Grand Variable Herz-Morrey-Hardy Spaces

• Grand Variable Herz-Morrey-Hardy spaces are Hardy-type spaces defined via a grand maximal function, incorporating dyadic Herz weights, Morrey localization, and variable exponent geometry.
• They enable precise atomic decompositions with central atoms, supporting boundedness proofs for singular integrals and fractional commutators.
• These spaces bridge classical Hardy theory and nonstandard growth frameworks, offering robust tools for operator estimates and anisotropic as well as variable-exponent settings.

Grand variable Herz-Morrey-Hardy spaces are Hardy-type function spaces that merge four ingredients in a single scale: dyadic Herz control, Morrey localization, variable-exponent Lebesgue geometry, and a grand-Lebesgue-type perturbation parameter. The concept is introduced in “Grand variable Herz-Morrey-Hardy Spaces and applications to the boundedness of integral operators” (Sultan et al., 24 Aug 2025). In that formulation, the underlying grand variable Herz-Morrey space is defined by a supremum over both a scale parameter and a perturbation parameter, while the Hardy version is obtained by replacing the function with its grand maximal function, exactly as in classical Hardy space theory (Sultan et al., 24 Aug 2025).

1. Defining framework and basic notation

The ambient setting is $\mathbb R^n$, with dyadic balls and annuli

$$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$

For a measurable exponent $p(\cdot):E\to[1,\infty)$ satisfying

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$

the variable Lebesgue space $L^{p(\cdot)}(E)$ is equipped with the Luxemburg norm

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$

The grand variable Herz-Morrey space is defined for

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$

by

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$

with quasi-norm

$$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$

When $\Gamma=0$, this reduces to a grand Herz-type space (Sultan et al., 24 Aug 2025).

The corresponding Hardy space is defined through the grand maximal function. For $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$0,

$$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$1

with

$$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$2

Then

$$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$3

where

$$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$4

The constituent mechanisms are explicit in the definition. Herz behavior is carried by the annular weights $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$5, Morrey behavior by the outer factor $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$6, Hardy behavior by the passage from $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$7 to $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$8, and the grand feature by the supremum over $$B_k=B(0,2^k),\qquad F_k=B_k\setminus B_{k-1},\qquad \chi_k=\chi_{F_k}.$$9 with exponent $p(\cdot):E\to[1,\infty)$0 (Sultan et al., 24 Aug 2025).

2. Structural characterization by central atoms

The core structural theorem for these spaces is an atomic decomposition in terms of central atoms. Under the assumptions

$p(\cdot):E\to[1,\infty)$1

with $p(\cdot):E\to[1,\infty)$2 log-Hölder continuous at $p(\cdot):E\to[1,\infty)$3 and $p(\cdot):E\to[1,\infty)$4, and

$p(\cdot):E\to[1,\infty)$5

a function $p(\cdot):E\to[1,\infty)$6 is a central $p(\cdot):E\to[1,\infty)$7-atom if:

1. $p(\cdot):E\to[1,\infty)$8,
2. $p(\cdot):E\to[1,\infty)$9,
3. moment cancellation holds:

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$0

A restricted-type atom is defined in the same way, except that one requires $$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$1 (Sultan et al., 24 Aug 2025).

The atomic decomposition theorem states that

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$2

if and only if

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$3

where each $$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$4 is a central $$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$5-atom supported in $$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$6, and the coefficients satisfy

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$7

Moreover,

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$8

where the infimum is taken over all such representations (Sultan et al., 24 Aug 2025).

The proof mechanism is also part of the theory. The construction proceeds by mollification, dyadic partition of unity, and polynomial subtraction to enforce cancellation; the converse direction estimates the grand maximal function of each atom and then sums the resulting contributions. The paper also records explicit support control in the decomposition, such as

$$1 \le p^-(E):=\operatorname*{ess\,inf}_{x\in E}p(x)\le p(x)\le p^+(E):=\operatorname*{ess\,sup}_{x\in E}p(x)<\infty,$$9

depending on the component under consideration (Sultan et al., 24 Aug 2025).

3. Maximal operators, singular integrals, and fractional commutators

The operator theory in this scale is organized around the atomic characterization. The paper first treats sublinear operators satisfying a standard size condition and bounded on $L^{p(\cdot)}(E)$0. Under

$L^{p(\cdot)}(E)$1

together with the critical growth conditions

$L^{p(\cdot)}(E)$2

sublinear operators satisfying the size condition

$L^{p(\cdot)}(E)$3

are bounded on $L^{p(\cdot)}(E)$4. The Hardy-space version then states that if $L^{p(\cdot)}(E)$5 is sublinear, bounded on $L^{p(\cdot)}(E)$6, and for compactly supported $L^{p(\cdot)}(E)$7 with vanishing moments up to order $L^{p(\cdot)}(E)$8,

$L^{p(\cdot)}(E)$9

whenever $$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$0, then

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$1

These are the basic singular-integral-type applications of the new Hardy scale (Sultan et al., 24 Aug 2025).

The same paper treats homogeneous fractional integrals

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$2

and their commutators

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$3

For $$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$4, under

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$5

with $$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$6 satisfying log-Hölder and decay conditions, $$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$7,

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$8

$$\|f\|_{L^{p(\cdot)}(E)}=\inf\left\{\eta>0:\int_E \left(\frac{|f(y)|}{\eta}\right)^{p(y)}dy\le1\right\}.$$9, $$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$0, $$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$1, and the $$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$2-Dini condition

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$3

the commutator is bounded

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$4

whenever

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$5

The proof is atomwise and uses sharp off-support estimates of the form

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$6

An analogous theorem replaces BMO by $$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$7, assumes

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$8

and yields boundedness under

$$p(\cdot)\in\mathcal P(\mathbb R^n),\qquad \alpha(\cdot)\in L^\infty(\mathbb R^n),\qquad 0\le \Gamma<\infty,\qquad u\in[1,\infty),\qquad \theta>0$$9

The abstract states these results as boundedness of commutators of homogeneous fractional integrals and their Lipschitz estimate on these spaces (Sultan et al., 24 Aug 2025).

4. Relation to adjacent grand, variable, and anisotropic scales

The Hardy construction did not appear in isolation. One immediate antecedent is the grand variable Herz-Morrey framework in “Grand variable Herz-Morrey type Besove spaces and Triebel-Lizorkin spaces,” which works with

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$0

and its non-homogeneous counterpart, proves vector-valued boundedness of sublinear operators, establishes the special case of the Hardy-Littlewood maximal operator, and then defines grand variable Herz-Morrey type Besov and Triebel-Lizorkin spaces with equivalent quasi-norms via Peetre maximal operators. That paper explicitly states that it is not about “Hardy spaces” in the classical sense, but it does provide Hardy-type maximal operator control and a natural smoothness-space analogue (Sultan et al., 19 Feb 2025).

A second adjacent development is anisotropic. “Anisotropic grand Herz type spaces with variable exponents and their applications” introduces anisotropic grand Herz spaces, anisotropic grand Herz-Morrey spaces, and anisotropic grand Herz-type Hardy spaces with variable exponents. In that setting the Euclidean dyadic annuli are replaced by annuli associated with an expansive dilation $$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$1, the determinant is

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$2

and the grand parameter enters via the sequence norm

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$3

The Hardy side is again described through a grand maximal function and central atomic decomposition, yielding an anisotropic counterpart of the Euclidean theory (Wang et al., 2024).

A third neighboring line is the non-grand variable exponent Herz-Morrey-Hardy theory. “Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces” defines

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$4

and proves

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$5

under

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$6

Its sequel on commutators proves

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$7

again by atomic decomposition (Gurbuz, 2024, Gurbuz, 2 Apr 2025).

These three directions suggest a clear synthesis, although that synthesis is only explicit in the later paper: grandification of the Herz-Morrey norm, variable-exponent Hardy control through a maximal function, and atom-based operator theory (Sultan et al., 24 Aug 2025).

5. Operator-theoretic precursors beyond the exact Hardy framework

A substantial part of the surrounding literature concerns Hardy-type operators on Herz or Herz-Morrey scales without defining the Hardy space itself. In the Euclidean variable-order setting, “Boundedness for fractional Hardy-type operator on Herz-Morrey spaces with variable exponent” and its later variable-$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$8 refinement prove boundedness of

$$M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n) =\left\{f\in L^{p(\cdot)}_{\mathrm{loc}}(\mathbb R^n\setminus\{0\}): \|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)}<\infty\right\},$$9

and its adjoint on variable-exponent Herz-Morrey spaces, with target weights of the form

$$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$0

and exponent relation

$$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$1

These papers are directly about Herz-Morrey spaces and Hardy-type operators, but not about grand norms or maximal-function Hardy spaces [(Wu, 2014); (Wu et al., 2015)].

The p-adic literature develops an analogous, though distinct, thread. “Bounds for $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$2-adic Hardy-type Operators and Commutator On $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$3-adic Variable Herz-Morrey Spaces” introduces p-adic variable Herz-Morrey spaces and proves boundedness of

$$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$4

together with commutator estimates controlled by a p-adic central bounded mean oscillation class $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$5 (Bashir et al., 2024). In a different p-adic direction, “Boundedness of weighted multilinear $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$6-adic Hardy operator on Herz type spaces” studies weighted multilinear $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$7-adic Hardy operators and their adjoints on products of homogeneous p-adic Herz spaces and homogeneous p-adic Morrey-Herz spaces, with exact operator norms determined by a single integral condition on the kernel $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$8 (Hussain et al., 2020).

Grand variable Herz-Morrey scales also appear without the Hardy-space layer. “The Commutators of $$\|f\|_{M\dot K^{\alpha(\cdot),u),\theta}_{\Gamma,p(\cdot)}(\mathbb R^n)} =\sup_{\vartheta>0}\sup_{m_0\in\mathbb Z} 2^{-m_0\Gamma} \left(\vartheta^\theta \sum_{k=-\infty}^{m_0}\|2^{k\alpha(\cdot)}f\chi_k\|_{p(\cdot)}^{\,u(1+\vartheta)}\right)^{1/(u(1+\vartheta))}.$$9-dimensional Rough Fractional Hardy Operators on Two Weighted Grand Herz-Morrey Spaces with Variable Exponents” works with the two-weight grand Herz-Morrey space

$\Gamma=0$0

and proves boundedness of $\Gamma=0$1th order commutators of rough fractional Hardy operators and their adjoints with BMO or Lipschitz symbols (Wang et al., 19 Feb 2025). Likewise, “Weighted estimates for commutators of multilinear Hausdorff operators on variable exponent Morrey-Herz type spaces” establishes boundedness of multilinear Hausdorff commutators on weighted variable exponent Herz and Morrey-Herz spaces, including symbol classes $\Gamma=0$2 and $\Gamma=0$3 (Chuong et al., 2017).

These works do not define grand variable Herz-Morrey-Hardy spaces in the maximal-function sense. They do, however, supply the operator estimates, dyadic-annulus technology, and variable-exponent hypotheses that the Hardy-space theory later exploits.

6. Scope, interpretation, and common distinctions

The exact object called a grand variable Herz-Morrey-Hardy space is the Hardy-type space

$\Gamma=0$4

introduced in (Sultan et al., 24 Aug 2025). This should be distinguished from several nearby notions.

First, it is not the same as a grand variable Herz-Morrey space. The latter is the underlying function space before the Hardy maximal-function lift, and it already supports vector-valued sublinear estimates and Peetre maximal characterizations of associated Besov and Triebel-Lizorkin scales (Sultan et al., 19 Feb 2025).

Second, it is not the same as a Herz-Morrey space on which a Hardy-type operator happens to be bounded. The Euclidean variable-order Hardy operators, the p-adic Hardy operators, and the rough fractional Hardy commutators all act on Herz or Herz-Morrey families, but those papers study operator mapping properties rather than a Hardy-space structure defined by grand maximal functions or atomic decompositions [(Wu, 2014); (Bashir et al., 2024); (Wang et al., 19 Feb 2025)].

Third, it is not merely a variable exponent Herz-Morrey-Hardy space. The non-grand theory already has a maximal-function definition and an atomic decomposition, and it supports Riesz potential and commutator inequalities. The grand theory adds the further supremum over perturbed exponents, encoded by $\Gamma=0$5 and the factor $\Gamma=0$6 (Gurbuz, 2024, Gurbuz, 2 Apr 2025, Sultan et al., 24 Aug 2025).

From the currently documented results, the atomic decomposition is the central engine of the theory. A plausible implication is that further boundedness theorems on these spaces will continue to depend on atomwise off-support decay, cancellation, and variable-exponent characteristic-function estimates, because those are the mechanisms used both in the Euclidean variable exponent Hardy theory and in the grand variable Herz-Morrey-Hardy construction (Sultan et al., 24 Aug 2025). The same evidence suggests that the subject sits at the intersection of classical Hardy space methods and nonstandard-growth function-space geometry rather than being a reformulation of either one alone.