Boundedness of Rough Integral Operators on Triebel-lizorkin Spaces

We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results.


Introduction
Throughout this paper, we let p denote the dual exponent to p defined by 1/p + 1/p = 1.Let n ≥ 2 and S n−1 represent the unit sphere in R n equipped with the normalized Lebesgue measure dσ = dσ(•).Let K be a kernel of Calderón-Zygmund type on R n given by where Ω is a homogeneous function of degree 0, integrable over S n−1 , and satisfies (1.1) It is well-known that the Triebel-Lizorkin space Ḟ β,q p (R n ) is a unified setting of many well-known function spaces including Lebesgue spaces L p (R n ), the Hardy spaces H p (R n ) and the Sobolev spaces L β p (R n ).Our main focus as the title of the paper suggests will be on studying the boundedness of three types of rough integral operators on Triebel-Lizorkin spaces.We start with the first type which concerns the homogeneous Calderón-Zygmund singular integral operator T Ω given by where f ∈ S(R n ), the space of Schwartz functions.
The investigation of the L p boundedness of T Ω was pioneered by Calderón and Zygmund in [3] and then continued by many authors.In [3], Calderón and Zygmund showed that the L p (1 < p < ∞) boundedness of T Ω holds if Ω ∈ L log L S n−1 and that this condition is essentially the weakest possible size condition on Ω for the L p (1 < p < ∞) boundedness of T Ω to hold.For endpoint results, A. Seeger in [20] proved that T Ω is of weak-type (1, 1) under the same L log L S n−1 condition, but in general T Ω with such an Ω is not bounded on H 1 (R n ) , as pointed out by M. Christ (see [21]).On the other hand, Connett [10] and Coifman and Weiss [9] independently showed that Here, H 1 S n−1 is the Hardy space on the unit sphere which contains the space L log L S n−1 as a proper space.In [13], Grafakos and Stefanov introduced the following condition: (1.2) sup and showed that it implies the L p boundedness of T Ω for p in a range dependent on the positive exponent α.For any α > 0, we let G α (S n−1 ) denote the family of Ω's which are integrable over S n−1 and satisfy (1.2).
This range of p was later improved to be ( 2+2α 1+2α , 2 + 2α) (see [11]).However, it is still unknown whether the latter range of p is sharp.We point out that Grafakos and Stefanov in [13] showed that In recent years, the investigation of boundedness of T Ω on Triebel-Lizorkin space Ḟ β,q p (R n ) has attracted the attention of many authors.For relevant results one may consult [5], [16], [6], [7], among others.For example, J. Chen and C. Zhang in [7] (see also [24]) proved the following: Theorem B. Suppose that Ω satisfies (1.1) and Ω ∈ G α (S n−1 ) for all α > 1.Then the operator T Ω is bounded on Ḟ β,q p (R n ) for all 1 < p, q < ∞ and β ∈ R.
We notice that the condition imposed on Ω is that Ω ∈ G α (S n−1 ) for all α > 1.The question that arises naturally is whether the operator T Ω is bounded on Ḟ β,q p (R n ) if Ω ∈ G α (S n−1 ) for some α > 0. We shall show that the condition Ω ∈ G α (S n−1 ) for all α > 1 is not necessary and we just need Ω ∈ G α (S n−1 ) for some α > 0. The condition that α > 1 is not necessary as described in the following theorem.
The question concerning the boundedness of T Ω when Ω ∈ H 1 S n−1 , which is separate from the problem addressed in Theorems A and B in light of (1.4), had been answered by Y. Chen and Y. Ding in [8].
The second type of our operators concerns a certain class of oscillatory singular integral operators.To state our second result, we need some preparation.Let P(n; m) denote the set of polynomials on R n which have real coefficients and degrees not exceeding m, and let H(n; m) denote the collection of polynomials in P(n; m) which are homogeneous of degree m.Also, let P(n; m, 0) be the class of all P ∈ P(n; m) with ∇P (0) = 0.For P (x) = |η|≤m a η x η , we set P = |η|≤m |a η | .Let n ≥ 2, m ∈ N and α > 0. An integrable function Ω on S n−1 is said to be in the space A(n; m; α) if (1.5) sup It was noted in [2] that A(n; 1; α) = G α (S n−1 ) and in the case n = 2, ∞ m=1 A(2; m; α) = G α (S 1 ).
For P ∈ P(n; d), let T Ω,P be the oscillatory singular integral operator defined by (1.6) T Ω,P f (x) = p.v.
While technically Theorem 1.1 can be subsumed in Theorem 1.2, we will first prove Theorem 1.1 and then use it in the proof of Theorem 1.2.
where C is a positive constant that depends on β, p, q, but not on d.
Throughout this paper, the letter C will stand for a positive constant that may vary at each occurrence.However, C does not depend on any of the essential variables.

Some definitions and lemmas
Now we recall the definition of the Triebel-Lizorkin spaces.Fix a radial Schwartz function It is well-known that S(R n ) is dense in Ḟ β,q p (R n ) and also the following hold: (1) We need the following result from [1].

Proof of main results
Before we start proving our main results we need some preparation.By the translation invariance of T Ω and T Ω,P , it suffices to establish their boundedness on the Triebel-Lizorkin spaces with β = 0. Choose a real valued, radial function 3 and for all ξ = 0, R φ2 t (ξ) and let First, it is easy to see that where is the Hardy-Littlewood maximal function of f in the direction of y.
Since M y is bounded on L p (R n ), 1 < p < ∞ with bound independent of y, by Minkowski's inequality we get Also, we shall need to study the boundedness of for any t ∈ R and for any g ∈ Ḟ 0,q p (R n ), by Hölder's inequality we have Taking supremum over g with g Ḟ 0,q p (R n ) ≤ 1 and by Hölder's inequality we have Now, since p > 1, by duality there exists a nonnegative function By the last inequality and (3.2) we have Also, by a similar argument as in the the proof of (3.2) we have By interpolation between (3.4) and (3.5) we get which when combined with (3.3) implies for any t ∈ R and 1 < p, q < ∞.
Proof of Theorem 1.1:It is easy to see that and hence we have where . By the same argument as proving (3.3) we get (3.9) .
Proof of Theorem 1.2: Let P (x) = |η|≤m a η x η with ∇P (0) = 0. Since the constant term in P (tx), if any, can be assimilated in the function f , we may assume without loss of generality that P (tx) does not have a constant term.Write P (rx) = d j=2 P j (x)r j , where P s (x) = |η|=s a η x η .We shall first consider the case d = 2 k for some k ≥ 1.The general case will be an easy consequence of this special case d = 2 k .Let m j = P j and Q be given by Q(rx) = d/2 j=2 P j (x)r j .By a dilation in r we may assume, without loss of generality, max d 2 <j≤d m j = 1 (see [6, p. 392]).Also, there is a j 0 , d 2 < j 0 ≤ d, such that m j0 = 1.It is easy to see that Decompose T Ω,P f (x) as where t 0 ∈ R is to be chosen later.We start with T 0 Ω,P (f ).We write T 0 Ω,P (f ) Ḟ 0,q p (R n ) .
By Lemma 2.1 we get By combining the last estimate with the trivial estimate By the last inequality and since (a + b) θ ≤ 2 θ−1 a θ + b θ (for θ ≥ 1 and a, b ≥ 0) we get Therefore, by (3.32)-(3.33)and by Plancherel's theorem we get (3.34)F t (f ) Ḟ 0,2 2 (R n ) ≤ C(t + 1) −(α+1) f Ḟ 0,2 2 (R n ) .As for the cases p = q and p > q, we follow the same argument as in the proof of Theorem 1.1 (in dealing with these cases the factor e iP (y) being harmless) to get (3.35)F t (f ) Ḟ 0,q p (R n ) ≤ C f Ḟ 0,q p (R n ) for p ≥ q.Now, the rest of the proof will follow by (3.30), (3.34)-(3.35)and the same argument as in the proof of (3.17).This completes the proof of Theorem 1.2.
a real polynomial of degree at most d, and let ψ ∈ C 1 [a, b].Then for any j 0 with 1 ≤ j 0 ≤ d, there exists a positive constant C independent of a, b, the coefficients of b 0 , . . ., b d and also independent of d such thatb a e ih(t) ψ(t) dt ≤ C |b j0 | dt holds for 0 < a < b ≤ 1.