Regularity of some nonlinear quantities on superharmonic functions in local Herz-type Hardy spaces

Publicacions Matem`atiques, Vol 42 (1998), 319–338. REGULARITY OF SOME NONLINEAR QUANTITIES ON SUPERHARMONIC FUNCTIONS IN LOCAL HERZ-TYPE HARDY SPACES Dashan Fan, Shanzhen Lu∗and Dachun Yang∗ Abstract In this paper, the authors introduce a kind of local Hardy spaces in Rnassociated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and M¨uller in a recent paper. 1. Introduction In recent years, Hardy space methods have lead to remarkable progress on nonlinear partial differential equations with critical growth; see [3], [15], [1] and [16]. The main reason is as follows: a typical difficulty when studying such equations is that the nonlinear term is a priori only known to be in L1(Rn) in which there is no good elliptic theory; however, there is a well-established regularity theory in a particular subspace of L1(Rn), namely the standard Hardy space H1(Rn). In other words, Calder´on-Zygmund singular integral operators are not bounded on L1(Rn); but, they are indeed bounded on H1(Rn) if they satisfy the vanishing moment condition; see [5]. Keywords. Hardy spaces, Herz space, superharmonic function. 1991 Mathematics subject classifications: Primary 42B30; Secondary 35B65. ∗The second and the third authors were supported in part by the NNSF and the SECF of China. 320 D. Fan, S. Lu, D. Yang Moreover, Coifman, Meyer, Lions and Semmes in [1] proved that certain nonlinear quantities mentioned above are in fact in H1(Rn). In particular, L. C. Evans and S. M¨uller in [3] established the local boundedness in standard Hardy space H1(Rn) of some nonlinear quantities on the weakly superharmonic functions. Applying their results to the two-dimensional Euler equations with nonnegative vorticity, Evans and M¨uller gave a new proof of Delort’s results in [2]. On the other hand, a theory of the Hardy spaces associated with Herz spaces has been developed considerably in recent years; see [6], [11] and [12]. These Herz-type Hardy spaces are good substitutes of the usual Hardy spaces when studying boundedness of non-translation invariant operators (see [13]) and can be regarded as the local version at the origin of the usual Hardy spaces. There is also a good regularity theory in these spaces; see [12] and [14]. However, it is still not clear how to apply the theory on these spaces to partial differential equations. The main purpose of this paper is to establish a local version at the origin of Evans and M¨uller result in [3]. That is, we will study the local boundedness on the Hardy spaces associated with Herz spaces of some nonlinear quantities on weakly superharmonic functions. Our main results also extend the corresponding results of Evans and M¨uller, Theorem 1.1 and Theorem 5.4 in [3]. We hope this will be helpful to finding more applications of the Herz-type Hardy spaces in the study on partial differential equations; see also [14]. Let us first introduce some definitions. For k∈Z, let Bk=B(0,2k)={x∈R n:|x|≤2 k }and Ak=Bk\Bk−1. We denote the characteristic function of Akby χk. Definition 1 ([8]). Let α∈Rand 0 <p,q<∞.f∈L q loc (Rn\{0}) is said to belong to Herz space ˙ Kα,p q(Rn), if kfk˙ Kα,p q(Rn)≡(∞ X k=−∞ 2kαpkfχkkp Lq(R n))1/p <∞. We introduce the local Herz spaces as follows. Definition 2. Let α∈Rand 0 <p,q<∞.f∈L q loc(Rn\{0}) is said to belong to local Herz space ˙ Kα,p q,loc(Rn\{0}) if for every φ∈C∞ c(Rn), φf ∈˙ Kα,p q(Rn). Supharmonic functions 321 Obviously, by the above definition, f∈˙ Kα,p q,loc(Rn) is equivalent to the fact that for any k∈Z,χBkf∈˙ Kα,p q(Rn). It is also easy to observe that f∈˙ Kα,p q,loc(Rn) if and only if for any k∈N,χBkf∈˙ Kα,p q(Rn). We easily verify that ˙ Kn(1/p−1/q),p q,loc (Rn)$Lp loc(Rn) when q>p≥1 and ˙ K0,p p,loc(Rn)=L p loc(Rn). Now choose η:Rn−→ Rto be any smooth function satisfying supp η⊆B(0,1) and ZB(0,1) η(x)dx =1. If f∈L1(Rn), we write f∗(x)≡sup r∈(0,∞)¯¯¯¯ 1 rnZRn f(y)ηµx−y r¶dy¯¯¯¯ . In [11] (and independently in [6]), S. Lu and D. Yang introduce the following Herz-type Hardy spaces, H˙ Kn(1−1/q),1 q(Rn), which can be regarded as the local version at the origin of the standard Hardy space, H1(Rn), studied by C. Fefferman and E. M. Stein in [5]. Definition 3 ([11]). Let 1 ≤q<∞.f∈L 1 (R n ) is said to belong to the space H˙ Kn(1−1/q),1 q(Rn)iff ∗∈˙ K n(1−1/q),1 q(Rn). Moreover, we define kfkH˙ Kn(1−1/q),1 q(Rn)≡kf ∗ k˙ K n(1−1/q),1 q(Rn). According to Lu-Yang [12], this definition does not depend on the particular choice of η. Obviously, when q=1,H˙ K n(1−1/q),1 q(Rn)= H 1 (R n ), the standard Hardy space studied by Fefferman and Stein in [5]. In [4], Fan and Yang introduce the local version, the space h˙ Kn(1−1/q),1 q(Rn), of the space H˙ Kn(1−1/q),1 q(Rn) in Goldberg’s sense [7]. For f∈L1 loc(Rn), we will hereafter set f∗∗(x) = sup 0<r≤1¯¯¯¯ 1 rnZRn f(y)ηµx−y r¶dy¯¯¯¯ . 322 D. Fan, S. Lu, D. Yang Definition 4 ([4]). Let 1 ≤q<∞. The space h˙ Kn(1−1/q),1 q(Rn)is defined by h˙ Kn(1−1/q),1 q(Rn)≡{f∈L 1 (R n ):f ∗∗ ∈˙ Kn(1−1/q),1 q(Rn)}. Moreover, in this case, we set kfkh˙ Kn(1−1/q),1 q(Rn)≡kf ∗∗k˙ Kn(1−1/q),1 q(Rn). It is clearly that when q=1,h˙ K n(1−1/q),1 q(Rn)=h 1 (R n ), the local Hardy space introduced by Goldberg in [7]. By the results in [4], we also know that the above definition does not depend on the special choice of η. Now, we introduce another local version HlocKn(1−1/q),1 q(Rn)ofthe space H˙ Kn(1−1/q),1 q(Rn) as follows. Definition 5. Let 1 ≤q<∞.f∈L 1 loc(Rn) is said to belong to the space Hloc ˙ Kn(1−1/q),1 q(Rn) if for any φ∈C∞ c(Rn), φf ∈h˙ Kn(1−1/q),1 q(Rn). In the next section, we will prove the following equivalent definition of the space Hloc ˙ Kn(1−1/q),1 q(Rn). Proposition 1. Let 1≤q<∞. Then f∈H loc ˙ Kn(1−1/q),1 q(Rn)if and only if for any k∈Z,χBkf∗∗ ∈˙ Kn(1−1/q),1 q(Rn), which is also equivalent to the fact that f∗∗ ∈˙ Kn(1−1/q),1 q,loc (Rn). We remark that when q=1,H loc ˙ Kn(1−1/q),1 q(Rn)=H 1 loc(Rn), the local Hardy space introduced by Evans and M¨uller in [3]. To state our main theorem, we still need to introduce the local Herztype Sobolev space H1 loc ˙ Kn(1/2−1/q),2 q(Rn); see also [14]. Definition 6. Let 1 ≤q<∞. We call u∈H1 loc ˙ Kn(1/2−1/q),2 q(Rn) if uand its distributional first partial derivatives ux1,... ,u x nbelong to the space ˙ Kn(1/2−1/q),2 q,loc (Rn). Obviously, when q=2,H 1 loc ˙ Kn(1/2−1/q),2 q(Rn)=H 1 loc(Rn), the standard local Sobolev space. Now, it is a position to state our main theorem. Supharmonic functions 323 Theorem 1. Let 1<q<∞,u∈H 1 loc ˙ K1−1/q,2 2q(R2)be a weak solution of the partial differential equation −4u=win R2. where w∈˙ K2(1−1/q),1 q,loc (R2)and (1.1) w≥0. Then ux1ux2,u 2 x 1−u 2 x 2∈H loc ˙ K2(1−1/q),1 q(R2). In addition, for each φ∈C∞ c(R2), we have the estimate (1.2) kφux1ux2kh˙ K2(1−1/q),1 q(R2)+kφ(u2 x1u2 x2)kh˙ K2(1−1/q),1 q(R2) ≤ckχB(0,R)|Du|k ˙ K1−1/q,2 2q(R2) for some constant cand some radius Rdepending only on φ. If we take q= 1, then Theorem 1 is just Theorem 1.1 in [3]. Thus, Theorem 1 is a generalization of Theorem 1.1 in [3]. In fact, Theorem 1 can be regarded as some kind local version of that theorem at the origin. See also Semmes [16] for several different versions of Evans and M¨uller’s theorem. It has been pointed by Evans and M¨uller in [3] that without the sign condition (1.1), Theorem 1 is false. However, in the radical case, the nonnegativity of wis not required. To be precise, we have Theorem 2. Let 1<q,q 1 ,q 2<∞,u∈H 1 loc ˙ K2(1/2−1/q1),2 q(R2)be a weak solution of the partial differential equation −4 u=win R2, with w∈˙ K2(1−1/q2),1 q2,loc (R2)and u(x)=u(r),w(x)=w(r)for r=|x|. Then ux1ux2,u2 x1−u2 x2∈H loc ˙ K2(1−1/q),1 q(R2). Furthermore, for each φ∈C∞ c(R2), we have the estimate (1.3) kφux1ux2kh˙ K2(1−1/q),1 q(R2)+kφ(u2 x1−u2 x2)kh˙ K2(1−1/q),1 q(R2) ≤c¡kχBk|Du|k2 ˙ K2(1/2−1/q1),2 q1(R2)+kχBkwk2 ˙ K2(1−1/q2),1 q2(R2)), where k∈Ndepends only on φ. 324 D. Fan, S. Lu, D. Yang If we choose q=1,q 1= 2 and q2= 1, we recover Theorem 5.4 in [3]. Finally, we point that we also obtain a local version on Herz-type Hardy spaces h˙ Kn(1−1/q),1 q(Rn) of Jones-Journ´e’s result on convergence a.e. and convergence in the distribution sense for a sequence bounded in the Hardy space H1(Rn); see [9], [1] and see also [3] for the local version on h1(Rn). Proposition 2. Let 1<q<∞. Suppose {fk}∞ k=1 is bounded in h˙ Kn(1−1/q),1 q(Rn)and fk→f, a.e., as k→∞, for f∈L1 loc(Rn). Then f∈h˙ Kn(1−1/q),1 q(Rn)and (1.4) fk→f in the sense of distributions. If we take q= 1, Proposition 2 is just Theorem 3.1 in [3]. 2. Proofs of Theorems We begin with the proof of Proposition 1. Proof of Proposition 1: When q= 1, this proposition is just Lemma 5.1 in [3]. Now we restrict that 1<q<∞. Suppose f∈Hloc ˙ Kn(1−1/q),1 q(Rn). For any k∈Z, we choose φ∈C∞ c(Rn) such that φ(x) = 1 when x∈ B(0,2k+ 1). Then, if x∈Bk, (φf)∗∗(x) = sup 0<r≤1¯¯¯¯ 1 rnZRn ηµx−y r¶φ(y)f(y)dy¯¯¯¯ = sup 0<r≤1¯¯¯¯ 1 rnZRn ηµx−y r¶f(y)dy¯¯¯¯ =f∗∗(x). Thus, kχBkf∗∗k˙ Kn(1−1/q),1 q(Rn)=kχBk(φf)∗∗k˙ Kn(1−1/q),1 q(Rn) ≤k(φf)∗∗k˙ Kn(1−1/q),1 q(Rn) <∞. That is, χBkf∗∗ ∈˙ Kn(1−1/q),1 q(Rn). Supharmonic functions 325 Now suppose χBkf∗∗ ∈˙ Kn(1−1/q),1 q(Rn) for each k∈Z. Fix any φ∈C∞ c(Rn) with supp φ⊆B(0,R). Choose l0∈Nsuch that 2l0−1≤ R+1<2 l 0. Observe first that (φf)∗∗(x) = sup 0<r≤1¯¯¯¯ 1 rnZRn ηµx−y r¶φ(y)f(y)dy¯¯¯¯ vanishes if x∈Rn\B(0,R+1). If x∈B(0,R+1), by (5.2) in [3, p. 215], we have (φf)∗∗(x)≤cÃf∗∗(x) + sup 0<r≤1 1 rn−1ZB(x,r) |f(y)|dy! ≤cÃf∗∗(x)+ l 0+1 X l=−∞ ZRn fl(x−y)|y|1−nχB0(y)dy!, where fl(x)=|f(x)|χ l (x). Let gl(x)≡RRnfl(x−y)|y|1−nχB0(y)dy and g(x)= l 0 +1 X l=−∞ gl(x). Then, k(φf)∗∗k˙ Kn(1−1/q),1 q(Rn)≤ckχBl0f∗∗k˙ Kn(1−1/q),1 q(Rn) +ckχBl0gk˙ Kn(1−1/q),1 q(Rn). By the hypothesis, we know that kχBl0f∗∗k˙ Kn(1−1/q),1 q(Rn)<∞. We still need to show that kχBl0gk˙ Kn(1−1/q),1 q(Rn)<∞. In fact, we have kχBl0gk˙ Kn(1−1/q),1 q(Rn)≤ l0+1 X l=−∞ l0 X k=−∞ 2kn(1−1/q)kglχkkLq(Rn) = l0+1 X l=−∞ min{l+2,l0} X k=−∞ ···+ l 0+1 X l=−∞ l0 X k=l+3 ··· ≡I 1+I 2. 326 D. Fan, S. Lu, D. Yang For I1, by the trivial estimate kglχkkLq(Rn)≤ckflkLq(Rn)and the hypothesis, we obtain I1≤c l0+1 X l=−∞ kflkLq(Rn)Ãl+2 X k=−∞ 2kn(1−1/q)! ≤c l0+1 X l=−∞ 2ln(1−1/q)kflkLq(Rn) ≤ckχBl0+1 f∗∗k˙ Kn(1−1/q),1 q(Rn) <∞. To estimate I2, we first note that if l≥0 and k≥l+ 2, then gl(x)≡0 when x∈Ak. Thus, I2= −1 X l=−∞ 2 X k=l+3 2kn(1−1/q)kglχkkLq(Rn). Now, since l≤−1 and k≥l+ 3, for x∈Ak,wehave g l (x)=ZR n |f l (x−y)||y|1−nχB0(y)dy ≤Z3|x|≤4|y|≤5|x| |fl(x−y)||y|1−nχB0(y)dy ≤c2k(1−n)kflkL1(Rn). By substituting this into I2, we obtain I2≤c −1 X l=−∞ 2 X k=l+3 2kkflkL1(Rn) ≤c −1 X l=−∞ 2ln(1−1/q)kfχlkLq(R n) ≤ckχB−1f∗∗k˙ Kn(1−1/q),1 q(Rn) <∞. This finishes the proof of Proposition 1. Supharmonic functions 327 Proof of Theorem 1: Fix R>8. Let B(0,R) denote the closed ball with center 0 and radius R, and set v(x)=−1 2πZB(0,R) w(y) log(|x−y|)dy. Then (2.1) θ=u−vis harmonic within B(0,R), and (2.2) vxi(x)= 1 2πZB(0,R) w(y)xi−yi |x−y|2dy, i =1,2. Choosing η∈C∞ c(R2) satisfying supp η⊆B(0,1),ZB(0,1) η(x)dx =1,η≥0. Let us also fix any point x0∈R2. Consider then for 0 <r≤1, the expression A=1 r2ZR2 vx1(x)vx2(x)ηµx−x0 r¶dx =1 4π2r2ZB(0,R)ZB(0,R) w(y)w(z) µZR2 x1−y1 |x−y|2 x2−z2 |x−z|2ηµx−x0 r¶dx¶dy dz, by (2.2). We change variables by replacing (x−x0)/r with x,(y−x 0 )/r with y, and (z−x0)/r with zto discover (2.3) A=r2 4π2ZB(−x0/r,R/r)ZB(−x0/r,R/r) w(x0+ry)w(x0+rz) ×µZR2 x1−y1 |x−y|2 x2−z2 |x−z|2η(x)dx¶dy dz. 334 D. Fan, S. Lu, D. Yang Proof of Theorem 2: We have the ordinary differential equation (2.16) −1 r(ru0)0(r)=w(r). We only prove the estimates for ux1ux2, because, similar to [3], those for u2 x1−u2 x2follow then by performing a rotation. Let η∈C∞ c(B(0,1)), η≥0 and RB(0,1) η(x)dx = 1. Define f(x)=(u x 1u x 2)(x)=x 1 x 2 r 2(u 0 ) 2 (r) and fi=fχifor i∈Z. We also write f∗∗(x) = sup 0<r≤1¯¯¯¯ 1 r2ZR2 ηµx−y r¶f(y)dy¯¯¯¯ and for i∈Z, f∗∗ i(x) = sup 0<r≤1¯¯¯¯ 1 r2ZR2 ηµx−y r¶fi(y)dy¯¯¯¯ . Since w∈˙ K2(1−1/q2),1 q2,loc (R2)⊂L1 loc(R2), the ordinary differential equation (2.16) implies that v(r)≡ru0(r) is absolutely continuous on each interval [0,R]. In particular, if 2i−1≤r≤2i, (2.17) ¯¯¯¯¯ v(r)−1 2i−1Z2i 2i−1 v(r)dr¯¯¯¯¯ ≤Z2i 2i−1 |v0(r)|dr ≤Z2i 2i−1 r|w(r)|dr. It is easy to verify that if h∈Lq(R2), supp h⊆B(0,r) and RR2h(x)dx=0, then |B(0,r)| 1/q−1khk−1 Lq(R2)his a central (2(1−1/q),q)-atom supported in B(0,r); see [11]. Thus, by Theorem 2.1 in [12] (see also [6]), kh∗∗k˙ K2(1−1/q),1 q(R2)≤kh ∗ k˙ K 2(1−1/q),1 q(R2) ≤cr2(1−1/q)khkLq(R2) ≤cr2khkL∞(R2), if h∈L∞(R2). Therefore, kf∗∗ ik˙ K2(1−1/q),1 q(R2)≤c22ikfikL∞(R2). Supharmonic functions 335 Moreover, f∗∗ i+2(x)=0onB(0,2i+1 −1) ⊃B(0,2i)ifi≥1. Now, by (2.17), 22ikfikL∞(R2)≤4 sup 2i−1<r≤2i r2|u0(r)|2 ≤c(2−iZ2i 2i−1 |ru0(r)|dr +Z2i 2i−1 |(ru0(r))0|dr)2 ≤c(2−i/q1Z2i 2i−1 |r1/q1u0(r)|dr +Z2i 2i−1 r|w(r)|dr)2 ≤c   2i(1−2/q1)ÃZ2i 2i−1 r|u0(r)|q1dr!1/q1 +ZAi |w(x)|dx   2 ≤c(2i(1−2/q1)µZAi |Du(x)|q1dx¶1/q1 +ZAi |w(x)|dx)2 . Therefore, kχBkf∗∗k˙ K2(1−1/q),1 q(R2) ≤ ∞ X j=−∞ kχBkf∗∗ jk˙ K2(1−1/q),1 q(R2) = k+2 X j=−∞ kχBkf∗∗ jk˙ K2(1−1/q),1 q(R2) ≤c k+2 X j=−∞ ½³2j(1−2/q1)k|Du|χjkLq1(R2)´2+k|w|χ jk 2 L 1(R 2)¾ ≤c k+2 X j=−∞ 24j(1/2−1/q1)k|Du|χjk2 Lq1(R2)+c k+2 X j=−∞ k|w|χ jk 2 L 1(R 2) ≤ckχ B k+2 |Du|k 2 ˙ K2(1/2−1/q1),2 q1(R2)+ckχBk+2 wk2 ˙ K2(1−1/q2),1 q2(R2) <∞. Hence ux1ux2∈H loc ˙ K2(1−1/q),1 q(R2). The estimation (1.4) now follows from Proposition 1. This finishes the proof of Theorem 2. 336 D. Fan, S. Lu, D. Yang Proof of Proposition 2: Since h˙ Kn(1−1/q),1 q(Rn)⊂h1(Rn), (1.4) holds by Theorem 3.1 in [3]. The remaining thing is to verify that f∈h˙ Kn(1−1/q),1 q(Rn). Choose φ∈C∞ c(Rn), supp φ⊂B(0,1) and RRnφ(x)dx =1. Fort>0, set φt(x)= 1 t nφ ¡x t ¢. Then by (1.4), for every x∈Rnand 0 <t≤1, we have φt∗fk(x)−→ φ t∗f( x ) ,as k−→ ∞ ; therefore, lim k→∞ sup 1≥t>0 |φt∗fk(x)|= sup k inf l≥ksup 1≥t>0 |φt∗fl(x)| ≥sup 1≥t>0 sup k inf l≥k|φt∗fl(x)| = sup 1≥t>0 lim k→∞ |φt∗fk(x)| = sup 1≥t>0 |φt∗f(x)|. On the other hand, since kfkkh˙ Kn(1−1/q),1 q(Rn)≤c, it follows from Theorem 2.1 in [4] that ° ° °sup 1≥t>0 |φt∗fk|° ° °˙ Kn(1−1/q),1 q(Rn)≤c. By Fatou’s lemmas of series and integration, we have c≥lim k→∞ ° ° °sup 1≥t>0 |φt∗fk|° ° °˙ Kn(1−1/q),1 q(Rn) = lim k→∞ (∞ X l=−∞ 2ln(1−1/q)° ° °χlsup 1≥t>0 |φt∗fk|° ° °Lq(Rn)) ≥ ∞ X l=−∞ 2ln(1−1/q)lim k→∞ ° ° °χlsup 1≥t>0 |φt∗fk|° ° °Lq(Rn) ≥ ∞ X l=−∞ 2ln(1−1/q)° ° °χllim k→∞ sup 1≥t>0 |φt∗fk|° ° °Lq(Rn) ≥ ∞ X l=−∞ 2ln(1−1/q)° ° °χlsup 1≥t>0 |φt∗f|° ° °Lq(Rn) =° ° °sup 1≥t>0 |φt∗f|° ° °˙ Kn(1−1/q),1 q(Rn). Supharmonic functions 337 Again, from Theorem 2.1 in [4], we deduce that f∈h˙ Kn(1−1/q),1 q(Rn). This finishes the proof of Proposition 2. References 1. R. 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Partial Differential Equations 19 (1994), 277–319. Dashan Fan: Anhui University and University of Wisconsin-Milwaukee Current Address: Department of Mathematical Sciences University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee WI 53201 U.S.A. e-mail: [email protected]wm.edu Shanzhen Lu and Dachun Yang: Department of Mathematics Beijing Normal University Beijing 100875 THE PEOPLE’S REPUBLIC OF CHINA e-mail: [email protected] e-mail: dcy[email protected] Primera versi´o rebuda el 26 de febrer de 1997, darrera versi´o rebuda el 17 de mar¸c de 1998