Factorization of Matrix Functions and Singular Integral Operators

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

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The Calder´on-Zygmund Theory I: Ellipticity

10.23943/princeton/9780691162515.003.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Classical Theory ◽

Singular Integral ◽

Singular Integrals ◽

Differential Operators ◽

Integral Operators ◽

Singular Integral Operators ◽

Single Parameter ◽

Partial Differential ◽

Translation Invariant ◽

Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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Definability of singular integral operators on Morrey–Banach spaces

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/81208120 ◽

2020 ◽

Vol 72 (1) ◽

pp. 155-170 ◽

Cited By ~ 1

Author(s):

Kwok-Pun HO

Keyword(s):

Banach Spaces ◽

Singular Integral ◽

Integral Operators ◽

Singular Integral Operators

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The Calderón–Zygmund Theorem with an $$L^1$$ Mean Hörmander Condition

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09810-9 ◽

2021 ◽

Vol 27 (2) ◽

Author(s):

Soichiro Suzuki

Keyword(s):

Singular Integral ◽

Integral Operators ◽

Acta Math ◽

Limited Range ◽

Singular Integral Operators ◽

Convolution Type ◽

Worst Case ◽

Hörmander Condition

AbstractIn 2019, Grafakos and Stockdale introduced an $$L^q$$ L q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the $$L^p$$ L p boundedness for the “limited-range” instead of $$1< p < \infty $$ 1 < p < ∞ . However, in this paper, we show that the $$L^q$$ L q mean Hörmander condition is actually enough to obtain the $$L^p$$ L p boundedness for all $$1< p < \infty $$ 1 < p < ∞ even in the worst case $$q=1$$ q = 1 . We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the $$L^2$$ L 2 boundedness for convolution type singular integral operators under the $$L^1$$ L 1 mean Hörmander condition.

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