Weighted Square Function Estimates in Harmonic Analysis

• Weighted square function estimates are quantitative inequalities for non-linear square functions acting on weighted spaces, using key characteristics such as Muckenhoupt A_p conditions.
• They extend classical Littlewood–Paley and Lusin area methods to capture both weak- and strong-type bounds with explicit dependence on weight constants.
• Advanced techniques like atomic decomposition and sparse domination underpin these results, enabling extensions to matrix weights, non-homogeneous filtrations, and fractal domains.

A weighted square function estimate refers to quantitative norm inequalities for square functions—central non-linear operators arising in real-variable harmonic analysis—when acting on weighted function spaces. The subject spans the classical Littlewood–Paley, Lusin area, and intrinsic (kernel-free) square functions, extending to contexts such as non-homogeneous filtrations, matrix-weighted spaces, non-integral elliptic square functions, and function spaces of Hardy or Herz type. This area incorporates delicate dependence on Muckenhoupt $A_p$ weights and often explicit control in terms of the $A_p$ characteristic, sometimes coupled with reverse Hölder, $A_\infty$, or testing-type constants.

1. Muckenhoupt Weights, Square Functions, and Weighted Hardy Spaces

Let $w(x)>0$ be locally integrable on $\mathbb{R}^n$ and recall the classical Muckenhoupt $A_p$ condition $(1\leq p<\infty)$: $$w\in A_p \Longleftrightarrow \sup_Q \Bigl(\frac{1}{|Q|}\int_Q w\Bigr)\Bigl(\frac{1}{|Q|}\int_Q w^{-1/(p-1)}\Bigr)^{p-1} < \infty$$ with $A_1$ given by the essential infimum property.

Weighted Hardy spaces $H^p_w$ $(0<p\leq 1)$ are defined by the finiteness of the maximal function $M_\varphi f(x)=\sup_{t>0}|f*\varphi_t(x)|$ in $L^p(w)$. When $p>1$, $H^p_w = L^p(w)$.

Intrinsic square functions, introduced by Wilson, are defined as follows: given $0<\alpha\leq 1$, set

$$A_\alpha(f)(y,t) = \sup_{\theta\in \mathcal{C}_\alpha}|f*\theta_t(y)|,$$

where $\mathcal{C}_\alpha$ is the class of compactly supported mean-zero functions satisfying the Hölder-$\alpha$ regularity.

The three canonical intrinsic square functions are:

• Lusin area: $S_\alpha(f)(x)$,
• Littlewood–Paley: $g_\alpha(f)(x)$,
• Maximal $g^*_{\lambda,\alpha}(f)(x)$, all defined using $A_\alpha(f)$ with specific integration domains and weights.

2. Weak-Type Estimates: Main Theorems and Exponents

The sharp weak-type ($L^{p,\infty}(w)$) inequalities for $S_\alpha$, $g_\alpha$, and $g^*_{\lambda,\alpha}$ on weighted Hardy spaces $H^p_w$ are as follows (for $w\in A_1$, $p=n/(n+\alpha)$):

$$w(\{x : S_\alpha(f)(x) > \lambda\}) \leq C \lambda^{-p}\|f\|_{H^p_w}^p$$

and identically for $g_\alpha$. For $g^*_{\lambda,\alpha}$, this bound holds when the aperture/moment satisfies $\lambda > 3 + 2\alpha/n$.

In the classical weighted Lebesgue scale, the following compressed summary captures the sharp results for all $1<p<\infty$:

• Strong-type: $$\|S\|_{L^p(w)\to L^p(w)} \lesssim [w]_{A_p}^{\max\{\frac12, 1/(p-1)\}}$$.
• Weak-type ($p>2$): $$\|S\|_{L^p(w)\to L^{p,\infty}(w)} \lesssim [w]_{A_p}^{1/2}$$, with no logarithmic correction (Hytönen et al., 2015).
• Weak-type ($1 $\|S\|_{L^{p,\infty}(w)}\lesssim [w]_{A_p}^{1/p}$.
• Critical ($p=2$): Current best is $$\|S\|_{L^{2,\infty}(w)} \lesssim [w]_{A_2}^{1/2}(1+\log[w]_{A_2})$$, with the logarithmic gap possibly being sharp (Lacey et al., 2012).

These exponents are optimal: no power less than $1/2$ can replace $1/2$ for weak-type $p>2$, and for $1

3. Techniques: Atomic Decomposition and Sparse Domination

The weighted Hardy space theory for weak-type bounds fundamentally employs atomic decomposition:

• Any $f\in H^p_w$ admits $f = \sum_j \lambda_j a_j$, where the $a_j$ are $(p,q,s)$-atoms supported in cubes/cells, satisfying moment cancellation and size $\|a_j\|_{L^q(w)} \leq w(Q_j)^{1/q-1/p}$. The $\ell^p$-sum of $\lambda_j$ is controlled by $\|f\|_{H^p_w}^p$.

A superposition principle converts atom-wise estimates into full space estimates: $$\text{if } \|f_j\|_{L^p(w)}\leq 1, \ \sum_j |c_j|^p\leq 1, \text{ then } w\left(\left|\sum_j c_j f_j\right|> \lambda\right) \leq C\lambda^{-p}.$$

Estimating the square function on a single atom then proceeds via “near/far" splitting:

• Near ($I_1$): Use Hölder and $L^q(w)$ boundedness, leveraging $A_1\subset A_q$, and Wilson's $L^q$ theory for $S_\alpha$.
• Far ($I_2$): Exploit atom cancellation to obtain pointwise decay

$$S_\alpha(a)(x)\leq C \frac{w(Q)^{1/p}}{|x-x_Q|^{n+\alpha}},$$

and sum over dyadic shells, reducing to a geometric series summing to $O(\lambda^{-p})$.

Sparse domination, central to modern weighted theory (Hytönen et al., 2015, Bailey et al., 2020), allows reduction of square functions (classical or intrinsic) to averages over sparse collections, so-called “sparse square functions,” whose weighted $L^p(w)$ norm is explicitly computable in terms of the $A_p$ characteristic: $$A^2_{\mathcal S}f(x) := \Bigl(\sum_{Q\in\mathcal S} \langle|f|\rangle_Q^2 \mathbf{1}_Q(x)\Bigr)^{1/2}$$ satisfies $L^p(w)$ and $L^{p,\infty}(w)$ bounds with the sharp $A_p^{1/2}$ exponent for $p>2$ (Hytönen et al., 2015).

4. Quantitative Dependence on Weights: Optimal Constants and Structure

Weighted square function inequalities are now available with complete quantitative dependence on the $A_p$ characteristic, sometimes augmented by $A_\infty$ or reverse Hölder constants for refined control:

• Sharp exponents: $[w]_{A_p}^{1/2}$ (strong and weak type, $p>2$), $[w]_{A_p}^{1/p}$ (weak, $1
• Mixed $A_p$–$A_\infty$ bounds: $$\|S\|_{L^p(w)\to L^{p,\infty}(w)} \lesssim [w]_{A_p}^{1/p} [w]_{A_\infty}^{1/2-1/p}$$, reducing to $[w]_{A_p}^{1/2}$ for $p>2$, with no logarithmic correction (Hytönen et al., 2015).
• Restricted weak-type: For characteristic functions $$\|S_w\chi_E\|_{L^{2,\infty}(w^{-1})} \leq C \sqrt{[w]_{A_2} \langle w\rangle_E |E|}$$, linear in $[w]_{A_2}$, sharp in this setting (Ivanisvili et al., 2018).

In non-homogeneous settings or for structures without classical doubling, exponents and the sufficiency of weight conditions can change dramatically:

• Non-homogeneous filtrations: Martingale $A_2$ is necessary, with optimal (linear) exponent, in contrast to homogeneous/dyadic $A_\infty$ where exponent $1/2$ suffices (Domelevo et al., 2017).
• Matrix weights: The sharp matrix $A_2$ bound for vector-valued square functions is $\|S_W f\|_{L^2} \lesssim [W]_{A_2} \|f\|_{L^2(W)}$, again linear and optimal (Hytönen et al., 2017).

5. Endpoint, Mixed, and Vector-Valued Extensions

The endpoint $p=2$ is subtle:

• In the classical theory (scalar weights), weak-type $(2,2)$ bounds carry an extra logarithmic factor: $$\|S\|_{L^{2,\infty}(w)} \lesssim [w]_{A_2}^{1/2}(1+\log[w]_{A_2})\|f\|_{L^2(w)}$$ with potential conjectural sharpness (Lacey et al., 2012, Ivanisvili et al., 2018). No known improvement removes the logarithm for general square functions, while for restricted weak type (on indicators), sharp $[w]_{A_2}^{1/2}$ is achieved.

For strong-type estimates, the Bellman function method produces explicit constants and also two-sided inequalities: $$C^{-1}[w]_{A_2}^{1/2}\|f\|_{L^2(w)} \leq \|S f\|_{L^2(w)} \leq C[w]_{A_2}^{1/2}\|f\|_{L^2(w)}$$ with the same method extending to heat/Poisson semigroup and Lusin area (Banuelos et al., 2016).

Mixed two-weight, bump-type, Fefferman–Stein, and Sawyer-type mixed weak-type inequalities for square functions, as well as local decay estimates, have all been developed for square functions associated with abstract operators satisfying Gaussian (Davies–Gaffney) estimates. These employ sparse domination, Orlicz bump techniques, and extrapolation arguments—often yielding sharp, explicit dependence on the $A_p$ and reverse Hölder constants (Cao et al., 2020, Bailey et al., 2020, Mena et al., 17 Jun 2025).

6. Extensions: Intrinsic Square Functions, Herz and Hardy Spaces, and Fractal Domains

The intrinsic square functions $S_\alpha$, $g_\alpha$, $g^*_{\lambda,\alpha}$ admit sharp strong- and weak-type results on weighted Hardy and Herz-type Hardy spaces:

• Endpoint mapping $H^p_w \to L^{p,\infty}(w)$ at $p=n/(n+\alpha)$, $w\in A_1$, and two-weight $A_1$ pairs in Herz contexts (Wang, 2010, Wang, 2010).

Atomic decomposition, decay estimates of the intrinsic kernel, and precise covering arguments (Calderón–Zygmund) are essential for endpoint weak-type mapping, while off-cube decay and doubling properties of $A_1$ are critical for both near and far contributions.

Further extensions comprise:

• Directional/anisotropic square functions: Embedding and maximal operator techniques address weighted square function estimates involving families of rectangles or multipliers in multiple directions, frequently yielding explicit dependence on parameters such as direction number ($N$), as in directional maximal (Carleson) theory (Accomazzo et al., 2020).
• Parabolic and fractal settings: Recent results obtain weighted square function (and local smoothing) estimates in variable coefficient or fractional-dimension scenarios with sharp dependence on geometric distribution parameters of the weight (Kim et al., 6 Nov 2025).

7. Open Problems and Limitations

• The possible removal of the logarithmic factor in the $p=2$ weak-type estimate for general classical or non-integral square functions remains unresolved.
• In non-homogeneous filtrations, $A_\infty$ is insufficient for lower square-function bounds; martingale $A_2$ is necessary, with unavoidable linear growth (Domelevo et al., 2017).
• Vector-valued and non-commutative square function regimes demand further investigation, particularly mixed-norm and endpoint estimates, though initial $A_1$-weighted weak-type (1,1) bounds are now available (Ray et al., 1 Oct 2024).
• Matrix-weighted extensions to more general singular integrals remain a largely open domain.

In summary, the theory of weighted square function estimates has achieved precision both in exponents and in dependence on weight characteristics through modern analytic techniques such as atomic decomposition, Bellman function methods, sparse domination, and sophisticated covering arguments. The landscape continues to evolve, with connections to operator theory, time-frequency analysis, and applications in PDE and signal processing.