An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

Remark 1.1

Although we will not be concerned with this issue in the present work it is worth mentioning that this subtle point has important applications to the Yamabe problem, as Moser did, and other related geometric analysis problems (cf. [11, 21] or the survey article [12] and the references therein for more details). Existence of extremizers for the sharp constant in the case Ω is an n-dimensional ball can be found in the work of Carleson and Chang [9].

In this paper we focus on the latter question, that of estimates for these potentials in the critical exponent. In particular, with simple proofs we establish some new exponential decay estimates in the spirit of (1.2). As we will see, our work extends a result of D. Adams in [1] and improves upon an estimate of H. Brezis and S. Wainger [7] (see also [31]). Here it is useful to change the perspective of the preceding inequality to a corresponding estimate for potentials in the critical exponent: Let Ω ⊂ ℝ^N be an open and bounded set. There exist constants c′ , C′ > 0 such that

for all f with supp f ⊂ Ω and ∥f∥_Lp(Ω) ≤ 1, where p = N/α and we have used I_αf to denote the Riesz potential of order α ∈ (0, N) of f , defined by

for γ(α) = π^N/22^αΓ(α/2)Γ((N − α)/2)⁻¹. The inequality (1.4) has an extensive history in the literature – it has been observed by Yudovich in [32], the case α = 1 is proved in Trudinger’s paper [30], a version for Bessel potentials is due to Strichartz [28], the proof of the statement we assert here is in Hedberg’s paper [18], while the optimal constant was established by Adams in [3] (see also [31] for the optimal constant on the Lorentz scale).

The consideration of (1.2) and (1.3), along with the comparison of (1.3) and (1.4) prompts one to wonder whether the improved exponential integrability found in (1.4) comes with a corresponding improved exponential decay estimate. The first result of this paper is the following theorem to this effect.

Theorem 1.2

Let Ω ⊂ ℝ^N be an open and bounded set and α ∈ (0, N). There exist constants c, C > 0 that depend on α, N, and Ω such that

for all f ∈ L^N/α(Ω) such that ∥f∥_L^N/α(Ω) ≤ 1 and supp f ⊆ Ω.

This is not surprising, since the proof amounts to relaxing the usual arguments in strong-type spaces to a weak-type setting. However, the technique is interesting as it suggests the possibility of other related inequalities. Indeed, when one examines the work of Yudovich [32], one finds that he asserts the result not only for integrals over a domain, but even for n-dimensional hyperplanes intersected with Ω, n ≤ N an integer (this bears a resemblance to the special case of Beurling mentioned at the introduction, who obtains the exponential decay on the circle). This corresponds to a property enjoyed by functions in the critical Sobolev space which is not true for general functions in BMO: the trace of such functions are BMO on the restriction. Our method can be adapted to this setting, and even to the setting of fractal sets. In order to state our next result let us first introduce the Hausdorff content of a set E ⊂ ℝ^N which is defined by

Here the infimum is taken over all possible coverings of arbitrary radii and ωβ:=πβ/2/Γβ2+1 is the volume of a β-dimensional sphere.

We can now state

Theorem 1.3

Let Ω ⊂ ℝ^N be an open and bounded set, α ∈ (0, N), and β ∈ (0, N]. There exist constants c, C > 0 that depend on α, N, β, and Ω such that

for all f ∈ L^N/α(Ω) such that ∥f∥_L^N/α(Ω) ≤ 1 and supp f ⊆ Ω.

This result is analogous to an estimate proved by D. Adams in [1] for the decay of level sets of the convolution of the Bessel kernel and a function in this critical space. In particular, in the proof of Theorem 3, Item (ii) on p. 913, Adams proves an exponential decay estimate of the μ measure of the level sets of such a convolution, where μ is a non-negative Radon measure which satisfies the ball growth condition μ(B(x, r)) ≤ r^β for some β > 0. One finds the analogy in the equivalence of H∞β and the supremum over all such measures.

From Theorem 1.3 we deduce the following improvement to the inequality (1.4), our

Corollary 1.4

Let Ω ⊂ ℝ^N be an open and bounded set, α ∈ (0, N), and β ∈ (0, N]. There exist constants c′ , C′ > 0 that depend on α, N, β, and Ω such that

for all f ∈ L^N/α(Ω) such that ∥f∥_L^N/α(Ω) ≤ 1 and supp f ⊆ Ω.

Here the integral is intended in the sense of Choquet, namely

which is a priori well-defined for φ ∈ C_c(ℝ^N), and which extends by completion to a space L1(H∞β) which consists of H∞β−quasi continuous functions, see [4] for details.

We conclude the introduction with an application of our techniques to improve the dimension on the estimate of H. Brezis and S. Wainger [7], which follows from an extension of our Theorem 1.3 to the Lorentz scale. To this end, let us recall that Brezis and Wainger [7, Theorem 3 (ii) on p. 784] proved a limiting case of a convolution inequality of O’Neil [22] which establishes that the second parameter in the Lorentz space L^p,q in this critical regime, while microscopic, is magnified in these inequalities^[1]. As in this paper we work exclusively with the Riesz kernels, we now state a version of their result in this context^[2] which has been proved by J. Xiao and Zh. Zhai [31, Theorem 3.1 (ii) on p. 364]: There exists constants c′ , C′ > 0 such that

for all f with supp f ⊂ Ω and ∥f∥_L^N/α,q(Ω) ≤ 1.

In particular, our method firstly enables us to establish an exponential decay of the level sets of such convolutions with respect to the Hausdorff content, which is our

Theorem 1.5

Let Ω ⊂ ℝ^N be an open and bounded set, α ∈ (0, N), β ∈ (0, N], and q ∈ (1,∞]. There exist constants c, C > 0 that depend on α, N, β, q, and Ω such that

for all f ∈ L^N/α,q(Ω) such that ∥f∥_L^N/α,q(Ω) ≤ 1 and supp f ⊆ Ω.

In the usual way this leads to an improved dimensional version of the inequality of [7] and [31], which we state as

Corollary 1.6

Let Ω ⊂ ℝ^N be an open and bounded set, α ∈ (0, N), β ∈ (0, N] and q ∈ (1,∞]. There exist constants c′ , C′ > 0 that depend on α, N, β, q, and Ω such that

for all f ∈ L^N/α,q(Ω) such that ∥f∥_L^N/α,q(Ω) ≤ 1 and supp f ⊆ Ω.

The paper is divided as follows. In Section 2 we provide some preliminaries about the Lorentz spaces we here require. In Section 3 we prove a variant of a technical result due to Hedberg as well as several other technical lemmata that will be used in the sequel. Finally, in Section 4 we prove the main results. The main point here is to prove Theorem 1.5, as Theorems 1.2 and 1.3 and Corollaries 1.4 and 1.6 will follow as immediate consequences, though we provide proofs for the convenience of the reader.

Definition 2.1

Let 1 < p < +∞and 1 ≤ q < +∞. We define

and for 1 ≤ p ≤ +∞and q = +∞

For these Banach spaces, one has a duality between L^p,q(ℝ^N) and L^p',q' (ℝ^N) for 1 < p < +∞and 1 ≤ q < +∞ (see, e.g. Theorem 1.4.17 on p. 52 of [17]). The Hahn-Banach theorem therefore gives

Let us observe that with this definition

where the spaces L¹(ℝ^N) and L^∞(ℝ^N) are intended in the usual sense. Note that the former equation is not standard, as L^1,∞(ℝ^N) has another possible definition, which is only possible through the introduction of a different object. In particular, for 1 < p < +∞, one has a quasi-norm on the Lorentz spaces L^p,q(ℝ^N) that is equivalent to the norm we have defined. What is more, this quasi-norm can be used to define the Lorentz spaces without such restrictions on p and q. Therefore let us introduce the following definition.

Definition 2.2

Let 1 ≤ p < +∞. If 0 < q < +∞we define

while if q = +∞we define

Then one has the following result on the equivalence of the quasi-norm on L~p,q(RN) and the norm on L^p,q(ℝ^N) (and so in the sequel we drop the tilde):

Proposition 2.3

Let 1 < p < +∞and 1 ≤ q ≤ +∞. Then

The proof for 1 ≤ q < +∞can be found as a variation of the one given for Lemma 2.2 in [22], which we record here as our

Lemma 2.4

(Hardy’s inequality). Let 1 < p < +∞. Then for any q ∈ [1,∞) one has

As the proof cited in [22] is a book of Zygmund which does not treat the case q > p, we here provide details for the convenience of the reader.

Proof of Lemma 2.4

By density it suffices to prove the result for functions f∈Cc∞(RN). For such functions the fundamental theorem of calculus implies

A computation of the derivative then yields

or

Letting I to denote the integral on the left-hand-side,Holder’s inequality on (0,∞) equipped with the measure dxx with exponents q, q′ yields

and the result follows from reabsorbing the term I^1−1/q.  □

It will be useful for our purposes to observe an alternative formulation of this equivalent quasi-norm in terms of the distribution function. In particular, Proposition 1.4.9 in [17] reads

Proposition 2.5

Let 1 ≤ p < +∞. If 0 < q < +∞, then

while if q = +∞

With these definitions, we are now prepared to state a version of Hölder’s inequality on the Lorentz scale. The following theorem is a slight strengthening of the statement in O’Neil’s paper [22, Theorem 3.4], as we observe that one actually can control the norm of the product with the product of the quasi-norms introduced above.

Theorem 2.6

Letf∈Lp1,q1(RN)andg∈Lp2,q2(RN), where

and

for some p > 1 and q ≥ 1. Then

As the paper of O’Neil does not contain a proof and our calculation leads to slightly different quantities and a different constant than the one claimed in his paper, we here provide one for completeness and the convenience of the reader. To this end, let us recall that O’Neil defines a product operator

as a bilinear operator on two measure spaces with values in a third measure space which additionally satisfies

and

Here ∥ · ∥_∞ and ∥ · ∥₁ denote the essential supremum and the Lebesgue integral on the corresponding measure spaces. For clarity of exposition we now restrict ourselves to Euclidean space and the notation we have previously introduced. We note, however, that these results also hold in this more general framework.

For such operators we require the estimate

Lemma 2.7

Assuming that we have established it, let us deduce Theorem 2.6.

Proof of Theorem 2.6

We have

By Lemma 2.7 one has

which by Hardy’s inequality (Lemma 2.4) implies

Now if

we have

and it suffices to apply Hölder’s inequality with exponents q₁, q₂ to obtain

For any different value of q which is admissible we define

Now Calderón’s Lemma implies

for ˜q ≤ q. This result can be found as Proposition 1.4.10 in [17] or Lemma 2.5 in [22] for the alternative norm, but with the same constant. This observation together with the previous case implies

Notice that the constant can be shown to be e^1/e, independent of the rest of parameters.  □

The proof of Lemma 2.7 will be argued from a variation of O’Neil’s Lemma 1.4:

Lemma 2.8

If |f| ≤ α and the support of f has measure at most x then one has

From this we can prove our Lemma 2.7 as follows.

Proof of Lemma 2.7

As in O’Neil’s proof of [22, Lemma 1.5] we pick a doubly infinite sequence {y_n} such that

and

From this we can express

where

This representation implies

and therefore

For the first we use the top equation in Lemma 2.8 and for the second we use the bottom equation:

For the second term we make the change of variables y = f ^*(u) to obtain

The same integration by parts performed in Lemma 1.5 then yields

In particular, for the first term of this second term we find

which precisely cancels the first term! Finally the second term is as desired, and thus we obtain the thesis.  □

Finally, we complete the proof of our Lemma 2.8.

Proof of Lemma 2.8

As in his proof of Lemma 1.4 in [22] we define the truncation of the function g at height u

and what remains above height u, g^u := g − g_u, we find

Note that being a product operator implies that if |f| ≤ α and has support on a set of measure at most x then

and

Thus we estimate

where we have used

which relies on the fact that h₁, h₂ have disjoint support.

Now by the estimates for h₁, h₂ (in the first the L^∞ estimate for h₁ and in the second the L¹ estimate) we find

The choice u = g^*(t) or g^*(x) and the equality

with a = t, x yield

 □

Theorem 2.9

(Theorem 3.5 in [22]). Let f ∈ L^p1,q1 (ℝ^N) and g ∈ L^p₂,q₂ (ℝ^N), where

Then

Proof

If we again define h = P(f , g) we have

and Hölder’s inequality implies

where ˜q₁ is chosen such that

The result then follows from Calderón’s Lemma as in the proof of theorem 2.6 above with the same constant e^1/e.  □

Lemma 3.1

Let α ∈ (0, N) and p ∈ (1, N/α). Then

where δ=Np−α>0.

Proof

We begin with the observation that

if t > r^α−N, while in the case t≤rα−N we have

Therefore we deduce that

which is the desired conclusion.  □

Lemma 3.2

Let f ∈ L^N/α,q(Ω) with q ∈ (1,∞] and suppose that 12N/α+1≤p<N/α. Then f ∈ L^p,1(Ω) and there exists a constant C₁ = C₁(α, N, q, Ω) > 0 such that

where 1q+1q′=1andδp:=N/p−α>0. Moreover, we can choose C₁ such that

Proof

By our slight variation of O’Neil’s version of Hölder’s inequality in Lorentz spaces, see Theorem 2.6 in Section 2 above, we have

where

We compute

which combined with the fact that r = N/δ_p yields

Define p0:=12N/α+1. Then the assumption p₀ ≤ p implies firstly that

and so

Therefore the estimate holds with

 □

Lemma 3.3

Under the hypothesis of Lemma 3.2 let f ∈ L^s(Ω) for some 1 < s < N/α. Then

where η:=N/s−α>0andC1′=C1′(α,N,s,|Ω|)>0.

Proof

The proof is analogous to the previous one. Indeed, given s ∈ (1, N/α) we define r by the relation

Note that for any choice of s in this range and any q ∈ (1,∞] one has

Therefore we can apply Theorem 2.6 to deduce the inequality

As above

and the result follows from the fact that r = N/η.  □

The following estimate is in the spirit of Hedberg’s lemma [18], while a variant has been argued by Adams in [2].

Lemma 3.4

(Hedberg). Under the hypothesis of Lemma 3.2, for every ε ∈ (0, α) one has the inequality

where we are using the fractional maximal function, i.e.

for some C₄ = C₄(N, α, ε) > 0 independent of δ.

Proof

We begin splitting the Riesz potential in two integrals as follows

We will estimate them separately and will conclude optimizing the choice of the parameter r. The first integral can be estimated as follows

On the other hand, the second integral can be estimated using Theorem 2.9 (Hölder’s inequality in the L¹ regime) and Lemma 3.1

where

One can then optimize in r, however for our purposes simply setting the upper bounds we have proved for J₁, J₂ is sufficient. In particular, from the choice

one deduces the inequality

where we have used Young’s inequality to estimate

which is independent of δ and a posteriori of p.  □

Let us next recall a weak-type estimate for the fractional maximal function with respect to the Hausdorff content.

Lemma 3.5

Let γ ∈ [0, N). There exists a constant C₅ = C₅(N, γ) > 0 such that

We provide a proof for the convenience of the reader (see also [8]).

Proof

Define

and note that by lower-semicontinuity of the fractional maximal function E_t is an open set. By the definition of the fractional maximal function,M_γ, for any x ∈ E_t there is a radii r_x such that