---
title: 'MoRight: Morrey-Space Regularity Theory'
url: https://www.emergentmind.com/topics/moright
type: topic
---

# MoRight: Morrey-Space Regularity Theory

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MoRight denotes the Morrey-space-based \(C^{1,\alpha}\)-regularity theory for degenerate elliptic equations of \(p\)-Laplacian type developed in "Higher Hölder regularity for degenerate elliptic PDEs with data in Morrey spaces" [2510.10359]. It concerns weak solutions of nonlinear elliptic equations with low-integrability right-hand sides controlled in Morrey spaces and establishes sharp local \(C^{1,\alpha}\)-regularity by combining a homogeneous comparison argument, the Fefferman–Phong inequality, and the Campanato–Meyers characterization of Hölder continuity. In the model case, the framework treats
\[
-\Delta_p u=f \quad \text{in } \Omega,
\qquad
\Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),
\]
for \(1<p<n\), with \(f \in L^{1,\lambda}(\Omega)\) and \(n-1<\lambda<n\), and it extends to general quasilinear operators with \(p\)-growth and degenerate or singular ellipticity.

## 1. PDE class and structural setting

The model equation underlying MoRight is the \(p\)-Poisson problem
\[
-\Delta_p u=f \quad \text{in } \Omega,
\]
equivalently written as
\[
-\operatorname{div}(A(x,\nabla u))=f
\quad\text{with}\quad
A(x,\xi)=|\xi|^{p-2}\xi.
\]
The general equation treated is
\[
-\operatorname{div} A(x,u,\nabla u)=B(x,u,\nabla u)\quad \text{in }\Omega,
\]
where \(A:\Omega\times \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is differentiable and satisfies \(p\)-growth, degenerate ellipticity, and controlled \(x,u\)-dependence [2510.10359].

The structural assumptions are stated in terms of the \(\xi\)-Jacobian of \(A\). For all \(\zeta\in \mathbb{R}^n\),
\[
\sum_{i,j=1}^n \frac{\partial A_j}{\partial \xi_i}(x,\mu,\xi)\zeta_i\zeta_j
\ge C_1(\kappa+|\xi|)^{p-2}|\zeta|^2,
\]
which is the uniform monotonicity condition in degenerate form. The \(\xi\)-derivatives obey
\[
\sum_{i,j=1}^n \left|\frac{\partial A_j}{\partial \xi_i}(x,\mu,\xi)\right|
\le C_2(\kappa+|\xi|)^{p-2},
\]
and the \(x,\mu\)-dependence is controlled by
\[
\sum_{i,j=1}^n \left|\frac{\partial A_j}{\partial x_i}(x,\mu,\xi)\right|
+\left|\frac{\partial A_j}{\partial \mu}(x,\mu,\xi)\right|
\le C_3(\kappa+|\xi|)^{p-2}|\xi|,
\]
with \(\kappa\in[0,1]\) and \(C_1,C_2,C_3>0\) depending only on structural data.

The lower-order term is assumed to satisfy Morrey-controlled growth,
\[
|B(x,\mu,\xi)|\le h(x)|\xi|^{p-1}+g(x)|\mu|^{p-1}+f(x),
\]
with
\[
h\in L^{p,\lambda}(\Omega),\qquad g,f\in L^{1,\lambda}(\Omega),
\qquad n-1<\lambda<n.
\]
This places the entire right-hand side within a Morrey-scale regime compatible with interior gradient Hölder estimates.

## 2. Morrey-space framework and Fefferman–Phong control

Morrey spaces are central to the MoRight formulation. For \(1\le q<\infty\) and \(\lambda\ge 0\), the space \(L^{q,\lambda}(\Omega)\) consists of all \(f\in L^q(\Omega)\) such that
\[
\|f\|_{L^{q,\lambda}(\Omega)}
:=
\sup_{x\in \Omega,\ 0<r<\operatorname{diam}(\Omega)}
\left(
\frac{1}{r^\lambda}\int_{\Omega\cap B_r(x)} |f(y)|^q\,dy
\right)^{1/q}
<\infty.
\]
The paper uses the basic facts \(L^{q,0}(\Omega)=L^q(\Omega)\), \(L^{q,n}(\Omega)=L^\infty(\Omega)\), and \(L^{q,\lambda}=\{0\}\) if \(\lambda>n\), as well as the embedding criterion: if \(1\le p\le q<\infty\) and \((n-\mu)/q\le (n-\lambda)/p\), then \(L^{q,\mu}(\Omega)\subset L^{p,\lambda}(\Omega)\) [2510.10359].

The route from Morrey data to elliptic regularity proceeds through Stummel–Kato classes and the Fefferman–Phong inequality. For \(1\le p<n\) and \(f\in L^1(\Omega)\), the Stummel-type modulus is
\[
\eta_p(f,r)
=
\sup_{x\in \Omega}
\int_{\Omega\cap B_r(x)}
\frac{|f(y)|}{|x-y|^{n-p}}\,dy.
\]
If \(f\in L^{1,\lambda}(\Omega)\) with \(n-p<\lambda<n\), then \(f\in S_p(\Omega)\) and
\[
\eta_p(f,r)\le c\, r^{\lambda-n+p}\|f\|_{L^{1,\lambda}(\Omega)}.
\]
This yields the Fefferman–Phong estimate in the form used in the paper: for any ball \(B\subset \Omega\) of radius \(r\),
\[
\int_B |f|\,|\phi|^p\,dx
\le c\,\eta_p(f,2r)\int_B |D\phi|^p\,dx,
\qquad \phi\in W^{1,p}_0(B),
\]
and, for Morrey data,
\[
\int_B |f|\,|\phi|^p\,dx
\le c\, r^{\lambda-n+p}\|f\|_{L^{1,\lambda}(\Omega)}
\int_B |D\phi|^p\,dx.
\]

In the model problem, a weak solution is \(u\in W^{1,p}(\Omega)\) such that
\[
\int_\Omega |Du|^{p-2}Du\cdot D\phi\,dx
=
\int_\Omega f\phi\,dx
\qquad\text{for all }\phi\in W^{1,p}_0(\Omega).
\]
The Fefferman–Phong mechanism is used to show that the right-hand side is finite even at this low integrability level. The paper also notes that the same mechanism applies to measure data of Morrey type, namely measures \(\mu\) satisfying \(|\mu(B_r(x))|\le C r^\lambda\).

## 3. Main local \(C^{1,\alpha}\)-regularity theorem

The principal result of MoRight is a sharp interior \(C^{1,\alpha}\)-estimate for weak solutions of the \(p\)-Poisson equation with Morrey data [2510.10359]. Let \(\Omega\subset \mathbb{R}^n\) be bounded Lipschitz, \(1<p<n\), \(n-1<\lambda<n\), and \(f\in L^{1,\lambda}(\Omega)\). If \(u\) is a weak solution of
\[
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=f \quad \text{in }\Omega,
\]
then
\[
u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega),
\]
with
\[
\alpha=
\begin{cases}
\min\!\left(\gamma,\ \lambda+1-\dfrac{2n}{p}\right),
& \text{if } \dfrac{2n}{\lambda+1}<p\le 2,\\[1.2ex]
\min\!\left(\gamma,\ \dfrac{\lambda+1-n}{p-1}\right),
& \text{if } 2\le p<n,
\end{cases}
\]
where \(\gamma=\gamma(n,p)\in(0,1)\) is the interior gradient Hölder exponent for the corresponding homogeneous equation.

The positivity constraints on \(\alpha\) are explicit. In the branch \(p\ge 2\), the quantity \((\lambda+1-n)/(p-1)\) lies in \((0,1)\) because \(\lambda\in(n-1,n)\). In the branch \(1<p\le 2\), one additionally requires
\[
p>\frac{2n}{\lambda+1},
\]
which ensures \(\lambda+1-2n/p>0\). Thus the singular regime admits local \(C^{1,\alpha}\)-regularity only above a dimensionally determined threshold.

The theorem is quantified through Campanato decay of the gradient oscillation. For balls \(B_r\Subset \Omega\), the argument yields an estimate of the form
\[
\fint_B |Du-(Du)_B|^p\,dx
\le C\big(r^{\gamma p}+\text{data-term}\big),
\]
and the Campanato–Meyers characterization then gives \(Du\in C^{0,\alpha}\), hence \(u\in C^{1,\alpha}\). The exponent \(\alpha\) is therefore determined by the competition between the homogeneous gradient exponent \(\gamma\) and the Morrey-scaling contribution of the forcing term.

## 4. Comparison scheme and proof mechanism

The proof is organized around homogeneous comparison. Fix a ball \(B_r\Subset \Omega\), enlarge it slightly to \(B^\varepsilon\), and let \(v\) solve the homogeneous comparison problem
\[
-\operatorname{div}(|Dv|^{p-2}Dv)=0 \quad \text{in } B^\varepsilon,
\qquad
v-u\in W^{1,p}_0(B^\varepsilon).
\]
Classical theory due to Ladyzhenskaya–Uraltseva, Evans, and Tolksdorf gives
\[
v\in C^{1,\gamma}_{\mathrm{loc}}
\]
for some \(\gamma=\gamma(n,p)\in(0,1)\) [2510.10359].

The oscillation of \(Du\) is decomposed as
\[
\fint_B |Du-(Du)_B|^p
\le
C\left[
\fint_B |Du-Dv|^p
+
\fint_B |Dv-(Dv)_B|^p
+
|(Du)_B-(Dv)_B|^p
\right].
\]
The term involving \(Dv\) is estimated by the homogeneous \(C^{1,\gamma}\)-regularity:
\[
\fint_B |Dv-(Dv)_B|^p \le C r^{\gamma p}.
\]
The mean-difference term is controlled by
\[
|(Du)_B-(Dv)_B|^p \le \fint_B |Du-Dv|^p.
\]
Accordingly, the core issue is to estimate
\[
I_1:=\int_B |Du-Dv|^p.
\]

At this point the proof bifurcates according to the degenerate and singular regimes. For \(p\ge 2\), strict \(p\)-monotonicity of the \(p\)-Laplacian yields
\[
\int |Du-Dv|^p \le C\int f(u-v).
\]
The term on the right is handled by Hölder combined with Fefferman–Phong:
\[
\int_B |f||u-v|
\le
\left(\int_B |f|\right)^{1/p'}
\left(\int_B |f||u-v|^p\right)^{1/p}
\le
C r^{\lambda+1-n/p}\|f\|_{L^{1,\lambda}}\|Du-Dv\|_{L^p(B)}.
\]
This leads to the Morrey-driven decay exponent that enters the \(p\ge 2\) formula for \(\alpha\).

For \(1<p<2\), the operator is singular where \(|Du|\) is small, and the argument uses the refined monotonicity inequality
\[
\big(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta\big)\cdot(\xi-\eta)
\ge
c\frac{|\xi-\eta|^2}{(|\xi|+|\eta|)^{2-p}}.
\]
After Hölder with exponents \(2/p\) and \(2/(2-p)\), together with the energy comparison and the same Fefferman–Phong step, one obtains
\[
\int_B |Du-Dv|^p
\le
C r^{(\lambda+1-n/p)p}\|f\|_{L^{1,\lambda}}^p,
\]
which after averaging over \(B\) yields the singular-branch decay. The common endpoint of both branches is a Campanato estimate
\[
\fint_B |Du-(Du)_B|^p \le C r^{\alpha p},
\]
and therefore \(Du\in C^{0,\alpha}\).

## 5. Extension to general quasilinear equations

MoRight extends the model \(p\)-Poisson theorem to the quasilinear equation
\[
-\operatorname{div} A(x,u,Du)=B(x,u,Du),
\]
under the structural assumptions on \(A\) and the Morrey-growth condition on \(B\). The conclusion is unchanged: any weak solution satisfies
\[
u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega)
\]
with the same exponent
\[
\alpha=
\begin{cases}
\min\!\left(\gamma,\ \lambda+1-\dfrac{2n}{p}\right),
& \text{if } \dfrac{2n}{\lambda+1}<p\le 2,\\[1.2ex]
\min\!\left(\gamma,\ \dfrac{\lambda+1-n}{p-1}\right),
& \text{if } 2\le p<n.
\end{cases}
\]
The comparison function \(v\) is now taken as the solution of
\[
-\operatorname{div}A(x,u,Dv)=0,
\]
and the same scheme applies [2510.10359].

The coercivity estimates are adapted to the nonlinear operator \(A\). For \(p\ge 2\),
\[
(A(x,\mu,\xi)-A(x,\mu,\eta))\cdot(\xi-\eta)\ge C|\xi-\eta|^p,
\]
while for \(1<p<2\),
\[
(A(x,\mu,\xi)-A(x,\mu,\eta))\cdot(\xi-\eta)
\ge
c\frac{|\xi-\eta|^2}{(\kappa+|\xi|+|\eta|)^{2-p}}.
\]
The \(B\)-term is controlled by the Morrey assumptions
\[
h\in L^{p,\lambda},\qquad g,f\in L^{1,\lambda},
\]
in conjunction with Fefferman–Phong and standard Young-type absorptions. In this sense, MoRight is not restricted to the prototype \(p\)-Laplacian but rather provides a full interior regularity scheme for degenerate and singular \(p\)-growth operators with mild coefficient dependence.

The paper makes a precise distinction between the degenerate and singular regimes. In the degenerate case \(p\ge 2\), the nonlinearity degenerates where \(|Du|\) is small, and strict \(p\)-monotonicity is sufficient to control the perturbation term. In the singular case \(1<p<2\), the operator is singular where \(|Du|\) is small, and the weighted quadratic inequality replaces strict \(p\)-monotonicity. The additional restriction \(p>2n/(\lambda+1)\) is therefore intrinsic to the singular-branch exponent.

## 6. Sharpness, thresholds, and position within the literature

A central feature of MoRight is the optimality of the Morrey threshold \(\lambda>n-1\). The paper invokes Serrin’s radial examples
\[
u(x)=|x|^\gamma,\qquad 0<\gamma\le 1,
\]
which solve
\[
-\Delta_p u=f
\]
with
\[
f(x)=\gamma^{p-1}\big[(\gamma-1)(p-1)-1+n\big]|x|^{(\gamma-1)(p-1)-1}.
\]
For these examples,
\[
f\in L^{1,\lambda}(B_1)
\quad\text{for}\quad
\lambda=(\gamma-1)(p-1)-1+n\le n-1.
\]
In that range, solutions are at most Lipschitz, which shows that the condition \(\lambda>n-1\) cannot be relaxed [2510.10359].

The exponent \(\alpha\) is likewise sharp in the sense stated by the paper: it is the minimum of the homogeneous gradient exponent \(\gamma\) and the Morrey-driven scaling exponents. Without improving either the data integrability or the homogeneous exponent, \(\alpha\) cannot be increased. This identifies the regularity threshold as a genuinely two-parameter phenomenon, depending simultaneously on the nonlinear homogeneous theory and the low-integrability scaling of the forcing term.

The paper also situates the result relative to classical linear theory. For \(p=2\) and smooth coefficients, Di Fazio (1992) proved \(u\in C^{1,\alpha}_{\mathrm{loc}}\) under \(f\in L^{1,\lambda}\), \(n-1<\lambda<n\), using Green’s function methods; MoRight recovers this linear case in the nonlinear degenerate and singular setting without Green’s function tools. More broadly, the paper states that prior Morrey-data results mainly yielded \(C^{0,\alpha}\) regularity or differentiability under stronger assumptions, whereas the present framework upgrades the conclusion to local \(C^{1,\alpha}\) in the full \(p\)-Laplacian-type regime.

The scope remains explicitly interior. Although \(\Omega\) is assumed bounded Lipschitz, no boundary regularity is claimed. The examples emphasized in the paper reflect the low-integrability reach of the theory: power-type singular data \(f(x)\approx |x|^{-s}\) in \(B_1\) belong to \(L^{1,\lambda}(B_1)\) if \(s\le n-\lambda\), so when \(n-1<\lambda<n\), one may take \(s\) just below \(1\) and still obtain
\[
u\in C^{1,\alpha}_{\mathrm{loc}}.
\]
This suggests that MoRight provides a near-endpoint interior regularity theory for degenerate elliptic equations with Morrey-controlled forcing.

Source: https://www.emergentmind.com/topics/moright