Tilde Closed Morrey Subspaces

• Tilde closed subspaces of Morrey spaces are defined as the closure of smooth, compactly supported functions in the Morrey norm, emphasizing approximation properties.
• They are characterized by triple vanishing conditions at the origin, infinity, and spatial tails, ensuring precise structural control.
• These subspaces exhibit operator invariance under classical singular and non-singular operators, playing a critical role in harmonic analysis and PDE applications.

A tilde closed subspace of a Morrey space is the norm closure of $C_c^\infty(\mathbb{R}^n)$ in the Morrey norm, consisting of all functions in the ambient Morrey space that can be approximated by smooth, compactly supported functions with respect to the Morrey topology. Recent theoretical developments provide precise characterizations of these subspaces, establish their invariance under a range of classical operators, and elucidate their functional-analytic and PDE-theoretic significance.

1. Definition and Fundamental Properties

For $1 < p < \infty$ and $0 \leq \lambda < n$, the (homogeneous) Morrey space $M^{p, \lambda}(\mathbb{R}^n)$ comprises all $f \in L^p_{\mathrm{loc}}(\mathbb{R}^n)$ such that

$$\|f\|_{M^{p,\lambda}} = \sup_{x \in \mathbb{R}^n,\, r > 0} r^{-\lambda} \left( \int_{B(x,r)} |f(y)|^p\,dy \right)^{1/p} < \infty,$$

where $B(x,r)$ denotes the Euclidean ball of radius $r$ centered at $x$.

The tilde closed subspace, denoted $\widetilde{M}^{p,\lambda}(\mathbb{R}^n)$, is defined as the closure of $C_c^\infty(\mathbb{R}^n)$ with respect to $\|\cdot\|_{M^{p,\lambda}}$: $$\widetilde{M}^{p,\lambda}(\mathbb{R}^n) := \overline{C_c^\infty(\mathbb{R}^n)}^{\,M^{p,\lambda}} = \left\{ f \in M^{p,\lambda}(\mathbb{R}^n) : \exists\,\varphi_k \in C_c^\infty,\, \varphi_k \to f \ \text{in}\ \|\cdot\|_{M^{p,\lambda}} \right\}.$$ Morrey spaces are non-separable for $\lambda>0$. Neither $C_c^\infty$ nor the Schwartz class $S(\mathbb{R}^n)$ is dense in $M^{p,\lambda}$ unless $\lambda = 0$ (the $L^p$ case) (Almeida et al., 2016).

2. Explicit Characterization via Vanishing Conditions

The central structural result, established in (Almeida et al., 2016, Alabalik et al., 2019), provides a threefold vanishing characterization: $f \in \widetilde{M}^{p,\lambda}$ if and only if $f$ satisfies

• Vanishing at the origin (Chiarenza–Franziosi):

$$V_0M^{p,\lambda} := \left\{ f\in M^{p,\lambda} : \lim_{r\to 0}\sup_{x\in \mathbb{R}^n} r^{-\lambda}\left( \int_{B(x,r)} |f|^p \right)^{1/p} = 0 \right\},$$

• Vanishing at infinity:

$$V_\infty M^{p, \lambda} := \left\{ f \in M^{p, \lambda} : \lim_{r \to \infty} \sup_{x\in \mathbb{R}^n} r^{-\lambda} \left( \int_{B(x, r)} |f|^p \right)^{1/p} = 0 \right\},$$

• Vanishing of tails:

$$V_*M^{p,\lambda} := \left\{ f \in M^{p,\lambda}: \lim_{N \to \infty} \sup_{x\in\mathbb{R}^n,\,0<r\leq 1} r^{-\lambda}\left( \int_{B(x, r)} |f(y)|^p \mathbf{1}_{|y|>N}(y) dy \right)^{1/p} = 0 \right\}.$$

The principal theorem asserts

$$\widetilde{M}^{p,\lambda}(\mathbb{R}^n) = V_0M^{p,\lambda} \cap V_\infty M^{p,\lambda} \cap V_* M^{p,\lambda}.$$

This triple vanishing criterion is both necessary and sufficient; membership in the tilde closed subspace is entirely controlled by vanishing at small scales, at large scales, and in the truncated spatial tails (Almeida et al., 2016, Alabalik et al., 2019).

3. Operator Invariance and Harmonic Analysis

Tilde closed subspaces are preserved by a wide class of singular and non-singular operators. For $1 < p < \infty$, $0 < \lambda < n$, the following invariance results hold (Alabalik et al., 2019):

• Hardy-Littlewood maximal operator $M$: $$M: \widetilde{M}^{p,\lambda} \to \widetilde{M}^{p,\lambda}$$.
• Calderón–Zygmund singular integral operators $S$: $$S: \widetilde{M}^{p,\lambda} \to \widetilde{M}^{p,\lambda}$$.
• Hardy operators $H$, $H^*$: Preserved.
• Riesz potential $I_\alpha$ and fractional maximal operator $M_\alpha$: For parameter restrictions $$0 < \alpha < n,~1<p<n/\alpha,~1/q=1/p-\alpha/n,~0<\lambda<n-\alpha p$$ with $\mu/q = \lambda/p$,

$$I_\alpha,~M_\alpha: \widetilde{M}^{p,\lambda} \longrightarrow \widetilde{M}^{q,\mu}.$$

These results ensure that the vanishing structure required for $C_c^\infty$-approximation is stable under standard operators of harmonic analysis—an essential property in PDE applications (Alabalik et al., 2019).

4. Functional-Analytic Decomposition and Duality

In wider generality, such as generalized Morrey spaces $L_{p,\phi}(\mathbb{R}^d)$ with variable growth $\phi$, the tilde closed subspace $\widetilde{L_{p,\phi}}$ is characterized by the vanishing of local norm quantities at small and large scales: $$A_{p, \phi}^0(f) = \limsup_{r \to 0} \sup_{x \in \mathbb{R}^d} \frac{M_p(f; B(x, r))}{\phi(x, r)};$$

$$A_{p, \phi}^\infty(f) = \limsup_{r \to \infty} \sup_{x \in \mathbb{R}^d} \frac{M_p(f; B(x, r))}{\phi(x, r)}.$$

The norm closure then admits a distance formula

$$\mathrm{dist}(f, \widetilde{L_{p, \phi}}) \asymp \max\{A_{p,\phi}^0(f), A_{p,\phi}^\infty(f), \|f\|_{L_{p, \phi}}\}$$

under standard hypotheses on $\phi$ (doubling, nearness, almost decreasing/increasing) (Yamaguchi, 29 Jan 2025).

Duality and bi-duality results identify

$$\left(\widetilde{L_{p,\phi}}\right)^* = B_{\phi, p'} \quad \text{(block space)},$$

$$\left(\widetilde{L_{p,\phi}}\right)^{**} = L_{p, \phi},$$

where the block space consists of finite or countable linear combinations of $\phi$-blocks, with norms computed via infimal decompositions (Yamaguchi, 29 Jan 2025).

5. Decomposition Theorems and Structural Examples

A functional decomposition exists for $\widetilde{\mathcal{M}}^p_q$ (alternative Morrey notation): $$f = f \cdot \chi_{|f| \leq R, |x| \leq R} + T^∞_R f + S_R f$$ where $T^∞_R f = f \cdot \chi_{|f| > R}$ and $S_R f = f \cdot \chi_{|x| > R}$. Membership in the tilde subspace corresponds to the tails $T^∞_R f$, $S_R f$ vanishing in norm as $R \to \infty$ (Takesako, 14 Jan 2026).

Illustrative examples:

• Constant functions $f \equiv 1$: Always fail the vanishing-at-infinity condition; not in $\widetilde{\mathcal{M}}^p_q$.
• Compactly supported $L^q$ functions: Always in $\widetilde{\mathcal{M}}^p_q$.
• Power growth/tails $f(x) = |x|^{-\alpha}$: Membership depends on both Morrey integrability and the vanishing of $S_R f$ (Takesako, 14 Jan 2026).

This decomposition is fundamental for compactness proofs and the fine structure of the tilde subspace.

6. Application to Commutators and Operator Theory

The tilde closed subspace is essential as a refined target for compactness of multilinear commutators generated by VMO symbols and fractional integral operators. For $a_1 \in \mathrm{VMO}$ and $a_2, \ldots, a_l \in \mathrm{BMO}$,

$$[f, I_\alpha]: \mathcal{M}^p_q \to \widetilde{\mathcal{M}}^s_t$$

is compact under the standard parameter regime. The proof exploits the structural decomposition, separately handling large-value truncations and spatial tails, and uses BMO and VMO approximation by smooth functions (Takesako, 14 Jan 2026). This improvement over boundedness results is only possible via the fine structure of the tilde subspace.

7. Weighted Embeddings and Further Generalizations

A general embedding

$$M^{p, \lambda}(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n, w)$$

holds for radial weights $w$, assuming integrability conditions and monotonicity of $t \mapsto t^\lambda w(t)^p$. For instance, with $w(x) = (1 + |x|)^{-\alpha}$ and $\alpha > \lambda/p$, this subsumes classical weighted results and extends them (Almeida et al., 2016).

These embedding results highlight how the tilde closed subspace provides the correct functional framework for analytic problems requiring both local and global decay.

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Summary Table: Characterization and Invariance of Tilde Closed Morrey Subspaces

Subspace Condition	Characterization in $M^{p, \lambda}$	Operator Invariance
$\widetilde{M}^{p, \lambda}$	Vanishing at $0$, $\infty$, and in tails ($V_0 \cap V_\infty \cap V_*$)	Hardy-Littlewood, Riesz, Calderón–Zygmund, Hardy operators
$\overline{\mathcal{M}}^p_q$ (bar)	$\lim_{R \to \infty} \|T^∞_R f\| = 0$
$*\mathcal{M}^p_q$ (star)	$\lim_{R \to \infty} \|S_R f\| = 0$

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The theory of tilde closed subspaces of Morrey spaces thus provides a comprehensive and robust framework for both structural analysis and applications, with broad implications in harmonic analysis, operator theory, and the theory of partial differential equations (Almeida et al., 2016, Alabalik et al., 2019, Takesako, 14 Jan 2026, Yamaguchi, 29 Jan 2025).