Riesz potentials #

Let E be a real normed space of finite dimension n ≥ 1 with an additive Haar measure μ, and let ω = μ(B(0, 1)). The Riesz kernel of order α ∈ (0, n) is ‖x - y‖ ^ (α - n), and the Riesz potential of a function is its integral against this kernel. This file proves the two kinds of Lᵖ-L^q estimates for Riesz potentials: the non-sharp ones on a set of finite measure, and the Hardy–Littlewood–Sobolev inequality on the whole space, at the exponent q forced by scaling. The constant there is only shown to exist; its best value is not computed.

Riesz potentials on sets of finite measure #

Let Ω ⊆ E be measurable. For 0 < κ ≤ 1 consider the potential

V f (x) = ∫_Ω ‖x - y‖ ^ (n (κ - 1)) f(y) dy,

whose kernel is the Riesz kernel of order n κ. Its two main estimates are those of Gilbarg–Trudinger, Lemma 7.12, which drive the Sobolev embeddings in the potential-theoretic approach:

the kernel integral ∫_Ω ‖x - y‖ ^ (n (κ - 1)) dy is at most κ⁻¹ ω ^ (1 - κ) μ(Ω) ^ κ, uniformly in x;
for 1 ≤ p ≤ q < ∞ with δ = 1/p - 1/q < κ, the potential maps Lᵖ(Ω) to L^q(Ω): ‖V f‖_q ≤ ((1 - δ) / (κ - δ)) ^ (1 - δ) ω ^ (1 - κ) μ(Ω) ^ (κ - δ) ‖f‖_p.

The kernel bound follows from a rearrangement inequality, valid in any metric measure space: a radially nonincreasing function of dist x y integrates over Ω to at most its integral over a closed ball about x of no smaller measure. The Lᵖ-L^q bound is then Young's inequality for integral kernels (TauCeti.lintegral_rpow_lintegral_mul_le_of_inv_add_inv_eq), applied to the kernel raised to the power r = 1 / (1 - δ), which is again a Riesz kernel, of order n (κ - δ) / (1 - δ).

The case κ = 1/n, p = n, where the kernel is ‖x - y‖ ^ (1 - n) and the bound holds for every finite q ≥ n with a constant of order q ^ (1 - 1/n), is the input to the borderline Sobolev embedding of W^{1,n}_0(Ω).

The Hardy–Littlewood–Sobolev inequality #

On the whole space, the Riesz potential of order α of a function f is

I_α f (x) = ∫ ‖x - y‖ ^ (α - n) ‖f y‖ dμ(y),

the convolution of ‖f‖ with the locally integrable kernel ‖·‖ ^ (α - n) (TauCeti.rieszPotential). It is taken in ℝ≥0∞, so it is defined for every f without any integrability hypothesis, and it dominates the potential of f itself for vector-valued f. No normalising constant is included.

The Hardy–Littlewood–Sobolev inequality (TauCeti.exists_eLpNorm_rieszPotential_le) says that I_α is bounded from Lᵖ to L^q whenever 1 < p and 1 / q = 1 / p - α / n:

‖I_α f‖_q ≤ C ‖f‖_p,

with C depending only on μ, α and p. Since q ∈ ℝ≥0 and 0⁻¹ = 0, the relation also admits q = 0, the case α p = n, and there the bound is degenerate. The exponent-zero seminorm eLpNorm g 0 vanishes when g is almost everywhere strongly measurable. If ‖f‖_p < ∞, then f is almost everywhere strongly measurable, hence so is I_α f, and the left side vanishes. If ‖f‖_p = ∞, the bound is trivial. It has content only for q ≠ 0, that is, α p < n.

This is the endpoint δ = κ excluded on sets of finite measure above, and the exponent q is forced by scaling. It is the fractional integration theorem behind the non-endpoint Strichartz estimates, where it is applied in the time variable, and behind the Sobolev embedding of the Bessel-potential spaces.

The proof is Hedberg's. Split the integral defining I_α f (x) into dyadic shells r ≤ ‖x - y‖ < 2 r with r = R 2ᵏ, k ∈ ℤ. On a shell the kernel is at most r ^ (α - n) and the shell lies in the ball B(x, 2 r). For the shells inside B(x, R) the integral of ‖f‖ over that ball is at most μ (B(x, 2 r)) M f (x), where M is the Hardy–Littlewood maximal function. Since α > 0 these shells sum to at most a multiple of R ^ α M f (x). For the shells outside B(x, R) Hölder's inequality bounds the same integral by μ (B(x, 2 r)) ^ (1 - 1 / p) ‖f‖_p. Since α < n / p these shells sum to at most a multiple of R ^ (α - n / p) ‖f‖_p. Choosing the radius R that balances the two terms gives Hedberg's inequality (TauCeti.exists_rieszPotential_le_mul_maximalFunction_rpow_mul_eLpNorm_rpow)

I_α f (x) ≤ C (M f (x)) ^ (1 - θ) ‖f‖_p ^ θ, where θ = α p / n,

valid for 1 ≤ p < n / α. Since (1 - θ) q = p, the strong-type (p, p) maximal inequality TauCeti.eLpNorm_maximalFunction_le turns this pointwise bound into the Hardy–Littlewood–Sobolev inequality when p > 1. The strong-type bound fails at p = 1.

Main declarations #

TauCeti.setLIntegral_le_setLIntegral_closedBall_of_antitone: balls maximise the integrals of radially nonincreasing kernels among sets of no larger measure.
TauCeti.setLIntegral_enorm_sub_rpow_le: the kernel bound.
TauCeti.setLIntegral_enorm_sub_rpow_le_finrank: the same bound for the kernel of exponent 1 - n, in terms of the dimension n.
TauCeti.eLpNorm_setLIntegral_enorm_sub_rpow_mul_le: the Lᵖ-L^q bound for the potential on a set of finite measure.
TauCeti.rieszPotential: the Riesz potential I_α f of order α on the whole space.
TauCeti.rieszPotential_zero, TauCeti.rieszPotential_add_le, TauCeti.rieszPotential_const_smul: I_α is sublinear.
TauCeti.exists_rieszPotential_le_mul_maximalFunction_rpow_mul_eLpNorm_rpow: Hedberg's inequality.
TauCeti.exists_eLpNorm_rieszPotential_le: the Hardy–Littlewood–Sobolev inequality.
TauCeti.enorm_integral_norm_sub_rpow_smul_le_rieszPotential, TauCeti.exists_eLpNorm_integral_norm_sub_rpow_smul_le: the potential of a vector-valued f is dominated by I_α f, so it satisfies the Hardy–Littlewood–Sobolev inequality too.

References #

D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Lemma 7.12.
L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter V, §1.2, Theorem 1.
L. Grafakos, Modern Fourier Analysis, 3rd ed., Theorem 1.2.3.