Riesz Potential Mapping in Grand Morrey Spaces

• Riesz potential mapping is a generalization of classical fractional integrals, defining operator continuity on both classical and grand Morrey spaces.
• It employs a reduction lemma and intermediate Morrey-norm techniques to control norm estimates under technical conditions on auxiliary functions.
• The framework significantly advances regularity theory for PDEs and singular integrals, addressing both homogeneous and nonhomogeneous measure settings.

The Riesz potential mapping concerns the boundedness and continuity properties of Riesz-type potential operators in various function space settings—both classical and generalized—and on a broad array of metric and measure structures, including spaces of homogeneous type, quasi-metric measure spaces, and their grandified analogs. These operators generalize classical fractional integrals and are central to the theory of function spaces and regularity of partial differential equations.

1. Generalized Grand Morrey Spaces and Riesz-Type Potentials

Let $(X,d,\mu)$ be a space of homogeneous type (quasimetric $d$, Borel measure $\mu$ with a doubling or upper Ahlfors–regular property). For $1<p<\infty$, $0<\lambda<1$, and auxiliary functions $\phi:(0,S)\to(0,\infty)$ and $A:(0,S)\to[0,\infty)$ with $S=\min\{p-1,\lambda\}$, generalized grand Morrey spaces $L^{p),\lambda)}_{\phi,A}(X)$ are defined via the norm

$\|f\|_{L^{p),\lambda)}_{\phi,A} := \sup_{0<\varepsilon<S} \phi(\varepsilon)^{1/(p-\varepsilon)} \|f\|_{L^{p-\varepsilon,\lambda-A(\varepsilon)}(X)},$

where

$$\|f\|_{L^{p-\varepsilon,\lambda-A(\varepsilon)}(X)} = \sup_{x\in X,\,0<r<\mathrm{diam}X} \mu(B(x,r))^{-\frac{\lambda-A(\varepsilon)}{p-\varepsilon}} \left( \int_{B(x,r)} |f(y)|^{p-\varepsilon}\, d\mu(y)\right)^{1/(p-\varepsilon)}.$$

Setting $A\equiv0$ recovers the grand Morrey spaces of Meskhi; with both $\phi,A$ trivial, the spaces reduce to classical Morrey spaces.

The Riesz-type potential is defined by

$$I_\alpha f(x) = \int_X \frac{f(y)}{d(x,y)^{B-\alpha}}\,d\mu(y),$$

where $0<\alpha<B$ and $B$ is the upper Ahlfors exponent.

2. Main Mapping Theorems: Boundedness Criteria

The principal result gives sharp boundedness criteria for the Riesz potential $I_\alpha$ between generalized grand Morrey spaces. Given $q$ determined by

$$\frac{1}{p} - \frac{1}{q} = \frac{\alpha}{(1-\lambda)B},$$

the mapping

$$I_\alpha: L^{p),\lambda)}_{\phi_1, A_1}(X) \rightarrow L^{q),\lambda)}_{\phi_2, A_2}(X)$$

is bounded provided technical conditions on the auxiliary functions hold:

• $A_2 \in C^1((0,q-1])$, with $\lim_{\varepsilon\to0}A_2(\varepsilon)=0$,
• $$B_2 := \lim_{\varepsilon\to0^+} \varepsilon A_2'(\varepsilon) < \frac{(1-\lambda)B}{\alpha q}$$,
• There is a smooth inverse $\theta(\varepsilon)$ solving $A_1(\theta(\varepsilon))=A_2(\varepsilon)$,
• Uniform control: $$\sup_{0<\varepsilon<S} \phi_2(\varepsilon)^{1/(q-\varepsilon)}/\phi_1(\theta(\varepsilon))^{1/(p-\theta(\varepsilon))}<\infty$$.

The explicit norm estimate aligns with an interpolated Morrey norm bound: $$\|I_\alpha f\|_{L^{q-\varepsilon, \lambda-A_2(\varepsilon)}(X)} \leq C(\varepsilon)\|f\|_{L^{p-\delta, \lambda-A_1(\delta)}(X)},$$ with $C(\varepsilon)$ uniform in $\varepsilon$ under the stated hypotheses.

3. Reduction Lemma and Methodological Framework

To transfer Morrey-norm boundedness to the grandified scale, the Reduction Lemma is used: if an operator $U$ is uniformly bounded on the family of approximating Morrey spaces and if the weights $\phi_1,\phi_2$ control the scaling, then $U$ extends to a bounded operator between the associated generalized grand Morrey spaces. Application to $I_\alpha$ leverages the classical theory and reduces the grand case to uniform control in $\varepsilon$.

This lemma enables a robust methodology. The proof for $I_\alpha$ proceeds by (1) establishing boundedness in the intermediate Morrey scale (using a precise analysis of kernel decay and measure growth), and (2) aggregating these bounds using the supremal structure of the grandified space norm, thus obtaining the desired global $L^{p),\lambda)}_{\phi,A}$ to $L^{q),\lambda)}_{\phi,A}$ estimate.

4. Corollaries, Non-Homogeneous Contexts, and Further Operators

The approach extends to measure-defined potential operators of the form

$$I_a f(x) = \int_X \left[\mu(B(x,d(x,y)))\right]^{a-1} f(y)\, d\mu(y),$$

with analogous results on mapping properties into corresponding generalized grand Morrey-type spaces. In Section 5, the doubling property is relaxed: modified Morrey spaces $L^{p,\lambda)}(X,\mu)_a$ utilize larger averaging balls, and a Hedberg-type pointwise estimate, together with the reduction lemma, yields boundedness results for $I_a$ in these nonhomogeneous settings as well.

5. Sharpness, Limitations, and Parameter Constraints

The index relation $\alpha< (1-\lambda)B / p$ (or, equivalently, $1/p - 1/q = \alpha / ((1-\lambda)B)$) is sharp for the Riesz potential, matching the optimality found in the classical Hardy–Littlewood–Sobolev theory. The structural condition on $A_2$, specifically $$\lim_{\varepsilon\to 0} \varepsilon A_2'(\varepsilon) < (1-\lambda)B / (\alpha q)$$, is necessary for invertibility and to avoid blow-up in norm estimates. The measure $\mu$ requires at least upper growth, $\mu(B(x,r)) \leq C r^B$; endpoint issues and the full extension to measures lacking any polynomial growth remain unresolved in this framework (Kokilashvili et al., 2012).

6. Comparative Context: Generalized Grand versus Classical Morrey Spaces

Space Type	Operator Mapping Property	Boundedness Condition
Classical Morrey	$I_\alpha:L^{p,\lambda}\to L^{q,\lambda}$	$\alpha< (1-\lambda)B/p$
Generalized Grand Morrey	$$I_\alpha:L^{p),\lambda)}_{\phi,A}\to L^{q),\lambda)}_{\phi,A}$$	Technical hypotheses on $A_i, \phi_i$; same $\alpha$, $p$, $q$ constraint

Generalized grand Morrey spaces, via "grandification" in both the integrability and Morrey parameters, subsume classical results and allow for robust treatment of limit-space and variable-exponent regimes. The reduction framework provides a systematic approach which also covers maximal, Calderón–Zygmund, and other singular operators under similar hypotheses (Kokilashvili et al., 2012).

7. Conclusion and Research Implications

The Riesz potential mapping in generalized grand Morrey spaces unifies and extends the boundedness theory for potential operators, providing a flexible analytical platform for non-standard function space settings. The reduction lemma methodology, explicit norm-control conditions, and adaptability to both homogeneous and nonhomogeneous measures deepen the connection between potential theory and harmonic analysis on metric measure spaces. This framework underpins further advances in regularity theory for PDEs, potential analysis, and the study of fractional and singular integrals in non-classical spaces (Kokilashvili et al., 2012).