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Singular integrals and related topics. (English) Zbl 1124.42011

Hackensack, NJ: World Scientific (ISBN 978-981-270-623-2/hbk). viii, 272 p. (2007).
Many other books related to singular integrals focus on general theory. In this book, however, the authors focus on the singular integrals of homogeneous kernels; \[ T_{\Omega}f(x) = \int_{R^n} \frac{\Omega(x-y)}{| x-y |^{n}} f(y)\,dy, \quad \text{where} \quad \int_{S^{n-1}}\Omega(x')\,d \sigma (x') = 0. \] It includes not only the case of smooth kernels, but also the case of rough kernels. Especially, they aim to establish a more perfect theory of singular integrals with rough kernels. For graduate students in analysis, this book is a nice introduction to the theory by M. Christ, J. L. Rubio de Francia and J. Duoandikoetxea [Invent. Math. 84, 541–561 (1986; Zbl 0568.42012) and Invent. Math. 93, 225–237 (1988; Zbl 0695.47052)].
This book consists of five chapters. In Chapter 1 basic results are reviewed with proofs and items well arranged. It contains the Hardy-Littlewood maximal operator, Calderón-Zygmund decomposition, Marcinkiewicz interpolation theorem, Fefferman-Stein inequality for the maximal operator, Kolmogorov inequality, \(A_p\) weights and Jones decomposition of \(A_p\) weights.
Chapter 2 is the main part of this book. First the authors prove the \(L^p\) and weak \(L^1\) estimates for \(T_{\Omega}\) when \(\Omega\) satisfies \(L^{\infty}\)-Dini condition, then they propose the following problem: can we weaken the condition of \(\Omega\) but still ensure the \(L^p\) and weak \((1,1)\) boundedness? They prove the following theorem.
Theorem 2.1. If \(\Omega\) {satisfies} \(L^{1}\)-Dini condition then \(T_{\Omega}\) is bounded on \(L^p\) and of type weak \((1,1)\).
Next they consider the following problem: can we still keep the \((p,p)\) and weak \((1,1)\) boundedness of \(T_{\Omega}\) without any smoothness of \(\Omega\) on the unit sphere? They show the following theorem.
Theorem 2.2 If \(\Omega \in L\log^{+}L(S^{n-1})\) then \(T_{\Omega}\) is bounded on \(L^p\).
Furhermore they prove the following theorem by Connett (1979) and Ricci and Weiss (1979).
Theorem 2.3. If \(\Omega \in H^1(S^{n-1})\) then \(T_{\Omega}\) is bounded on \(L^p\).
They state the following theorem by Seeger (1996) without proof.
Theorem 2.4. If \(\Omega \in L\log^{+}L(S^{n-1})\) then \(T_{\Omega}\) is of weak type \((1,1)\).
Next they investigate weighted estimates and prove the following theorem by Duoandikoetxea (1993) and Watson (1990).
Theorem 2.5. If \(\Omega \in L^q(S^{n-1}) \;(q>1)\) then \(T_{\Omega}\) is bounded on \(L^p(w)\) for some suitable weight functions \(w\).
Their explanation is detailed enough. To prove Theorem 2.5 they use the technique of good-\(\lambda\) inequality, Littlewood-Paley theory and Stein-Weiss interpolation method with change of measure. These ideas have already been explained in the proofs of previous theorems.
Chapter 3 discusses fractional integrals with homogeneous kernels; \[ T_{\Omega, \alpha}f(x) = \int_{\mathbb R^n} \frac{\Omega(x-y)}{| x-y |^{n - \alpha }} f(y)\,dy. \] In Chapter 4 they study a class of oscillatory singular integrals with polynominal phases; \[ Tf(x) = \int_{\mathbb R^n}e^{i P(x,y)} K(x-y) f(y) \,dy. \] First they prove Ricci and Stein’s theorem (1987).
Theorem 4.1. Assume that \(K\) satisfies the following conditions: (i) \(K\) is a \(C^1\) function away from the origin, (ii) \(K\) is homogeneous of degree \(-n\), and (iii) the mean value of \(K\) on the unit sphere vanishes.
Then \(T\) is bounded on \(L^p\) where \(P\) is a polynomial.
Next they prove Chanillo and Christ’s theorem (1987).
Theorem 4.2. By the same assumptions as above, \(T\) is of weak type \((1,1)\).
Next they consider the case of rough kernels. Let \(K(x) = \Omega (x) / | x |^n\), where \(\Omega\) is homogeneous of degree 0. They prove the following theorem by Lu and Zhang (1992).
Theorem 4.3. If \(\Omega \in L^q (S^{n-1}) \;(q>1)\) then \(T\) is bounded on \(L^p\).
Chapter 5 deals with the Littlewood-Paley theory. The Littlewood-Paley \(g\) function \(g_{\varphi}\) is defined by \[ g_{\varphi} (x) = \biggl( \int_{0}^{\infty} | \varphi_t * f (x) |^2\, \frac{dt}{t} \biggr)^{1/2}. \] First they prove the following classical theorem.
Theorem 5.1. If \(\varphi\) satisfies (i) \(\int \varphi (x)\, dx =0 \), (ii) \(| \varphi (x) | \leq C ( 1 + | x |)^{-n-1}\) and
(iii) \( \int | \varphi (x+h) - \varphi (x) | \,dx \leq C | h |^{\alpha} \), then \(g_{\varphi}\) is bounded on \(L^p\).
The Marcinkiewicz integral is defined by \[ \mu_{\Omega}(f)(x)= \biggl( \int_0^{\infty} \biggl| \int_{ | x-y | \leq t} \frac{\Omega (x-y)}{ | x-y |^{n-1}} f(y)\,dy \biggr|^2 \frac{dt}{t^3} \biggr)^{1/2}, \] which is a generalization of \(g\) function. They prove Ding, Fan and Pan’s theorem (2000).
Theorem 5.2. If \(\Omega\) is homogeneous of degree 0, \(\int_{S^{n-1}} \Omega (x') \,d\sigma (x')= 0\) and \(\Omega \in H^{1}(S^{n-1})\), then \(\mu_{\Omega}\) is bounded on \(L^p\).
Finally they prove the following.
Theorem 5.3. If \(\Omega\) is homogeneous of degree 0, \(\int_{S^{n-1}} \Omega (x') \,d\sigma (x')= 0\) and \(\Omega \in L ( \log^{+}L )^{1/2}(S^{n-1})\), then \(\mu_{\Omega}\) is bounded on \(L^p\).
They also consider commutators of singular integral operators and multilinear oscillatory singular integrals.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
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