Besov-Morrey Spaces Associated with Hermite Operators and Applications to Fractional Hermite Equations

Electronic Journal of Diﬀerential Equations, Vol. 2018 (2018), No. 187, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

BESOV-MORREY SPACES ASSOCIATED WITH HERMITE OPERATORS AND APPLICATIONS TO FRACTIONAL HERMITE EQUATIONS

NGUYEN ANH DAO, NGUYEN NGOC TRONG, LE XUAN TRUONG

Communicated by Jesus Ildefonso Diaz

Abstract. The purpose of this article is to establish the molecular decomposition of the homogeneous Besov-Morrey spaces associated with the Hermite 2 n operator H = −∆ + |x| on the Euclidean space R . Particularly, we obtain some estimates for the operator H on the Hermite-Besov-Morrey spaces and the regularity results to the fractional Hermite equations (−∆ + |x|2)su = f, and (−∆ + |x|2 + I)su = f. Our results generalize some results by Anh and Thinh [1].

1. Introduction In this article, we study the Besov-Morrey spaces associated with the Hermite operator H = −∆ + |x|2 on Rn, n ≥ 1. It is known that the classical theory of the Besov and Triebel-Lizorkin spaces plays a crucial role not only in the theory of function spaces, but also in the theory of partial diﬀerential equations and harmonic analysis, see e.g. [7, 9, 10, 11, 12, 14, 15], and the references therein. Recently, the theory of the Besov and Triebel-Lizorkin spaces associated with the operators has been developed by many authors when one observed that the classical Besov and Triebel-Lizorkin spaces are not always the most suitable to investigate a number of operators, see [1, 2, 3, 4, 18, 10, 11, 19], and their references. For example, Petrusev and Xu [13] studied the characterization of the inhomogeneous Besov and Triebel-Lizorkin spaces in terms of Littlewood-Paley decomposition in the context of Hermite expansions that the frame elements have almost exponential localization. Note that these frame elements can be viewed as an analogue of the ϕ-transform of Frazier and Jawerth [7]. Another approach introduced by Anh and Thinh [1] is of deﬁning the Besov and Triebel-Lizorkin spaces in terms of the heat kernels via square functions. Their approach adapted to the study of the theory of both homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces. This allows them to extend the range of indices 1 ≤ p, q ≤ ∞ of the homogeneous Besov

2010 Mathematics Subject Classiﬁcation. 42B35, 42B20. Key words and phrases. Fractional Hermite equations; Hermite-Besov-Morrey space; molecular decomposition. c 2018 Texas State University. Submitted September 26, 2018. Published November 20, 2018. 1 2 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

α,H α,H space BMp,q (resp. Triebel-Lizorkin spaces FMp,q ) to 0 < p, q ≤ ∞, compare to the results in [8]. One of the most interesting studies of the theory of Besov spaces is the Besov- Morrey spaces, introduced first by Kozono and Yamazaki [9] to investigate time- local solutions of the Navier-Stokes equations with the initial data in the spaces of this type. As a matter of fact, the Besov-Morrey spaces share several features of Besov and Morrey spaces. They represent the local oscillations and singularities of functions more precisely than the classical Besov spaces. Thus, they behav better in many aspects, particularly under the action of singular integrals and pseudo- differential operators. In addition, Mazzucato [11, 12] established the wavelet decompositions to characterize the homogeneous and inhomogeneous Besov-Morrey spaces. For more results on the Besov-Morrey spaces, we refer the reader to [9, 10, 11, 12, 14, 15, 17, 19] and the references therein. Inspired by the above results, we would like to generalize the theory of the α,H homogeneous Besov spaces associated with the Hermite operator BMp,q to the one of the homogeneous Besov-Morrey spaces associated with the Hermite operator α,H α,H BMp,q,r in this paper. To study BMp,q,r, we use the results in [1], specifically, the estimates on the heat kernels via the square functions. Beside, we also establish the α,H molecular decompositions for BMp,q,r. As applications, we obtain the regularity of solutions to the fractional Hermite equations:

s H u = f, and s (H + I) u = f. We organize this paper as follows: Section 2 contains some preliminary results and deﬁnitions of functional spaces. Section 3 is devoted to the study of the molecular decomposition for the Hermite-Besov-Morrey space. Finally, we investigate the regularity of solutions on Hermite-Besov-Morrey spaces to the fractional Hermite equations in Section 4. Throughout this paper, we always use C and c to denote positive constants that are independent of the main parameters involved but whose values may diﬀer from line to line. We write A . B if there is a universal constant C such that A ≤ CB; and A ∼ B if A . B and B . A. We use the following notation: N = {0, 1, 2,... }, − − N+ = {1, 2, 3,... }, Z = {−1, −2,... }, Z0 = {0, −1, −2,... } a ∧ b = min{a, b}, a ∨ b = max{a, b}, and int[a] is the integer part of a.

2. Preliminaries 2.1. Dyadic cube. The set of all dyadic cubes D in Rn is deﬁned by n Y k k D = [mj2 , (mj + 1)2 ): m1, m2, . . . , mn, k ∈ Z . j=1

Qn h k k For a dyadic cube Q := j=1 mj2 , (mj + 1)2 , for some m1, m2, . . . , mn, k ∈ Z we denote by `(Q) and xQ the length and the center of the dyadic cube Q. In this k kn case, `(Q) = 2 and xQ = (mj + 1/2)2 j=1. Moreover, for every ν ∈ Z, we set ν Dν = {Q ∈ D : `(Q) = 2 }. EJDE-2018/187 BESOV-MORREY SPACES 3

2.2. Morrey space. Let us ﬁrst recall the deﬁnition of the Morrey spaces.

r Deﬁnition 2.1. For every 0 < p ≤ r < ∞, the Morrey space Mp is deﬁned by

r p n n( 1 − 1 ) M ≡ f ∈ L ( ): kfk r = sup sup R r p kfk p < ∞ . p loc R Mp L (B(x0,R)) n x0∈R R>0 Next, we point out some known results about the Morrey norms. Proposition 2.2. Let 0 < p ≤ r < ∞. Then

1 − 1 kfk r ∼ sup |Q| r p kfk p , (2.1) Mp L (Q) Q∈D θ θ kf kMr = kfk rθ , ∀θ > 0, (2.2) p Mpθ Z b 1/q Z b 1/q q dt q dt |F (·, t)| r ≤ kF (·, t)k r , for 0 < q ≤ p. (2.3) M Mp a t p a t Proof. Note that (2.1) and (2.2) follow from the deﬁnition of the Morrey spaces. While, (2.3) can be obtained by using Minkowski integral inequality, see also [8, (2.20)].

For θ > 0, we denote by Mθ the Hardy-Littlewood maximal function Z 1/θ 1 θ n Mθf(x) = sup |f(y)| dy , x ∈ R , x∈B |B| B where the supremum is taken over all balls B ⊂ Rn containing x. Then, we have a version of the Feﬀerman-Stein vector-valued maximal inequality for the Morrey spaces, see [16, Proposition 2.1]. Proposition 2.3. Let 0 < q ≤ ∞, 0 < p ≤ r < ∞, and 0 < θ < min{p, q}. Then

1/q 1/q X q X q |Mθfk| r |fk| r . Mp . Mp k∈Z k∈Z Remark 2.4. As a consequence of Proposition 2.3, the Hardy-Littlewood maximal r operator Mθ is bounded on Mp. Next, we put 1−p/r 1/p 1 X 1−p/r p Av = sup |Q| |sQ| . J∈D,`(J) 2v |J| > Q∈Dv ,Q⊂J

We borrow a result of Wang [19, p.779] involving the characterization of Av in the Morrey norms.

Lemma 2.5. Let 0 < p ≤ r < ∞, and ν ∈ Z. Assume that the sequence {sQ : Q ∈ Dν } satisﬁes

X −1/r k |Q| |s |χ k r < ∞. Q Q Mp Q∈Dν Then X −1/r k |Q| |s |χ k r ∼ A . Q Q Mp ν Q∈Dv 4 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

2.3. Kernel estimates on Hermite operators. For any k ≥ 0 and for t > 0, we √ √ k −t denote the kernel associated with (t H) e H by pt,k(x, y). We recall here the results of [1, Lemma 2.1 and Propisition 2.2].

Proposition 2.6. For k ∈ N, there exist C > 0 and δ > 0 so that tk n (1) |pt,k(x, y)| ≤ C (t+|x−y|)n+k , for x, y ∈ R . (2) for any |h| < t, we have |h|δ tk |p (x + h, y) − p (x, y)| ≤ C , for x, y ∈ n. t,k t,k t (t + |x − y|)n+k R n Proposition 2.7. For every y ∈ R , we have pt,k(·, y) ∈ S. 2.4. Calder´onreproducing formulas. In this part we recall Calder´on’sformula from [1], that is useful for studying the homogeneous Besov-Morrey spaces.

+ 0 Proposition 2.8. Let m1, m2 ∈ N and f ∈ S . Then 1 Z ∞ √ √ √ √ dt m1 −t H m2 −t H 0 f = − m−1 (t H) e (t H) e f in S , 2 (m − 1)! 0 t 0 where m = m1 + m2, and S is the dual space of the Schwartz functions S as usual.

3. Besov-Morrey Spaces associated with the Hermite operators It is convenient for us to introduce first the homogeneous Besov-Morrey spaces corresponding to the Hermite operator H. Definition 3.1. Let α ∈ R, 0 < p, q ≤ ∞, p ≤ r ≤ ∞, and for every positive 1 integer m > n + max{α, 0} + int[n( − 1)] + 1, with θ0 = min{1, p, q}. Then, we θ0 α,H,m define the homogeneous Hermite-Besov-Morrey space BMp,q,r as follows n Z ∞ √ √ q dt1/q α,H,m 0 −α m −t H BM := f ∈ S : kfk α,H,m = t k(t ) e fkMr p,q,r BMp,q,r H p 0 t o < ∞ .

α,H,m α,H,m Remark 3.2. If r = p, then the space BMp,q,r is exactly the space BMp,q in [1].

α,H,m We will show that BMp,q,r is independent of the choice of m when m is large enough. Precisely, we have the following result.

Theorem 3.3. Let α ∈ R, 0 < p, q ≤ ∞, and p ≤ r ≤ ∞. Let m1, m2 be the positive integers such that 1 m1, m2 > n + max{α, 0} + int[n( − 1)] + 1, θ0 α,H,m1 α,H,m2 with θ0 = min{1, p, q}. Then, the spaces BMp,q,r and BMp,q,r coincide with equivalent norms.

α,H As a consequence of Theorem 3.3, we can deﬁne the Besov space BMp,q,r as any space BMα,H,m, for any positive integer m > n + max{α, 0} + int[n( 1 − 1)] + 1. p,q,r θ0 We now recall the deﬁnition of the molecules associated with the Hermite operator in [1]. EJDE-2018/187 BESOV-MORREY SPACES 5

Deﬁnition 3.4. Let 0 < r ≤ ∞, α ∈ R, and N,M ∈ N+. A function u is said√ to M be an (H, M, N, α, r) molecule if there exist√ a function b from the domain ( H) and a dyadic cube Q ∈ D so that (i) u = ( H)M b, and (ii) √ |x − x |−n−N |( )kb(x)| ≤ `(Q)M−k|Q|α/n−1/r 1 + Q , for k = 0,..., 2M. H `(Q)

Brieﬂy, we denote u = mQ, for every dyadic cube Q ∈ D. Next, we have some elementary estimates.

n Lemma 3.5. Let N ∈ N+ and a > t > 0. For any x, z ∈ R , Z −n−N −n−N −n−N |x − y| |z − y| n |x − z| 1 + 1 + dy . t 1 + . n t a t R For a proof of the above lemma, we refer to [1, Lemma 3.6]. Next, we have a α,H,m result of the molecular decomposition for BMp,q,r .

Theorem 3.6. Let α ∈ R, 0 < p, q ≤ ∞, p ≤ r ≤ ∞, and θ0 = min{1, p, q}. 1 (i) For every M,N ∈ + and m > n + max{α, 0} + int[n( − 1)] + 1, if f ∈ N θ0 α,H,m BMp,q,r , then there exist a sequence of (H, M, N, α, r) molecules {mQ}Q∈Dv ,v∈Z and a sequence of coeﬃcients {sQ}Q∈Dv ,v∈Z so that X X 0 f = sQmQ, in S . v∈Z Q∈Dv Moreover, 1/q X q A kfk α,H,m . (3.1) v . BMp,q,r v∈Z (ii) Conversely, if X X 0 f = sQmQ, in S , v∈Z Q∈Dv where {mQ}Q∈Dv ,v∈Z is a sequence of (H, M, N, α, r) molecules and {sQ}Q∈Dv ,v∈Z 1/q is a sequence of coeﬃcients satisfying P Aq < ∞, then f ∈ BMα,H,m, and v∈Z v p,q,r 1/q X q kfk α,H,m A , (3.2) BMp,q,r . v v∈Z n n provided that N,M ∈ + such that < θ0, M > max{ − α, m}, with m > N n+N θ0 max{α, 0} + N + n.

α,H,m Proof of part (i). For every f ∈ BMp,q,r , it follows from Proposition 2.8 that Z ∞ √ √ √ √ M+N −t m −t dt 0 f = cm,M,N (t H) e H(t H) e Hf , in S , 0 t 1 with cm,M,N = − 2m+M+N−1(m+M+N−1)! . Thus, Z 2v+1 √ √ √ √ X M+N −t m −t dt f = cm,M,N (t H) e H(t H) e Hf 2v t v∈Z Z 2v+1 √ √ √ √ X X M+N −t m −t dt = cm,M,N (t H) e H[(t H) e Hf.χQ] . 2v t v∈Z Q∈Dv 6 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

For any v ∈ Z and Q ∈ Dv, we set √ √ −v(α−n/r) m −t sQ = 2 sup |(t H) e Hf(y)|, (3.3) (y,t)∈Q×[2v ,2v+1)

M/2 and mQ = H bQ, with

Z 2v+1 √ √ √ √ 1 M N −t m −t dt bQ = t (t H) e H[(t H) e Hf.χQ] . sQ 2v t Obviously, we have

X X 0 f = sQmQ, in S . v∈Z Q∈Dv

Thus, it remains to show that mQ is an (H, M, N, α, r) molecule. Indeed, for k = 0,..., 2M, and for any x ∈ Rn, from Proposition 2.6 we have

Z 2v+1 √ √ √ √ k/2 1 M−k N+k −t m −t dt |H bQ(x)| = t (t H) e H[(t H) e Hf.χQ] sQ 2v t Z 2v+1 Z √ √ 1 M−k m −t dt ≤ t |pt,N+k(x, y)| (t H) e Hf(y) dy s v t Q 2 Q (3.4) √ √ 1 m −t . sup (t H) e Hf(z) sQ (z,t)∈Q×[2v ,2v+1) Z 2v+1 Z N M−k t dt × t n+N dy . 2v Q (t + |x − y|) t On the other hand, it is not diﬃcult to verify that

Z N −n−N t |x − xQ| v v+1 n+N dy ≤ C(n, N) 1 + v , ∀t ∈ [2 , 2 ). (3.5) Q (t + |x − y|) 2

Combination (3.3), (3.4) and (3.5) yields

|x − x |−n−N | k/2b (x)| 2v(α+M−k−n/r) 1 + Q . H Q . 2v This implies that m is an ( , M, N, α, r) molecule. Q H √ Next, we prove (3.1). We observe that w(x, t) ≡ Hm/2e−t Hf(x) is a solution of the equation 2 −(∆x,t + |x| )w = 0, with ∆x,tw = wtt + ∆w. So, w is a subharmonic function. Thanks to [5, Lemma 5.2], for every θ ∈ (0, ∞) we obtain

Note that |Qe| ∼ 2v|Q| and t ∼ 2v, for any (y, t) ∈ Qe. Hence, it follows from the last inequality that 9 v+1 √ √ Z 8 2 Z √ √ 1/θ m −t 1 m −t θ dt sup |(t H) e Hf(y)| . |(t H) e Hf(y)| dy |Q| 3 v 3 t (y,t)∈Qe 4 2 2 Q 9 v+1 Z 8 2 √ √ 1/θ m −t θ dt . [Mθ(|(t H) e Hf|)(x)] , 3 v t 4 2 (3.6) for any x ∈ Q. From (3.3) and (3.6), we obtain

9 v+1 Z 8 2 √ √ 1/θ −v(α−n/r) m −t θ dt |sQ|χQ(x) . 2 [Mθ(|(t H) e Hf|)(x)] χQ(x), 3 v t 4 2 or 9 v+1 Z 8 2 √ √ 1/θ X −1/r −vα m −t θ dt |Q| |sQ|χQ(x) . 2 [Mθ(|(t H) e Hf|)(x)] . 3 2v t Q∈Dv 4 Thanks to Lemma 2.5, we have 9 2v+1 Z 8 √ √ dt1/θ −vα m −t H θ Av . 2 [Mθ(|(t H) e f|)] Mr . 3 v t p 4 2 Next, Minkowski integral inequality (see (2.3)) yields 9 2v+1 h Z 8 √ √ dti1/θ −vα m −t H θ Av 2 k θ |(t ) e f| k r . . M H Mp 3 v t 4 2 r At the moment, for a ﬁxed θ ∈ (0, θ0), then Mθ is a bounded operator on Mp, likewise 9 2v+1 h Z 8 √ √ dti1/θ −vα m −t H θ Av 2 k(t ) e fk r . H Mp 3 v t 4 2 9 v+1 Z 8 2 √ √ θ 1/θ h −α m −t dti t k(t ) e Hfk r . H Mp 3 v t 4 2 9 v+1 Z 8 2 √ √ q 1/q h −α m −t dti t k(t ) e Hfk r , . H Mp 3 v t 4 2 where the last inequality is obtained by using H¨older’sinequality. Therefore, 9 v+1 1/q Z 8 2 √ √ q 1/q X q h X −α m −t dti A t k(t ) e Hfk r . v . H Mp 3 2v t v∈Z v∈Z 4 P By noting that χ 3 v 9 v+1 ≤ 2, we obtain v∈Z ( 4 2 , 8 2 ) 9 v+1 Z 8 2 √ √ q Z ∞ √ √ q X −α m −t dt −α m −t dt t k(t ) e Hfk r ≤ 2 t k(t ) e Hfk r , H Mp H Mp 3 2v t 0 t v∈Z 4 which implies 1/q h Z ∞ √ √ q dti1/q X q −α m −t H A t k(t ) e fkMr = kfk α,H,m . v . H p BMp,q,r 0 t v∈Z This completes the proof of part (i). 8 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

To prove (ii) of Theorem 3.6, we need the following auxiliary lemmas.

Lemma 3.7. Let N > 0, and let η, v ∈ Z be such that v ≤ η. Let {fQ}Q∈Dv be a sequence of functions satisfying

−η −n−N |fQ(x)| . 1 + 2 |x − xQ| . n Then, for any θ ∈ ( n+N , ∞) and for a sequence of numbers {sQ}Q∈Dv , we have X (η−v)n X |sQ||fQ(x)| . 2 θ Mθ |sQ|χQ (x). Q∈Dv Q∈Dv The proof of the above lemma can be found in [7, p.147]. Next, we recall [1, Lemma 3.6]. Lemma 3.8. Under the assumptions as in (ii) of Theorem 3.6, we have

√ √ m−N−n −n−N m −t α − 1 t |x − xQ| v |(t ) e Hm (x)| |Q| n r 1 + , ∀t < 2 , H Q . 2v 2v √ √ v M −n−N m −t α − 1 2 |x − xQ| v |(t ) e Hm (x)| |Q| n r 1 + , ∀t ≥ 2 . H Q . t t Proof of part (ii) of Theorem 3.6. We begin by writing

k+1 Z 2 √ √ q dt q X −α X X m −t H kfk α, ,m = t k sQ(t ) e mQkMr BM H H p p,q,r 2k t k∈Z v∈Z Q∈Dv √ √ q X −kα X X m −t 2 k |s | sup |(t ) e Hm |k r . Q H Q Mp t∈[2k,2k+1) k∈Z v>k Q∈Dv √ √ q X −kα X X m −t + 2 k |s | sup |(t ) e Hm |k r Q H Q Mp t∈[2k,2k+1) k∈Z v≤k Q∈Dv := I1 + I2. Thus, the proof is complete if we can demonstrate that

X q I1,I2 . Av. (3.7) v∈Z n We ﬁrst prove (3.7) for I1. Keep in mind that v ≥ k+1 in this case. Since θ0 > n+N n n and M > max{ − α, m}, we can choose a real number θ ∈ ( , θ0) such that θ0 n+N n v k+1 M > θ − α. By noting that 2 ≥ 2 > t, Lemma 3.8 implies √ √ m −t α − 1 (k−v)(m−N−n) −v −n−N sup |(t H) e HmQ(x)| . |Q| n r 2 1 + 2 |x − xQ| . t∈[2k,2k+1) Thus, √ √ X m −t |sQ| sup |(t H) e HmQ(x)| t∈[2k,2k+1) Q∈Dv X α − 1 (k−v)(m−N−n) −v −n−N . |Q| n r 2 |sQ| 1 + 2 |x − xQ| (3.8) Q∈Dv vα (k−v)(m−N−n) X −1/r −v −n−N . 2 2 |Q| |sQ| 1 + 2 |x − xQ| . Q∈Dv EJDE-2018/187 BESOV-MORREY SPACES 9

−v −n−N Now, we apply Lemma 3.7 with η = v and fQ(x) = (1 + 2 |x − xQ|) to obtain X −1/r −v −n−N X −1/r |Q| |sQ| 1 + 2 |x − xQ| . Mθ |Q| |sQ|χQ (x), (3.9) Q∈Dv Q∈Dv n for θ ∈ ( n+N , θ0). Inserting (3.9) into (3.8) yields √ √ X m −t |sQ| sup |(t H) e HmQ(x)| t∈[2k,2k+1) Q∈Dv vα (k−v)(m−N−n) X −1/r . 2 2 Mθ |Q| |sQ|χQ (x). Q∈Dv Then q X h −kα X αv (k−v)(m−N−n) X −1/r i I1 2 2 2 Mθ |Q| |sQ|χQ r . Mp k∈Z v>k Q∈Dv X X (k−v)(m−N−n−α) X −1/r q = 2 Mθ |Q| |sQ|χQ r Mp (3.10) k∈Z v>k Q∈Dv q X h X (k−v)(m−N−n−α) X −1/r i 2 Mθ |Q| |sQ|χQ r . . Mp k∈Z v>k Q∈Dv r Again the fact that Mθ is bounded on Mp implies

X −1/r X −1/r Mθ |Q| |sQ|χQ r k |Q| |sQ|χQkMr ∼ Av. (3.11) Mp . p Q∈Dv Q∈Dv Combination (3.10) and (3.11) yields q X h X (k−v)(m−N−n−α) i I1 . 2 Av . k∈Z v>k Applying Young’s inequality yields

X (k−v)(m−N−n−α) 2 Av v>k q−1 1/q X (k−v)(m−N−n−α)q q X (k−v)(m−N−n−α)q 2(q−1) 2 q ≤ 2 2 Av . v>k v>k

(k−v)(m−N−n−α)q P 2(q−1) Since m > N + n + α, v>k 2 is then bounded by a constant independent of k, v. Thus, X X (k−v)(m−N−n−α)q 2 q I1 . 2 Av k∈Z v>k ! X X (k−v)(m−N−n−α)q X 2 q q = 2 Av . Av. v∈Z k

It remains to show that estimate (3.7) holds for I2. Actually, the proof for I2 is most likely to the one for I1, with only one diﬀerent point that we use Lemma 3.8 for v ≤ k, i.e. √ −n−N √ α 1 |x − xQ| m −t H n − r (v−k)M sup |(t H) e mQ(x) . |Q| 2 1 + v . t∈[2k,2k+1) 2 10 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

Proceed similarly to the proof (from (3.8) to (3.11)) above, we obtain q X h X (v−k)(M+α) i I2 . 2 Av . k∈Z v≤k By noting that M + α > 0, apply Young’s inequality yields the result. This completes the proof of Theorem 3.6. 1 n Proof of Theorem 3.3. Let N = int[n( − 1)] + 1, and M > max{m1, m2, − α}. θ0 θ0 α,H,m1 Because m1 and m2 play the same role, it then suﬃces to prove that BMp,q ,→ α,H,m2 BMp,q . α,H,m1 In fact, for f ∈ BMp,q,r , thanks to (i) of Theorem 3.6, there exist a sequence n o of (H, M, N, α, r) molecules mQ : Q ∈ Dv, v ∈ Z , and a sequence of coeﬃcients n o sQ : Q ∈ Dv, v ∈ Z so that

X X 0 f = sQmQ, in S , v∈Z Q∈Dv and 1/q X q A kfk α,H,m1 . v . BMp,q,r v∈Z In other words, (P Aq )1/q is ﬁnite. v∈Z v α,H,m2 By (ii) of Theorem 3.6, we obtain f ∈ BMp,q,r . Furthermore, f satisﬁes 1/q X q kfk α,H,m2 A . BMp,q,r . v v∈Z Or, we obtain the result.

4. Regularity on Besov-Morrey spaces for fractional Hermite equations In this part, we study the regularity results of solutions of the two fractional Hermite equations: s s n H u = f, and (I + H) = f, on R , α,H for any s > 0, and for f ∈ BMp,q,r. To solve the indicated equations, it is necessary to investigate the operators H−s and (I + H)−s, named by the Riesz potential of Hermite operator and the Bessel potential of Hermite operator respectively. In fact, by following [1, Proposition 2.5], we can deﬁne the operators H−s : S0 → S0 and (I + H)−s : S0 → S0 by setting −s −s −s −s hH f, φi = hf, H φi, and h(I + H) f, φi = hf, (I + H) φi, for any f ∈ S0, and for φ ∈ S. Note that h·, ·i is the pair between a linear function in S0 and a function in S. Moreover, for any φ ∈ S we have Z ∞ −s 1 s −t dt H φ = t e Hφ ∈ S, Γ(s) 0 t Z ∞ −s 1 s −t −t dt (I + H) φ = t e e Hφ ∈ S. Γ(s) 0 t EJDE-2018/187 BESOV-MORREY SPACES 11

−t k −t Let Kt(x, y) (resp. Kt,k(x, y)) be the kernel of e H (resp. (tH) e H). Thanks to [6, Lemma 2.5], and [1, Lemma 2.4], we have the following results.

Lemma 4.1. For k ∈ N, there exists c, C > 0 so that for all y ∈ R ( k+1 |x−y|2 − 2 k Ct exp − c t , 0 < t ≤ 1; |∂ Kt(x, y)| ≤ 2 (4.1) x e−te−|x−y| , t > 1. C |x − y|2 K (x, y) ≤ exp − c , (4.2) t,k tn/2 t Our regularity results are as follows.

α,H Theorem 4.2. Let α ∈ R, 0 < q ≤ ∞, 0 < p ≤ r ≤ ∞, and f ∈ BMp,q,r. Assume that u is a solution of equation Hsu = f, Then, there exists a constant C > 0 such that

kuk α+2s,H ≤ Ckfk α,H . BMp,q,r BMp,q,r

α,H Theorem 4.3. Let α ∈ R, 0 < q ≤ ∞, 0 < p ≤ r ≤ ∞, and f ∈ BMp,q,r. Assume that u is a solution of equation (H + I)su = f. Then, there exists a constant C > 0 such that

kuk α+2s,H ≤ Ckfk α,H . BMp,q,r BMp,q,r Theorems 4.2 and 4.3 are just a consequence of the theorem below.

Theorem 4.4. Let α ∈ R, 0 < p ≤ r < ∞, and 0 < q ≤ ∞. For any s > 0, the −s −s α,H α+2s,H operator H (resp. (I + H) ) is bounded from BMp,q,r to BMp,q,r .

Proof of Theorem 4.4. Let {mQ : Q ∈ Dv, v ∈ Z} be a sequence of (H, 4M, N, α, r) molecules, with M,N ∈ N, and M > s + n/2 + N/2. −s We first prove that H (mQ) is an (H, 2M, N, α+2s, r) molecule associated with 2M −s M the cube Q. Indeed, let mQ = H bQ as in Definition 3.4, and put yQ = H H bQ. Then √ −s M 2M H mQ = H yQ = ( H) yQ. Thus, it suffices to show that √ −n−N k 2M−k α+2s − 1 |x − xQ| |( ) y (x)| `(Q) |Q| n r 1 + , (4.3) H Q . `(Q) for k = 0,..., 4M. In fact, we have Z ∞ −s M 1 s −t M dt yQ(x) = H H bQ = t e HH bQ(x) . Γ(s) 0 t Therefore, √ Z 4v √ k 1 s −t 2M+k dt |( H) yQ(x)| ≤ |t e H( H) bQ(x)| Γ(s) 0 t Z ∞ √ 1 s −t 2M+k dt + |t e H( H) bQ(x)| := I1 + I2. Γ(s) 4v t First, we estimate I1. Thanks to Lemma 4.1, we have √ Z √ −t 2M+k 2M+k |e H( H) bQ(x)| = |Kt(x, y)( H) bQ(y)|dy n R Z 1 |x − y|2 √ exp − 2c |( )2M+kb (y)|dy. . n/2 H Q n t t R 12 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

Taking Deﬁnition 3.4 into account, we obtain √ Z 1 |x − y|2 |x − y|−n−N |e−tH( )2M+kb (x)| exp − c 1 + √ H Q . n/2 n t t R t −n−N α − 1 v(2M−k) |y − xQ| × |Q| n r 2 1 + dy. 2v Next, we apply the inequality (1 + a + b) ≤ (1 + a)(1 + b), for all a, b ≥ 0 and the fact t < 4v to the right hand side of the above inequality to obtain √ −t 2M+k |e H( H) bQ(x)| −n−N Z 2 α − 1 v(2M−k) |x − xQ| 1 |x − y| |Q| n r 2 1 + exp − c dy; . v n/2 2 n t t R thus √ −n−N −t 2M+k α − 1 v(2M−k) |x − xQ| |e H( ) b (x)| |Q| n r 2 1 + . H Q . 2v This implies v −n−N Z 4 α 1 |x − xQ| dt n − r v(2M−k) s I1 . |Q| 2 1 + v t 2 0 t (4.4) −n−N α+2s − 1 v(2M−k) |x − xQ| |Q| n r 2 1 + . . 2v It remains to consider I2. By (4.2), we have √ √ M −t k −M M −t k |H e H( H) bQ(x)| = t |(tH) e H( H) bQ(x)| Z √ −M k = t |Kt,M (x, y)( H) bQ(y)|dy n R Z 1 |x − y|2 √ t−M exp −c |( )kb (y)|dy. . n/2 H Q n t t R In similar to the above proof, we also have √ M −t k |H e H( H) bQ(x)| Z −n−N −n−N −M 1 |x − y| α − 1 v(4M−k) |y − xQ| t 1 + √ |Q| n r 2 1 + dy. . n/2 v n t 2 R t By Lemma 3.5, and noting that t ≥ 4v, we obtain √ M −t k |H e H( H) bQ(x)| −n−N −M α − 1 v(4M−k) |x − xQ| . t |Q| n r 2 1 + √ t −n−N t (n+N)/2 −M α − 1 v(4M−k) |x − xQ| t |Q| n r 2 1 + . . 4v 2v Thus −n−N Z ∞ α 1 |x − xQ| n+N dt n − r v(4M−k−N−n) s+ 2 −M I2 . |Q| 2 1 + v t 2 v t 4 (4.5) −n−N α+2s − 1 v(2M−k) |x − xQ| |Q| n r 2 1 + . . 2v −s Hence, (4.3) follows from (4.4) and (4.5). Thus H (mQ) is an (H, 2M, N, α+2s, r) molecule associated with the cube Q. By Theorem 3.6 and a suitable choice of M,N, −s α,H α+2s,H we obtain the boundedness of H from BMp,q,r to BMp,q,r . EJDE-2018/187 BESOV-MORREY SPACES 13

Similarly, we can also establish the boundedness of the Bessel potential (I +H)−s α,H α+2s,H from BMp,q,r to BMp,q,r . We leave the proof to the reader.

Acknowledgments. We would like to thank the anonymous referees for their valuable comments.

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Nguyen Anh Dao Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, HoChiMinh City, Vietnam E-mail address: [email protected] 14 N. A. DAO, N. N. TRONG, L. X. TRUONG EJDE-2018/187

Nguyen Ngoc Trong (corresponding author) Faculty of Mathematics and Computer Science, VUNHCM - University of Science, HoChiMinh city, Vietnam. Department of Primary Education, HoChiMinh City University of Education, Vietnam E-mail address: [email protected]

Le Xuan Truong Department of Mathematics and Statistics, University of Economics HoChiMinh City, Vietnam E-mail address: [email protected]