On potentials of distributions in Orlicz-Hardy type spaces on the Heisenberg group

Abstract: In this work, we introduce Orlicz-Hardy type spaces and Orlicz-Calderón Hardy type spaces on the Heisenberg group $\mathbb{H}<sup>{n}$ and study the relationship between them by means of the Heisenberg sub-Laplacian $\mathcal{L}$. More precisely, we show, under suitable assumptions, that every distribution in the Orlicz-Hardy space $H<sup>Φ(\mathbb{H}<sup>{n})$ can be represented uniquely as the sub-Laplacian of a function in an appropriate Orlicz-Calderón Hardy space. In this way, for any $f \in H<sup>Φ(\mathbb{H}<sup>{n})$, we obtain a uniqueness and solvability result for the equation $\mathcal{L}F=f$.

• The paper proves that the sub-Laplacian is a bijection from the Orlicz-Calderón-Hardy space HΦq,2(Hⁿ) onto the Orlicz-Hardy space HΦ(Hⁿ) when Q(2 + Q/q)⁻¹ < i(Φ) ≤ I(Φ) < ∞ and 1 < q < (n + 1)/n.
• It establishes equivalent maximal-function characterizations and an atomic decomposition for HΦ(Hⁿ), enabling potential solutions to be constructed from atoms and controlled through Luxemburg norms, maximal inequalities, and singular-integral estimates.
• It shows the threshold is essentially sharp because HΦq,2(Hⁿ) is trivial when the upper type of Φ falls below Q(2 + Q/q)⁻¹, while the endpoint and higher-order equations remain open problems.

Background and problem

The paper studies the inhomogeneous equation $\mathcal{L}F = f$ on the Heisenberg group $\mathbb{H}^n$, where $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$ is the sub-Laplacian associated with the left-invariant horizontal vector fields, $f \in \mathcal{S}'(\mathbb{H}^n)$ is given data, and $F$ is the unknown. The homogeneous dimension is $Q = 2n+2$. The program originates with Gatto–Jiménez–Segovia, who solved $\Delta^m F = f$ for $f \in H^p(\mathbb{R}^n)$ by introducing Calderón–Hardy spaces; subsequent extensions covered non-isotropic dilations (Durán), variable exponents and weights (Rocha), and the Heisenberg group for $f \in H^p(\mathbb{H}^n)$ and $f \in H^{p(\cdot)}(\mathbb{H}^n)$ (Rocha). Liu–He–Mo recently treated the Orlicz case on $\mathbb{H}^n$0. This paper completes the parallel development on $\mathbb{H}^n$1: it introduces Orlicz-Hardy spaces $\mathbb{H}^n$2 and Orlicz-Calderón-Hardy spaces $\mathbb{H}^n$3 and proves that, under a type condition on $\mathbb{H}^n$4, the sub-Laplacian is an isomorphism between them.

The Orlicz framework interpolates between the fixed-exponent and variable-exponent settings: while $\mathbb{H}^n$5 adapts the exponent to position, $\mathbb{H}^n$6 adapts integrability to the size of $\mathbb{H}^n$7. The admissible class consists of Orlicz functions of positive lower type $\mathbb{H}^n$8 and positive upper type $\mathbb{H}^n$9, defined via $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$0 for $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$1 (resp. $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$2). Examples include $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$3, $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$4, $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$5, and $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$6.

Orlicz-Hardy spaces and atomic decomposition

The space $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$7 is defined via the grand maximal function $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$8, built from convolutions of $\mathcal{L} = -\sum_{i=1}^{2n} X_i^2$9 with dilates of Schwartz functions. A key structural result establishes that, for large enough $f \in \mathcal{S}'(\mathbb{H}^n)$0 and $f \in \mathcal{S}'(\mathbb{H}^n)$1, the Luxemburg norms of the grand maximal function, the discrete maximal function $f \in \mathcal{S}'(\mathbb{H}^n)$2, and the tangential maximal function $f \in \mathcal{S}'(\mathbb{H}^n)$3 are mutually comparable for every radial $f \in \mathcal{S}'(\mathbb{H}^n)$4 with nonzero integral. This well-definedness relies on boundedness of the Hardy-Littlewood maximal operator $f \in \mathcal{S}'(\mathbb{H}^n)$5 on $f \in \mathcal{S}'(\mathbb{H}^n)$6 when $f \in \mathcal{S}'(\mathbb{H}^n)$7, together with a Fefferman-Stein vector-valued inequality for $f \in \mathcal{S}'(\mathbb{H}^n)$8 on $f \in \mathcal{S}'(\mathbb{H}^n)$9 proved here via Muckenhoupt weights on $F$0.

The paper then proves an atomic decomposition: every $F$1 admits $F$2 in $F$3, where each $F$4 is a $F$5-atom supported on a Korányi ball $F$6 — bounded by $F$7 in $F$8 and vanishing moments up to homogeneous degree $F$9 — with the square-function-type control

$Q = 2n+2$0

The proof follows the Calderón-Zygmund/Whitney machinery of Folland-Stein, adapted through the auxiliary estimate that controls $Q = 2n+2$1-sums of maximal functions of atoms (Proposition on maximal functions over atom families), which itself rests on the vector-valued maximal inequality and a pointwise domination $Q = 2n+2$2.

Orlicz-Calderón-Hardy spaces

For $Q = 2n+2$3 and $Q = 2n+2$4, the space $Q = 2n+2$5 consists of classes $Q = 2n+2$6 in the quotient $Q = 2n+2$7 ($Q = 2n+2$8) whose maximal function $Q = 2n+2$9 belongs to $\Delta^m F = f$0. Two facts are established that make the space usable as a solution space:

Well-defined action of $\Delta^m F = f$1: if $\Delta^m F = f$2, any representative $\Delta^m F = f$3 lies in $\Delta^m F = f$4, and since distinct representatives differ by a polynomial of homogeneous degree at most $\Delta^m F = f$5, the distribution $\Delta^m F = f$6 is independent of the choice — so $\Delta^m F = f$7 is intrinsically defined.

Completeness and injectivity: $\Delta^m F = f$8 is complete (via the generalized Riesz-Fischer property, using Fatou's lemma and continuity of $\Delta^m F = f$9), and $f \in H^p(\mathbb{R}^n)$0 is injective: $f \in H^p(\mathbb{R}^n)$1 forces $f \in H^p(\mathbb{R}^n)$2. Injectivity uses the pointwise bound $f \in H^p(\mathbb{R}^n)$3 at infinity together with the positive upper type of $f \in H^p(\mathbb{R}^n)$4.

The central technical tool is a pointwise estimate for potentials of atoms: if $f \in H^p(\mathbb{R}^n)$5 is a $f \in H^p(\mathbb{R}^n)$6-atom supported on $f \in H^p(\mathbb{R}^n)$7 and $f \in H^p(\mathbb{R}^n)$8, where $f \in H^p(\mathbb{R}^n)$9 is Folland's fundamental solution of $f \in H^p(\mathbb{H}^n)$0, then

$f \in H^p(\mathbb{H}^n)$1

where $f \in H^p(\mathbb{H}^n)$2 are truncated singular integral operators associated with second-order derivatives of the fundamental solution. The decay exponent $f \in H^p(\mathbb{H}^n)$3 is precisely what makes the first term summable in $f \in H^p(\mathbb{H}^n)$4 under the hypothesis below.

Main theorem: solvability and uniqueness

The principal result states that for $f \in H^p(\mathbb{H}^n)$5 and any Orlicz function satisfying

$f \in H^p(\mathbb{H}^n)$6

the sub-Laplacian is a bijection from $f \in H^p(\mathbb{H}^n)$7 onto $f \in H^p(\mathbb{H}^n)$8, with two-sided norm equivalence $f \in H^p(\mathbb{H}^n)$9. Consequently, for every $f \in H^{p(\cdot)}(\mathbb{H}^n)$0 there exists a unique $f \in H^{p(\cdot)}(\mathbb{H}^n)$1 solving $f \in H^{p(\cdot)}(\mathbb{H}^n)$2.

Surjectivity is constructive: decomposing $f \in H^{p(\cdot)}(\mathbb{H}^n)$3 into $f \in H^{p(\cdot)}(\mathbb{H}^n)$4-atoms, one forms the potential series $f \in H^{p(\cdot)}(\mathbb{H}^n)$5 with $f \in H^{p(\cdot)}(\mathbb{H}^n)$6. The three terms arising from the pointwise estimate are each controlled in $f \in H^{p(\cdot)}(\mathbb{H}^n)$7 by $f \in H^{p(\cdot)}(\mathbb{H}^n)$8: the decay term via Lemma on power rescaling plus the vector-valued maximal inequality (using $f \in H^{p(\cdot)}(\mathbb{H}^n)$9 so that $\mathbb{H}^n$00 has upper type exceeding $\mathbb{H}^n$01); the local maximal term via Proposition on atom families; and the singular integral term via the $\mathbb{H}^n$02-boundedness of $\mathbb{H}^n$03 on $\mathbb{H}^n$04 combined with the same atomic square-function estimate. Convergence of the potential series in $\mathbb{H}^n$05 follows from completeness, and continuity of $\mathbb{H}^n$06 yields $\mathbb{H}^n$07.

The complementary regime is degenerate: if $\mathbb{H}^n$08, then $\mathbb{H}^n$09. The proof shows any nonzero class would force $\mathbb{H}^n$10 at infinity, whose $\mathbb{H}^n$11-modular diverges precisely because $\mathbb{H}^n$12 near zero with $\mathbb{H}^n$13. Thus the threshold $\mathbb{H}^n$14 is sharp in the sense that no nontrivial solution space exists below it, mirroring the classical condition $\mathbb{H}^n$15 of Gatto-Jiménez-Segovia.

Limitations and open questions

The paper restricts attention to the first-order equation $\mathbb{H}^n$16. Although fundamental solutions of the iterated operator $\mathbb{H}^n$17 are known for all $\mathbb{H}^n$18 (Benson-Dooley-Ratcliff), the author notes that the equation $\mathbb{H}^n$19 on $\mathbb{H}^n$20 is substantially more complicated and is not treated here; extending the Calderón-space method to $\mathbb{H}^n$21 remains open. The results also require the strict inequality $\mathbb{H}^n$22; the endpoint case $\mathbb{H}^n$23 is not addressed. Finally, the analysis depends on the availability of Folland's explicit fundamental solution and on the $\mathbb{H}^n$24-boundedness of the specific truncated operators $\mathbb{H}^n$25 on $\mathbb{H}^n$26, so the argument does not immediately transfer to more general sub-Laplacians or stratified groups without analogous ingredients.

Conclusion

The paper extends the Calderón-Hardy space approach to inhomogeneous sub-Laplacian equations to the Orlicz scale on the Heisenberg group. Its contributions are the definition of $\mathbb{H}^n$27 with equivalent maximal characterizations, an atomic decomposition with Luxembug-norm square-function control, the construction and completeness of $\mathbb{H}^n$28, and the bijectivity of $\mathbb{H}^n$29 between these spaces under the sharp-in-spirit type condition $\mathbb{H}^n$30, with the degeneracy of the space below the threshold confirming the role of this constant.