Pseudodifferential Operators on Weighted Hardy Spaces

We study two sufficient conditions for the boundedness of a class of pseudodifferential operators T with symbols in the Hölmander class Smρ,δ(R ) on weighted Hardy spaces Hω(R ), where ω belongs to Muckenhoupt class Ap. /e first one is an estimate from Hω(R ) into Lω(R ). We get a better range of admissible p and m. /e second one is a weighted version bounded for the operators T on Hω(R ), and it is an addition to the literature.


Introduction
e purpose of this paper is to study some sufficient conditions for the boundedness of pseudodifferential operators T on weighted Hardy space H 1 ω (R n ), where the operators T have symbols in the Hölmander class S m ρ,δ (R n ). As in [1], for m ∈ R and ρ, δ ∈ [0, 1], a symbol a(x, ξ) ∈ S m ρ,δ (R n ) is a smooth function defined on R n × R n such that holds for all multi-indices α, β ∈ N n , where C α,β is independent of x and ξ. We now assume that the symbol a(x, ξ) is smooth in both the spatial variable x and the frequency variable ξ.
Given f ∈ C ∞ 0 (R n ), the pseudodifferential operator T ∈ L m ρ,δ associated with the symbol a(x, ξ) ∈ S m ρ,δ (R n ) is given by where f denotes the Fourier transform of f. Moreover, we can express T by a kernel K(x, y) as (see, e.g., [2]) Tf(x) � K(x, y)f(y)dy.
Pseudodifferential operators play an important role in the theory of partial differential equations. It is well known that the Hardy spaces H p (R n ) coincide with the Lebesgue spaces L p (R n ) when p > 1.
e L p and weighted L p boundedness of the operator T ∈ L m ρ,δ have been extensively studied. We refer to [1,2,3,4] for the L p bounds and [5,6,7,8] for the weighted L p bounds.
For p ∈ (0, 1], there is an estimate from L 1 (R n ) into weak L 1 (R n ) for the pseudodifferential operator T ∈ L m ρ,δ (cf. [5]). As known, the Hardy space H p (R n ) is an advantageous substitute for L p (R n ). e behavior of the pseudodifferential operator T on H p (R n ) has attracted a lot of interest. For example, Alvarez and Hounie [5] have found that the pseudodifferential operator T with symbol in Hounie and Kapp [9] have shown that the operator T with 0 ≤ δ ≤ ρ < 1 and m � − (n(1 − ρ))/2 is bounded from the local Hardy space h 1 (R n ) into L 1 (R n ). Yabuta [10] has proved the operator T involving a modulus of continuity w(t) is bounded from e bounds of the pseudodifferential operator T from the weighted Hardy space H 1 ω (R n ) into the weighted Lebesgue space L 1 ω (R n ) have also been studied. Yabuta [11] has found that the operator T is bounded from 1,δ and ω ∈ A 1 . In view of this, it is natural to look for a wide range of operator T in L m ρ,δ to study the bounds on the weighted Hardy space H 1 ω (R n ).
e second one is an estimate on weighted Hardy spaces H 1 ω (R n ) for the pseudodifferential operator T. It is well known that under certain conditions of m, ρ, δ, the operator T is bounded on h 1 (R n ) (cf. [9,12]). Alvarez and Hounie [5] have found that the pseudodifferential operator T is bounded on It is natural to look for a weighted version estimate on H 1 ω (R n ). We now state our second main result.
e remainder of this paper is organized as follows. In Section 2, we present some definitions and well-known results we use later. e aim of Section 3 is to set up the estimate from H 1 Proposition 1). e aim of Section 4 is to establish the estimate on weighted Hardy spaces H 1 ω (R n ) for pseudodifferential operators T in L m ρ,δ . Most of the notations we use are standard. C denotes a constant that may change from line to line and we write a ≲ b as shorthand for a ≤ Cb. If a ≲ b and b ≲ a, we mean a ∼ b. For a measurable set A, |A| denotes the Lebesgue measure of A and χ A the characteristic function. B will always denote a ball, and tB(t > 0) denotes the ball B dilated by t.

Notations and Auxiliary Lemma
In this section, we first present an auxiliary lemma about the pseudodifferential operator T associated with the kernel K(x, y). Let S(R n ) be the class of Schwartz functions and S ′ (R n ) be its dual space. e space of C ∞ -function with compact support is denoted by C ∞ 0 (R n ). Pseudodifferential operators are bounded from S(R n ) to S(R n ) and so possess distribution kernels K(x, y) ∈ S ′ (R n × R n ). en, the following formula for the kernel is useful (cf. Proposition 3.1 in [9]; see also [5]).
, and associate with the pseudodifferential operator T ∈ L m ρ,δ . en, the distribution kernel K(x, y) of T is smooth away from the diagonal (x, x): x ∈ R n { } and is given by where ψ ∈ C ∞ 0 (R n ) satisfies ψ(ξ) � 1 for |ξ| ≤ 1 and the limit is taken in S ′ (R n ) and independent of the choice of ψ. If Moreover, for any multi-index α, β ∈ N n and N ∈ N, e following useful L p ω bound for the pseudodifferential operator T is obtained by Michalowski et al. [7].
Remark 1. Obviously, the L 2 ω bounds of pseudodifferential operators T are established automatically. e following useful L 2 bound of the pseudodifferential operator T ∈ L m ρ,δ is obtained by Alvarez and Hounie [5].

Remark 3.
e range of m, p here is m < − n(1 − ρ)| (1/p) − (1/2)| + min 0, (n(ρ − δ)/2) and 1 < p < ∞, respectively. Let p ∈ [1, ∞). A nonnegative locally integrable function ω belongs to Muckenhoupt class A p , if there exists a constant C > 0, such that for all balls B ⊂ R n , We denote A ∞ � ∪ p≥1 A p . It is well known that ω ∈ A p implies ω ∈ A q for all q > p. Also, if ω ∈ A p , then ω ∈ A q for some q ∈ [1, p). We thus write q ω � inf p ≥ 1: ω ∈ A p to denote the critical index of ω. For a measurable set E, Journal of Function Spaces following lemma provides a way to compare |E| and ω(E) of a set E (see [13]).

Lemma 4.
Let ω ∈ A p and p ≥ 1. en, there exists a constant C > 0 such that for all balls B and measurable subsets E ⊂ B.
Given a weight function ω on R n , we denote by L p ω (R n ) the weighted Lebesgue space of all functions f satisfying Analogous to the classical Hardy space, the weighted Hardy space H 1 ω (R n ) can be defined in terms of maximal functions.

Definition 2.
Let ω be a weight with the critical index q ω . An (1, ∞)-atom with respect to ω is a function a satisfying and a(x)x α dx � 0 for every multi-index α with |α| ≤ [n(q ω − 1)].
e Hardy space H 1 ω (R n ) is a linear space spanned by all of (1, ∞)-atoms with respect to ω. Namely, f ∈ H 1 ω (R n ) if and only if f can be written as (see [13]) in the sense of S ′ , where each a j is an (1, ∞)-atom with respect to ω and λ j satisfies Definition 3. Let T be a pseudodifferential operator in L m ρ,δ . We say T * 1 � 0 if R n Ta(x)dx � 0 for all a ∈ L ∞ (R n ) with compact support and R n a(x)dx � 0.

The Proof of Theorem 1
In this section, we prove that the pseudodifferential oper-
Proof. Inspired by the proof of Lemma 3.2 in [15], we consider two cases about the radius r.
Proof. e proof of eorem 1 is motivated by the atomic decomposition for . We obtain an atomic decomposition of f satisfying (15) and (16). So, to prove that the pseudodifferential operators T are bounded from , it suffices to show that for each (1, ∞)-atom a with respect to ω, we have Ta L 1 ω (R n ) ≤ C. Recall that an (1, ∞)-atom a with respect to ω is a function satisfying supp(a) ⊂ B, for some ball B � B(x 0 , r). Now, let a be such an atom and write |Ta|ω � 2B |Ta|ω + (2B) c |Ta|ω � I 1 + I 2 . (28) It is easy to estimate the term I 1 . Using Hölder inequality and L 2 ω -boundedness for the pseudodifferential operator T (see Remark 1), we get where C is independent of a.

The Proof of Theorem 2
In this section, we establish the weighted norm inequality on weighted Hardy spaces H 1 ω (R n ) for pseudodifferential operators T in L m ρ,δ .
To estimate I 2 , we first estimate (Ta) * (x) for |x − x 0 | > 4r. For any t > 0, since T * 1 � 0(see Definition 3), we have For the term E 1 , by the mean value theorem and Hölder's inequality, we have