Lorentz-Type Endpoint Estimates

• Lorentz-type endpoint estimates are specialized mapping theorems that capture borderline integrability when classical Lᵖ bounds fail.
• They utilize refined spaces like L^(p,q) and weak-Lᵖ to achieve optimal results in settings including harmonic analysis, PDEs, and geometric measure theory.
• Applications span sharp Strichartz estimates, Sobolev embeddings, and Calderón–Zygmund operators, elucidating critical regularity and operator rigidity.

Lorentz-type endpoint estimates refer to a class of sharp mapping theorems, inequalities, and regularity results in analysis where strong $L^p$-type bounds fail at critical exponents, but weak or refined bounds can still be obtained in Lorentz, Lorentz–Morrey, or related spaces. These Lorentz-scale substitutions are ubiquitous across harmonic analysis, PDE, geometric measure theory, operator theory, and dispersive equations. Lorentz spaces $L^{p,q}$ refine the classical $L^p$ scale by distinguishing levels of integrability and providing endpoint spaces (e.g., weak-$L^p$, $L^{p,\infty}$) crucial for quantifying borderline regularity and mapping properties. Lorentz-type endpoint phenomena manifest in restriction theorems, Strichartz estimates, Sobolev embeddings, Calderón–Zygmund theory, commutator bounds, eigenfunction estimates, fractional and singular integral operators, and more. They play a key role in understanding sharpness, rigidity, interpolation, and the limits of functional analysis tools under criticality.

1. Lorentz Space Fundamentals and Endpoint Sharpness

Lorentz spaces $L^{p,q}(\Omega)$ are defined via the distribution function and non-increasing rearrangement $f^*(t)$ of $|f|$: $$\|f\|_{L^{p,q}} = \left( q \int_0^\infty [t^{1/p} f^*(t)]^q \frac{dt}{t} \right)^{1/q}, \quad q<\infty;\quad \|f\|_{L^{p,\infty}} = \sup_{t>0} t^{1/p} f^*(t).$$ They interpolate between $L^p$ and $L^{p,\infty}$ (weak-$L^p$), capturing finer distinctions in regularity and integrability. Lorentz spaces admit extended Hölder-type inequalities: $$\|fg\|_{L^{q,r}} \leq C \|f\|_{L^{q_1,r_1}} \|g\|_{L^{q_2,r_2}},$$ given $1/q=1/q_1+1/q_2,\, 1/r=1/r_1+1/r_2$. In endpoint regimes—either for maximal operators, fractional integrals, or nonlinear commutators—the Lorentz target is often the smallest possible rearrangement-invariant function space supporting boundedness, e.g., $L^{n/(n-\gamma),\infty}$ for Riesz potential $I_\gamma$ acting on $L^{1,q}$ for any $q\in(0,1]$; no strictly smaller r.i. target is possible under general conditions (Mihula et al., 1 Oct 2025).

2. Hardy–Littlewood, Riesz Potentials, and Non-improvability

A canonical Lorentz endpoint result is the Riesz potential mapping $$I_\gamma f(x)=c_\gamma\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\gamma}}dy$$. For $f\in L^1(\mathbb{R}^n)$,

$$I_\gamma : L^1(\mathbb{R}^n) \longrightarrow L^{n/(n-\gamma),\infty}(\mathbb{R}^n)$$

is sharp. This optimality persists for $f$ in any Lorentz $L^{1,q}(\mathbb{R}^n)$, $q\in(0,1]$—the weak target $L^{n/(n-\gamma),\infty}$ is minimal among rearrangement-invariant quasi-Banach spaces. Two general abstract methods establish such endpoint rigidity: via Lorentz-endpoint domain functionals, and via Marcinkiewicz–type rearrangement estimates exploiting extremal behaviour of the operator on characteristic functions and their non-increasing rearrangements. These principles extend to fractional maximal operators, singular integrals, and trace embeddings (Mihula et al., 1 Oct 2025).

3. Lorentz-Type Endpoint Strichartz Estimates for Dispersive PDE

Endpoint Strichartz estimates are critical for dispersive equations such as Schrödinger and wave equations, particularly in the presence of singular or critical potentials. In the critical inverse-square Schrödinger setting ($H=-\Delta-\sigma|x|^{-2}$, $\sigma=(n-2)^2/4$, $n\geq 3$), the solution splits into radial and non-radial parts (Mizutani, 2016):

• Radial part: The endpoint $(p,q)=(2,2n/(n-2))$ does not admit a strong $L^2_t L^{2n/(n-2)}_x$ bound, but a weak-type Lorentz bound holds: $$\|e^{-itH}P_{\text{rad}}f\|_{L^2_t L^{2n/(n-2),\infty}_x} \lesssim \|f\|_{L^2_x}$$ No stronger $L^2_t L^{2n/(n-2),q}_x$ estimate is possible for finite $q<\infty$. This closely reflects 2D free Schrödinger endpoint rigidity. The inhomogeneous version holds between $L^{2n/(n-2),1}$ and $L^{2n/(n-2),\infty}$.
• Non-radial part: All admissible Strichartz bounds persist, including the strong endpoint.

These endpoint Lorentz phenomena elucidate the influence of symmetry, critical spectral effects, and dimension-reduction on dispersive regularity. They also extend to heat and wave equations in geometric settings.

4. Lorentz-Type Endpoint Regularity in Geometric Analysis and PDE

Lorentz-type refinements critically enhance endpoint estimates in elliptic and parabolic regularity theory. For the quasilinear Dirichlet problem in non-smooth domains (Reifenberg flat, BMO coefficients) (Adimurthi et al., 2014), sharp weighted endpoint estimates are proved for gradients: $$\int_\Omega |\nabla u(x)|^p\,w(x)\,dx \leq C \int_\Omega |f(x)|^p\,w(x)\,dx,\quad w\in A_1,$$ with sub-natural exponents handled via Lorentz–Morrey spaces. Gradient estimates in Lorentz-Morrey and weighted Lorentz scales below the natural exponent $p$ are achieved using advanced good-$\lambda$ methods, characterizing sharpness in terms of the underlying geometry and coefficient regularity.

The endpoint Alexandrov–Bakelman–Pucci estimate also illustrates Lorentz sharpness: in $n\ge3$,

$$\sup_{x\in\Omega}|u(x)| \leq \sup_{x\in\partial\Omega}|u(x)| + c_n\|\Delta u\|_{L^{n/2,1}(\Omega)},$$

where $L^{n/2,1}$ is sharp. For $n=2$, a logarithmic integral surrogate is necessary (Steinerberger, 2018).

5. Restriction, Oscillatory Integrals, and Spectral Lorentz Endpoints

In Fourier analysis, endpoint Lorentz estimates refine classical $L^p$ restriction and oscillatory integral results. Consider the Stein–Tomas theorem and extensions (Bak et al., 2010). For measures $\mu$ with size/decay: $$\|Ff\|_{L^{p_0,2}(\mathbb{R}^d)} \leq C \|f\|_{L^2(d\mu)},$$ the Lorentz secondary exponent $2$ is optimal; larger values fail by Knapp–type counterexamples. This sharp Lorentz endpoint persists in restriction, oscillatory integral operators of Carleson–Sjölin–Hörmander type, spectral projectors, and operators with fold singularities.

Spectral estimates on manifolds (e.g., Laplace eigenfunctions on non–positive curvature) achieve improved endpoint Lorentz bounds with logarithmic corrections (Sogge, 2015): $$\|e_j\|_{L^{p_c}(M)} \lesssim \lambda_j^{\frac{n-1}{2(n+1)}} (\log\log \lambda_j)^{-\frac{2}{(n+1)^2}},$$ with $p_c = \frac{2(n+1)}{n-1}$.

6. Bilinear, Multilinear, and Commutator Endpoint Theory

Endpoint behavior in multilinear operators is delicate, especially for bilinear Hilbert transform (BHT) and Calderón–Zygmund commutators. In BHT, the boundary of the Lacey–Thiele region $\mathcal H$ supports only restricted weak-type estimates with Lorentz–Orlicz logarithmic refinements: $$|\langle BHT(f_1,f_2),f_3 \rangle| \lesssim |F_1|^{3/4}|F_2|^{3/4}|F_3|^{-1/2}\log\log(e^e+|F_3|/\min\{|F_1|,|F_2|\}),$$ valid for major subsets (Plinio et al., 2014). The double-logarithmic factor is sharp and parallels the Walsh “quartile” model, where all log factors can be removed.

For multilinear Calderón–Zygmund commutators with Dini or log-Dini moduli, endpoint boundedness from product Hardy spaces to weak Lorentz spaces ($L^{1/m,\infty}$) is established, and generalized Christ–decomposition and cube selection drive summability (Li et al., 2016).

7. Endpoint Estimates for Riesz Transform and Hodge Systems

In non-doubling geometric settings, endpoint Lorentz boundedness for the Riesz transform is realized on connected sums of manifolds: $$\|Rf\|_{(n^*,1)} \leq C\|f\|_{(n^*,1)},\quad n^* = \min_i n_i.$$ This sharp result fails for $L^{n^*,p}$, $p>1$—Lorentz endpoint is optimal (He, 2023). Similar optimal Lorentz–Besov endpoint bounds hold for fractional integration, cocanceling operators, and Hodge systems, e.g.,

$$\|I_\alpha F\|_{\dot{B}^{0,1}_{d/(d-\alpha),1}} \leq C\|F\|_{L^1},$$

with duality pairing yielding sharp Lorentz–Besov bounds for test functions (Hernandez et al., 2021).

8. Lorentz Endpoints in Fractional Sobolev, Gagliardo–Nirenberg, and Interpolation Theory

Classical $L^p$-type Sobolev and inter/extrapolation estimates fail at critical exponents where Lorentz-space endpoints survive. For instance, the one-dimensional Sobolev embedding at critical smoothness yields only weak $L^p$: $$\|(u(x)-u(y))/(x-y)\|_{L^{p,\infty}(\mathbb{R}\times\mathbb{R})} \leq C\|u'\|_{L^p(\mathbb{R})},$$ with analogous higher-dimensional, fractional, and mixed exponent generalizations (Brezis et al., 2021). Logarithmic and product-type refinements quantify the precise limiting behavior at criticality. Optimality in the Lorentz scale is established: no improvement to $L^{p,q}$, $q<\infty$ is possible.

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Lorentz-type endpoint estimates thus encapsulate a recurrent theme in modern analysis: at critical regularity, strict $L^p$ bounds can fail, but endpoint substitutions in Lorentz or related scales preserve mapping or regularity, often with sharpness, rigidity, and precise understanding of operator structure and functional interpolation. This structural paradigm is irreducible in restriction theory, dispersive PDE, operator theory, geometric analysis, and beyond.