Weighted Morrey Spaces in Harmonic Analysis

• Weighted Morrey spaces are refinements of classical Morrey spaces that incorporate weight functions to measure local Lᵖ-integrability against ball or cube sizes.
• They interpolate between weighted Lebesgue control and localized regularity, supporting analyses of maximal operators, singular integrals, and fractional operators.
• These spaces underpin critical applications in PDE regularity, embedding theory, and operator extrapolation, with various normalization models such as Samko and Komori–Shirai types.

Weighted Morrey spaces are weighted refinements of Morrey’s classical scale that measure local $L^p$-integrability against a normalization depending on the size of the underlying ball or cube and, in many formulations, on the weight itself. They interpolate between weighted Lebesgue control and finer local regularity control, and they appear in harmonic analysis, singular integrals, fractional operators, square functions, commutators, and PDE regularity. The modern literature does not use a single canonical normalization: Komori–Shirai type, Samko type, generalized Morrey, local Morrey, central Morrey, mixed-Morrey, one-sided, multilinear, and discrete variants all coexist, often with different weight conditions and different operator theories (Fu et al., 2012).

1. Basic definitions and principal models

A standard weighted Morrey space is defined, for $1<p<\infty$, $0<\kappa<1$, and a weight $w$, by

$$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$

where the supremum runs over all balls $B\subset \mathbb R^n$. In the Komori–Shirai normalization, this is also written $M_{p,\lambda}(w)$ with $0<\lambda<1$, and when $w=1$ one recovers the classical Morrey scale after the usual parameter identification (Wang, 2014).

A different but closely related normalization is the Samko-type space

$$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$

where the normalization uses the Lebesgue size of the cube rather than $1<p<\infty$0. A broader formulation is

$1<p<\infty$1

which includes Samko type, Komori–Shirai type, and Poelhuis–Torchinsky type by choosing different control functions $1<p<\infty$2 (Duoandikoetxea et al., 2020).

Several later generalizations are explicitly designed to unify different weighted Morrey models.

Variant	Prototype norm	Representative source
Komori–Shirai type	$1<p<\infty$3	(Fu et al., 2012)
Samko type	$1<p<\infty$4	(Lappas, 2021)
Unified two-parameter scale	$1<p<\infty$5	(Duoandikoetxea et al., 2019)

The unified scale

$1<p<\infty$6

contains Komori–Shirai type, Samko type, and Poelhuis–Torchinsky type as special cases. For power weights $1<p<\infty$7, it admits explicit equivalences between parameter choices, and triviality or $1<p<\infty$8-type collapse occurs at boundary parameter values such as $1<p<\infty$9 in the stated framework (Duoandikoetxea et al., 2019).

The literature also contains weighted local Morrey spaces

$0<\kappa<1$0

with the supremum over balls centered at the origin, generalized weighted Morrey spaces $0<\kappa<1$1, mixed-Morrey spaces in space-time, central Morrey spaces over local fields, and discrete weighted Morrey spaces on $0<\kappa<1$2 (Duoandikoetxea et al., 2021).

2. Weight classes and structural conditions

The basic ambient weights are Muckenhoupt weights. For $0<\kappa<1$3, $0<\kappa<1$4 means

$0<\kappa<1$5

for all balls $0<\kappa<1$6. For fractional operators, the relevant class is $0<\kappa<1$7, and in multilinear settings one uses the multiple-weight classes $0<\kappa<1$8 and $0<\kappa<1$9 with

$w$0

These weight conditions drive most strong and weak Morrey estimates for maximal, Calderón–Zygmund, fractional, oscillatory, and multilinear operators (Fu et al., 2011).

A central feature of weighted Morrey theory is that classical $w$1-theory does not transfer verbatim. In the Samko-type setting, the $w$2-condition “does not suffice,” and the boundedness theory for the maximal operator and singular integral operators requires additional Morrey-specific conditions. One such condition is the class $w$3, defined through

$w$4

together with the weighted integral condition

$w$5

The same source also emphasizes a genuine difference between maximal and singular integral operators on weighted Morrey spaces, including distinct endpoint behavior for power weights (Nakamura et al., 2016).

A second structural development is the Morrey-adapted Muckenhoupt condition defined through Köthe duality: $w$6 When $w$7, this reduces to the classical $w$8 condition. In the Morrey setting it provides a sharp criterion for several maximal and Calderón-type operators, particularly in local and origin-centered settings, and yields a reverse doubling property for weights (Duoandikoetxea et al., 2020).

Power weights are the main explicit test case throughout the subject. The condition

$w$9

reappears across the theory, while Morrey-specific boundedness regions are typically shifted by the Morrey parameters. This suggests that the interaction between scaling and weighting, rather than the weight alone, is the primary organizing principle.

3. Maximal and singular operators

For the Hardy–Littlewood maximal operator, weighted Morrey boundedness is available in several formulations. In the Komori–Shirai setting, if $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$0 and $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$1, then

$$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$2

and weak type at $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$3 holds when $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$4. In the broader two-parameter scale $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$5, sharp boundedness for power weights is given by explicit inequalities in $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$6, and these conditions are necessary and sufficient for the maximal operator in the power-weighted case (Gerhold, 5 Aug 2025).

The Köthe-dual formulation yields exact characterizations in local and origin-centered settings. For the maximal operator over balls centered at the origin,

$$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$7

boundedness on $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$8, weak boundedness, and membership in $$L^{p,\kappa}(w) = \left\{ f\in L^p_{\mathrm{loc}}(w): \|f\|_{L^{p,\kappa}(w)}<\infty \right\}, \qquad \|f\|_{L^{p,\kappa}(w)} = \sup_B \left( \frac{1}{w(B)^\kappa} \int_B |f(x)|^p w(x)\,dx \right)^{1/p},$$9 are equivalent. For weighted local Morrey spaces $B\subset \mathbb R^n$0, the usual Hardy–Littlewood maximal operator is bounded if and only if

$B\subset \mathbb R^n$1

For global Morrey spaces, this condition is necessary, while sufficiency requires the Morrey Muckenhoupt condition plus a local Lebesgue $B\subset \mathbb R^n$2 condition (Duoandikoetxea et al., 2020).

For singular integrals, the theory is subtler. If the maximal operator is bounded on $B\subset \mathbb R^n$3 and the weighted integral condition holds, then a Calderón–Zygmund singular integral operator $B\subset \mathbb R^n$4 is bounded on $B\subset \mathbb R^n$5. Commutators $B\subset \mathbb R^n$6 with $B\subset \mathbb R^n$7 satisfy analogous estimates under the same Morrey-specific hypotheses. The same work makes explicit that, unlike the Lebesgue case $B\subset \mathbb R^n$8, the domains of boundedness for $B\subset \mathbb R^n$9 and $M_{p,\lambda}(w)$0 differ in Morrey spaces; for power weights $M_{p,\lambda}(w)$1, the maximal operator allows the endpoint $M_{p,\lambda}(w)$2, while singular integrals exclude it (Nakamura et al., 2016).

A recurring proof pattern is the decomposition

$M_{p,\lambda}(w)$3

combined with weighted Lebesgue bounds for the local term and kernel decay plus weight properties for the nonlocal term. This local/nonlocal splitting appears in maximal, square, oscillatory, multilinear, and commutator estimates, indicating a stable methodological core across the theory (Fu et al., 2012).

4. Fractional, square, oscillatory, and multilinear theories

For fractional maximal and fractional integral operators on weighted Morrey spaces, the weight conditions are not merely the Lebesgue-space $M_{p,\lambda}(w)$4 conditions. In the global Morrey setting, the boundedness of $M_{p,\lambda}(w)$5 is governed by a testing condition involving the Morrey norm of $M_{p,\lambda}(w)$6 and of $M_{p,\lambda}(w)$7 in a dual Morrey-type space $M_{p,\lambda}(w)$8. For $M_{p,\lambda}(w)$9, one needs in addition the dilation condition

$0<\lambda<1$0

For power weights $0<\lambda<1$1, these abstract criteria reduce to the explicit inequalities

$0<\lambda<1$2

yielding a complete characterization in that case (Nakamura et al., 2016).

In weighted local Morrey spaces, the theory becomes sharper. For Riesz transforms on $0<\lambda<1$3, a necessary and sufficient condition is

$0<\lambda<1$4

and the same sufficient condition yields boundedness for all suitably defined Calderón–Zygmund operators. For $0<\lambda<1$5 and $0<\lambda<1$6, the characterizations hold in the Sobolev range

$0<\lambda<1$7

and the corresponding work states that, unlike global Morrey spaces, the boundedness range cannot be extended to the Adams range in the local setting (Duoandikoetxea et al., 2021).

Intrinsic square functions and their vector-valued analogues form another major branch. For $0<\lambda<1$8, Wilson’s intrinsic square function is

$0<\lambda<1$9

and for sequences $w=1$0,

$w=1$1

If $w=1$2, $w=1$3, and $w=1$4, then

$w=1$5

and for $w=1$6, $w=1$7, the corresponding weak estimate holds. Related boundedness and commutator results were also proved for intrinsic $w=1$8-functions and $w=1$9-functions on weighted Morrey-type and amalgam spaces (Wang, 2014).

Oscillatory integral operators and their fractional analogues are bounded on weighted Morrey spaces under $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$0 or $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$1 assumptions, and the associated BMO commutators remain bounded. The operator norm depends only on the degree of the polynomial phase, not its coefficients. In the multilinear direction, multilinear Calderón–Zygmund operators and multilinear fractional integrals act on products of weighted Morrey spaces with multiple weights, while multilinear Littlewood–Paley square operators and their commutators satisfy both strong Morrey estimates and weak-type $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$2 endpoint inequalities in weighted Morrey spaces (Fu et al., 2011).

5. Generalizations, extrapolation, embeddings, and non-Euclidean variants

A major unifying theme is extrapolation from weighted Lebesgue spaces. If an operator satisfies a Rubio de Francia type weighted estimate on $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$3 for all $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$4, then it extends to weighted Morrey spaces $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$5 or related models for all admissible $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$6 and $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$7, including weak-type inequalities at $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$8. The key mechanism is a family of continuous embeddings of weighted Morrey spaces into suitable $$L^{p,\lambda}(w) = \left\{ f\in L^1_{\mathrm{loc}}(w): \sup_Q |Q|^{-\lambda/d} \left(\int_Q |f|^p w\,dx\right)^{1/p}<\infty \right\},$$9-weighted $1<p<\infty$00 spaces, which also addresses the extension problem for operators whose pointwise representation may be unavailable. Vector-valued inequalities and $1<p<\infty$01-weighted Morrey inequalities then follow automatically in the same framework (Duoandikoetxea et al., 2016).

The unified scale $1<p<\infty$02 further systematizes operator theory. In general, results are obtained for weights of the form $1<p<\infty$03 with $1<p<\infty$04 and nonnegative $1<p<\infty$05, and extrapolation yields abstract theorems transferring weighted Lebesgue-space boundedness to the entire Morrey scale. For the Hardy–Littlewood maximal operator, the power-weighted criteria are sharp (Duoandikoetxea et al., 2019).

Weighted Morrey spaces also support an interpolation-compactness theory. If a linear operator is bounded on $1<p<\infty$06 throughout a weighted range and compact on one weighted Morrey space $1<p<\infty$07, then compactness extrapolates to all spaces in the corresponding scale. Applications include compact commutators of Calderón–Zygmund operators, rough singular integrals, and Bochner–Riesz multipliers (Lappas, 2021).

Beyond Euclidean global spaces, several nonstandard geometries have been treated. One-sided weighted Morrey spaces extend Nakai’s generalized Morrey framework to one-sided Muckenhoupt weights $1<p<\infty$08, enabling weak and strong type estimates for one-sided sublinear operators and compactness of Riemann–Liouville integral operators on local one-sided weighted Morrey spaces (Hou et al., 2018). Over local fields $1<p<\infty$09, generalized weighted central Morrey spaces

$1<p<\infty$10

support quantitative boundedness estimates for Hardy–Hilbert-type operators and the Hardy–Littlewood–Pólya operator via an explicit kernel constant $1<p<\infty$11 (Ashraf et al., 13 Jun 2025). On $1<p<\infty$12, the discrete space

$1<p<\infty$13

admits a full maximal-operator theory; for power weights $1<p<\infty$14, boundedness is characterized by

$1<p<\infty$15

This suggests that Morrey-weight interactions persist robustly across continuous, local, non-Archimedean, and discrete settings (Hao et al., 2023).

6. Embeddings, applications to PDE, and conceptual issues

Embedding theory clarifies both the scale structure and the distinction from weighted Lebesgue spaces. For fixed $1<p<\infty$16, one has

$1<p<\infty$17

and $1<p<\infty$18. In the unweighted case,

$1<p<\infty$19

For different weights, sufficient embedding conditions involve Muckenhoupt assumptions and size conditions on balls, and explicit criteria are available for piecewise power weights $1<p<\infty$20 (Gerhold, 5 Aug 2025).

Applications to PDE are prominent. A broad class of sublinear operators satisfying size conditions is bounded on weighted Morrey spaces, including Hardy–Littlewood maximal operators, Calderón–Zygmund singular integrals, fractional integrals, Bochner–Riesz means at the critical index, oscillatory singular operators, singular integrals with oscillating kernels, and commutators with BMO functions. These bounds are then used to establish regularity in weighted Morrey spaces for strong solutions to nondivergence elliptic equations with VMO coefficients (Fu et al., 2012).

Generalized weighted Morrey and generalized weighted mixed-Morrey spaces support two-weight estimates for sublinear operators generated by fractional integrals and Calderón–Zygmund operators, together with BMO commutator estimates and weak-$1<p<\infty$21 endpoint bounds. The same framework yields regularity properties for solutions of elliptic and parabolic equations, including mixed space-time Morrey estimates (Ramadana et al., 2024).

A related boundary-regularity direction appears in weighted Morrey–Sobolev inequalities on the half-space, where the Sobolev weight is a power of the distance to the boundary. The resulting pointwise oscillation estimates explicitly distinguish the regimes $1<p<\infty$22, $1<p<\infty$23, $1<p<\infty$24, and $1<p<\infty$25, and the paper states that these estimates are optimal up to a multiplicative constant. It also notes that the oscillation scaling for balls approaching the boundary reflects the influence of the weight in Morrey-type space norms (Schaftingen et al., 22 Oct 2025).

Several conceptual points recur across the subject. First, “weighted Morrey space” is not a single definition but a family of related spaces with different normalizations and geometric emphases. Second, the classical $1<p<\infty$26 paradigm is often necessary but frequently not sufficient in Morrey settings; additional testing, doubling, local, or duality conditions may be indispensable. Third, maximal, singular, and fractional operators can exhibit genuinely different boundedness regimes on Morrey spaces, especially for power weights and local spaces. These are not pathologies but structural consequences of the local, non-rearrangement-invariant character of the Morrey framework (Nakamura et al., 2016).