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Hilbert and Riesz Transforms
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Analyticity of Holomorphic Functions; Liouville and Morera
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Bounded Variation and the Riemann–Stieltjes Integral
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Lp Spaces and Test-Function Conventions
- Complex Power Series and Analytic Functions
- Complex Riesz–Thorin Endpoint Interpolation
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Contour Integration
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Darboux, L'Hôpital, and Taylor's Theorem
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Dirichlet Kernel Localisation and Pointwise Fourier Convergence
- Distributions Test Functions and Differentiation
- Divergence and Almost Everywhere Convergence of Fourier Series
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Equivalent Forms of Completeness
- Fejer and Poisson Summability of Fourier Series
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Probability and the Probabilistic Method
- Foundations of the Real Numbers for Analysis
- Fourier Multipliers and Sobolev Characterisations
- Fourier Transform Convolution and Approximate Identities
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Reflexivity and Eberlein Smulian
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Schwartz Space and the Plancherel Theorem
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Sequential Uniform Boundedness with Countable Choice
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tempered Distributions and the Fourier Transform
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Duality of Lᵖ and L^q
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Real Gamma and Beta Functions
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Trigonometric and Oscillatory Examples in One Variable
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Volumes of Elementary Solids and Solids of Revolution
2 · Summary
This page proves the strict-range mapping theory of the periodic conjugate function and the theory of the Hilbert and Riesz transforms on . Every argument that needs a choice principle assumes Countable Choice and names the step that spends it; the remaining proofs are choice-free.
On the circle the conjugate function is first defined as the coefficient multiplier on trigonometric polynomials, so constants lie in its kernel and no extension is built into the definition. The conjugate Dirichlet kernel is computed exactly and identified with the cotangent principal value under local regularity; the finite kernel convolution is kept distinct from the limiting truncation. The square identity for real mean-zero polynomials, Riesz interpolation of the bounded L² action, and complex duality give the Marcel Riesz conjugate-function theorem: extends uniquely to a bounded operator on for every , while the exact operator norm of the partial sums grows logarithmically and rules out compatible strong or extensions. The same Lebesgue-constant lower bound, together with the Fejér means which converge in , yields uniform bounds for the Fourier partial sums and their norm convergence in the strict range.
On the line the page defines the truncated Hilbert transform and its principal value, proves the uniform sine-integral bounds and the value under Countable Choice, and shows that on Schwartz functions the principal value is the tempered convolution with , whose Fourier multiplier is . The multiplier produces the extension with , , and skew-adjointness, with no zero-frequency exception on the line. For the Riesz transforms are defined by the multipliers ; polar coordinates identify that multiplier with the principal-value kernel , giving the bound, the finite square-sum identity , and the kernel size, first-difference and spherical cancellation estimates needed by the later singular-integral theory.
The closing remark maps the endpoint statements this page does not claim — weak , real-line strict-range , almost-everywhere truncation convergence, real Hardy space and BMO — to the later pages that own them, and the companion page carries the interval, Poisson-kernel and endpoint counterexamples that do not refute those weaker conclusions.
3 · Logical flowchart
4 · Definitions, theorems and proofs
Conjugate function on the circle
Definition
Work on the torus with the conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus: the characters are for , the Fourier coefficients of a one-period integrable are , and a trigonometric polynomial is a finite complex linear combination of characters.
Let be a trigonometric polynomial, so that for and for . The conjugate function of is the trigonometric polynomial
Equivalently, is the unique trigonometric polynomial whose Fourier coefficients are
Consequently : every constant trigonometric polynomial lies in the kernel of .
The assignment is complex-linear on the space of trigonometric polynomials. It also preserves real-valuedness: if is real, then for every , and , so has the conjugate symmetry that characterizes a real trigonometric polynomial.
Two conventions are fixed by this definition. The factor and the sign refer to the characters and to the coefficient convention above; conjugating real functions, as in , is the opposite sign convention. And is defined on trigonometric polynomials alone: no bound on any norm and no extension to arbitrary integrable functions is asserted here. The extension to for is proved on this page after the kernel formula below.
The conjugate Dirichlet kernel, and the periodic principal-value formula
Statement
Assume Countable Choice, and work on with the conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus. Let and let . Put
the conjugate Dirichlet kernel, and write for the conjugate partial sum. Then:
- is the convolution of with : for almost every , .
- Extend to by the square-summable coefficient family ; the resulting class is the limit of the partial sums .
- If a representative of is on an open interval containing , then
the limit existing for almost every such , and being the symmetric principal value about the singularity.
The finite kernel is not itself a cotangent truncation: equals minus the oscillatory remainder , and only the limit of the convolutions recovers the principal value.
Facts & Assumptions
Given: Countable Choice, , , and the characters .
is defined on trigonometric polynomials coefficientwise by , it is complex-linear, kills constants, and preserves real-valuedness. Conjugate function on the circle
Fourier coefficients, partial sums , characters, and the torus convolution are as defined there, and the torus integral is invariant under the reflections used below. Period-one Fourier coefficients, partial sums, and convolution on the torus
The Dirichlet kernel is . Dirichlet and Fejer kernels
For one-period integrable , for every . Fourier partial sums are Dirichlet convolutions
For and , . Finite sums of the sine harmonics
Parseval: in the finite-subset-supremum sense, so the tails over tend to . The Parseval identity for Fourier series
Every square-summable coefficient family in is the Fourier coefficient family of a unique class, realized as the limit of its symmetric partial sums. Riesz–Fischer: the Fourier coefficient map is onto the space of square-summable families
Riemann-Lebesgue: if is integrable on one period then as . Riemann-Lebesgue lemma for Fourier coefficients
Norm-convergent sequences in have subsequences converging almost everywhere to a representative of the limit. Complex Lp completeness and almost-everywhere subsequences
The real mean value theorem bounds the increment of a real function by the supremum of its derivative times the interval length. Applied separately to the real and imaginary parts, it gives for a complex function on a compact interval about , with . The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with
Dominated convergence. Dominated convergence
Cosine addition formula: . The addition formulas for sine and cosine
Proof
By [F1] and [F3], is the trigonometric polynomial with coefficients on and elsewhere, so , using . Applying [F5] with gives for , while ; in particular is odd, one-periodic, and .
By [F1], the conjugate partial sum is the trigonometric polynomial . Expanding from 1.1 and substituting in each finite sum as in [F2] and [F4], for every . The bounded finite kernel makes the integral exist for each translate of the representative, and the finite coefficient calculation is exact; this keeps the finite- object a polynomial-level convolution and makes no claim on any cotangent kernel.
Fix a point at which some representative of is on an open interval containing , and put for , so that for a constant and all small by [F10]. Step 1.2 gives ; since is one-periodic, odd, and has by 1.1, that integral equals . The closed form of 1.1 splits the kernel as , so with and , the integrands being defined and measurable off the null point .
Put . Since for every (with ), [F6] gives . Thus [F7] supplies a unique class whose symmetric partial sums are exactly the of 1.2 and which is their limit; by [F9] there is an increasing sequence with for almost every .
For the point of 2.1, [F12] writes , so . Both and are integrable on : the second because is on the finite torus and the quotient is bounded near by ; away from , is bounded and , so the quotient is integrable there as well. Extending them by zero to one period, [F8] gives for these integrable functions, hence as ; the convergence is at every such , and no uniformity in is claimed.
Also at the point of 2.1, for the oddness of gives , and by [F10] the function is integrable on ; [F11] therefore gives as . So the symmetric principal value exists at and equals .
Combining 3.1 and 3.2, for every at which is near the finite convolutions satisfy , so the full sequence converges to the principal value at every such ; by 2.2 it also converges to along a subsequence for almost every . Therefore for almost every in the open set where is near , as asserted.
The periodic conjugate square identity for real mean-zero polynomials
Statement
Let be a real-valued trigonometric polynomial on with zero mean, and let be the conjugate function of Conjugate function on the circle. Then
The identity is asserted for trigonometric polynomials only; no extension to arbitrary inputs is claimed here, and the mean-zero hypothesis is not removable: for the constant polynomial one has and , so the right-hand side equals while the left-hand side vanishes.
Facts & Assumptions
Given: A real-valued trigonometric polynomial on with zero mean; the conjugate function on trigonometric polynomials.
On trigonometric polynomials acts coefficientwise by ; it is complex-linear, kills constants, and preserves real-valuedness, so is real-valued for real . Conjugate function on the circle
Characters satisfy and a trigonometric polynomial is a finite complex linear combination of characters. Fourier coefficients and trigonometric polynomials on the torus
If is a trigonometric polynomial, then its Fourier coefficient at is and for . The trigonometric characters are orthonormal in of the torus
Proof
Write the finite expansion given by [F2]. Since has mean zero and , [F3] gives ; since is real-valued, [F1] makes real-valued, so has zero mean as well. Adding the two expansions and using complex-linearity of from [F1] gives : a trigonometric polynomial whose only frequencies are strictly positive.
By [F2], , so expanding the finite square and collecting terms shows that is a trigonometric polynomial whose only frequencies are strictly positive. On a polynomial carried by the characters with , the coefficient rule of [F1] gives , and complex-linearity gives .
Expanding, ; by 1.1 both and are real-valued trigonometric polynomials with real-valued conjugate transforms, and . By complex-linearity of , .
By 2.1 and 2.2, . Taking real and imaginary parts of this identity of trigonometric polynomials, whose four real and imaginary parts are real-valued by 2.2, gives and .
Substituting and into gives , hence , which is the asserted identity.
The Marcel Riesz conjugate-function theorem on the circle
Statement
Assume Countable Choice and the conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus on the torus with normalized Haar measure , so that : characters , Fourier coefficients , trigonometric polynomials as finite complex linear combinations of characters, and the conjugate function of Conjugate function on the circle, with and .
- For every the operator extends uniquely from the trigonometric polynomials to a bounded complex-linear operator , and the extensions are mutually consistent: almost everywhere for every .
- Constants lie in the kernel: for every .
- No compatible endpoint extension exists: there is no bounded operator with for every trigonometric polynomial , and no bounded operator with for every trigonometric polynomial . The failure is of strong-type boundedness; assertions about weak type , maximal truncations or a bounded mean-oscillation range are not made here.
Facts & Assumptions
Given: Countable Choice; the torus with normalized Haar measure ; the conjugate function on trigonometric polynomials, .
On trigonometric polynomials is complex-linear, kills constants and preserves real-valuedness; the characters satisfy and a trigonometric polynomial has only finitely many nonzero Fourier coefficients. Conjugate function on the circle Period-one Fourier coefficients, partial sums, and convolution on the torus
For every real mean-zero trigonometric polynomial one has . The periodic conjugate square identity for real mean-zero polynomials
Parseval: for , and , the sums being finite-subset-net limits whose value is also the limit of the symmetric partial sums . The Parseval identity for Fourier series
Hölder and Minkowski for complex : for conjugate exponents and complex measurable functions, the integral of a product is bounded by the product of the norms, and the norm of a sum by the sum of the norms. Complex Holder, Minkowski, and the quotient norm
Cesàro means: ; for and one has ; for continuous one-periodic one has . Cesaro and Abel means of a Fourier series Fejer means converge in L^p for 1 <= p < infinity Fejer means converge uniformly for continuous periodic functions
The Fejér kernel satisfies and . The Fejer kernel is a positive approximate identity
For one-period integrable , at every , where is real-valued, even and . Fourier partial sums are Dirichlet convolutions Dirichlet and Fejer kernels
On the continuous periodic functions with the supremum norm, , and for this number is at least . Fourier partial-sum operator norm equals the Lebesgue constant
For every measure space and the complex space is complete. Complex Lp completeness and almost-everywhere subsequences
Riesz–Thorin: a complex-linear map defined on the complex finite simple functions with finite-measure support which is bounded with constants between and , , , extends uniquely to a bounded operator with norm at most . Riesz-Thorin interpolation theorem
For a finite measure space and , every bounded complex-linear functional on is integration against a unique with the bilinear pairing, and the norms agree. Complex Lp duality from real Lp duality
Norm recovery: for (and only for sigma-finite measures, which includes ), . The norm is the supremum of pairings against unit functions
On a finite measure space, for and . Finite-measure includes into for
Tonelli/Fubini for functions on sigma-finite product spaces. Fubini's theorem for L^1 functions on a sigma-finite product
Proof
Let be a real mean-zero trigonometric polynomial. By [F3] and the coefficient rule of [F1], , where is used in the middle equality; again by [F1], is real-valued with , so is real with zero mean. Moreover [F3] with the coefficient rule gives , which is times the real number ; since is real-valued, its integral is real, so and is a real mean-zero trigonometric polynomial.
For a trigonometric polynomial put , and for put . Then is complex-linear, for every , and for every trigonometric polynomial and every one has . Indeed the definitions and of [F1] give and for every , and subtracting gives for every , which identifies the two trigonometric polynomials.
For and , . Indeed [F7] gives and with real and even, so the double integral of is at most and [F14] applies; the substitution and evenness of turn into .
For , each is a trigonometric polynomial with and ; for , each is a trigonometric polynomial with and . This is [F5] with together with [F6]: from and one gets , hence and, for , for every .
For and every real one has by [F7], hence : on the unit ball of , every is bounded by .
Put a real mean-zero trigonometric polynomial, . Then by 1.1, and for every , so every is finite. Indeed fix , put and let be real with zero mean; the square identity [F2], the identification of as a real mean-zero trigonometric polynomial in 1.1, and the definition of give , while [F4] applied with exponents to the functions and gives . Dividing by and writing yields , hence ; taking the supremum over gives the recursion.
Suppose a bounded linear satisfies for every trigonometric polynomial ; put and for . Then and for every trigonometric polynomial , by 1.2: the multiplier of is on positive frequencies, on negative frequencies and at zero, and the half-mean term supplies the remaining at zero. For with let : by 1.4 these are trigonometric polynomials with and , and 1.2 gives because is again a trigonometric polynomial on which acts as . Hence , while 1.5 with gives , so . Therefore for every , and in particular the continuous functions give .
By [F8], for every . Write . Given , choose a continuous one-periodic with and . Choose with , let have modulus one with (take if this value is zero), and set . Then by [F6], and because is continuous and its Cesaro means converge uniformly by [F5]. By pairing symmetry from 1.3 and , for all sufficiently large we have . Since , this gives for every , hence .
Suppose a bounded linear satisfies for every trigonometric polynomial ; put and define on by the same formula . Then and for every trigonometric polynomial , by the frequency check in 2.2. For with and its Cesàro means , which are trigonometric polynomials with by 1.4, identity 1.2 gives and hence ; by 1.5, , so . Therefore for every .
No bounded satisfies on trigonometric polynomials: such a would give for every by 2.2, while grows without bound by 2.3, a contradiction for large.
For every there is a finite constant with for every complex trigonometric polynomial : writing and applying 2.1 to the real mean-zero parts gives , and likewise for the imaginary part, so works; here uses and [F4]. Since the trigonometric polynomials are dense in by [F5] and that space is complete by [F9], therefore has a unique extension to a bounded complex-linear operator on with ; uniqueness holds because two continuous extensions of one map agree on the dense polynomial core.
No bounded satisfies on trigonometric polynomials: such a would give for every by 3.1, while grows without bound by 2.3, a contradiction for large.
For all one has for the bilinear pairing. Indeed, for trigonometric polynomials the coefficient identity and the rule of [F1] give , since ; both pairings are bounded bilinear functionals on (bounded by times the norm of ), and they agree on the dense polynomial core, so they agree everywhere.
Let . If , set . Otherwise choose with and let be restricted to the complex finite simple functions on . For and , approximate a simple function by trigonometric polynomials in using [F5]; since and , [F13] gives convergence in as well. The extensions therefore satisfy and as classes, so has the endpoint bounds . Applying [F10] gives an extension with , where . It agrees with on polynomials: uniformly approximate a polynomial by finite simple functions ; then in , hence in by [F13], while boundedness gives in .
Let and , so that 4.3 gives a bounded operator on with norm . For the formula defines a complex-linear functional on , bounded by because [F4] bounds ; by [F11] there is a unique with for all and . Put : then is complex-linear and bounded with . It extends the polynomial core, because for a trigonometric polynomial and any the defining identity, the consistency of 4.3 and the skew-adjointness of 4.2 give , so by the norm recovery [F12] applied to the difference in .
Collecting: for every the operator of 4.3 (for ) and of 5.1 (for ) is a bounded complex-linear extension of to , and it is the only such extension because trigonometric polynomials are dense in for and continuous extensions of one map agree on a dense set. If and , choose trigonometric polynomials in using [F5]; [F13] gives convergence in , and the bounded extensions agree on polynomials, so their images converge to the same limit. Thus . Also because kills constants by [F1]. By 3.2 there is no bounded compatible extension and by 4.1 no bounded compatible extension. This proves all three assertions.
Uniform Lp bounds for periodic Fourier partial sums
Statement
Assume Countable Choice and let . With the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus, write for the operator norm of the -th Fourier partial sum. Then
Moreover the bound is explicit: with the norm of the extension of the conjugate function supplied by The Marcel Riesz conjugate-function theorem on the circle, one has .
Facts & Assumptions
Given: Countable Choice, , and the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus: characters , coefficients , partial sums , trigonometric polynomials, and convolution.
The torus carries the normalized translation-invariant Haar integral with , and for . The one-dimensional torus and its normalized Haar integral Period-one Fourier coefficients, partial sums, and convolution on the torus
The conjugate function is defined on trigonometric polynomials coefficientwise by ; it is complex-linear, kills constants, and the characters satisfy . Conjugate function on the circle
For every the operator extends uniquely to a bounded complex-linear on with , and . The Marcel Riesz conjugate-function theorem on the circle
For and , the Cesaro means are trigonometric polynomials and ; hence the trigonometric polynomials are dense in . Fejer means converge in L^p for 1 <= p < infinity Cesaro and Abel means of a Fourier series
On the finite measure space one has for . Finite-measure includes into for
Complex carries the norm structure of Complex Holder, Minkowski, and the quotient norm, so the triangle inequality applies to finite sums.
A trigonometric polynomial whose Fourier coefficients all vanish is the zero polynomial; equivalently, a finite family of distinct characters is linearly independent. The trigonometric characters are orthonormal in of the torus
Proof
Define for , with the constant function. Then is complex-linear and bounded with , where : indeed the triangle inequality of [F6] gives , and by [F1] and [F5].
For a trigonometric polynomial one has . Indeed [F2] gives , so has coefficients for , at , and for ; the constant function has coefficients at and elsewhere, and , so adding yields exactly the coefficients for and for .
For define the modulation , a complex-linear map on . Since , one has pointwise and hence : each is an isometry. For a trigonometric polynomial and every , the coefficients satisfy , because in [F2] gives .
is bounded on : for each by [F1] and [F5], so the defining finite sum gives .
For every trigonometric polynomial and every , . Indeed, by 1.2 and 1.3 the left-hand side of the identity has coefficients , and ; their difference has coefficients for every , and two trigonometric polynomials with equal coefficients are equal by [F7].
The right-hand side of 2.1 defines a bounded operator on with norm at most : by 1.3 each modulation is an isometry and by 1.1 , so the triangle inequality of [F6] bounds the difference of the two composites by .
The identity of 2.1 holds for every , not only for trigonometric polynomials: both sides are bounded operators on by 1.4 and 2.2, they agree on the set of trigonometric polynomials, and that set is dense in by [F4]; two bounded operators agreeing on a dense set agree everywhere.
By 3.1 and 2.2, for every and every , while for step 1.4 gives . Hence for every and , which is the assertion.
Periodic Fourier partial sums converge in the strict Lp range
Statement
Assume Countable Choice, and use the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus and The one-dimensional torus and its normalized Haar integral: the torus carries normalized Haar measure with , and for .
- For every and every ,
- At the endpoints the operator norms grow at least as the Lebesgue constants. For every , Hence both families and are unbounded, and there exist and such that fails to converge in and fails to converge in .
No failure of weak-type or of any endpoint mapping weaker than norm convergence is asserted.
Facts & Assumptions
Given: Countable Choice, the torus conventions of Period-one Fourier coefficients, partial sums, and convolution on the torus and The one-dimensional torus and its normalized Haar integral, and the norms of Complex Lp classes and Euclidean test-function conventions.
The torus integral is normalized, , and translation invariant: for integrable . The character , the coefficient , the partial sum , trigonometric polynomials and torus convolution are defined as in the cited definition, as is for the normalized measure. Period-one Fourier coefficients, partial sums, and convolution on the torus The one-dimensional torus and its normalized Haar integral
For every one-period integrable , every and every , . Fourier partial sums are Dirichlet convolutions
is real, even, continuous and bounded, , and . The Fejer kernel satisfies and , so . Dirichlet and Fejer kernels The Fejer kernel is a positive approximate identity
For and the Cesaro means satisfy , are trigonometric polynomials, and ; hence trigonometric polynomials are dense in . For a trigonometric polynomial one has whenever . Fejer means converge in L^p for 1 <= p < infinity Cesaro and Abel means of a Fourier series
For every one has . Uniform Lp bounds for periodic Fourier partial sums
For integrable complex one has and ; Tonelli's theorem applies to nonnegative measurable functions on the finite product . The modulus of an integral is bounded by the integral of the modulus Complex Holder, Minkowski, and the quotient norm Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
For every the space is complete. Complex Lp completeness and almost-everywhere subsequences
(Sequential uniform boundedness.) If is a Banach space, a normed space over the same field, and , , are bounded linear maps with for every and every , then . Sequential uniform boundedness under countable choice
Proof
For , the convolution formula [F2] and the integral triangle inequality [F7] give for every , so is bounded on with . For , Tonelli and the translation invariance of [F1] give , so is bounded on . Both bounds are finite by [F3].
For all one has : the first equality is [F2], the second is the substitution in the absolutely convergent torus convolution, and the third is [F4]. Since by [F3], the identity and the Fejer convergence of [F4] give as .
For , the unit ball of is contained in the unit ball of , and for continuous the partial sum is a trigonometric polynomial, whose essential supremum equals its supremum; hence by [F6].
Let , and . By [F4] choose a trigonometric polynomial with , where is the finite bound of [F5]. For one has by [F4], so the triangle inequality [F7] and the bound [F5] give . Hence in for every .
For fixed and every , step 1.2 and [F3] give , while ; therefore .
By step 1.3 and step 2.1, for every both endpoint norms satisfy , and . Hence , which is the norm-growth assertion of part 2.
Suppose no failed to converge. Then each would have , and since is Banach by [F8] and each is a bounded linear operator on it by step 1.1, the sequential uniform boundedness principle [F9] would give , contradicting step 3.1. Hence there is with ; if converged to some in , then for all large , a contradiction. So does not converge in .
The same argument with in place of : each is bounded on by step 1.1, this space is Banach by [F8], and by step 3.1, so [F9] supplies with , and does not converge in .
Step 1.4 proves part 1, and steps 4.1 and 4.2 together with the norm lower bounds of steps 1.3 and 2.1 prove part 2.
Truncated Hilbert transform and principal value
Definition
Fix and a function , with the conventions of Complex Lp classes and Euclidean test-function conventions. For and define the truncated Hilbert transform
Each truncation is an ordinary Lebesgue integral over the complement of an interval of length around , and it is absolutely convergent. For this follows from the pointwise bound on the domain of integration; for it follows from Hölder's inequality applied to the two half-lines and , where has finite norm, with conjugate to (Complex Holder, Minkowski, and the quotient norm). Changing on a null set changes no integral, so is a well-defined number attached to the class of ; and is itself a measurable function of .
The Hilbert transform in the principal-value sense is defined only where the truncations converge:
whenever this limit exists in . No almost-everywhere existence of this limit, and no bound of in any norm, is asserted by this definition. The definition also does not extend to : for the tail the bound is finite but the integral over an unbounded domain is not controlled, so the truncation of a merely bounded need not converge absolutely at any .
Three distinctions are recorded here for later use. First, is an integral of a truncated singular kernel, while the pairing of a test function with the principal-value distribution of is a separate object; the two agree only under the convergence just defined. Second, the limit is taken symmetrically in about the singularity , and unsymmetric truncations are a different object. Third, is a pointwise partial function, whereas the extension constructed later on this page is a single bounded operator agreeing with where the latter exists on a dense class.
The sine integral under Countable Choice: uniform bounds and the value pi/2
Statement
Assume Countable Choice. Put for and , and write for . Then:
- as ; that is, the improper integral converges and equals .
- The partial integrals are uniformly bounded: for every , moreover for , and more precisely for all .
- For every real and every , reading the integrand at as ,
so that and for every and every real .
The argument uses Countable Choice only; it does not invoke the published full-AC sine-integral lemma of the same name on the Dirichlet-kernel page.
Facts & Assumptions
Given: Countable Choice (The Axiom of Countable Choice ()) and the functions and of the statement.
Integration by parts on a compact interval for differentiable factors with integrable derivatives. If are differentiable on with integrable, then
The second fundamental theorem: an integrable derivative integrates to the endpoint increment. The second fundamental theorem: if is differentiable on with and is integrable, then
Under Countable Choice a bounded Riemann integrable function on a compact interval is Lebesgue measurable, and its Riemann and Lebesgue integrals agree. A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
Fubini's theorem for L^1 functions on a sigma-finite product. Fubini's theorem for L^1 functions on a sigma-finite product
Tonelli's theorem for nonnegative product-measurable functions on a sigma-finite product. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
Dominated convergence. Dominated convergence
Sine and cosine have derivatives cosine and minus sine, and , . The derivatives of sine and cosine are cosine and minus sine
The real exponential is its own derivative. The exponential function is smooth and
for every real , hence for and as . for every real , hence
A continuous function on a compact interval is Riemann integrable. A continuous function on is Riemann integrable, by Heine-Cantor and Riemann's criterion
Sine and cosine are 1-Lipschitz: and . Sine and cosine are -Lipschitz on
Parity and the Pythagorean identity: , , , hence and . Parity and the Pythagorean identity for sine and cosine
Continuous maps on Euclidean spaces are Borel measurable, so the product integrands below are measurable. Continuous functions on Euclidean spaces are Borel measurable
Change of variable for improper integrals: for a monotone differentiable surjection satisfying the proper hypotheses on compact truncations, the two improper integrals converge simultaneously and are equal, with orientation retained for decreasing parametrizations. Change of variable in an improper integral
Principal arctangent: and . Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series
Principal arctangent is a continuous strictly increasing bijection from onto , and on the principal interval. The principal inverse tangent
A convergent nonnegative improper Riemann integral on a half-line agrees with the Lebesgue integral of its integrand, under Countable Choice. A nonnegative improper Riemann integral on a half-line agrees with the Lebesgue integral
Additivity of the integral over subintervals, in the oriented form. For : is integrable on if and only if it is integrable on and on , and then ; with the oriented form for arbitrary
Proof
By [F7], and , so as ; and [F12] with gives for , while [F13] gives . Thus is bounded by on and continuous at from the right.
For and , [F8] and [F9] give , and [F10] bounds , so as ; [F2] applied to therefore gives for every , and this tends to as when . By [F18] the nonnegative continuous function is Lebesgue integrable on with .
For , [F2] applied to , whose derivative is by [F7] and [F9], gives , and at both sides equal . For and put ; [F8], [F9] and [F7] give , while [F10], [F12] and [F13] give and as .
By 1.1 the quotient is continuous on and extends continuously to with value , and it is bounded by there; by [F11] it is Riemann integrable on every compact interval , .
Let and , and put on . By [F8] and [F9], is continuously differentiable with , so is nonincreasing, holds nowhere, and [F2] gives . Since by [F7], [F1] applies with factors and and, using from 1.1, yields ; at this is .
Under the given Countable Choice, [F3] applies on every compact interval: for the continuous integrands , and of steps 2.1 and 2.2, the proper Riemann integral on or equals the corresponding Lebesgue integral.
By 2.1 and [F19], for one has with and by 2.2; for the bound follows from in 2.1, and gives . Hence for every and on .
Fix and . By [F6] applied on to the functions as , which converge pointwise to and are dominated by the integrable function of 1.2, and by 3.1 and 1.3, .
Let and . The functions converge pointwise as to and are dominated by the integrable function , so [F6] with 2.2 and 3.1 gives . For , the bound in 2.2 makes Cauchy as and bounds the resulting improper tail by .
For fixed , as for every , with and of finite measure, so [F6] and 3.1 give as .
Fix and put on . By [F14] the integrand is product measurable, and [F5] with 1.2 gives , so of the product and [F4] may be applied. By 1.3, 3.1 and 4.1, the outer -integration of [F4] turns the -inner integral into , while the outer -integration turns the -inner integral into ; hence satisfies , and [F15] with the substitution followed by [F16] gives .
Let and . By 4.2 and 4.3, , so letting and using 5.1 gives : indeed since [F17] makes strictly increasing onto , whence for every one has for all , while always. Letting yields , so the improper integral converges to .
If , the integrand with its assigned value at is identically zero, and the identity, bound and limit follow directly, with . If , both finite integrals vanish. For and , put for , ; by [F7] and [F13], is continuous and even, so [F15] with the substitution on and [F19] give . By [F15] with the substitution (orientation retained, and by [F13]) and [F7], , so ; step 3.2 bounds this by , and step 6.1 gives the limit .
The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier
Statement
Assume Countable Choice and use the Fourier convention of Fourier transform of a tempered distribution. Define the tempered distribution by its pairing with a Schwartz test function, equal to
Then, for every Schwartz function :
- the principal value of Truncated Hilbert transform and principal value exists at every , and equals for the tempered convolution of Convolution of a tempered distribution with a schwartz function;
- hence is a tempered distribution, and , where and is the Schwartz transform of .
The principal value is taken symmetrically about the singularity, and the statement is made for Schwartz functions only; no mapping property and no almost-everywhere statement for general is asserted.
Facts & Assumptions
Given: Countable Choice, the Schwartz space and its seminorms , and the Fourier convention .
The truncated Hilbert transform is , and the principal-value transform is its symmetric limit wherever it exists; the definition asserts no almost-everywhere existence by itself. Truncated Hilbert transform and principal value
The sine integral satisfies , its partial integrals obey for all and for , and in absolute value for . The sine integral under Countable Choice: uniform bounds and the value pi/2
The Fourier transform of a tempered distribution is defined by ; the pairing is bilinear with no conjugation. Fourier transform of a tempered distribution
For and Schwartz one has , the product being the product of a tempered distribution with a smooth polynomially bounded function. Fourier transform converts allowed tempered convolutions to products
The tempered convolution is defined by , a scalar function of . Convolution of a tempered distribution with a schwartz function
A tempered distribution is a continuous complex-linear functional on Schwartz space. Tempered distribution
Schwartz seminorms are finite for . Schwartz space and its seminorms
If then for every , with norm bounded by a finite sum of Schwartz seminorms. Schwartz derivatives are integrable
The Fourier transform is a topological automorphism of Schwartz space, so for . Fourier transform is a topological automorphism of Schwartz space
Mean value theorem: for differentiable , on . The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with
Fubini for L^1 functions on a sigma-finite product. Fubini's theorem for L^1 functions on a sigma-finite product
Dominated convergence. Dominated convergence
Substitution for improper integrals, with orientation retained for decreasing parametrizations. Change of variable in an improper integral
Proof
For both integrals in the definition of converge absolutely: on the bound is integrable, and on the bound from [F10] is integrable on a set of length two. Hence and is a tempered distribution by [F6]. Moreover by oddness of , so for the truncated pairing equals the defining two-piece pairing with the local piece integrated over ; consequently .
Fix and . By [F11] applied on the product of the finite-measure annulus with , using the integrable majorant from [F8], with . Writing the exponential in cosine and sine, the cosine term is odd and integrates to zero, while [F13] with gives for the partial sine integral of [F2]; in particular .
Fix and . For , [F1] gives ; since , this equals . The tail is absolutely convergent by [F8], and the first integral converges as by [F12], the integrand tending pointwise to and being dominated on by thanks to [F10]. The resulting limit is exactly by the defining formula of in 1.1 and the convolution definition [F5]. Hence exists at every and equals .
By 1.1 and [F3], . Holding fixed, [F2] gives as , with . Thus [F12] against yields as . Then for by [F2] makes pointwise as , and a second application of [F12] gives . Combining the two limits with the pairing identity gives for every , that is, as tempered distributions.
By 3.1 and [F4] applied to the tempered distribution and the Schwartz function , , the product being that of the distribution with the Schwartz function supplied by [F9] and [F3]. By 2.2 the same is the pointwise principal-value transform of ; thus the principal value defines the tempered convolution with and has the signum Fourier multiplier, as claimed.
The Hilbert transform is an L2 isometry and squares to minus the identity
Statement
Assume Countable Choice, use the convention, and let with . The Hilbert transform of The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier defines, on Schwartz functions, the operator with . Then extends uniquely to a bounded operator on , still denoted , and for every ,
In particular the single point , where vanishes, is a Lebesgue-null set and creates no zero-mode exception. Nonzero constant functions are not in , so there is no constant mode in the domain to transform.
Facts & Assumptions
Given: Countable Choice and the multiplier of the Schwartz Hilbert transform.
For Schwartz the principal-value Hilbert transform satisfies as tempered distributions. The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier
A measurable multiplier with finite essential supremum defines the bounded operator on ; its Schwartz-core action extends uniquely to , it depends only on the almost-everywhere class of , and . Exact L2 Fourier multiplier norm
Plancherel: is a surjective linear isometry of , so and and . Plancherel theorem
Fourier transformation is injective on tempered distributions. Fourier transform is a topological automorphism of tempered distributions
Proof
The symbol satisfies for every , for every , and the exceptional set is the Lebesgue-null singleton .
For Schwartz , [F2] identifies with an class whose regular tempered distribution has Fourier transform . By [F1], the tempered distribution has the same transform. Injectivity [F4] gives equality of these distributions, so is represented by the class . Thus no membership of the principal value is assumed in this identification.
By [F2] the Schwartz-core action of extends uniquely to a bounded operator on ; by 2.1 the Schwartz action of is that core action, so on , and for , because almost everywhere by 1.1.
Likewise, on the Schwartz core by 1.1 and [F3]; both and are bounded on and agree on the dense Schwartz core, so on all of .
The Hilbert transform is skew-adjoint on L2
Statement
Assume Countable Choice and use the first-variable-linear pairing on . Then for all ,
Equivalently : the Hilbert transform is skew-adjoint, and the statement is a statement about the extension of the Schwartz principal-value operator, not about pointwise values.
Facts & Assumptions
Given: Countable Choice, the first-variable-linear pairing, and the Hilbert transform with symbol .
The Hilbert transform is the operator with multiplier , extending the Schwartz principal-value operator; and . The Hilbert transform is an L2 isometry and squares to minus the identity
For a measurable symbol with essential supremum at most one the operator acts on , and the Schwartz-core action of extends uniquely to it. Exact L2 Fourier multiplier norm
Plancherel: is a surjective linear isometry that preserves the first-variable-linear inner product, . Plancherel theorem
Proof
The symbol satisfies for every : indeed , and both sides vanish at .
Since preserves the inner product by [F3] and is the multiplier operator of [F1] with , one has for all , the last expression being an absolutely convergent integral because and .
Applying 2.1 with the roles of and interchanged and conjugating the symbol by 1.1, , which is the asserted skew-adjointness.
Riesz transforms on Euclidean space
Definition
Assume Countable Choice, let , and let . On define the -th Riesz transform by its Fourier multiplier
where the Fourier transform is the unitary Plancherel extension of Plancherel theorem and the multiplier acts by . The symbol is measurable and for every on account of , so the published multiplier theorem Exact L2 Fourier multiplier norm applies with essential supremum at most one: is a well-defined bounded complex-linear operator on , it depends only on the almost-everywhere class of , and in particular the assigned value has no effect on the operator. The theorem also identifies on Schwartz functions with the regular distribution of , and gives .
The Riesz kernel attached to this definition is the function on
with the Euler integral. Since , the published convergence theorem Euler's Gamma integral converges exactly for positive real parameters gives , so is a positive finite constant and is a smooth function on , odd under and homogeneous of degree : for .
This definition asserts only the multiplier description. It does not assert that the principal value exists for any particular or ; that statement is proved separately for Schwartz functions, as is the identification of the limit with the class . In dimension the constant collapses to , by the value of The real Gamma functional equation , and is the line Hilbert kernel; the comparison of with the Hilbert transform of the line is worked out on the examples page. The Fourier convention is the convention of . Replacing its phase by leaves both and unchanged: the frequency rescaling preserves .
The Riesz transform is the principal value of its kernel, with the matching constant
Statement
Assume Countable Choice, use the convention, and let with be the Riesz kernel of Riesz transforms on Euclidean space. Then for every Schwartz function :
- the truncated integrals converge as for every , with a limit that is continuous in ; and
- that continuous function is a representative of the class , whose Fourier multiplier is .
Existence of the principal value is asserted only for Schwartz , pointwise in ; no almost-everywhere convergence for general or inputs is claimed.
Facts & Assumptions
Given: Countable Choice, , , the Riesz kernel , the symbol for with , and the operator on .
The Riesz kernel is with , smooth and odd on , and ; the operator is the bounded operator with symbol . Riesz transforms on Euclidean space
For , one has , for all , and when . Thus for all : use if , and the tail bound if . The sine integral under Countable Choice: uniform bounds and the value pi/2
Polar coordinates: for nonnegative Borel and, by splitting real and imaginary parts into their positive and negative parts, for integrable complex Borel , and the finite Borel measure is uniquely determined by this property. Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma
for , and volumes scale as . The closed form for the volume of the unit -ball
Fubini for L^1 functions on a sigma-finite product. Fubini's theorem for L^1 functions on a sigma-finite product
Dominated convergence. Dominated convergence
defines the tempered convolution for and Schwartz . Convolution of a tempered distribution with a schwartz function
for and Schwartz . Fourier transform converts allowed tempered convolutions to products
Fourier transformation is a topological automorphism of , hence injective. Fourier transform is a topological automorphism of tempered distributions
with no conjugate on the right-hand side. Fourier transform of a tempered distribution
For a continuous curve differentiable on , the bound implies . Identify with when applying this inequality. The mean value inequality: if is continuous and differentiable on with , then
The real one-variable chain rule applies to compositions of real scalar functions; below it is applied separately to the real and imaginary parts of each coordinate section of . The chain rule, in one line from Carathéodory: if is differentiable at and is differentiable at , then is differentiable at with
Schwartz seminorms: for every integer there is a finite constant with . Schwartz space and its seminorms
Linear change of variables for Lebesgue measure. A linear map of sends Lebesgue measurable sets to Lebesgue measurable sets, with when is invertible and Lebesgue null when it is not
The Gamma function satisfies . The real Gamma functional equation
Proof
Fix and . Write , and put for by [F13], and set . Join to by the coordinate segments with successive endpoints , where . On the -th segment, the real one-variable chain rule [F12] on both components gives for ; this follows from the definition of the coordinate partial derivative and is valid also when , when the curve is constant. Applying [F11] to this complex curve viewed in bounds its increment by . Telescoping gives for every . Hence on the bound is integrable in dimensions, while and the Schwartz bound [F13] with give . Thus on , is integrable. Since by oddness of and symmetry of the annulus, , and in the first term yields the absolutely convergent limit . For , the sequence is bounded, so the tail constants have a common finite bound. This and the common small-ball bound supply integrable dominators for [F6]; continuity of gives pointwise convergence in both integrals, hence .
For and put . The cosine part of the integrand is odd in , so it integrates to zero on the symmetric annulus, and gives . Polar coordinates [F3] turn this into .
For one has with : by [F3] the measure is invariant under the orthogonal map , so substituting for an orthogonal map with (take if , and otherwise take with ) and reflecting for (which preserves and kills the other components by oddness) leaves only .
: compute twice. Polar coordinates [F3] give ; for , slicing at gives, by [F5], [F14] and [F4], , hence . For the sphere is : the defining identity of [F3], applied to functions supported in the annulus , shows that the measure is the counting measure , so , while by of [F15]. Hence for every , and by cancellation of and .
In 1.2 let and . For each fixed with , the substitution (with orientation, [F2]) gives ; when the integral is zero. In all cases [F2] bounds its absolute value by , uniformly in and . Since the sphere has finite measure, [F6] on gives by 1.3 and the constant identity of 1.4.
Define the tempered distribution by the symmetric principal-value pairing for ; the two-piece bound of 1.1 shows the limit exists, is finite, and is Schwartz-continuous. By [F10], ; the double integrand is absolutely integrable since , so [F5] applies, giving . By 2.1 the bracket converges to pointwise off the null set , and by the uniform bound of 2.1 it is dominated by a constant times ; [F6] therefore yields , i.e. as tempered distributions.
By [F8] and 3.1, ; by [F1] the class has Fourier transform as well, so the two tempered distributions agree and [F9] gives . By [F7] and the definition of in 3.1, the value is exactly the limit of 1.1; the continuity in 1.1 therefore makes a continuous representative of the class , and the truncated integrals converge to it at every .
Riesz transforms are L2 contractions and square to minus the identity in sum
Statement
Assume Countable Choice and let . For the Riesz transforms of the multiplier definition,
and
Both statements are statements only; no bound for is asserted, and the operators are the operators of the definition, so all identities hold as classes (no pointwise statement is made).
Facts & Assumptions
Given: Countable Choice, the dimension , and the Riesz transforms with symbols for and .
Each is defined as the bounded operator with multiplier ; the symbol is measurable with everywhere, the value at the origin is immaterial, and the definition asserts no more than the multiplier description. Riesz transforms on Euclidean space
A measurable multiplier with finite essential supremum defines the bounded operator with , and depends only on the almost-everywhere class of . Exact L2 Fourier multiplier norm
Plancherel: is a surjective complex-linear isometry of , so . Its inverse is complex-linear, hence and for every class . Plancherel theorem
Proof
For every the symbol values satisfy , while ; the single point is Lebesgue null. Hence the function is measurable, bounded with , and equals the constant almost everywhere.
By [F2] applied to the bounded measurable symbol of [F1], ; consequently, for and using the isometry of [F3], .
Since , composition gives in the notation of [F2], and summing the finitely many bounded operators gives for the almost-everywhere- symbol of 1.1.
By [F2] the operator depends only on the almost-everywhere class of , which by 1.1 is the class of the constant ; hence , and by the linearity and isometry of [F3]. Therefore for every .
Riesz kernel size, difference and spherical-cancellation bounds
Statement
Assume Countable Choice, let and , and let with be the Riesz kernel of Riesz transforms on Euclidean space. Then:
- for every ;
- with one has whenever and ; and
- for every , where is the polar surface measure of Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma on the unit sphere .
The constant is explicit and depends only on ; at it reads . These are the raw size, first-difference and cancellation estimates that a later singular-integral treatment consumes; no Calderón–Zygmund kernel definition is invoked here.
Facts & Assumptions
Given: Countable Choice, , , and the Riesz kernel with .
The Riesz kernel has , is smooth, odd and homogeneous of degree on , so whenever and . Riesz transforms on Euclidean space
If is a norm on a real vector space and are vectors, then . The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for
For and the Euclidean norm satisfies , hence for every coordinate . The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for
Mean value theorem: a real function continuous on a closed interval and differentiable on its interior has a point whose derivative equals the average rate of change. The mean value theorem, as the case of Cauchy's: for continuous on with and differentiable on there is with
For every natural the function is differentiable on with derivative . For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
Polar coordinates: for and every Borel , , and is a finite Borel measure on . Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma
Linear change of variables for Lebesgue measure, in particular for invertible linear . A linear map of sends Lebesgue measurable sets to Lebesgue measurable sets, with when is invertible and Lebesgue null when it is not
Proof
Fix . By [F1] the kernel is , so by the coordinate bound of [F3] and the positivity .
Fix and with and put . By [F2] applied to the Euclidean norm, , so ; by [F3], and . In particular and the kernel is defined at both arguments.
Let and . The function is Borel and ; the map is a linear bijection with , so [F7] gives , while gives , that is, . On the other hand [F6] applied to the nonnegative and the negative part of gives with , so . Hence for every the homogeneity [F1] gives .
Keep and as in 1.2, put and ; by [F5] with one has on . The interval with endpoints and lies in by 1.2. If , then and the following bound is immediate. If , [F4] on the interval with ordered endpoints gives a point with and therefore . Insert into the difference and expand: , so by 1.2 and the preceding bound, and by , with , since and .
The three assertions are proved: for is 1.1; the difference bound with the stated constant is 2.1, whose hypothesis keeps both arguments nonzero as recorded in 1.2; and the vanishing of every spherical integral , , is 1.3.
Endpoint map for Hilbert and Riesz transforms
Statement
This page proves the strict-range facts for the periodic conjugate operator and the facts for the line Hilbert transform and the Euclidean Riesz transforms: The Marcel Riesz conjugate-function theorem on the circle bounds the conjugate operator on for and records the failure of compatible strong-type and extensions; The Hilbert transform is an L2 isometry and squares to minus the identity identifies the line Hilbert transform as the multiplier by with ; and Riesz transforms are L2 contractions and square to minus the identity in sum gives the Euclidean contractions with .
Three distinct endpoints are deliberately not settled here, and the reader should not read this page as a negative statement about them. First, weak bounds, the real-line strict-range theory for the line and Riesz transforms, and the almost-everywhere convergence of truncated integrals are deferred to the later Calderón–Zygmund decomposition and singular-integrals material, which supplies the covering and Calderón–Zygmund kernel estimates this page stops short of. Second, the real Hardy space endpoint belongs to the later real Hardy space and maximal-function material. Third, the bounded mean-oscillation endpoint belongs to the later BMO material. Each of those later pages is named here by title only; no result from them is used as a premise anywhere on this page.
Two further distinctions are recorded. The periodic conjugate operator is presented through the circle's zero mode: constants lie in its kernel and the multiplier vanishes at frequency , whereas on the line the corresponding signum multiplier vanishes on a Lebesgue-null singleton and the square identity holds with no zero-mode exception. The line Hilbert and Riesz endpoint questions are distinct from the circle's partial-sum operator norms. For the periodic conjugate operator itself, however, the Lebesgue-constant lower bound above is used to rule out compatible strong and extensions.
Finally, the companion examples page constructs the interval indicator whose Hilbert transform is and uses it to refute a bounded strong-type action and a bounded action compatible with the transform. Those computations refute strong-type mapping only: they are consistent with a weak bound and with a bounded BMO-valued endpoint, and they say nothing against the deferred results named above.
5 · Examples, counterexamples and false statements
None yet.