An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
aa r X i v : . [ m a t h . F A ] J u l AN IMPROVEMENT TO THEJOHN-NIRENBERG INEQUALITY FOR FUNCTIONSIN CRITICAL SOBOLEV SPACES ´ANGEL D. MART´INEZ AND DANIEL SPECTOR
Abstract.
It is known that functions in a Sobolev space with criticalexponent embed into the space of functions of bounded mean oscillation,and therefore satisfy the John-Nirenberg inequality and a correspondingexponential integrability estimate. While these inequalities are optimalfor general functions of bounded mean oscillation, the main result ofthis paper is an improvement for functions in a class of critical Sobolevspaces. Precisely, we prove the inequality H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ Ce − ct q ′ for all k f k L N/α,q (Ω) ≤ β ∈ (0 , N ], where Ω ⊂ R N , H β ∞ isthe Hausdorff content, L N/α,q (Ω) is a Lorentz space with q ∈ (1 , ∞ ], q ′ = q/ ( q −
1) is the H¨older conjugate to q , and I α f denotes the Rieszpotential of f of order α ∈ (0 , N ). Introduction
In 1933, A. Beurling observed that if f ( z ) is an analytic function for | z | ≤ ˆ ˆ | z | < | f ′ ( z ) | dxdy ≤ π, then the set of angles θ for which | f ( e iθ ) | ≥ s has Lebesgue measure at most e − s +1 (cf. [5, 21]). This type of decay implies integrability of exp( α | f ( z ) | )for α <
1, while the sharp result, for α = 1, was not proved until 1985 in thework of Chang and Marshall (cf. [10]). This exponential decay estimate ofBeurling can be seen as an instance of a Sobolev embedding in the criticalexponent with target a subspace of the space of functions of bounded meanoscillation ( BM O ). Remarkably, this work precedes Sobolev’s work on em-beddings, the subsequent numerous contributors for results in the criticalexponent, and even the introduction of
BM O . In this paper we are interested in such improvements to the
BM O em-bedding for functions in a critical Sobolev space. To this end, let us recallthat the space of functions of bounded mean oscillation was introduced byF. John and L. Nirenberg in their seminal paper [19]. Given Q ⊂ R N afinite cube it can be defined as follows BM O ( Q ) := (cid:26) u ∈ L ( Q ) : k u k BMO ( Q ) := sup Q ⊂⊂ Q Q | u − u Q | dx < + ∞ (cid:27) . Here the supremum is computed with respect to cubes Q with sides parallelto the coordinate axes and u Q denotes the mean of u in the cube Q .While the original motivation of such a definition arose from the consid-eration of problems in elasticity, the influence of this space on harmonicanalysis and its applications is far reaching. A central aspect of this impor-tance is its rˆole as a replacement for L ∞ , e.g. in the theory of concerningthe boundedness of translation invariant singular integrals [23, Theorem 1.1and Remark 1.3], as an endpoint in interpolation [14, Section III], in theregularity theory of elliptic equations [20] (which provided a celebrated al-ternative to the original - and independent - work of De Giorgi and Nashon Hilbert’s 19th problem), and, the main point of this paper, as a targetfor Sobolev embeddings in the critical exponent.As noted by John and Nirenberg, any bounded function has boundedmean oscillation, though the space BM O ( Q ) is strictly larger than L ∞ ( Q ).In particular on p. 416 in [19] they give a class of examples of the form u ( x ) := ˆ Q log | x − y | f ( y ) dy (1.1)for some f ∈ L ( Q ). The uniting idea for these examples, and a funda-mental result proved by them for this space, is that functions in BM O ( Q )enjoy an exponential decay estimate for their level sets (what is now knownas the John-Nirenberg inequality): |{ x ∈ Q : | u ( x ) − u Q | > t }| ≤ C exp( − ct/ k u k BMO ) | Q | (1.2)for certain constants c, C >
0. From the inequality (1.2) one easily deducesan embedding into the Orlicz space of exponentially integrable functions:There exists c ′ , C ′ > Q exp ( c ′ | u ( x ) − u Q | ) dx ≤ C ′ (1.3)for any function k u k BMO ≤ JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 3
The proof of (1.2), and therefore (1.3), employs a Calder´on-Zygmund de-composition. However, as was observed by N. Trudinger [31, Theorem 1 onp. 476], for certain
BM O functions one has a simpler proof of the inequal-ity (1.3). In particular, for functions which posses a weak derivative in anappropriate space, Trudinger shows how (1.3) follows from representationformulae for these functions in terms of potentials applied to the Euclideannorm of their weak derivatives along with corresponding estimates for thesepotentials. This perspective brings into focus two important problems toconsider – that of optimal representations of functions as potentials of theirderivatives and that of the mapping properties of these potentials. The for-mer plays a rˆole in the determination of sharp constants, e.g. of the bestvalue of β in the inequality Ω exp( β | u ( x ) | p ′ ) dx ≤ C ′ for all u ∈ W k,p (Ω) such that k∇ k u k L p (Ω) ≤
1, where p = N/k – in the case k = 1 the sharp constant is due to J. Moser [21], for k ≥ p = 2 itis due to D. Adams [3], while for other values of k, p the result is due to I.Shafrir and the second author [28] (which builds on the foundational workof D. Adams [3], see also [15, 16] and [27]). Remark 1.1.
Although we will not be concerned with this issue in thepresent work it is worth mentioning that this subtle point has important ap-plications to the Yamabe problem, as Moser did, and other related geometricanalysis problems (cf. [11, 21] or the survey article [12] and the referencestherein for more details). Existence of extremizers for the sharp constant inthe case Ω is an n -dimensional ball can be found in the work of Carlesonand Chang [9] . In this paper we focus on the latter question, that of estimates for thesepotentials in the critical exponent. In particular, with simple proofs weestablish some new exponential decay estimates in the spirit of (1.2). As wewill see, our work extends a result of D. Adams in [1] and improves uponan estimate of H. Brezis and S. Wainger [7] (see also [33]). Here is it usefulto change the perspective of the preceding inequality to a correspondingestimate for potentials in the critical exponent: Let Ω ⊂ R N be an openand bounded set. There exists constants c ′ , C ′ > Ω exp (cid:16) c ′ | I α f ( x ) | p ′ (cid:17) dx ≤ C ′ (1.4) A. D. MART´INEZ AND D. SPECTOR for all f with supp f ⊂ Ω and k f k L p (Ω) ≤
1, where p = N/α and we haveused I α f to denote the Riesz potential of order α ∈ (0 , N ) of f , defined by I α f ( x ) := 1 γ ( α ) ˆ R N f ( y ) | x − y | N − α dy for γ ( α ) = π N/ α Γ( α/ N − α ) / − . The inequality (1.4) has an ex-tensive history in the literature – it has been observed by Yudovich in [34],is implicit in [25] for N = 2, the case α = 1 is proved in Trudinger’s paper[31], a version for Bessel potentials is due to Strichartz [29], the proof of thestatement we assert here is in Hedberg’s paper [18], while the optimal con-stant was established by Adams in [3] (see also [33] for the optimal constanton 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 inte-grability found in (1.4) comes with a corresponding improved exponentialdecay estimate. The first result of this paper is the following theorem tothis effect. Theorem 1.2.
Let Ω ⊂ R N be open and α ∈ (0 , N ) . There exist constants c, C > that depend on α, N, and Ω such that |{ x ∈ Ω : | I α f ( x ) | > t }| ≤ Ce − ct NN − α for all f ∈ L N/α (Ω) such that k f k L N/α (Ω) ≤ . This is not surprising, since the proof amounts to relaxing the usual argu-ments in strong-type spaces to a weak-type setting. However, the techniqueis interesting as it suggests the possibility of other related inequalities. In-deed, when one examines the work of Yudovich [34], one finds that he assertsthe result not only for integrals over a domain, but even for n -dimensionalhyperplanes intersected with Ω, n ≤ N an integer (this bears a resemblanceto the special case of Beurling mentioned at the introduction, who obtainsthe exponential decay on the circle). This corresponds to a property en-joyed by functions in the critical Sobolev space which is not true for generalfunctions in BM O : the trace of such functions are
BM O on the restriction.Our method can be adapted to this setting, and even to the setting of fractalsets. In order to state our next result let us first introduce the Hausdorffcontent of a set E ⊂ R N which is defined by H β ∞ ( E ) := inf ( ∞ X i =1 ω β r βi : E ⊂ ∞ [ i =1 B ( x i , r i ) ) . JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 5
Here the infimum is taken over all possible coverings of arbitrary radii and ω β := π β/ / Γ (cid:0) β + 1 (cid:1) is the volume of a β -dimensional sphere.We can now state Theorem 1.3.
Let Ω ⊂ R N be open, α ∈ (0 , N ) , and β ∈ (0 , N ] . Thereexist constants c, C > that depend on α, N, β, and Ω such that H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ Ce − ct NN − α for all f ∈ L N/α (Ω) such that k f k L N/α (Ω) ≤ . This result is analogous to an estimate proved by D. Adams in [1] for thedecay of level sets of the convolution of the Bessel kernel and a function inthis critical space. In particular, in the proof of Theorem 3, Item (ii) onp. 913, Adams proves an exponential decay estimate of the µ measure of thelevel sets of such a convolution, where µ is a non-negative Radon measurewhich satisfies the ball growth condition µ ( B ( x, r )) ≤ r β for some β > H β ∞ and the supremum over allsuch measures.From Theorem 1.3 we deduce the following improvement to the inequality(1.4), our Corollary 1.4.
Let Ω ⊂ R N be open, α ∈ (0 , N ) , and β ∈ (0 , N ] . Thereexist constants c ′ , C ′ > that depend on α, N, β, and Ω such that ˆ Ω exp (cid:16) c ′ | I α f | p ′ (cid:17) d H β ∞ ≤ C ′ (1.5) for all f ∈ L N/α (Ω) such that k f k L N/α (Ω) ≤ . We conclude the introduction with an application of our techniques toimprove 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) onp. 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 inthis critical regime, while microscopic, is magnified in these inequalities .As in this paper we work exclusively with the Riesz kernels, we now state aversion of their result in this context which has been proved by J. Xiao and We here paraphrase the expression used by H. Brezis in [6], which he conveyed tothe second author during a discussion of the rˆole of the second exponent in the Lorentzspaces. In [7], as before with [1], their results are stated for the Bessel kernel.
A. D. MART´INEZ AND D. SPECTOR
Zh. Zhai [33, Theorem 3.1 (ii) on p. 364]: There exists constants c ′ , C ′ > ˆ Ω exp (cid:16) c ′ | I α f ( x ) | q ′ (cid:17) dx ≤ C ′ (1.6)for all f with supp f ⊂ Ω and k f k L N/α,q (Ω) ≤ Theorem 1.5.
Let Ω ⊂ R N be open, α ∈ (0 , N ) , β ∈ (0 , N ] , and q ∈ (1 , ∞ ] .There exist constants c, C > that depend on α, N, β, q, and Ω such that H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ Ce − ct q ′ for all f ∈ L N/α,q (Ω) such that k f k L N/α,q (Ω) ≤ and supp f ⊆ Ω . In the usual way this leads to an improved dimensional version of theinequality of [7] and [33], which we state as
Corollary 1.6.
Let Ω ⊂ R N be open, α ∈ (0 , N ) , β ∈ (0 , N ] and q ∈ (1 , ∞ ] .There exist constants c ′ , C ′ > that depend on α, N, β, q, and Ω such that ˆ Ω exp (cid:16) c ′ | I α f | q ′ (cid:17) d H β ∞ ≤ C ′ (1.7) for all f ∈ L N/α,q (Ω) such that k f k L N/α,q (Ω) ≤ . The paper is divided as follows. In Section 2 we provide some prelimi-naries about the Lorentz spaces we here require. In Section 3 we prove avariant of a technical result due to Hedberg as well as several other technicallemmata that will be used in the sequel. Finally, in Section 4 we prove themain results. The main point here is to prove Theorem 1.5, as Theorems 1.2and 1.3 and Corollaries 1.4 and 1.6 will follow as immediate consequences,though we provide proofs for the convenience of the reader.2.
Preliminaries on Lorentz spaces
Let us now introduce several equivalent quasi-norms that can be used todefine the Lorentz spaces L p,q ( R N ). We begin with the development of R.O’Neil in [22]. For f a measurable function on R N , we define m ( f, y ) := |{| f | > y }| . JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 7
As this is a non-increasing function of y , it admits a left-continuous inverse,called the non-negative rearrangement of f , and which we denote f ∗ ( x ).Further, for x > f ∗∗ ( x ) := 1 x ˆ x f ∗ ( t ) dt. With these basic results, we can now give a definition of the Lorentz spaces L p,q ( R N ). Definition 2.1.
Let < p < + ∞ and ≤ q < + ∞ . We define k f k L p,q ( R N ) := (cid:18) ˆ ∞ (cid:2) t /p f ∗∗ ( t ) (cid:3) q dtt (cid:19) /q , and for ≤ p ≤ + ∞ and q = + ∞k f k L p, ∞ ( R N ) := sup t> t /p f ∗∗ ( t ) . 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 k f k L p,q ( R N ) = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ˆ R N f g dx (cid:12)(cid:12)(cid:12)(cid:12) : g ∈ L p ′ ,q ′ ( R N ) k g k L q ′ ,r ′ ( R N ) ≤ (cid:27) . Let us observe that with this definition k f k L , ∞ ( R N ) = k f k L ( R N ) k f k L ∞ , ∞ ( R N ) = k f k L ∞ ( R N ) , where the spaces L ( R N ) and L ∞ ( R N ) are intended in the usual sense. Notethat the former equation is not standard, as L , ∞ ( R N ) has another possibledefinition, which is only possible through the introduction of a differentobject. In particular, for 1 < p < + ∞ , one has a quasi-norm on the Lorentzspaces L p,q ( R N ) that is equivalent to the norm we have defined. What ismore, this quasi-norm can be used to define the Lorentz spaces without suchrestrictions on p and q . Therefore let us introduce the following definition. Definition 2.2.
Let ≤ p < + ∞ . If < q < + ∞ we define ||| f ||| ˜ L p,q ( R N ) := (cid:18) ˆ ∞ (cid:0) t /p f ∗ ( t ) (cid:1) q dtt (cid:19) /q , while if q = + ∞ we define ||| f ||| ˜ L p, ∞ ( R N ) := sup t> t /p f ∗ ( t ) . A. D. MART´INEZ AND D. SPECTOR
Then one has the following result on the equivalence of the quasi-normon ˜ L p,q ( R N ) and the norm on L p,q ( R N ) (and so in the sequel we drop thetilde): Proposition 2.3.
Let < p < + ∞ and ≤ q ≤ + ∞ . Then ||| f ||| ˜ L p,q ( R N ) ≤ k f k L p,q ( R N ) ≤ p ′ ||| f ||| ˜ L p,q ( R N ) . The proof for 1 ≤ q < + ∞ can be found as a variation of the one givenfor Lemma 2.2 in [22], which we record here as our Lemma 2.4 (Hardy’s inequality) . Let < p < + ∞ . Then for any q ∈ [1 , ∞ ) one has (cid:18) ˆ ∞ (cid:20) x /p x f ( t ) dt (cid:21) q dxx (cid:19) /q ≤ pp − (cid:18) ˆ ∞ (cid:2) x /p f ( x ) (cid:3) q dxx (cid:19) /q As the proof cited in [22] is a book of Zygmund which does not treat thecase 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 ∈ C ∞ c ( R N ). For such functions the fundamental theorem of calculusimplies ˆ ∞ ddx (cid:20) x /p x f ( t ) dt (cid:21) q dx = 0 . A computation of the derivative then yields ˆ ∞ q (cid:20) x /p x f ( t ) dt (cid:21) q − (cid:18) (1 /p − x /p − ˆ x f ( t ) dt + x /p − f ( x ) (cid:19) dx = 0 , or ˆ ∞ (cid:20) x /p x f ( t ) dt (cid:21) q dxx = pp − ˆ ∞ (cid:20) x /p x f ( t ) dt (cid:21) q − x /p f ( x ) dxx . Letting I to denote the integral on the left-hand-side, Holder’s inequalityon (0 , ∞ ) equipped with the measure dxx with exponents q, q ′ yields I ≤ pp − I − /q (cid:18) ˆ ∞ (cid:2) x /p f ( x ) (cid:3) q dxx (cid:19) /q and the result follows from reabsorbing the term I − /q . (cid:3) It will be useful for our purposes to observe an alternative formulation ofthis equivalent quasi-norm in terms of the distribution function. In partic-ular, Proposition 1.4.9 in [17] reads
JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 9
Proposition 2.5.
Let ≤ p < + ∞ . If < q < + ∞ , then ||| f ||| L p,q ( R N ) ≡ p /q (cid:18) ˆ ∞ (cid:0) t |{| f | > t }| /p (cid:1) q dtt (cid:19) /q , while if q = + ∞ ||| f ||| L p, ∞ ( R N ) ≡ sup t> t |{| f | > t }| /p . With these definitions, we are now prepared to state a version of H¨older’sinequality on the Lorentz scale. The following theorem is a slight strength-ening of the statement in O’Neil’s paper [22, Theorem 3.4], as we observethat one actually can control the norm of the product with the product ofthe quasi-norms introduced above.
Theorem 2.6.
Let f ∈ L p ,q ( R N ) and g ∈ L p ,q ( R N ) , where p + 1 p = 1 p and q + 1 q ≥ q for some p > and q ≥ . Then k f g k L p,q ( R N ) ≤ e /e p ′ ||| f ||| L p ,q ( R N ) ||| g ||| L p ,q ( R N ) . As the paper of O’Neil does not contain a proof and our calculation leadsto slightly different quantities and a different constant than the one claimedin his paper, we here provide one for completeness and the convenience ofthe reader. To this end, let us recall that O’Neil defines a product operator h = P ( f, g )as a bilinear operator on two measure spaces with values in a third measurespace which additionally satisfies k h k ∞ ≤ k f k ∞ k g k ∞ , k h k ≤ k f k k g k ∞ and k h k ≤ k f k ∞ k g k . Here k·k ∞ and k·k denote the essential supremum and the Lebesgue integralon the corresponding measure spaces. For clarity of exposition we nowrestrict ourselves to Euclidean space and the notation we have previouslyintroduced. We note, however, that these results also hold in this moregeneral framework. For such operators we require the estimate
Lemma 2.7. xh ∗∗ ( x ) ≤ ˆ x f ∗ ( t ) g ∗ ( t ) dt. Assuming that we have established it, let us deduce Theorem 2.6.
Proof of Theorem 2.6.
We have k h k L p,q ( R N ) = (cid:18) ˆ ∞ ( x /p h ∗∗ ( x )) q dxx (cid:19) q . By Lemma 2.7 one has h ∗∗ ( x ) ≤ x ˆ x f ∗ ( t ) g ∗ ( t ) dt. which by Hardy’s inequality (Lemma 2.4) implies k h k L p,q ( R N ) ≤ p ′ (cid:18) ˆ ∞ ( x /p f ∗ ( x ) g ∗ ( x )) q dxx (cid:19) q . Now if 1 p = 1 p + 1 p q = 1 q + 1 q we have x /p f ∗ ( x ) g ∗ ( x ) = x /p f ∗ ( x ) x /p g ∗ ( x )and it suffices to apply H¨older’s inequality with exponents q , q to obtain k h k L p,q ( R N ) ≤ p ′ ||| f ||| L p ,q ( R N ) ||| g ||| L p ,q ( R N ) . For any different value of q which is admissible we define1˜ q = 1 q + 1 q . Now Calder´on’s Lemma implies k h k L p,q ( R N ) ≤ p ′ (cid:18) ˜ qp (cid:19) / ˜ q − /q || h || L p, ˜ q ( R N ) JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 11 for ˜ q ≤ q . This result can be found as Proposition 1.4.10 in [17] or Lemma2.5 in [22] for the alternative norm, but with the same constant. Thisobservation together with the previous case implies k h k L p,q ( R N ) ≤ p ′ (cid:18) ˜ qp (cid:19) / ˜ q − /q ||| f ||| L p ,q ( R N ) ||| g ||| L p ,q ( R N ) ≤ e /e p ′ ||| f ||| L p ,q ( R N ) ||| g ||| L p ,q ( R N ) . Notice that the constant can be shown to be e /e , independent of the restof parameters. (cid:3) The proof of Lemma 2.7 will be argued from a variation of O’Neil’s Lemma1.4:
Lemma 2.8. If | f | ≤ α and the support of f has measure at most x thenone has h ∗∗ ( t ) ≤ αg ∗∗ ( t ) h ∗∗ ( t ) ≤ α xt g ∗∗ ( x ) . 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 adoubly infinite sequence { y n } such that y = f ∗ ( t ) ,y n ≤ y n +1 , lim n →∞ y n = + ∞ and lim n →−∞ y n = 0 . From this we can express f ( z ) = ∞ X n = −∞ f n ( z )where f n ( z ) := | f ( z ) | ≤ y n − , f ( z ) − y n − sgn f ( z ) if y n − < | f ( z ) | ≤ y n , y n sgn f ( z ) if | f ( z ) | > y n . This representation implies h = P X n = −∞ f n , g ! + P ∞ X n =1 f n , g ! =: X n = −∞ h n + ∞ X n =1 h n and therefore h ∗∗ ( t ) ≤ X n = −∞ h ∗∗ n ( t ) + ∞ X n =1 h ∗∗ n ( t ) . For the first we use the top equation in Lemma 2.8 and for the second weuse the bottom equation: h ∗∗ ( t ) ≤ X n = −∞ ( y n − y n − ) g ∗∗ ( t ) + ∞ X n =1 ( y n − y n − ) m ( f, y n − ) t g ∗∗ ( m ( f, y n − ))= f ∗ ( t ) g ∗∗ ( t ) + 1 t ˆ ∞ f ∗ ( t ) m ( f, y ) g ∗∗ ( m ( f, y )) dy =: I + II For the second term we make the change of variables y = f ∗ ( u ) to obtain II = − t ˆ t ug ∗∗ ( u ) df ∗ ( u ) . The same integration by parts performed in Lemma 1.5 then yields II = − t ug ∗∗ ( u ) f ∗ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t + 1 t ˆ t g ∗ ( u ) f ∗ ( u ) du. In particular, for the first term of this second term we find − t ug ∗∗ ( u ) f ∗ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = − g ∗∗ ( t ) f ∗ ( t ) , which precisely cancels the first term! Finally the second term is as desired,and thus we obtain the thesis. (cid:3) Finally, we complete the proof of our Lemma 2.8.
JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 13
Proof of Lemma 2.8.
As in his proof of Lemma 1.4 in [22] we define thetruncation of the function g at height ug u ( z ) := ( g ( z ) if | g ( z ) | ≤ uu sgn g ( z ) if | g ( z ) | > u and what remains above height u , g u := g − g u , we find h = P ( f, g ) = P ( f, g u ) + P ( f, g u )= h + h . Note that being a product operator implies that if | f | ≤ α and has supporton a set of measure at most x then k h k L ∞ ( R N ) ≤ k f k L ∞ ( R N ) k g u k L ∞ ( R N ) ≤ αu, k h k L ( R N ) ≤ k f k L ( R N ) k g u k L ∞ ( R N ) ≤ αxu and k h k L ( R N ) ≤ k f k L ∞ ( R N ) k g u k L ( R N ) ≤ α ˆ ∞ u m ( g, y ) dy. Thus we estimate h ∗∗ ( t ) = 1 t ˆ t h ∗ ( z ) dz ≤ t ˆ t h ∗ ( z ) + h ∗ ( z ) dz, where we have used h ∗ ( z ) ≤ h ∗ ( z ) + h ∗ ( z ) , which relies on the fact that h , h have disjoint support.Now by the estimates for h , h (in the first the L ∞ estimate for h andin the second the L estimate) we find h ∗∗ ( t ) ≤ t α (cid:18) tu + ˆ ∞ u m ( g, y ) dy (cid:19) h ∗∗ ( t ) ≤ t α (cid:18) xu + ˆ ∞ u m ( g, y ) dy (cid:19) . The choice u = g ∗ ( t ) or g ∗ ( x ) and the equality ag ∗ ( a ) + ˆ ∞ g ∗ ( a ) m ( g, y ) dy = ag ∗∗ ( a )with a = t, x yield h ∗∗ ( t ) ≤ αg ∗∗ ( t ) h ∗∗ ( t ) ≤ α xt g ∗∗ ( t ) . (cid:3) Finally we require the L endpoint of H¨older’s inequality stated in [22]. Theorem 2.9 (Theorem 3.5 in [22]) . Let f ∈ L p ,q ( R N ) and g ∈ L p ,q ( R N ) ,where p + 1 p = 11 q + 1 q ≥ Then k f g k L ( R N ) ≤ e /e ||| f ||| L p ,q ( R N ) ||| g ||| L p ,q ( R N ) . Proof.
If we again define h = P ( f, g ) we have k h k L ( R N ) = lim x →∞ ˆ x h ∗ ( t ) dt = lim x →∞ xh ∗∗ ( x ) ≤ lim x →∞ ˆ x f ∗ ( t ) g ∗ ( t ) dt = ˆ ∞ t /p f ∗ ( t ) t /p g ∗ ( t ) dtt , and H¨older’s inequality implies k h k L ( R N ) ≤ ||| f ||| L p , ˜ q ( R N ) ||| g ||| L p ,q ( R N ) where ˜ q is chosen such that 1˜ q + 1 q = 1 . The result then follows from Calder´on’s Lemma as in the proof of theorem2.6 above with the same constant e /e . (cid:3) Auxiliary results
In this section we expose some technical results that will be used in thesequel. We will need the following estimate for the weak- L p quasi-norm oftruncated potentials: JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 15
Lemma 3.1.
Let α ∈ (0 , N ) and p ∈ (1 , N/α ) . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ B (0 ,r ) c | · | N − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p ′ , ∞ ( R N ) = | B (0 , | /p ′ r − δ where δ = Np − α > .Proof. We begin with the observation that (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B (0 , r ) c : 1 | x | N − α > t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = 0if t > r α − N , while in the case t ≤ r α − N we have t (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B (0 , r ) c : 1 | x | N − α > t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) /p ′ = t − N ( N − α ) p ′ | B (0 , | /p ′ . Therefore we deduce thatsup t> t (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ B (0 , r ) c : 1 | x | N − α > t (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) /p ′ = r ( α − N ) (cid:16) − N ( N − α ) p ′ (cid:17) | B (0 , | /p ′ = r − δ | B (0 , | /p ′ , which is the desired conclusion. (cid:3) Lemma 3.2.
Let f ∈ L N/α,q (Ω) and suppose that ( N/α + 1) ≤ p < N/α .Then f ∈ L p, (Ω) and there exists a constant C = C ( α, N, q, Ω) > suchthat k f k L p, (Ω) ≤ C δ − /q ′ p k f k L N/α,q (Ω) , where δ p := N/p − α > . Moreover, we can choose C such that C δ − /q ′ p ≥ . Proof.
By our slight variation of O’Neil’s version of H¨older’s inequality inLorentz spaces, see Theorem 2.6 in Section 2 above, we have k f χ Ω k L p, ( R N ) ≤ e /e p ′ ||| f χ Ω ||| L N/α,q ( R N ) ||| χ Ω ||| L r,q ′ ( R N ) where 1 p = αN + 1 r q + 1 q ′ . We compute ||| χ Ω ||| q ′ L r,q ′ ( R N ) = | Ω | q ′ /r r ˆ t q ′ − dt = | Ω | q ′ /r rq ′ , which combined with the fact that r = N/δ p yields k f k L p, (Ω) ≤ e /e p ′ | Ω | δ p /N (cid:18) Nq ′ δ p (cid:19) /q ′ k f k L N/α,q (Ω) . Define p := ( N/α + 1). Then the assumption p ≤ p implies firstlythat p ′ ≤ ( N/α + 1) ( N/α + 1) − N/α + 1
N/α + 1 − N + αN − α , and so k f k L p, (Ω) ≤ e /e N + αN − α max {| Ω | , } (cid:18) Nq ′ (cid:19) /q ′ δ − /q ′ p k f k L N/α,q (Ω) . Therefore the estimate holds with C := max ( e /e N + αN − α max {| Ω | , } (cid:18) Nq ′ (cid:19) /q ′ , δ /q ′ p ) . (cid:3) Lemma 3.3.
Under the hypothesis of Lemma 3.2 let f ∈ L s (Ω) for some < s < N/α . Then k f k L s (Ω) ≤ C ′ k f k L N/α,q (Ω) , where η := N/s − α > and C ′ = C ′ ( α, N, s, | Ω | ) > .Proof. The proof is analogous to the previous one. Indeed, given s ∈ (1 , N/α ) we define r by the relation1 s = αN + 1 r . JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 17
Note that for any choice of s in this range and any q ∈ (1 , ∞ ] one has1 s ≤ q + 1 . Therefore we can apply Theorem 2.6 to deduce the inequality ||| f χ Ω ||| L s ( R N ) ≤ k f χ Ω k L s,s ( R N ) ≤ e /e s ′ ||| f χ Ω ||| L N/α,q ( R N ) ||| χ Ω ||| L r, ( R N ) . As above ||| χ Ω ||| L r, ( R N ) = r | Ω | ˆ dt = r | Ω | , and the result follows from the fact that r = N/η . (cid:3) The following estimate is in the spirit of Hedberg’s lemma [18], while avariant 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 | I α f ( x ) | ≤ C M α − ε f ( x ) δδ + ε k f k εε + δ L p, ( R N ) for some C = C ( N, α, ε ) > independent of δ .Proof. We begin splitting the Riesz potential in two integrals as follows I α f ( x ) = 1 γ ( α ) ˆ R N f ( y ) | x − y | N − α dy = 1 γ ( α ) ˆ B ( x,r ) f ( y ) | x − y | N − α dy + 1 γ ( α ) ˆ B ( x,r ) c f ( y ) | x − y | N − α dy = J ( x ) + J ( x ) . We will estimate them separately and will conclude optimizing the choiceof the parameter r . The first integral can be estimated as follows | J ( x ) | ≤ γ ( α ) ∞ X n =0 ˆ B ( x,r − n ) \ B ( x,r − n − ) | f ( y ) || x − y | N − α dy ≤ γ ( α ) ∞ X n =0 ( r − n ) N ( r − n ) α − ε ( r − n ) α − ε B ( x,r − n ) | f ( y ) | r − n − ) N − α dy ≤ r ε N − α γ ( α ) ∞ X n =0 (2 − n ) ε M α − ε ( f )( x ) ≤ C ( N, α, ε ) r ε M α − ε f ( x ) , where we are using the fractional maximal function, i.e. M β f ( x ) = sup r> r β B ( x,r ) | f ( y ) | dy. On the other hand, the second integral can be estimated using Theorem 2.9(H¨older’s inequality in the L regime) and Lemma 3.1 | J ( x ) | ≤ e /e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ B (0 ,r ) c | · | N − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L p ′ , ∞ ( R N ) ||| f ||| L p, ( R N ) ≤ e /e | B (0 , | /p ′ r − δ k f k L p, ( R N ) ≤ C r − δ k f k L p, ( R N ) . where C = e /e max {| B (0 , | , } . One can then optimize in r , however for our purposes simply setting theupper bounds we have proved for J , J is sufficient. In particular, from thechoice r ( x ) = (cid:18) C C k f k L p, ( R N ) M α − ε f ( x ) (cid:19) ε + δ one deduces the inequality | I α f ( x ) | ≤ C M α − ε f ( x ) δδ + ε k f k εε + δ L p, ( R N ) where we have used Young’s inequality to estimate2 C δδ + ε C ε ( δ + ε ) ≤ C + C ) =: C which is independent of δ and a posteriori of p . (cid:3) Let us next recall a weak-type estimate for the fractional maximal functionwith respect to the Hausdorff content.
Lemma 3.5.
Let γ ∈ [0 , N ) . There exists a constant C = C ( N, γ ) > such that H N − γ ∞ ( { x : M γ f ( x ) > t } ) ≤ C t k f k L ( R N ) We provide a proof for the convenience of the reader (see also [8]).
Proof.
Define E t := { x : M γ f ( x ) > t } , JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 19 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 r γx B ( x,r x ) | f ( y ) | dy > t. (3.1)Then E t ⊂ [ x ∈ E t B ( x, r x ) , while the inequality (3.1) implies thatsup x ∈ E t r x < + ∞ . Therefore, we may apply Vitali’s covering theorem (see, e.g. [13, Theorem1 on p. 27]) to find a countable subcollection of disjoint balls such that E ⊂ ∞ [ i =1 B ( x i , r i )From this and the definition of the Hausdorff content we find H N − γ ∞ ( E ) ≤ ∞ X i =1 ω N − γ (5 r i ) N − γ ≤ t ω N − γ N − β | B (0 , | ∞ X i =1 ˆ B ( x i ,r i ) | f ( y ) | dy ≤ C t k f k L ( R N ) , the last inequality holds because the selected balls are disjoint. This com-pletes the proof, with C := ω N − γ N − β | B (0 , | . (cid:3) Proofs of the Main Results
We are now prepared to prove the main result of this paper, Theorem 1.5,from which we will deduce Theorems 1.2 and 1.3, as well as Corollaries 1.4and 1.6.
Proof of Theorem 1.5.
We begin with the elementary inequality, for r > M α − ε f ( x ) ≤ ( M r ( α − ε ) | f | r ( x )) /r which together with Lemma 3.4 implies that { x : | I α f ( x ) | > t } ⊂ { x : t < C M r ( α − ε ) | f | r ( x ) δ ( δ + ε ) r k f k εε + δ L p, ( R N ) } . It is convenient to rewrite this inclusion as { x : | I α f ( x ) | > t } ⊂ x : tC k f k εε + δ L p, ( R N ) ( δ + ε ) rδ < M r ( α − ε ) | f | r ( x ) in order to invoke Lemma 3.5. In particular, for any β ∈ (0 , N ], we maychoose ε ∈ (0 , α ], r ∈ (1 , N/α ) such that N − β = r ( α − ε ) and N/r − α = β/r − ε >
0, from which we deduce H β ∞ ( { x : | I α f ( x ) | > t } ) ≤ C C k f k εε + δ L p, ( R N ) t ( δ + ε ) rδ ˆ R N | f | r . We recall the fact that supp f ⊂ Ω to write f = f χ Ω and utilize Lemma3.2 to obtain the inequality k f k L p, (Ω) ≤ C δ − /q ′ k f k L N/α,q (Ω) , and Lemma 3.3 with s = r > k f k L r (Ω) ≤ C ′ ( α, N, r, | Ω | ) k f k L N/α,q (Ω) , which combined yield the estimate H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ C ( C ′ ) r (cid:18) C C δ − /q ′ t (cid:19) ( δ + ε ) rδ k f k r + εrδ L N/α,q (Ω) ≤ C (cid:18) C C δ − /q ′ t (cid:19) ( δ + ε ) rδ , where we have used the fact that C δ − /q ′ ≥
1, the assumption that k f k L N/α,q (Ω) ≤
1, and C := C ( C ′ ) r .For t sufficiently large, we will choose δ = δ ( t ) > C C δ − /q ′ t = exp( − . (4.1) JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 21
This is possible whenever 0 < δ ( t ) ≤ δ ( α, N, q, | Ω | , ε ) with δ chosen suf-ficiently small, which is to say that p must be chosen sufficiently close to N/α . In particular, recalling p = ( N/α + 1), we can do so for all t ≥ t := C C δ − /q ′ p exp( − t this implies H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ C exp (cid:18) − ( δ + ε ) rδ (cid:19) ≤ C exp (cid:16) − εrδ (cid:17) . The choice of δ from equation (4.1) thus yields the estimate H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ C exp( − ct q ′ )where c = εr (cid:18) exp( − C C (cid:19) q ′ . This concludes the proof for t ≥ t ( α, N, q, | Ω | , ε ), while in the case t ∈ (0 , t ) we have H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ H β ∞ (Ω)= H β ∞ (Ω) exp( ct q ′ ) exp( − ct q ′ ) ≤ H β ∞ (Ω) exp( ct q ′ ) exp( − ct q ′ ) . In particular, the theorem holds with c chosen as above and C = max { C , H β ∞ (Ω) exp( ct q ′ ) } . (cid:3) We next show how one can deduce Theorems 1.2 and 1.3 from Theorem1.5.
Proof of Theorems 1.2 and 1.3.
First we observe that k f k L N/α,N/α (Ω) ≤ NN − α ||| f ||| L N/α,N/α (Ω) ≡ NN − α k f k L N/α (Ω) so that if k f k L N/α (Ω) ≤ k f k L N/α,N/α (Ω) ≤ NN − α . Therefore by rescaling f by this factor, Theorem 1.5 implies H β ∞ ( { x ∈ Ω : | I α f ( x ) | > t } ) ≤ Ce − ct NN − α with c = c ( N − α ) N . This completes the proof of Theorem 1.3. Theorem 1.2also follows in the case β = N , up to a new constant C , by the equivalenceof H N ∞ and the Lebesgue measure L N . (cid:3) We conclude with the proofs of Corollaries 1.4 and 1.6.
Proof of Corollaries 1.4 and 1.6.
We compute, for c ′ > ˆ Ω exp( c ′ | I α f | q ′ ) d H β ∞ = ˆ ∞ H β ∞ ( { x ∈ Ω : exp( c ′ | I α f ( x ) | q ′ ) > t } ) dt = ˆ ∞ H β ∞ ( x ∈ Ω : | I α f ( x ) | > (cid:18) ln( t ) c ′ (cid:19) /q ′ )! dt. The integral for t ∈ (0 ,
1) can be estimated above by H β ∞ (Ω), while for t ∈ (1 , ∞ ) we utilize Theorem 1.5 to obtain ˆ Ω exp( c ′ | I α f | q ′ ) d H β ∞ ≤ H β ∞ (Ω) + ˆ ∞ C exp (cid:18) − c ln( t ) c ′ (cid:19) dt = H β ∞ (Ω) + C ˆ ∞ t c/c ′ dt < + ∞ as soon as c ′ < c . The result follows with C ′ := H β ∞ (Ω) + C ˆ ∞ t c/c ′ dt. This completes the proof of Corollary 1.6. Corollary 1.4 follows with arescaling of the norm, as computed in the proof of Theorems 1.2 and 1.3. (cid:3) Acknowledgments
This work was initiated while the first named author was visiting theNonlinear Analysis Unit in the Okinawa Institute of Science and Technol-ogy Graduate University. He warmly thanks OIST for the invitation andhospitality. The first named author is supported by the National ScienceFoundation under Grant No. DMS-1638352.
JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 23
References [1] D. R. Adams,
Traces of potentials. II , Indiana Univ. Math. J. (1972/73), 907–918,DOI 10.1512/iumj.1973.22.22075. MR313783 ↑
3, 5[2] ,
A note on Riesz potentials , Duke Math. J. (1975), no. 4, 765–778.MR458158 ↑ A sharp inequality of J. Moser for higher order derivatives , Ann. of Math.(2) (1988), no. 2, 385–398, DOI 10.2307/1971445. MR960950 ↑
3, 4[4] ,
A note on Choquet integrals with respect to Hausdorff capacity , Functionspaces and applications (Lund, 1986), Lecture Notes in Math., vol. 1302, Springer,Berlin, 1988, pp. 115–124, DOI 10.1007/BFb0078867. MR942261 ↑ [5] Beurling, A., Etudes sur une probleme de majoration,
These, Almqvist and Wiksell,Uppsala, 1933. pages 1[6] H. Br´ezis,
Laser beams and limiting cases of Sobolev inequalities , Nonlinear par-tial differential equations and their applications. Coll`ege de France Seminar, Vol. II(Paris, 1979/1980), Res. Notes in Math., vol. 60, Pitman, Boston, Mass.-London,1982, pp. 86–97. MR652508 ↑ A note on limiting cases of Sobolev embeddings and con-volution inequalities , Comm. Partial Differential Equations (1980), no. 7, 773–789,DOI 10.1080/03605308008820154. MR579997 ↑
3, 5, 6[8] T. Bagby and W. P. Ziemer,
Pointwise differentiability and absolute continuity ,Trans. Amer. Math. Soc. (1974), 129–148, DOI 10.2307/1996986. MR344390 ↑ On the existence of an extremal function for an in-equality of J. Moser,
Bull. Sci. Math., 2e serie, 110 (1985), pp. 113-127. pages 3[10] Chang, S.-Y. A.; Marshall, D. E.,
A sharp inequality concerning the Dirichlet integral ,Amer. J. Math., 107 (1985), pp. 1015–1033. pages 1[11] Chang, S.-Y. A.; Yang, P. C. P,
Prescribing Gaussian curvature on S , ActaMath.159(1987), no. 3-4, pp. 215-259. pages 3[12] S.-Y. A. Chang and P. C. Yang, he inequality of Moser and Trudinger and applica-tions toconformal geometry , Comm. Pure Appl. Math. (2003), 1135-1150 129–148.MR1989228 ↑ Measure theory and fine properties of functions ,Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR1158660 ↑ H p spaces of several variables , Acta Math. (1972),no. 3-4, 137–193, DOI 10.1007/BF02392215. MR447953 ↑ On the regularity of solutions to Poisson’s equation , C. R.Math. Acad. Sci. Paris (2015), no. 9, 819–823, DOI 10.1016/j.crma.2015.07.001.MR3377679 ↑ On the role of Riesz potentials in Poisson’s equation and Sobolev embeddings ,Indiana Univ. Math. J. (2015), no. 6, 1697–1719, DOI 10.1512/iumj.2015.64.5706.MR3436232 ↑ Classical Fourier analysis , 3rd ed., Graduate Texts in Mathematics,vol. 249, Springer, New York, 2014. ↑
7, 8, 11[18] L. I. Hedberg,
On certain convolution inequalities , Proc. Amer. Math. Soc. (1972),505–510, DOI 10.2307/2039187. MR312232 ↑
4, 17 [19] F. John and L. Nirenberg,
On functions of bounded mean oscillation , Comm. PureAppl. Math. (1961), 415–426, DOI 10.1002/cpa.3160140317. MR131498 ↑ A new proof of De Giorgi’s theorem concerning the regularity problem forelliptic differential equations , Comm. Pure Appl. Math. (1960), 457–468, DOI10.1002/cpa.3160130308. MR170091 ↑ A sharp form of an inequality by N. Trudinger , Indiana Univ. Math. J. (1970/71), 1077–1092, DOI 10.1512/iumj.1971.20.20101. MR301504 ↑
1, 3[22] R. O’Neil,
Convolution operators and L ( p, q ) spaces , Duke Math. J. (1963), 129–142. ↑
5, 6, 8, 9, 11, 13, 14[23] J. Peetre,
On convolution operators leaving L p, λ spaces invariant , Ann. Mat. PuraAppl. (4) (1966), 295–304, DOI 10.1007/BF02414340. MR209917 ↑ Espaces d’interpolation et th´eor`eme de Soboleff , Ann. Inst. Fourier (Greno-ble) (1966), no. fasc., fasc. 1, 279–317 (French). MR221282 ↑ [25] Pohozaev, S. I., The Sobolev embedding in the case pl = n , Proc. Tech. Sci. Conf. onAdv. Sci. Research 1964-1965, Mathematics Section, Moskov. Energet. Inst. Moscow(1965), pp. 158-170. pages 4[26] E. M. Stein, Singular integrals and differentiability properties of functions , Prince-ton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.MR0290095 ↑ [27] I. Shafrir, The best constant in the embedding of W N, ( R N ) into L ∞ ( R N ), PotentialAnal. (2019), no. 4, 581–590, DOI 10.1007/s11118-018-9695-5. MR3938494 ↑ Best constants for two families of higher order criticalSobolev embeddings . part B, Nonlinear Anal. (2018), no. part B, 753–769, DOI10.1016/j.na.2018.04.027. MR3886600 ↑ A note on Trudinger’s extension of Sobolev’s inequalities , IndianaUniv. Math. J. (1971/72), 841–842, DOI 10.1512/iumj.1972.21.21066. MR293389 ↑ Introduction to Fourier analysis on Euclidean spaces ,Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series,No. 32. MR0304972 ↑ [31] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications , J. Math.Mech. (1967), 473–483, DOI 10.1512/iumj.1968.17.17028. MR0216286 ↑
3, 4[32] Orobitg, J.; Verdera, J.,
Choquet Integrals, Hausdorff Content and the Hardy-Littlewood Maximal Operator , Bulletin of the London Mathematical Society, Volume30, Issue 2, March 1998, pp. 145-150. pages[33] J. Xiao and Zh. Zhai,
Fractional Sobolev, Moser-Trudinger Morrey-Sobolev inequal-ities under Lorentz norms , J. Math. Sci. (N.Y.) (2010), no. 3, 357–376, DOI10.1007/s10958-010-9872-6. Problems in mathematical analysis. No. 45. MR2839038 ↑
3, 4, 6[34] V. I. Judoviˇc,
Some estimates connected with integral operators and with solu-tions of elliptic equations , Dokl. Akad. Nauk SSSR (1961), 805–808 (Russian).MR0140822 ↑ JOHN-NIRENBERG TYPE INEQUALITY FOR CRITICAL SOBOLEV SPACES 25
Institute for Advanced Study, Fuld Hall 412, 1 Einstein Drive, Prince-ton, NJ 08540, United States of America
E-mail address : [email protected] Okinawa Institute of Science and Technology Graduate University, Non-linear Analysis Unit, 1919–1 Tancha, Onna-son, Kunigami-gun, Okinawa,Japan
E-mail address ::