Boundedness of vector-valued intrinsic square functions in Morrey type spaces

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spaces $L^{p,\kappa}(w)$ when $1\leq p<\infty$, $0<\kappa<1$, and in the generalized Morrey spaces $L^{p,\Phi}$ for $1\le p<\infty$, where $\Phi$ is a growth function on $(0,\infty)$ satisfying the doubling condition.


Introduction and main results
The intrinsic square functions were first introduced by Wilson in [17,18]; they are defined as follows. For 0 < α ≤ 1, let C α be the family of functions ϕ defined on R n such that ϕ has support containing in {x ∈ R n : |x| ≤ 1}, R n ϕ(x) dx = 0, and for all x, x ′ ∈ R n , For (y, t) ∈ R n+1 + = R n × (0, ∞) and f ∈ L 1 loc (R n ), we set Then we define the intrinsic square function of f (of order α) by the formula A α (f )(y, t) 2 dydt t n+1 where Γ(x) denotes the usual cone of aperture one: Γ(x) = (y, t) ∈ R n+1 + : |x − y| < t .
Let f = (f 1 , f 2 , . . .) be a sequence of locally integrable functions on R n . For any x ∈ R n , Wilson [18] also defined the vector-valued intrinsic square functions of f by In [18], Wilson has established the following two theorems.
If we take w ∈ A 1 , then M (w)(x) ≤ C · w(x) for a.e.x ∈ R n by the definition of A 1 weights (see Section 2). Hence, as a straightforward consequence of TheoremB, we obtain Theorem B. Let 0 < α ≤ 1, p = 1 and w ∈ A 1 . Then there exists a constant In particular, if we take w to be a constant function, then we immediately get the following Theorem C. Let 0 < α ≤ 1 and 1 < p < ∞. Then there exists a constant Theorem D. Let 0 < α ≤ 1 and p = 1. Then there exists a constant C > 0 On the other hand, the classical Morrey spaces L p,λ were originally introduced by Morrey in [10] to study the local behavior of solutions to second order elliptic partial differential equations. Since then, these spaces play an important role in studying the regularity of solutions to partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [1,2,13]. In [9], Mizuhara introduced the generalized Morrey spaces L p,Φ which was later extended and studied by many authors (see [4][5][6]8,11]). In [7], Komori and Shirai defined the weighted Morrey spaces L p,κ (w) which could be viewed as an extension of weighted Lebesgue spaces, and then discussed the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [14][15][16], we have established the strong type and weak type estimates for intrinsic square functions on L p,Φ and L p,κ (w).

Notations and definitions 2.1 Generalized Morrey spaces
Let Φ = Φ(r), r > 0, be a growth function, that is, a positive increasing function in (0, ∞) and satisfy the following doubling condition.
Obviously, when Φ(r) = r λ with 0 < λ < n, L p,Φ is just the classical Morrey spaces introduced in [10]. We also denote by W L 1,Φ = W L 1,Φ (R n ) the generalized weak Morrey spaces of all measurable functions f for which for every x 0 ∈ R n and all r > 0. The smallest constant C > 0 satisfying (2.3) is also denoted by f W L 1,Φ .

Weighted Morrey spaces
A weight w is a nonnegative, locally integrable function on R n , B = B(x 0 , r B ) denotes the ball with the center x 0 and radius r B . For 1 < p < ∞, a weight function w is said to belong to A p , if there is a constant C > 0 such that for every ball B ⊆ R n , It is well known that if w ∈ A p with 1 ≤ p < ∞, then for any ball B, there exists an absolute constant C > 0 such that Moreover, if w ∈ A ∞ , then for all balls B and all measurable subsets E of B, there exists δ > 0 such that Given a ball B and λ > 0, λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by |E| and the weighted measure Given a weight function w on R n , for 1 ≤ p < ∞, the weighted Lebesgue space L p w (R n ) defined as the set of all functions f such that We also denote by W L 1 w (R n ) the weighted weak space consisting of all measurable functions f such that (2.7) In particular, for w equals to a constant function, we shall denote L p w (R n ) and W L 1 w (R n ) simply by L p (R n ) and W L 1 (R n ).

Definition 2.2 ([7]
). Let 1 ≤ p < ∞, 0 < κ < 1 and w be a weight function on R n . Then the weighted Morrey space is defined by and the supremum is taken over all balls B in R n .
For p = 1 and 0 < κ < 1, we also denote by W L 1,κ (w) the weighted weak Morrey spaces of all measurable functions f satisfying (2.9) Throughout this paper, the letter C always denote a positive constant independent of the main parameters involved, but it may be different from line to line.
Let us now turn to estimate the other term I 2 . For any ϕ ∈ C α , 0 < α ≤ 1, j = 1, 2, . . . and (y, t) ∈ Γ(x), we have For any x ∈ B, (y, t) ∈ Γ(x) and z ∈ 2 ℓ+1 B\2 ℓ B ∩ B(y, t), then by a direct computation, we can easily see that Thus, by using the above inequalities (3.1) and (3.2) together with Minkowski's inequality for integrals, we deduce Then by duality and Cauchy-Schwarz inequality, we get Furthermore, it follows from Hölder's inequality, (3.3) and the A p condition that where we denote the conjugate exponent of p > 1 by p ′ = p/(p − 1). Since w ∈ A p ⊂ A ∞ for all 1 < p < ∞. Hence, we apply the inequality (2.5) to obtain where the last series is convergent since 0 < κ < 1 and δ > 0. Summarizing the above two estimates for I 1 and I 2 , and then taking the supremum over all balls B ⊆ R n , we complete the proof of Theorem 1.1.

Proof of Theorem 1.2. Let
. . . Then for any given λ > 0, one writes Theorem B and the inequality (2.4) imply that .
We now turn to deal with the other term I ′ 2 . In the proof of Theorem 1.1, we have already showed that for any x ∈ B (see (3.3)), It follows directly from the A 1 condition that In addition, since w ∈ A 1 ⊂ A ∞ , then by the inequality (2.5), we can see that where in the last inequality we have used the fact that δ * · (1 − κ) > 0. If > λ/2 = ∅, then the inequality holds trivially. Now if instead we suppose that Then by the pointwise inequality (3.4), we have which is equivalent to . Therefore .
Summing up the above estimates for I ′ 1 and I ′ 2 , and then taking the supremum over all balls B ⊆ R n and all λ > 0, we finish the proof of Theorem 1.2.

Proofs of Theorems 1.3 and 1.4
Proof of Theorem 1.3. Let Applying Theorem C and the doubling condition (2.1), we obtain We now turn to estimate the other term J 2 . We first use the inequality (3.3) and Hölder's inequality to obtain Since 1 ≤ D(Φ) < 2 n , then by using the doubling condition (2.1) of Φ, we know that Combining the above estimates for J 1 and J 2 , and then taking the supremum over all balls B = B(x 0 , r) ⊆ R n , we complete the proof of Theorem 1.3.
Note that 1 ≤ D(Φ) < 2 n . Arguing as in the proof of (4.1), we can get Hence, for any x ∈ B(x 0 , r), > λ/2 = ∅, then the inequality holds trivially. Now we may suppose that Then by the pointwise inequality (4.3), we can see that which is equivalent to Summing up the above estimates for J ′ 1 and J ′ 2 , and then taking the supremum over all balls B = B(x 0 , r) ⊆ R n and all λ > 0, we conclude the proof of Theorem 1.4.