Abstract
For oscillatory singular integrals with polynomial phases and Hölder class kernels, we establish their uniform boundedness on \(L^p\) spaces as well as a sharp logarithmic bound on the Hardy space \(H^1\). These results improve the ones in (Pan in Forum Math 31: 535–542, 2019) by removing the restriction that the phase polynomials be quadratic.
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1 Introduction
Let \(n, d \in \mathbb {N}\), \(x=(x_1, \ldots , x_n) \in \mathbb {R}^n\). Let
where \(\alpha = (\alpha _1, \ldots , \alpha _n)\), \(|\alpha | = \alpha _1 + \cdots + \alpha _n\), \(x^\alpha =x_{1}^{\alpha _1}\cdots x_{n}^{\alpha _n}\) and \(a_\alpha \in \mathbb {R}\). For each nonconstant polynomial P(x) we let
When \(\deg (P)=1\), the value of \(\Vert P\Vert _o\) shall be interpreted as \(\infty \).
For a Calderón–Zygmund type singular kernel K(x), let \(T_{P, K}\) be the oscillatory singular integral operator defined by
In [7] the author proved the following:
Theorem 1.1
Suppose that \(\deg (P) = 2\) and there exist \(q > 2\) and \(\delta >0\) such that
Then,
(i) For \( 1< p < \infty \), there exists a positive constant \(C_p\) such that
for all \(f \in L^p(\mathbb {R}^n)\). The constant \(C_p\) may depend on n, p, \(\delta \), q and A, but is otherwise independent of K and the coefficients of P;
(ii) There exists a positive constant C such that
for all \(f \in H^1(\mathbb {R}^n)\). The constant C may depend on n, \(\delta \), q and A, but is otherwise independent of K and the coefficients of P. The bound given in (5) is the best possible in the sense that the logarithmic function cannot be replaced by any function with a slower rate of growth.
The above conditions (a)–(c) are commonly referred to as the size, smoothness and cancellation conditions for singular kernels, respectively. In classical Calderón–Zygmund theory of singular integrals, one assumes that condition (a) holds for \(q= \infty \) together with the \(C^1\) condition \(|\nabla K(x)| \le C|x|^{-n-1}\) instead of the weaker Hölder continuity condition (b), as well as (c).
The restriction that P(x) be quadratic (i.e. \(\deg (P)=2\)) in the above theorem is clearly a severe one. For the operator \(T_{P,K}\) with a \(C^1\) kernel K, both the \(L^p\) bound in (1) and the \(H^1\) bound in (2) have been known to be true when the phase polynomial P is of arbitrary degree (for \(L^p\) see [8]; for \(H^1\) see [1]). The main purpose of this paper is to show that, for \(T_{P,K}\) with a kernel K in the Hölder class, the same \(L^p\) and \(H^1\) bounds are true when the degree of the phase polynomial P is arbitrary. We have the following:
Theorem 1.2
Let P(x) be a real-valued polynomial of any positive degree. Suppose that K(x) satisfies CZ(\(q,\delta \))(a)–(c) for some \(q > 1\) and \(\delta >0\). Then,
(i) For \( 1< p < \infty \), there exists a positive constant \(C_p\) such that
for all \(f \in L^p(\mathbb {R}^n)\). The constant \(C_p\) may depend on n, p, \(\delta \), q, \(\deg (P)\) and A, but is otherwise independent of K and the coefficients of P;
(ii) There exists a positive constant C such that
for all \(f \in H^1(\mathbb {R}^n)\). The constant C may depend on n, \(\delta \), q, \(\deg (P)\) and A, but is otherwise independent of K and the coefficients of P. The bound given in (7) is the best possible in the sense that the logarithmic function cannot be replaced by any function with a slower rate of growth.
We point out that, aside from lifting the restriction that \(\deg (P)=2\) from Theorem 1.1, Theorem 1.2 also improves the range of q from \(q > 2\) to the more natural range \(q >1\).
In the rest of the paper we shall use \(A \lesssim B\) to mean that \(A \le c B\) for a certain constant c which depends on some essential parameters only. A subscript may be added to the symbol \(\lesssim \) to indicate a particular dependence as appropriate.
2 \(L^p\) Boundedness
In this section we will establish part (i) of Theorem 1.2. For \(u \in \mathbb {R}^n\) and \(r > 0\) we let B(u, r) denote the ball \(\{x \in \mathbb {R}^n:|x - u| \le r\}\). An important tool will be the following lemma from [6].
Lemma 2.1
Let P(x) be given as in (1). Let \(R > 0\) and let \(\psi : \mathbb {R}^n \rightarrow \mathbb {C}\) be an integrable function supported in B(0, R/2). Then
where \(\Lambda := \sum _{1 \le |\alpha | \le d} |a_\alpha | R^{|\alpha |}\).
For \(h > 0\), we let \(T_{P, K, h}\) denote the truncation of \(T_{P, K}\) given by
We will establish the following uniform \(L^p\) boundedness theorem:
Theorem 2.1
Let P(x) be a real-valued polynomial of any degree and \(h > 0\). Suppose that K(x) satisfies CZ(\(q,\delta \))(a)–(c) for some \(q > 1\) and \(\delta >0\). Then, for \( 1< p < \infty \), there exists a positive constant \(C_p\) such that
for all \(f \in L^p(\mathbb {R}^n)\). The constant \(C_p\) may depend on n, p, \(\delta \), q, \(\deg (P)\) and A, but is otherwise independent of h, K and the coefficients of P.
Proof
Without loss of generality we may assume that P(x) is nonconstant and \(P(0)=0\). In order to prove (10) we shall use induction on \(\deg (P)\). When \(\deg (P)=1\), by \(P(x-y) = P(x)-P(y)\), (10) follows from the \(L^p\) boundedness of singular integrals ([4], page 300). Suppose that for a \(d \ge 2\), (10) holds for all P with \(\deg (P) \le d-1\).
We now assume that \(\deg (P) = d\), i.e.
with \(\sum _{|\alpha |= d} |a_\alpha | \ne 0\).
It is easy to see that for \(t > 0\), \(t^{n}K(t x)\) satisfies conditions CZ(\(q,\delta \))(a)–(c) with the same q, \(\delta \) and A. Thus, by rescaling if necessary we may assume that \(\sum _{|\alpha |= d} |a_\alpha | =1\).
Let
Then \(\deg (R) \le d-1\) and for \( 0 < h \le 8\)
By (12), Hölder’s inequality and CZ(\(q,\delta \))(a),
Thus, for \( 0 < h \le 8\)
For the rest of this proof we assume that \(h > 8\). For \(j \le 2\) let \(I_j = [2^{j}, 2^{j+1}]\),
and let \(S_j\) denote the following operator
Then
where
For each j, we will now apply Lemma 2.1 to the estimate of \(\Vert m_j\Vert _\infty \) with \(R = 2^{j+2}\). By
and
we have \(R \Lambda ^{-1/d} \le 1\). Thus, for any \(\xi \in \mathbb {R}^n\),
For any \( v \in B(0, 1)\), \(j \ge 2\) and \(x \in B(0, 2^{j+1})\backslash B(0, 2^j)\) we have \(|x| \ge 2|v|\) and
Thus, by CZ(\(q,\delta \))(b),
Since \((B(0, 2^j)\Delta B(v, 2^j))\cup (B(0, 2^{j+1})\Delta B(v, 2^{j+1})) \subseteq B(v, 2^{j+2})\backslash B(v, 2^{j-1})\), by (17)–(19) and CZ(\(q,\delta \))(a) we have
where \(\mu = \min \{\delta , 1/q^\prime \}\). It follows from Plancherel’s theorem that
By CZ(\(q,\delta \))(a), for any \(t_1, t_2 \) satisfying \( 0< t_1 < t_2 \) and \( t_2/t_1 \lesssim 1\),
Thus,
By the Riesz–Thorin interpolation theorem, for \(1< p < \infty \),
where
Let \(m= [\log _2 h]\). Then
It follows from (13), (21) and (23) that
The proof of Theorem 2.1 is now complete. \(\square \)
By using
interpreted in the distributional sense, we obtain (6) for all test functions f. Part (i) of Theorem 1.2 then follows by standard arguments.
3 \(H^1 \rightarrow H^1\) Estimates
As for the \(L^p\) boundeness, the \(H^1\) arguments in [7] relied both on the phase being quadratic as well as the condition CZ(\(q,\delta \)) with a \(q > 2\). To prove part (ii) of Theorem 1.2, we will let the degree of the phase polynomial be any positive integer while assuming that K satisfies CZ(\(q,\delta \)) with a \(q > 1\).
Lemma 3.1
Let \(d \ge 2\), \(P(x) =\sum _{|\alpha |\le d} a_\alpha x^\alpha \) and K(x) satisfy
for some \(q > 1\). Then,
(i) for any \(0< a < b\) and \(\nu \ge 1\),
(ii) for any \(\lambda \ge 1\),
where \(\gamma = \min \{q, 2\}\).
Proof
(i) Let \(N = [\log _2(b/a)]\). Then
which proves (25). The proof of (26) is simpler and will be omitted.
(ii) Since \(1 < \gamma \le q\), we have for any \(s > 0\),
For any \(\lambda \ge 1\), by Hölder’s inequality and applying Lemma 2.3 in [1] (taking p to be \(\gamma ^\prime \ge 2\)),
\(\square \)
Lemma 3.2
Let K(x) be given as in Lemma 3.1 and Q(x) be a polynomial satisfying \(\nabla Q(0) = 0\). Let f be a Lebesgue measurable function satisfying
Then, there exists a \(C > 0\) such that
The constant C may depend on \(\deg (Q)\) but is otherwise independent of the coefficients of Q(x).
Proof
When \(\deg (Q) \le 1\), by \(\nabla Q(0) = 0\), (31) follows from (30) trivially.
Suppose that \(d \ge 2\) and (31) holds for all Q(x) satisfying \(\deg (Q) \le d-1\) and \(\nabla Q(0) = 0\).
Assume that \(\deg (Q) = d \) and \(\nabla Q(0) = 0\). Then
where \(\deg (R) \le d-1\) and \(\nabla R(0) = 0\). Thus,
Let
Then
Let \( \gamma = \min \{q, 2\}\). By Hölder’s inequality, (29) and Lemma 4.3 in [5] (after interpolating between the \(L^2 \rightarrow L^2\) bound there and a trivial \(L^1 \rightarrow L^\infty \) bound), we have
Now (31) follows from (32) and (33). \(\square \)
We will now prove part (ii) of Theorem 1.2.
Proof
Let P(x) be a real-valued polynomial of any positive degree. Suppose that K(x) satisfies CZ(\(q,\delta \))(a)–(c) for some \(q > 1\) and \(\delta >0\). Let \(\gamma = \min \{q, 2\}\). For P with \(\deg (P) = 1\), we have \(\Vert P\Vert _o = \infty \) in which case (7) holds trivially. Thus we may assume that \(d = \deg (P) \ge 2\).
Since \(T_{P, K}\) is translation invariant, by the standard atomic theory of Hardy spaces, it suffices to prove that
holds for every \(H^1(\mathbb {R}^n)\) atom \(f(\cdot )\) which is supported in a ball centered at the origin (see [2, 3, 9, 10]). Additionally, due to the invariance of the CZ(\(q,\delta \)) conditions under \(K(x) \rightarrow t^n K(tx)\) and the invariance of \(\Vert \Vert _o\) under \(P(x) \rightarrow P(tx)\), we may assume that f satisfies (28)–(30).
First we will prove that
Let
Let
Thus, by (36) and (37), (35) would follow if we can prove that
Since (38) holds trivially when \( a = b\), we may assume that \( a < b\). Then
For \( y \in B(0,1)\),
This proves (38) and, in turn, (35).
By part (i) of Theorem 1.2, CZ(\(q,\delta \))(b), (35) and (29),
The proof of part (ii) of Theorem 1.2 is now complete. \(\square \)
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Pan, Y. \(L^p\) and \(H^1\) Boundedness of Oscillatory Singular Integral Operators with Hölder Class Kernels. Integr. Equ. Oper. Theory 93, 42 (2021). https://doi.org/10.1007/s00020-021-02659-z
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DOI: https://doi.org/10.1007/s00020-021-02659-z