in \mathbb R^n.\end{align*}$$
In particular,
$ h_{\frac {t_0}{4}} * f(0) < \infty $
, that is,
$$\begin{align*}\int h_{\frac{t_0}{4}}(y) f(y) \, dy < \infty. \end{align*}$$
By duality,
$ h_{\frac {t_0}{4}} \in M_{v^{-1}}^{p', q'} $
, we can conclude that
$ v \in D^h_{p, q} $
.
Now, we are ready to prove our main result.
Proof of Theorem 2.1 (Laplacian case)
Assume that
$ v \in D^h_{p,q} $
and consider the following expression:
$$ \begin{align} \lim_{t \rightarrow 0} \left( h_{t} * (f * M_{\xi} {\phi^*})\right)(x) &= \lim_{t \rightarrow 0} \left(( h_{t} * f) * M_{\xi} {\phi^*}\right)(x) \nonumber\\ &= \lim_{t \rightarrow 0} \int_{\mathbb{R}^n} (h_{t} * f)(y) M_{\xi} {\phi^*}(x - y) \, dy. \end{align} $$
We aim to use the DCT on the right-hand side of equation (5.5) and interchange the limit and integration. To do this, we define a sequence of functions by fixing x as follows:
$$\begin{align*}F_t(y) := (h_{t} * f) (y) M_{\xi} {\phi^*}(x - y). \end{align*}$$
Now, applying Theorem 4.3, we obtain the inequality
$$ \begin{align} |F_t(y)| \leq A\, |\mathcal{M}f(y) M_{\xi} {\phi^*}(x - y)| \leq A\, \mathcal{M}f(y) | {\phi^*}|(x - y) \end{align} $$
for all
$ t> 0 $
. In order to utilize the DCT, we need the right-hand side of equation (5.6) to be an integrable function. Employing Lemma 5.3, we observe that
$$\begin{align*}\int_{\mathbb{R}^n} \mathcal{M}f(y) | {\phi^*}|(x - y) \, dy = \mathcal{M}f * | {\phi^*}|(x) < \infty \end{align*}$$
for almost every
$ x $
. Consequently, we can apply the DCT in equation (5.5) to obtain
$$ \begin{align} \lim_{t \rightarrow 0} \left( h_{t} * (f * M_{\xi} {\phi^*})\right)(x) &= \int_{\mathbb{R}^n} \lim_{t \rightarrow 0} \left( h_{t} * f\right)(y) M_{\xi} {\phi^*}(x - y) \, dy \nonumber\\ &= \left(\lim_{t \rightarrow 0} ( h_{t} * f)\right) * M_{\xi} {\phi^*}(x). \end{align} $$
However, by Lemma 5.3, which states that
$v \in D^h_{p,q}$
implies
$v_\xi \in D^h_p$
for almost every
$\xi \in \mathbb {R}^n$
, it follows that
$f \in M_v^{p, q}$
implies
$f * M_{\xi } {\phi ^*} \in L_{v_\xi }^{p}$
for almost every
$\xi \in \mathbb {R}^n$
. Consequently, by applying Theorem 1.1, which states that
$ v \in D^h_p(\mathbb {R}^{n}) $
if and only if
$\lim _{t \to 0} (h_t * f)(x) = f(x) \quad \text {a.e. for all } f \in L^p_v(\mathbb {R}^n),$
we obtain
$$ \begin{align} \lim_{t \rightarrow 0} \left( h_{t} * \left(f * M_{\xi} {\phi^*}\right)\right)(x) = f * M_{\xi} {\phi^*}(x). \end{align} $$
Now, comparing equations (5.7) and (5.8), we derive
$$\begin{align*}\left(\lim_{t \rightarrow 0} \left( h_{t} * f\right) - f\right) * M_{\xi} {\phi^*}(x) = 0 \end{align*}$$
for almost every
$ x \in \mathbb {R}^n $
. Therefore, we can conclude that
$ \lim _{t \rightarrow 0} ( h_{t} * f) = f $
almost everywhere.
Conversely, let us assume
$ \lim _{t \rightarrow 0} h_{t} * f(x) = f(x) $
almost everywhere for all nonnegative
$ f \in M_v^{p, q} $
. By Proposition 5.4, there exists
$ t_0>0 $
such that
$ h_{t_0} * f(x) < \infty $
almost everywhere for all nonnegative
$ f \in M_v^{p, q} $
. Fix
$ x_0 $
such that
$ h_{t_0} * f(x_0) < \infty $
. Following the same approach as in the converse part of Proposition 5.4, we can similarly obtain that
$h_{\frac {t_0}{4}} * f(x) < \infty $
for all
$ x \in \mathbb {R}^n $
, that is,
$$\begin{align*}\int_{\mathbb{R}^n} h_{\frac{t_0}{4}}(y) f(y) \, dy < \infty. \end{align*}$$
By duality,
$ h_{\frac {t_0}{4}} \in M_{v^{-1}}^{p', q'} $
, which implies that
$ v \in D^h_{p,q} $
.
6 Heat equation with Hermite operator
In this section, we consider the heat semigroup
$e^{-t H} f$
and derive their simplified expressions. We then proceed to prove our main result, Theorem 2.1 (Hermite part).
Consider the heat equation (1.1) associated with the harmonic oscillator
$ H $
, whose heat semigroup is given by (1.5). The spectral decomposition of the Hermite operator is expressed in (1.4), where the eigenfunctions are the normalized Hermite functions
$$\begin{align*}\Phi_\alpha(x) = \prod_{j=1}^n \tilde{h}_{\alpha_j}(x_j)\end{align*}$$
and each
$ \tilde {h}_k $
denotes the one-dimensional normalized Hermite function, given by
$$\begin{align*}\tilde{h}_k(x) = \left( \sqrt{\pi}, 2^k k! \right)^{-1/2} (-1)^k e^{\frac{x^2}{2}} \frac{d^k}{dx^k}\big( e^{-x^2} \big).\end{align*}$$
These functions form an orthonormal basis for
$L^2$
and satisfy the eigenvalue relation
$$ \begin{align*}H \Phi_\alpha = (2|\alpha| + n)\Phi_\alpha,\end{align*} $$
where
$|\alpha | = \alpha _1 + \dots + \alpha _n$
. A fundamental identity involving Hermite functions is Mehler’s formula (see [Reference Thangavelu42, Lemma 1.1.1]), which for
$|w| < 1$
takes the form:
$$ \begin{align} \sum_{k=0}^{\infty} \frac{\tilde{h}_k(x) \tilde{h}_k(y)}{2^k k!} w^k=\left(1-w^2\right)^{-\frac{1}{2}} e^{-\frac{1}{2} \frac{1+w^2}{1-w^2}\left(x^2+y^2\right)+\frac{2 w}{1-w^2} x y}. \end{align} $$
Using Mehler’s formula (6.1), we can rewrite the heat semigroup (1.5) as follows:
$$ \begin{align} e^{-t H} f(x) = \int_{\mathbb{R}^n} h_t^H(x, y) f(y) \, dy = \int_{\mathbb{R}^n} \frac{e^{-\left[\frac{1}{2}|x-y|^2 \coth 2t + x \cdot y \tanh t\right]}}{(2 \pi \sinh 2t)^{\frac{n}{2}}} f(y) \, dy \end{align} $$
as shown in [Reference Thangavelu42, Equation (4.1.2)]. By applying Stefano Meda’s change of parameters
$$\begin{align*}t = \frac{1}{2} \log \frac{1+s}{1-s} \quad \text{for } t \in (0, \infty) \text{ and } s \in (0,1),\end{align*}$$
equivalently
$s=\tanh {t},$
we obtain
$$ \begin{align} h_{t}^H(x, y) = \left( \frac{1-s^2}{4 \pi s} \right)^{\frac{n}{2}} e^{-\frac{1}{4}\left[s|x+y|^2 + \frac{1}{s}|x-y|^2\right]}. \end{align} $$
In the limit as
$ s \to 0^{+} $
, it follows that
$ t \to 0^{+} $
as well. From equation (6.3), we observe that the following inequality holds:
$$ \begin{align} h_{t}^H(x, y) \leq (1 - s^2) h_s(x - y), \end{align} $$
where
$ h_s $
denotes the classical heat kernel (1.3). Furthermore, using the conditions,
$$\begin{align*}\begin{cases} 2|x| < |y| & \Rightarrow |x + y| \leq 3|x - y| \\ 2|x| \geq |y| & \Rightarrow |x + y| \leq 3|x|, \end{cases} \end{align*}$$
we can deduce the following estimates:
$$\begin{align*}\begin{aligned} & \left\{ \begin{array}{@{}l} e^{-\frac{1}{4}\left[s |x + y|^2 + \frac{1}{s} |x - y|^2 \right]} \geq e^{-\left( \frac{9 s^2 + 1}{s} \right) \frac{|x - y|^2}{4}} \\ e^{-\frac{1}{4}\left[s |x + y|^2 + \frac{1}{s} |x - y|^2 \right]} \geq e^{-\frac{9s}{4} |x|^2} e^{-\left( \frac{9 s^2 + 1}{s} \right) \frac{|x - y|^2}{4}}. \end{array} \right. \end{aligned} \end{align*}$$
Thus, for any
$ (x, y) $
and
$ 0 < s < 1 $
, with the relation
$ s = \tanh t $
, the following inequality holds:
$$ \begin{align} h_{t}^H(x, y) \geq e^{-\frac{9s}{4} |x|^2} \left( \frac{1 - s^2}{1 + 9 s^2} \right)^{\frac{n}{2}} h_{\frac{s}{1 + 9 s^2}}(x - y). \end{align} $$
In the following, we present the proof of the Hermite heat part of Theorem 2.1.
Proof of Theorem 2.1 (Hermite case)
Let
$ f $
be a nonnegative function in
$ M^{p, q}_v $
. By Lemma 5.3, there exists a test function
$ \phi \in \mathcal {S} $
such that
$ f * |\phi | \in L_v^p $
for some weight
$ v \in D^h_p $
. To prove that
$ \lim _{t \to 0} e^{-t H} f(x) = f(x) $
almost everywhere, it suffices to show that
$$ \begin{align} \left( \lim_{t \to 0} e^{-t H} f \right) * |\phi|(x) = f * |\phi|(x) \quad \text{almost everywhere}. \end{align} $$
Now, consider the left-hand side. By applying a method similar to that used in (5.5) and (5.6) and invoking the DCT, we can express
$$\begin{align*}\left(\lim_{t \rightarrow 0} e^{-t H} f\right) * |\phi|(x) = \lim_{t \rightarrow 0} \left(e^{-t H} f * |\phi|\right)(x)\quad\mbox{almost everywhere}.\end{align*}$$
Now, we have
$$ \begin{align*} \begin{aligned} \lim_{t \rightarrow 0} (e^{-t H} f * |\phi|)(x) &= \lim_{t \rightarrow 0}\int_{\mathbb{R}^n} e^{-t H} f(y) |\phi|(x - y) \, dy\\ &= \lim_{t \rightarrow 0}\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} h_{t}^H(y, y') f(y') \, |\phi|(x - y) \, dy'dy. \end{aligned} \end{align*} $$
Using upper bound (6.4) applying the associativity of convolution, we obtain
$$ \begin{align*} \begin{aligned} \lim_{t \rightarrow 0} (e^{-t H} f * |\phi|)(x) \leq&\lim_{s \rightarrow 0}\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} (1 - s^2) h_s(y - y') f(y') \, |\phi|(x - y) \, dy'dy \\ =&\lim_{s \rightarrow 0}(1 - s^2)\left((h_s\ast f)\ast |\phi|\right)(x)\\ =&\lim_{s \rightarrow 0}\left(h_s\ast (f\ast |\phi|)\right)(x). \end{aligned} \end{align*} $$
However, by Lemma 5.2, we know that
$ f * |\phi | \in L_{v}^{p} $
for some weight
$ v \in D^h_p $
. Consequently, using Theorem 1.1, we arrive
$$ \begin{align} \left(\lim_{t \rightarrow 0} e^{-t H} f\right) * |\phi|(x)\leq f * |\phi|(x)\text{ a.e. }x\in\mathbb{R}^n. \end{align} $$
Proceeding in a similar manner and utilizing the lower bound from equation (6.5), we obtain the inequality
$$ \begin{align} \left( \lim_{t \to 0} e^{-t H} f \right) * |\phi|(x) \geq f * |\phi|(x) \quad \text{a.e. } x \in \mathbb{R}^n. \end{align} $$
Thus, combining (6.7) and (6.8), we obtain the conclusion (6.6), and hence establish the pointwise convergence almost everywhere.
Conversely, let us assume
$ \lim _{t \rightarrow 0} e^{-t H} f(x) = f(x) $
almost everywhere for all nonnegative
$ f \in M_v^{p, q} $
. Then, by (6.4), there exists
$ t_0>0 $
such that
$$\begin{align*}e^{-t H} f(x)\lesssim h_{t_0} * f(x) < \infty \end{align*}$$
almost everywhere for all nonnegative
$ f \in M_v^{p, q} $
. Following the same approach as in the converse part of Proposition 5.4, we can similarly deduce that
$ h_{\frac {t_0}{4}} \in M_{v^{-1}}^{p', q'} $
, which implies
$ v \in D^h_{p,q} $
.
7 Concluding remarks
Our main Theorems 2.1 and 2.5 naturally open the door for the further investigation. In view of ongoing research interest mentioned in the introduction, we raise some interesting open questions in the following points:
-
(1) Can we relax the the non-negativity assumption on the initial data in Theorem 2.1? The present proofs rely on positivity arguments and maximal function techniques. However, it would be of independent interest to determine whether the equivalence remains valid for general (possibly sign-changing) functions in weighted modulation spaces (see Remark 2.2).
-
(2) It would be worthwhile to extend the analysis to other semigroups, such as Poisson-type semigroups or semigroups generated by more general differential operators. See, for example, the discussion following Theorem 1.1. In such settings, additional analytical difficulties may arise due to weaker kernel regularity or the absence of explicit representations.
-
(3) Up to now, our knowledge of maximal operator on modulation spaces is quite limited (see Theorem 2.5). The systematic investigation of the boundedness properties of maximal operators in modulation spaces remains an important open direction.
-
(4) The combination of semigroup kernel estimates with time–frequency techniques provides a flexible method for deriving pointwise convergence theorems in non-classical function spaces. This sheds new light on how large (rough) initial data one can take (see Remark 2.3). We expect that the ideas developed here may serve as a foundation for further results on almost everywhere convergence and related fine properties of other evolution equations in the modulation space setting.
Acknowledgements
We sincerely thank the referees for their careful reading of our manuscript and for their valuable comments and suggestions, which have significantly improved the presentation of the article.
