{"history":[],"article":{"id":"571e5fc3-c421-47dc-94a4-86573783596c","title":"Singular integral","slug":"singular-integral","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **singular integrals** are central to [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) and are intimately connected with the study of partial differential equations.\nBroadly speaking a singular integral is an [integral operator](https://bliptext.com/articles/integral-operator)\nwhose kernel function K : ******R******n×******R******n → ******R****** is [singular](https://bliptext.com/articles/mathematical-singularity) along the diagonal x = y.\nSpecifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0.\nSince such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality.\nUsually further assumptions are required to obtain results such as their boundedness on Lp(**R**n).\n\n# The Hilbert transform\nThe archetypal singular integral operator is the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) H.\nIt is given by convolution against the kernel K(x) = 1/(πx) for x in **R**.\nMore precisely,\nThe most straightforward higher dimension analogues of these are the [Riesz transforms](https://bliptext.com/articles/riesz-transform), which replace K(x) = 1/x with\nwhere i = 1, ..., n and x_i is the i-th component of x in **R**n. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.\n\n# Singular integrals of convolution type\nA singular integral of convolution type is an operator T defined by convolution with a kernel K that is [locally integrable](https://bliptext.com/articles/locally-integrable-function) on **R**n\\{0}, in the sense that\nSuppose that the kernel satisfies:\nThen it can be shown that T is bounded on Lp(**R**n) and satisfies a weak-type (1, 1) estimate.\nProperty 1.\nis needed to ensure that convolution with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics) p.v. K given by the [principal value integral](https://bliptext.com/articles/cauchy-principal-value)\nis a well-defined [Fourier multiplier](https://bliptext.com/articles/fourier-multiplier) on L2.\nNeither of the properties 1.\nor 2.\nis necessarily easy to verify, and a variety of sufficient conditions exist.\nTypically in applications, one also has a cancellation condition\nwhich is quite easy to check.\nIt is automatic, for instance, if K is an [odd function](https://bliptext.com/articles/odd-function).\nIf, in addition, one assumes 2.\nand the following size condition\nthen it can be shown that 1.\nfollows.\nThe smoothness condition 2.\nis also often difficult to check in principle, the following sufficient condition of a kernel K can be used:\nObserve that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.\n\n# Singular integrals of non-convolution type\nThese are even more general operators.\nHowever, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on Lp.\n\n## Calderón–Zygmund kernels\nA function K : Rn×Rn → R is said to be a [Calderón](https://bliptext.com/articles/alberto-calder-n)–[Zygmund](https://bliptext.com/articles/antoni-zygmund) kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.\n

\n\n\n## Singular integrals of non-convolution type\nT is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel K if\nwhenever f and g are smooth and have disjoint support.\nSuch operators need not be bounded on Lp\n\n## Calderón–Zygmund operators\nA singular integral of non-convolution type T associated to a Calderón–Zygmund kernel K is called a Calderón–Zygmund operator when it is bounded on L2, that is, there is a C > 0 such that\nfor all smooth compactly supported ƒ.\nIt can be proved that such operators are, in fact, also bounded on all Lp with 1 < p < ∞.\n\n## The T(b) theorem\nThe T(b) theorem provides sufficient conditions for a singular integral operator to be a Calderón–Zygmund operator, that is for a singular integral operator associated to a Calderón–Zygmund kernel to be bounded on L2.\nIn order to state the result we must first define some terms.\nA normalised bump is a smooth function φ on **R**n supported in a ball of radius 1 and centred at the origin such that |∂α φ(x)| ≤ 1, for all multi-indices |α| ≤ n + 2.\nDenote by τx(φ)(y) = φ(y − x) and φr(x) = r−nφ(x/r) for all x in **R**n and r > 0.\nAn operator is said to be weakly bounded if there is a constant C such that\nfor all normalised bumps φ and ψ.\nA function is said to be accretive if there is a constant c > 0 such that **R**e(b)(x) ≥ c for all x in **R**.\nDenote by Mb the operator given by multiplication by a function b.\nThe T(b) theorem states that a singular integral operator T associated to a Calderón–Zygmund kernel is bounded on L2 if it satisfies all of the following three conditions for some bounded accretive functions b1 and b2:\n

is weakly bounded;

T(b_1) is in [BMO](https://bliptext.com/articles/bounded-mean-oscillation);

T^t(b_2), is in [BMO](https://bliptext.com/articles/bounded-mean-oscillation), where Tt is the transpose operator of T.

\n","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:56:28.355Z","updated_at":"2025-01-26T09:56:28.355Z","search_vector":"'integr':2 'singular':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **singular integrals** are central to [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) and are intimately connected with the study of partial differential equations.\nBroadly speaking a singular integral is an [integral operator](https://bliptext.com/articles/integral-operator)\nwhose kernel function K : ******R******n×******R******n → ******R****** is [singular](https://bliptext.com/articles/mathematical-singularity) along the diagonal x = y.\nSpecifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0.\nSince such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality.\nUsually further assumptions are required to obtain results such as their boundedness on Lp(**R**n).\n\n# The Hilbert transform\nThe archetypal singular integral operator is the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) H.\nIt is given by convolution against the kernel K(x) = 1/(πx) for x in **R**.\nMore precisely,\nThe most straightforward higher dimension analogues of these are the [Riesz transforms](https://bliptext.com/articles/riesz-transform), which replace K(x) = 1/x with\nwhere i = 1, ..., n and x_i is the i-th component of x in **R**n. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.\n\n# Singular integrals of convolution type\nA singular integral of convolution type is an operator T defined by convolution with a kernel K that is [locally integrable](https://bliptext.com/articles/locally-integrable-function) on **R**n\\{0}, in the sense that\nSuppose that the kernel satisfies:\nThen it can be shown that T is bounded on Lp(**R**n) and satisfies a weak-type (1, 1) estimate.\nProperty 1.\nis needed to ensure that convolution with the [tempered distribution](https://bliptext.com/articles/distribution-mathematics) p.v. K given by the [principal value integral](https://bliptext.com/articles/cauchy-principal-value)\nis a well-defined [Fourier multiplier](https://bliptext.com/articles/fourier-multiplier) on L2.\nNeither of the properties 1.\nor 2.\nis necessarily easy to verify, and a variety of sufficient conditions exist.\nTypically in applications, one also has a cancellation condition\nwhich is quite easy to check.\nIt is automatic, for instance, if K is an [odd function](https://bliptext.com/articles/odd-function).\nIf, in addition, one assumes 2.\nand the following size condition\nthen it can be shown that 1.\nfollows.\nThe smoothness condition 2.\nis also often difficult to check in principle, the following sufficient condition of a kernel K can be used:\nObserve that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.\n\n# Singular integrals of non-convolution type\nThese are even more general operators.\nHowever, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on Lp.\n\n## Calderón–Zygmund kernels\nA function K : Rn×Rn → R is said to be a [Calderón](https://bliptext.com/articles/alberto-calder-n)–[Zygmund](https://bliptext.com/articles/antoni-zygmund) kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.\n \n \n \n \n \n \n \n\n## Singular integrals of non-convolution type\nT is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel K if\nwhenever f and g are smooth and have disjoint support.\nSuch operators need not be bounded on Lp\n\n## Calderón–Zygmund operators\nA singular integral of non-convolution type T associated to a Calderón–Zygmund kernel K is called a Calderón–Zygmund operator when it is bounded on L2, that is, there is a C > 0 such that\nfor all smooth compactly supported ƒ.\nIt can be proved that such operators are, in fact, also bounded on all Lp with 1 0.\nAn operator is said to be weakly bounded if there is a constant C such that\nfor all normalised bumps φ and ψ.\nA function is said to be accretive if there is a constant c > 0 such that **R**e(b)(x) ≥ c for all x in **R**.\nDenote by Mb the operator given by multiplication by a function b.\nThe T(b) theorem states that a singular integral operator T associated to a Calderón–Zygmund kernel is bounded on L2 if it satisfies all of the following three conditions for some bounded accretive functions b1 and b2:\n is weakly bounded; \n T(b_1) is in [BMO](https://bliptext.com/articles/bounded-mean-oscillation); \n T^t(b_2), is in [BMO](https://bliptext.com/articles/bounded-mean-oscillation), where Tt is the transpose operator of T. \n ","revisionCount":0},"hasMore":false,"nextCursor":null}

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