$L^p$ bounds, $1<p<2$ , for convolution with an $L^2$ -bounded Hörmander kernel on $\mathbb R^d$ , uniform in the kernel

Proved

DbarRegularity.eLpNorm_integral_mul_sub_le_of_hormander_of_lt_two

by Mazecto · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

harmonic-analysislp-spacessingular-integrals

Let $d\ge1$ , $1<p<2$ and $B_1,B_2\ge0$ . There is a constant $A\ge0$ , depending only on $d,p,B_1,B_2$ , with the following property. Let $K\in L^2(\mathbb R^d)$ be complex valued, and suppose that

$K$ satisfies the Hörmander condition with constant $B_1$ :

\int_{\{x:\ |x|\ge2|y|\}}|K(x-y)-K(x)|\,dx\le B_1\qquad(y\ne0);

convolution with $K$ is bounded on $L^2$ with norm at most $B_2$ : $\|K*g\|_{L^2}\le B_2\|g\|_{L^2}$ for every $g\in L^2(\mathbb R^d)$ .

Then for every $f\in L^p(\mathbb R^d)$ ,

\|K*f\|_{L^p}\le A\,\|f\|_{L^p},\qquad K*f(x)=\int K(x-y)f(y)\,dy .

This is the range $1<p<2$ of the $L^p$ theory of singular convolution operators, with a constant uniform over all kernels satisfying the two bounds.

Formalization Note. $\mathbb R^d$ is EuclideanSpace ℝ (Fin d) with Lebesgue measure volume, $d\ge1$ is 0 < d, and $p$ is in ℝ≥0∞. The constants $A,B_1,B_2$ are in ℝ≥0. The Hörmander condition is the lower integral ∫⁻ x in {x | 2 * ‖y‖ ≤ ‖x‖}, ‖K (x - y) - K x‖ₑ ≤ B₁. Norms are eLpNorm. $K*f(x)$ is the Bochner integral ∫ y, K (x - y) * f y (set to $0$ where not integrable). The quantifiers are ordered $\exists A\ \forall K\ \forall f$ .

Preamble

import Definitions.Def_DbarRegularity_Setting
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2

Formal statement

namespace DbarRegularity

open MeasureTheory Set
open scoped ENNReal NNReal

/-- `L^p` bound, `1 < p < 2`, for convolution with an `L²` kernel `K` on `ℝ^d` that satisfies the
Hörmander condition with constant `B₁` and is bounded on `L²` with norm `B₂`; the constant
depends only on `d`, `p`, `B₁`, `B₂`. -/
theorem eLpNorm_integral_mul_sub_le_of_hormander_of_lt_two {d : ℕ} (hd : 0 < d) (p : ℝ≥0∞)
    (hp : 1 < p) (hp2 : p < 2) (B₁ B₂ : ℝ≥0) :
    ∃ A : ℝ≥0, ∀ K : EuclideanSpace ℝ (Fin d) → ℂ, MemLp K 2 volume →
      (∀ y, y ≠ 0 → ∫⁻ x in {x | 2 * ‖y‖ ≤ ‖x‖}, ‖K (x - y) - K x‖ₑ ≤ B₁) →
      (∀ g, MemLp g 2 volume →
        eLpNorm (fun x => ∫ y, K (x - y) * g y) 2 volume ≤ B₂ * eLpNorm g 2 volume) →
      ∀ f, MemLp f p volume →
        eLpNorm (fun x => ∫ y, K (x - y) * f y) p volume ≤ A * eLpNorm f p volume := by sorry

end DbarRegularity

Source

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970), Chapter II, § 2.2, Theorem 1, pp. 29–31

View graph

LpL^pLp bounds, 1<p<21<p<21<p<2, for convolution with an L2L^2L2-bounded Hörmander kernel on Rd\mathbb R^dRd, uniform in the kernel

$L^p$ bounds, $1<p<2$ , for convolution with an $L^2$ -bounded Hörmander kernel on $\mathbb R^d$ , uniform in the kernel