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An $ L^ 1$-Type Estimate for Riesz Potentials

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An $ L^ 1$-Type Estimate for Riesz Potentials An L1-type estimate for Riesz potentials 1 2 3 Armin Schikorra∗ , Daniel Spector† , and Jean Van Schaftingen‡ 1Department of Mathematics, University of Basel, Basel, Switzerland 2Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 3Institut de Recherche en Mathématique et en Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium Abstract In this paper we establish new L1-type estimates for the classical Riesz potentials of order α (0,N): ∈ Iαu N/(N α) N C Ru 1 N N . k kL − (R ) ≤ k kL (R ;R ) This sharpens the result of Stein and Weiss on the mapping properties of 1 N Riesz potentials on the real Hardy space (R ) and provides a new fam- H ily of L1-Sobolev inequalities for the Riesz fractional gradient. 1 Introduction and Main Results Let N 2 and define the Riesz potential I of order α (0,N) by its action on ≥ α ∈ a measurable function u via the convolution 1 u(y) Iαu(x) (Iα u)(x) := N α dy, ≡ ∗ γ(α) ˆ N x y R | − | − N/2 α α N α whenever it is well-defined. Here, γ(α) = π 2 Γ( 2 )/Γ( −2 ) is a normal- arXiv:1411.2318v4 [math.FA] 1 Sep 2016 ization constant [24, p. 117] that ensures that the Riesz potentials satisfy the semigroup property Iα+βu = IαIβu, for α, β > 0, such that α + β < N, for u in a suitable class of functions. p N The study of the mapping properties of Iα on L (R ) was initiated by Sobolev, who proved the following fundamental theorem about integrals of the potential type in 1938 [21, p. 50]. ∗[email protected], A.S. supported by SNF †[email protected], D.S. supported by MOST 103-2115-M-009-016-MY2 ‡[email protected] 1 Theorem 1.1 (Sobolev). Let 0 < α < N and 1 < p < N/α. Then there exists a constant C = C(p, α, N) > 0 such that Iαu Np/(N αp) N C u p N (1.1) k kL − (R ) ≤ k kL (R ) for all u Lp(RN ). ∈ In particular we see that Sobolev’s result concerns Lp estimates for Riesz N potentials when 1 < p < α and strictly excludes the case p = 1. Indeed, it is well-known that no such inequality as (1.1) can hold in this regime - one may consider, for example (cf. [24, p. 119]), an approximation of the identity (for explicit construction one can see Section 3 of this paper). Then the right-hand- side of the inequality in Theorem 1.1 stays bounded while pointwise 1 1 I ρ (x) I (x) = , α ε → α γ(α) x N α | | − N N α N which does not belong to the Lebesgue space L − (R ), and Fatou’s lemma gives the desired contradiction. It is then natural to ask if there is a substitute for the inequality (1.1). One possibility was given by Stein and Weiss [25, p. 31], where they demonstrated that if one replaces Lp(RN ) with the real Hardy space p N p N p N N (R ) := u L (R ): Ru L (R ; R ) H ∈ ∈ (where Ru := DI1u is the vector-valued Riesz transform), one can extend the validity of Theorem 1.1 to the regime p = 1. For p (1, ), p(RN ) = Lp(RN ), ∈ ∞ H but for p = 1 the Hardy space 1(RN ) is strictly contained in L1(RN ). Their H result implies the following theorem. Theorem 1.2 (Stein-Weiss). Let 0 < α < N and 1 p < N/α. Then there exists a ≤ constant C = C(p, α, N) > 0 such that Iαu Np/(N αp) N C u p N + Ru p N N k kL − (R ) ≤ k kL (R ) k kL (R ;R ) for all u p(RN ). ∈ H Actually the approach to Sobolev inequalities due to Gagliardo [10, p. 120] and Nirenberg [17, p. 128] gives another replacement to Theorem 1.1 for 1 ≤ α < N. Indeed, written in the language of potentials, one sees that the results [10, 17] assert the existence of a constant C > 0 such that I1u N/(N 1) N C Ru 1 N N , k kL − (R ) ≤ k kL (R ;R ) N 1 N N for all u C∞(R ) such that Ru L (R ; R ). Therefore, if 1 α < N, the ∈ c ∈ ≤ preceding inequality and Theorem 1.1 applied to Iαu = Iα 1I1u allows us to − deduce that Iαu N/(N α) N C I1u N/(N 1) N k kL − (R ) ≤ k kL − (R ) C Ru 1 N N , ≤ k kL (R ;R ) N 1 N N for all u C∞(R ) such that Ru L (R ; R ). ∈ c ∈ The main result of this paper is the following theorem demonstrating that this L1-type estimate holds for the Riesz potential of any order α (0,N). ∈ 2 Theorem A. Let N 2 and 0 < α < N. Then there exists a constant C = ≥ C(α, N) > 0 such that Iαu N/(N α) N C Ru 1 N N k kL − (R ) ≤ k kL (R ;R ) N 1 N N for all u C∞(R ) such that Ru L (R ; R ). ∈ c ∈ Remark 1.3. Theorem A is false when N = 1, see Counterexample 3.2 in Sec- tion 3. Our motivation for such an inequality can be found in the study of cer- tain fractional partial differential equations introduced in [20], where existence results are demonstrated for a continuous spectrum of such equations param- eterized by the Riesz fractional gradient α D u := DI1 αu, − for 0 < α < 1. With this notation, an alternative formulation of Theorem A is the following. Theorem A0. Let N 2 and 0 < α < 1. Then there exists a constant C = ≥ C(α, N) > 0 such that α u N/(N α) N C D u 1 N N (1.2) k kL − (R ) ≤ k kL (R ;R ) N for all u C∞(R ). ∈ c Theorem A0 is a natural analogy to the Sobolev inequalities known for the fractional Laplacian when p > 1 and integer order derivatives for p 1, ≥ though one might have guessed such a theorem from several additional fac- tors. Firstly, related results for Besov spaces with the same degree of fractional differentiability have long been known in the literature (see e.g. [6, Lemma D.2; 8, Theorem 1.4; 13, Theorem 4; 22, Theorem 2; 32, Theorem 8.3]). A sec- ond factor suggesting such an inequality is the observation that the asymp- totics of the constant in Theorem 1.1 are O(1/(p 1)) as p 1, which agrees − → with the asymptotics of the operator norm of the vector-valued Riesz trans- form R : Lp(RN ) Lp(RN ; RN ). Finally, there is the more recent work of the → second author and R. Garg [11, 12] which shows that the logarithmic potential N IN u, defined for u C∞(R ) by ∈ c 1 1 IN u(x) = N 1 log u(y) dy, S ˆ N x y | − | R | − | 1 has, for any u with N u = 0, the representation R ´ 1 x y IN u(x) = N 1 − Ru(y) dy. S ˆ N x y · | − | R | − | Therefore, when α = N one has the corresponding estimate I u N C Ru 1 N N . k N kL∞(R ) ≤ k kL (R ;R ) 1For this class of functions, this is equivalent to asking Ru L1(RN ; RN ). ∈ 3 Our proof of Theorem A is quite direct, and relies only on the boundedness of the Riesz transform and of the classical maximal function operator on Lp for 1 < p < + . We do not rely upon any Sobolev type embedding nor ∞ any multiplier theorem that goes beyond the Riesz transform. Here the crucial observation is that the vector-valued Riesz transform is curl-free, i.e. ∂R u ∂R u j = i ∂xi ∂xj for all i, j 1,...,N . In fact, an interesting point to note is that the same ∈ { } proof shows that one has IαF N/(N α) N N C F 1 N N (1.3) k kL − (R ;R ) ≤ k kL (R ;R ) for vector fields F L1(RN ; RN ) that satisfy either curl F = 0 or div F = 0, the ∈ pair of which is reminiscent of the conditions for inclusion in the real Hardy space [25]. The remainder of the paper is organized as follows. In Section 2 we give proofs of the main results. In Section 3 we discuss several more intricate ques- tions in greater detail, including connections of our result with more technical results from the literature, an open question in regard to a sharp result in the scale of Lorentz spaces, and the details of the counterexample mentioned in Remark 1.3. 2 Proofs of the Main Results We now prove Theorem A. In the course of the proof, we will use C to des- ignate a constant that may depend upon α and N, though the constant may change from line to line. N 1 N N Proof of Theorem A. Let u C∞(R ) be such that Ru L (R ; R ). We claim ∈ c ∈ it suffices to show that, for j 1,...,N , one has the existence of a uniform ∈ { } constant C = C(α, N) > 0 such that RjuIαϕ C Ru 1 N N ϕ N (2.1) L (R ;R ) α N ˆ N ≤ k k k kL (R ) R N for every ϕ Cc∞ (R ). ∈ N 2 Indeed, utilizing the identity v = i=1 Ri v, the boundedness of Ri : N/(N α) N N/(N α) N − L − (R ) L − (R ), and duality we have → P N Iαu N/(N α) N C RiIαu N/(N α) N k kL − (R ) ≤ k kL − (R ) i=1 X N = C sup RiIαu ψi.
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