Caderón–Zygmund Estimates for Weak Solutions

• The paper establishes quantitative regularity for weak solutions by proving Calderón–Zygmund-type bounds in strong function spaces like Sobolev and Besov.
• It leverages classical decomposition methods, sharp maximal function techniques, and nonhomogeneous analysis to control oscillatory behaviors in elliptic, parabolic, and nonlocal PDEs.
• The findings extend to complex systems including nonlinear elliptic equations, fractional Laplacian problems, and double-phase systems, setting benchmarks in modern PDE theory.

A Calderón–Zygmund-type estimate for weak solutions is a fundamental regularity assertion stating that solutions of singular integral equations or elliptic/parabolic PDEs with rough data or coefficients satisfy quantitative bounds in strong function spaces (e.g., Sobolev, Besov, Morrey), thereby promoting regularity and integrability beyond what is a priori available. For weak solutions, these estimates undergird both linear and nonlinear regularity theory, form the backbone for multilinear extension, and serve as canonical benchmarks for modern PDE analysis.

1. Foundational Weak-Type Estimates for Multilinear Calderón–Zygmund Operators

Let $T$ be an $m$-linear Calderón–Zygmund operator on $\mathbb{R}^n$ associated with kernel $$K:\mathbb{R}^n \times (\mathbb{R}^n)^m \to \mathbb{C}$$, satisfying the standard size and Hölder-smoothness conditions on kernel variables. The endpoint weak-type bound asserts

$$\sup_{t > 0} t\,\left|\{x \in \mathbb{R}^n : |T(f_1, \ldots, f_m)(x)| > t\}\right|^{1/m} \leq C \prod_{j=1}^{m} \|f_j\|_{L^1(\mathbb{R}^n)}$$

for $f_j \in L^1(\mathbb{R}^n)$ and $C$ depending only on $n, m$, kernel constants, and the $L^2$–norm of $T$. This is proved via a variant of the classical function decomposition, leveraging Nazarov–Treil–Volberg nonhomogeneous techniques and a “good/bad” decomposition. The bad parts are controlled through Whitney decompositions and reduction to point-mass measures, with Hörmander-type cancellation (geometric lemma) providing a key estimate (Stockdale et al., 2018).

For $m = 2$, the bilinear analogue holds: $$\sup_{t > 0} t \left|\{x: |T(f, g)(x)| > t\}\right|^{1/2} \leq C \|f\|_1 \|g\|_1$$ By multilinear interpolation and duality, this yields strong-type bounds for all $L^{p_j}$ spaces with $1 < p_j < \infty$ and $1/p = \sum 1/p_j$.

For PDE applications, such estimates enable a priori control in weak spaces for nonlinear terms, e.g., bounding bilinear forms involving gradients in elliptic equations when data are merely integrable.

2. Calderón–Zygmund Theory for Nonlinear Elliptic Equations and Besov Regularity

Recent advances establish Besov-scale Calderón–Zygmund-type estimates for weak solutions to general nonlinear elliptic equations: $$\text{div}~A(x, \nabla u) = \text{div}[P(|F|) F] \quad \text{in}~\Omega \subset \mathbb{R}^n$$ where $A(x, \xi)$ is a Carathéodory map satisfying coercivity/monotonicity, $p$-growth, zero-order bounds, and VMO-type continuity in $x$. Under sharp maximal function techniques and higher integrability results, one obtains

$$\|V_P(\nabla u)\|_{B^{\alpha}_{2, q}(\Omega')} \leq C\Bigl\{ \|V_P(\nabla u)\|_{L^2(\Omega)} + \|P(|F|)F\|_{B^{\alpha}_{2, q}(\Omega)} + 1 \Bigr\}$$

where $$V_P(\nabla u) := \sqrt{P'(M^2 + |\nabla u|^2)} \nabla u$$. This generalizes Calderón–Zygmund regularity to Orlicz–Sobolev and Besov function space frameworks, with maximal function bounds mediating the oscillatory structure of the problem (Cheng et al., 2024).

Higher integrability and oscillation controls (via sharp maximal functions) allow pointwise control of the gradient, and sharp maximal/Fefferman–Stein inequalities establish fractional smoothness inherited from the data.

3. Global Calderón–Zygmund Estimates for Nonlocal and Fractional Laplacian Equations

For equations driven by fractional Laplacians

$$(-\Delta)^s u = f \quad \text{in}~\Omega, \qquad u=0~\text{outside}~\Omega$$

boundaries pose critical regularity questions. For kernels $A(x, y)$ with varying assumptions—VMO, Dini, or Hölder continuity—a hierarchy of global regularity estimates for $u/d^s$ (where $d(x) = \text{dist}(x,\partial \Omega)$) is derived (Byun et al., 2024):

• VMO kernel: Higher $L^p$–integrability
• Dini kernel: Boundedness and $C^s$ continuity up to $\partial \Omega$
• Hölder kernel: Fractional–Sobolev and Hölder bounds

The exponent range and constant dependencies are shown to be sharp through explicit barrier examples and function space embeddings. Nonlocal sharp maximal function estimates and Wolff potential bounds yield pointwise estimates.

4. Nonlocal Parabolic Calderón–Zygmund-Type Estimates with Discontinuous Kernels

Analyzing non-homogeneous nonlocal parabolic equations

$$\partial_t u(x, t) + \mathcal{L}_A u(x, t) = (-\Delta)^s f(x, t) + g(x, t)$$

with measurable, possibly discontinuous kernels $A(x, y, t)$, one proves genuine increments of spatial differentiability for weak solutions, surpassing local theory outcomes:

$$\int_{Q_{r/2}(z_0)} \int_{B_{r/2}(x_0)\times B_{r/2}(x_0)} |d_{s+\sigma}u(x, y, t)|^q\,\frac{dx\,dy\,dt}{|x-y|^{n+q\sigma}} \leq C\{\cdots\}$$

where $q>2$, $\sigma>0$. The gain from $s$ to $s+\sigma$ leverages nonlocal covering arguments, tail estimates, and reverse Hölder iteration. Diagonal/off-diagonal level-set decompositions and fractional Caccioppoli-type energy estimates drive the proof (Byun et al., 2023).

5. Calderón–Zygmund Estimates in Quasilinear and Double-Phase Systems

For $p$-Laplacian or double-phase systems—either in elliptic or parabolic form—the existence of Calderón–Zygmund-type bounds hinges on sharp conditions for data regularity and coefficient oscillation. For the $p$-Laplacian (Brasco et al., 2016): $$\int_{B_r}|D_jV(u)|^2\,dx \leq \frac{C}{(R-r)^2}\int_{B_R}|\nabla u|^p\,dx + \frac{C}{(R-r)^{p-2-2}}[f]_{W^{s,p'}(B_R)}^{p'}$$ where $s > (p-2)/p$ is sharp. Uhlenbeck's second derivative trick, fractional duality, and real interpolation scaffold the argument.

For double-phase parabolic systems, local and global CZ-type estimates are obtained via intrinsic scaling, stopping-time arguments, and approximation/reflection methods. For all exponents $\theta > 1$: $$\int_{Q_{\rho}(z_0)} [H(z, |Du|)]^{\theta} dz \leq C\{(\int_{Q_{2\rho}(z_0)} H(z, |Du|) dz)^{\theta} + \int_{Q_{2\rho}(z_0)} [H(z, |F|)]^{\theta} dz + 1\}$$ with $H(z, s) = s^p + a(z) s^q$ (Kim, 2023).

6. Calderón–Zygmund Gradient Estimates for $p$-Laplace Systems with BMO Complex Coefficients

Extensions to vector-valued weak solutions in the elliptic $p$-Laplace setting with complex coefficients $a(x)$ only in small-BMO are established (Quach et al., 24 Dec 2025): $$\|\mathrm{M}_{\beta}(|Du|^p)\|_{L^q(\Omega)} \leq C(\|\mathrm{M}_{\beta}(|F|^p)\|_{L^q(\Omega)} + \mu^p)$$ where $\mathrm{M}_{\beta}$ is the fractional maximal operator and $\mu\in[0,1]$ tunes (degenerate) ellipticity. Strong accretivity and sector conditions ensure monotonicity. Covering/level-set decay arguments and uniform estimates in Reifenberg-flat domains produce full $L^q$ and Morrey space regularity for $|Du|$.

7. Internal Calderón–Zygmund Theory for General Parabolic $p$-Laplacian Equations

Fully nonlinear parabolic PDEs of the form

$$u_t = \mathrm{div}~A(x, t, u, \nabla u) + \mathrm{div}(|F|^{p-2} F) + f$$

are handled by freezing/perturbation techniques for small mean oscillation in $x$, rescaling strategies, and compactness arguments, together with DiBenedetto's intrinsic geometry method and maximal-function-free covering. The main outcome: $$\int_{Q_3} |\nabla u|^{p q} dz \leq C\{\text{energy and data terms}\}$$ valid for $p > 2n/(n+2)$ and $q > 1$ (Nguyen, 2017). The proof unifies covering technique, rescaling, reverse Hölder, and Lipschitz approximation to patch local regularity estimates globally.

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In summary, Calderón–Zygmund-type estimates for weak solutions provide a flexible, quantitative regularity framework spanning singular integrals, nonlinear PDEs (both local and nonlocal), variable regularity coefficients, vector-valued unknowns, and systems with degenerate or double-phase structure. Sharpness of exponents, reliance on maximal function technology, and modern decomposition/cancellation methods underpin the technical machinery at the forefront of this theory.