The weak-type (1,1) bound for the Hardy--Littlewood maximal function is $O(\sqrt{n} \log n)$

For f∈L1(R^n), Mf(x)=sup_{r>0}|B(x,r)|^{-1}∫_{B(x,r)}|f|. Covering arguments yield 3^n, and Stein–Strömberg obtained O(n). Heat propagation H_t=e^{tΔ} and ergodic averages \widetilde P_t=t^{-1}∫_0^tH_sds offer dimension-friendly semigroup tools, but the classical comparison from ball averages to ergodic averages incurs a linear dimensional loss.

The goal is to reduce the dimensional constant C_n in ||Mf||_{1,∞}≤C_n||f||_1. Ball averages have sharp geometric boundaries, whereas heat kernels have smooth Gaussian tails; converting between them without accumulating dimensional losses is difficult. Moreover, the heat maximal function takes a supremum over all times and absolute values, so it is harder than the positive ergodic maximal operator.

The paper has three linked innovations. First, it proves Mf≲√nH*, reducing the geometric loss to √n. Second, it uses classical Laplacian partial balayage, ν=κ1_Ω−Δu, to create a controlled domain and residual measure. Third, it analyzes the residual in logarithmic time: Gamma-function estimates control high Mellin frequencies, while Abel-power square functions control low frequencies, producing O(log n) rather than O(√n).

• �� Atomic reduction: approximate a general positive L1 datum by a finite atomic measure ν of mass m.
• �� Partial balayage: construct u≥0 and Ω with ν=κ1_Ωdx−Δu and |Ω|=m/κ.
• �� Residualization: set σ=ν−κ1_Ωdx and R_tσ=H_tσ−\widetilde P_tσ; use the sign of the ergodic term on Ω^c.
• �� Logarithmic time: write t=e^v and Fourier transform the orbit F_{σ,x}(v)=H_{e^v}σ(x).
• �� Frequency split: define Q_Λ for |ξ|>Λ and T_Λ for |ξ|≤Λ, obtaining sup_t|R_tσ|≤Q_Λ+T_Λ.
• �� High frequencies: prove ∫Q_Λ≲C nΛ^2e^{-cΛ}m and choose Λ=An.
• �� Low frequencies: use A_s=s(s−Δ)^{-1}, Abel powers A_s^N, Theorem 1.3, and dyadic summation over O(logΛ) scales.
• �� Final transfer: obtain the O(log n) heat bound and multiply by the √n pointwise loss.

There are no datasets, numerical experiments, learned models, or empirical training baselines. The proof is tested on finite positive atomic measures for n≥3 and then extended by approximation. The relevant comparisons are the covering bound 3^n, the Stein–Strömberg O(n) estimate, the earlier O(√n)-type heat estimate, and the new O(log n) heat estimate. The high- and low-frequency estimates function as the paper's main internal decomposition checks.

The heat maximal operator has weak constant C log(1+n), and the ball maximal operator has C√n log(1+n). Theorem 1.3 gives a C√Nκm square-function bound. The high-frequency integral becomes O(m) after Λ=An, while the low-frequency estimate costs log^2(eΛ)κm before the weak-type argument. Thus log n comes from summing frequency scales, whereas √n comes from the ball-to-heat pointwise transfer.

The result can support dimension-sensitive estimates for singular integrals, heat equations, interpolation arguments, and diffusion semigroups. The partial-balayage framework may also inform obstacle problems and related potential-theoretic estimates. These are theoretical uses rather than demonstrated engineering applications; transferring the method requires an appropriate diffusion kernel, positivity, and an analogue of the balayage structure.

The proof is formulated first for n≥3, where the Newtonian kernel and classical Laplacian balayage have the required form. The logarithm may be an artifact of dyadic frequency summation, but no removal is known. The √N square-function dependence is a central bottleneck, and no lower bound establishes necessary dimensional divergence. The paper also does not provide non-Euclidean, nonlocal, or sharp low-dimensional extensions.