Singular integral operators of convolution type

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rⁿ and Tⁿ through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L² is evident because the Fourier transform converts them into multiplication operators. Continuity on L^p spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on L^p spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

L² theory

Hilbert transform on the circle

Hilbert transform on the real line

Riesz transforms in the complex plane

Main article: Riesz transform

The complex Riesz transforms R and R* in the complex plane are the unitary operators on L²(C) defined as multiplication by z/|z| and its conjugate on the Fourier transform of an L² function f:

\displaystyle{\widehat{Rf}(z)={\overline{z}\over |z|} \widehat{f}(z),\,\,\, \widehat{R^*f}(z)={z\over |z|} \widehat{f}(z).}

Idenitifying C with R², R and R* are given by

\displaystyle{R=-iR_1 + R_2,\,\,\, R^*=-iR_1 - R_2,}

where R₁ and R₂ are the Riesz transforms on R² defined below.

On L²(C), the operator R and its integer powers are unitary. They can also be expressed as singular integral operators:^[15]

\displaystyle{R^kf(w)=\lim_{\varepsilon\rightarrow 0} \int_{|z-w|\ge \varepsilon} M_k(w-z)f(z)\,dx\, dy,}

where

\displaystyle{M_k(z)={k\over 2\pi i^k} {z^k \over |z|^{k+2}} \,\,\,\, (k\ge 1), \,\,\,\, M_{-k}(z) =\overline{M_k(z)}.}

Defining the truncated higher Riesz transforms as

\displaystyle{R^{(k)}_\varepsilon f(w)=\int_{|z-w|\ge \varepsilon} M_k(w-z)f(z)\,dx\, dy,}

these operators can be shown to be uniformly bounded in operator norm. For odd powers this can be deduced by the method of rotation of Calderón and Zygmund, described below.^[16] If the operators are known to be bounded in operator norm it can also be deduced using the Poisson operators.^[17]

The Poisson operators T_s on R² are defined for s > 0 by

\displaystyle{T_sf(x) ={1\over 2\pi}\int_{\mathbf{R}^2} {s f(x)\over (|x-t|^2 + s^2)^{3/2}}\, dt.}

They are given by convolution with the functions

\displaystyle{P_s(x)= {s\over 2\pi(|x|^2 + s^2)^{3/2} }.}

P_s is the Fourier transform of the function e^{− s|x|}, so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L²(R²). Since P_y is positive and integrable with integral 1, the operators T_s also define a contraction semigroup on each L^p space with 1 < p < ∞.

The higher Riesz transforms of the Poisson kernel can be computed:

\displaystyle{R^kP_s(z)={k\over 2\pi i^k} {z^k \over (|z|^2 +s^2)^{k/2+1}}}

for k ≥ 1 and the complex conjugate for − k. Indeed, the right hand side is a harmonic function F(x,y,s) of three variable and for such functions^[18]

\displaystyle{T_{s_1} F(x,y,s_2) = F(x,y,s_1+s_2).}

As before the operators

\displaystyle{T_\varepsilon R^k - R^{(k)}_\varepsilon}

are given by convolution with integrable functions and have uniformly bounded operator norms. Since the Riesz transforms are unitary on L²(C), the uniform boundedness of the truncated Riesz transforms implies that they converge in the strong operator topology to the corresponding Riesz transforms.

The uniform boundedness of the difference between the transform and the truncated transform can also be seen for odd k using the Calderón-Zygmund method of rotation.^[19]^[20] The group T acts by rotation on functions on C via

\displaystyle{U_\theta f(z)=f(e^{i\theta}z).}

This defines a unitary representation on L²(C) and the unitary operators R_θ commute with the Fourier transform. If A is a bounded operator on L²(R) then it defines a bounded operator A⁽¹⁾ on L²(C) simply by making A act on the first coordinate. With the identification L²(R²) = L²(R) ⊗ L²(R), A⁽¹⁾ = A ⊗ I. If φ is a continuous function on the circle then a new operator can be defined by

\displaystyle{B ={1\over 2\pi} \int_0^{2\pi} \varphi(\theta) U_\theta A^{(1)} U_\theta^* \, d\theta.}

This definition is understood in the sense that

\displaystyle{(Bf,g) ={1\over 2\pi} \int_0^{2\pi} \varphi(\theta) (U_\theta A^{(1)} U_\theta^*f,g) \, d\theta}

for any f, g in L²(C). It follows that

\displaystyle{\| B \| \le {1\over 2\pi} \int_0^{2\pi} |\varphi(\theta)| \cdot\|A\|\, d\theta.}

Taking A to be the Hilbert transform H on L²(R) or its truncation H_ε, it follows that

\begin{align} R &={1\over 2\pi} \int_0^{2\pi} e^{-i\theta} U_\theta H^{(1)} U_\theta^* \, d\theta,\\ R_\varepsilon &={1\over 2\pi} \int_0^{2\pi} e^{-i\theta} U_\theta H^{(1)}_\varepsilon U_\theta^* \, d\theta. \end{align}

Taking adjoints gives a similar formuls for R* and its truncation. This gives a second way to verify estimate the norms of R, R* and their truncations. It has the advantage of being applicable also for L^p spaces.

The Poisson operators can also be used to show that the truncated higher Riesz transforms of a function tend to the higher Riesz transform at the common Lebesgue points of the function and its transform. Indeed, (R^kT_ε − R^(k)_ε)f → 0 at each Lebesgue point of f; while (R^k − R^kT_ε)f → 0 at each Lebesgue point of R^kf.^[21]

Beurling transform in the complex plane

Riesz transforms in higher dimensions

Main article: Riesz transform

For f in the Schwartz space of Rⁿ, the jth Riesz transform is defined by

R_j f(x) =c_n\lim_{\varepsilon \to 0} \int_{|y|\ge \varepsilon} f(x-y){y_j\over |y|^{n+1}}dy= \frac{c_n}{n-1}\int \partial_j f(x-y){1\over |y|^{n-1}} dy,

where

c_n=\Gamma\left(\tfrac{n+1}{2}\right)\pi^{-\frac{n+1}{2}}.

Under the Fourier transform:

\widehat{R_j f}(t)={it_j\over |t|}\widehat{f}(t).

Thus R_j corresponds to the operator ∂_jΔ^−1/2, where Δ = −∂₁² − ... −∂_n² denotes the Laplacian on Rⁿ. By definition R_j is a bounded and skew-adjoint operator for the L² norm and

R_1^2 + \cdots + R_n^2 = -I.

The corresponding truncated operators

R_{j,\varepsilon} f(x) =c_n\int_{|y|\ge \varepsilon} f(x-y){y_j\over |y|^{n+1}} dy

are uniformly bounded in the operator norm. This can either be proved directly or can be established by the Calderón−Zygmund method of rotations for the group SO(n).^[24] This expresses the operators R_j and their truncations in terms of the Hilbert transforms in one dimension and its truncations. In fact if G = SO(n) with normalised Haar measure and H⁽¹⁾ is the Hilbert transform in the first coordinate, then

\begin{align} R_j &=\int_G \varphi(g) gH^{(1)}g^{-1} \, dg, \\ R_{j,\varepsilon} &=\int_G \varphi(g) gH_\varepsilon^{(1)} g^{-1} \, dg, \\ R_{j,\varepsilon,R} &=\int_G \varphi(g) gH_{\varepsilon,R}^{(1)} g^{-1} \, dg. \end{align}

where φ(g) is the (1,j) matrix coefficient of g.

In particular for f ∈ L², R_j,εf → R_jf in L². Moreover, R_j,εf tends to R_j almost everywhere. This can be proved exactly as for the Hilbert transform by using the Poisson operators defined on L²(Rⁿ) when Rⁿ is regarded as the boundary of a halfspace in Rⁿ⁺¹. Alternatively it can be proved directly from the result for the Hilbert transform on R using the expression of R_j as an integral over G.^[25]^[26]

The Poisson operators T_y on Rⁿ are defined for y > 0 by^[27]

T_yf(x) =c_n\int_{\mathbf{R}^n} \frac{y f(x)}{\left (|x-t|^2 + y^2 \right )^{\frac{n+1}{2}}} dt.

They are given by convolution with the functions

P_y(x)=c_n \frac{y}{\left (|x|^2 + y^2 \right )^{\frac{n+1}{2}}}.

P_y is the Fourier transform of the function e^−y|x|, so under the Fourier transform they correspond to multiplication by these functions and form a contraction semigroup on L²(Rⁿ). Since P_y is positive and integrable with integral 1, the operators T_y also define a contraction semigroup on each L^p space with 1 < p < ∞.

The Riesz transforms of the Poisson kernel can be computed

R_j P_\varepsilon(x)= c_n \frac{x_j}{\left(|x|^2 + \varepsilon^2 \right)^{\frac{n+1}{2}}}.

The operator R_jT_ε is given by convolution with this function. It can be checked directly that the operators R_jT_ε − R_j,ε are given by convolution with functions uniformly bounded in L¹ norm. The operator norm of the difference is therefore uniformly bounded. We have (R_jT_ε − R_j,ε)f → 0 at each Lebesgue point of f; while (R_j − R_jT_ε)f → 0 at each Lebesgue point of R_jf. So R_j,εf → R_jf on the common Lebesgue points of f and R_jf.

L^p theory

Elementary proofs of M. Riesz theorem

The theorem of Marcel Riesz asserts that singular integral operators that are continuous for the $L 2$ norm are also continuous in the $L p$ norm for $1 < p < \infty$ and that the operator norms vary continuously with $p$ .

Bochner's proof for Hilbert transform on the circle ^[28]

Once it is established that the operator norms of the Hilbert transform on $L p (T)$ are bounded for even integers, it follows from the Riesz–Thorin interpolation theorem and duality that they are bounded for all $p$ with $1 < p < \infty$ and that the norms vary continuously with $p$ . Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms $H ε$ are uniformly bounded in operator norm and converge in the strong operator topology to $H$ .

It is enough to prove the bound for real trigonometric polynomials without constant term:

f \left (e^{i\theta} \right ) = \sum_{m=1}^N a_m e^{im\theta} + a_{-m} e^{-im\theta}, \qquad a_{-m}=\overline{a_m}.

Since f + iHf is a polynomial in $e iθ$ without constant term

\frac{1}{2\pi}\int_0^{2\pi} (f+iHf)^{2n} \, d\theta = 0.

Hence, taking the real part and using Hölder's inequality:

\|Hf\|_{2n}^{2n} \le \sum_{k=0}^{n-1} {2n\choose 2k} \left | \left ((Hf)^{2k},f^{2n-2k} \right ) \right |\le \sum_{k=0}^{n-1} {2n\choose 2k} \|Hf\|_{2n}^{2k}\cdot\|f\|_{2n}^{2n-2k}.

So the M. Riesz theorem follows by induction for $p$ an even integer and hence for all $p$ with $1 < p < \infty$ .

Cotlar's proof for Hilbert transform on the line ^[29]

Once it is established that the operator norms of the Hilbert transform on $L p (R)$ are bounded when $p$ is a power of 2, it follows from the Riesz–Thorin interpolation theorem and duality that they are bounded for all $p$ with $1 < p < \infty$ and that the norms vary continuously with $p$ . Moreover, the arguments with the Poisson integral can be applied to show that the truncated Hilbert transforms $H ε$ are uniformly bounded in operator norm and converge in the strong operator topology to $H$ .

It is enough to prove the bound when f is a Schwartz function. In that case the following identity of Cotlar holds:

(Hf)^2= f^2 + 2H(fH(f)).

In fact, write f = f₊ + f₋ according to the $\pm i$ eigenspaces of H. Since f ± iHf extend to holomorphic functions in the upper and lower half plane, so too do their squares. Hence

f^2 -(Hf)^2= \left (f_+ + f_- \right )^2 + \left (f_+-f_- \right )^2 =2 \left (f_+^2 + f_-^2 \right )=-2iH \left (f_+^2 -f_-^2 \right )=-2H(f(Hf)).

(Cotlar's identity can also be verified directly by taking Fourier transforms.)

Hence, assuming the M. Riesz theorem for $p = 2 n$ ,

\|Hf\|^2_{2^{n+1}} = \left \|(Hf)^2 \right \|_{2^n} \le \left \|f^2 \right \|_{2^n} + 2 \|H(fH(f))\|_{2^n} \le \|f\|_{2^{n+1}}^2 + 2 \|H\|_{2^n} \|f\|_{2^{n+1}} \|Hf\|_{2^{n+1}}.

Since

R^2 > 1 + 2 \|H\|_{2^n} R

for $R$ sufficiently large, the M. Riesz theorem must also hold for $p = 2 n +1$ .

Exactly the same method works for the Hilbert transform on the circle.^[30] The same identity of Cotlar is easily verified on trigonometric polynomials f by writing them as the sum of the terms with non-negative and negative exponents, i.e. the $\pm i$ eigenfunctions of $H$ . The $L p$ bounds can therefore be established when $p$ is a power of 2 and follow in general by interpolation and duality.

Calderón–Zygmund method of rotation

The method of rotation for Riesz transforms and their truncations applies equally well on $L p$ spaces for $1 < p < \infty$ . Thus these operators can be expressed in terms of the Hilbert transform on $R$ and its truncations. The integration of the functions $Φ$ from the group $T$ or $SO(n)$ into the space of operators on $L p$ is taken in the weak sense:

\left (\int_G \Phi(x)\, dx\, f,g \right ) =\int_G (\Phi(x)f,g)\, dx

where f lies in $L p$ and $g$ lies in the dual space $L q$ with $1 / p + 1 / q .$ It follows that Riesz transforms are bounded on $L p$ and that the differences with their truncations are also uniformly bounded. The continuity of the $L p$ norms of a fixed Riesz transform is a consequence of the Riesz–Thorin interpolation theorem.

Pointwise convergence

Maximal functions

Much of the L^p theory has been developed using maximal functions and maximal transforms. This approach has the advantage that it also extends to L¹ spaces in an appropriate "weak" sense and gives refined estimates in L^p spaces for p > 1. These finer estimates form an important part of the techniques involved in Lennart Carleson's solution in 1966 of Lusin's conjecture that the Fourier series of L² functions converge almost everywhere.^[39] In the more rudimentary forms of this approach, the L² theory is given less precedence: instead there is more emphasis on the L¹ theory, in particular its measure-theoretic and probabilistic aspects; results for other L^p spaces are deduced by a form of interpolation between L¹ and L^∞ spaces. The approach is described in numerous textbooks, including the classics Zygmund (1977) and Katznelson (2004). Katznelson's account is followed here for the particular case of the Hilbert transform of functions in L¹(T), the case not covered by the development above.^[40]

If f is an L¹ function on the circle its maximal function is defined by^[41]

\displaystyle{f^*(t)=\sup_{0<h\le \pi} {1\over 2h} \int_{t-h}^{t+h} |f(s)|\, ds.}

f* is finite almost everywhere and is of weak L¹ type. In fact for λ > 0 if

\displaystyle{E_f(\lambda)=\{x:\,|f(x)| > \lambda\}, \,\, f_\lambda =\chi_{E(\lambda)} f,}

then^[42]

m(E_{f^*}(\lambda))\le {8\over \lambda}\int_{E_f(\lambda)} |f|\le {8\|f\|_1\over \lambda},

where m denotes Lebesgue measure.

The Hardy−Littlewood inequality above leads to a proof that almost every point x of T is a Lebesgue point of an integrable function f, so that

\displaystyle{\lim_{h\to 0} \frac{\int^{x+h}_{x-h}|f(t)-f(x)|\, dt}{2h} \to 0.}

In fact let

\omega(f)(x)=\limsup_{h\to 0} \frac{\int^{x+h}_{x-h}|f(t)-f(x)|\, dt}{2h} \le f^*(x) +|f(x)|.

If g is continuous, then the ω(g) =0, so that ω(f − g) = ω(f). On the other hand, f can be approximated arbitrarily closely in L¹ by continuous g. Then, using Chebychev's inequality,

m\{x: \, \omega(f)(x)> \lambda\} =m\{x: \, \omega(f-g)(x)> \lambda\} \le m\{x: \, (f-g)^*(x)> \lambda\} + m\{x: \, |f(x)-g(x)|> \lambda\} \le C\lambda^{-1}\|f-g\|_1.

The right hand side can be made arbitrarily small, so that ω(f) = 0 almost everywhere.

The Poisson integrals of an L¹ function f satisfy^[43]

\displaystyle{|T_rf|\le f^*.}

It follows that T_r f tends to f pointwise almost everywhere. In fact let

\displaystyle{\Omega(f)=\limsup_{r\rightarrow 1} |T_rf -f|.}

If g is continuous, then the difference tends to zero everywhere, so Ω(f − g) = Ω(f). On the other hand, f can be approximated arbitrarily closely in L¹ by continuous g. Then, using Chebychev's inequality,

m\{x: \, \Omega(f)(x)> \lambda\} =m\{x: \, \Omega(f-g)(x)> \lambda\} \le m\{x: \, (f-g)^*(x)> \lambda\} + m\{x: \, |f(x)-g(x)|> \lambda\} \le C\lambda^{-1}\|f-g\|_1.

The right hand side can be made arbitrarily small, so that Ω(f) = 0 almost everywhere. A more refined argument shows that convergence occurs at each Lebesgue point of f.

If f is integrable the conjugate Poisson integrals are defined and given by convolution by the kernel Q_r. This defines Hf inside |z| < 1. To show that Hf has a radial limit for almost all angles,^[44] consider

\displaystyle{F(z)=\exp (-f(z) -iHf(z)),}

where f(z) denotes the extension of f by Poisson integral. F is holomorphic in the unit disk with |F(z)| ≤ 1. The restriction of F to a countable family of concentric circles gives a sequence of functions in L^∞(T) which has a weak g limit in L^∞(T) with Poisson integral F. By the L² results, g is the radial limit for almost all angles of F. It follows that Hf(z) has a radial limit almost everywhere. This is taken as the definition of Hf on T, so that T_rH f tends pointwise to H almost everywhere. The function Hf is of weak L¹ type.^[45]

The inequality used above to prove pointwise convergence for L^p function with 1 < p < ∞ make sense for L¹ functions by invoking the maximal function. The inequality becomes

\displaystyle{|H_\varepsilon f - T_{1-\varepsilon}Hf|\le 4f^*.}

Let

\displaystyle{\omega(f)=\limsup_{\varepsilon\rightarrow 0} |H_\varepsilon f - T_{1-\varepsilon}Hf|.}

If g is smooth, then the difference tends to zero everywhere, so ω(f − g) = ω(f). On the other hand, f can be approximated arbitrarily closely in L¹ by smooth g. Then

m\{x: \, \omega(f)(x)> \lambda\} =m\{x: \, \omega(f-g)(x)> \lambda\} \le m\{x: \, 4(f-g)^*(x)> \lambda\} \le C\lambda^{-1}\|f-g\|_1.

The right hand side can be made arbitrarily small, so that ω(f) = 0 almost everywhere. Thus the difference for f tends to zero almost everywhere. A more refined argument can be given^[46] to show that, as in case of L^p, the difference tends to zero at all Lebesgue points of f. In combination with the result for the conjugate Poisson integral, it follows that, if f is in L¹(T), then H_εf converges to Hf almost everywhere, a theorem originally proved by Privalov in 1919.

General theory

Calderón & Zygmund (1952) introduced general techniques for studying singular integral operators of convolution type. In Fourier transform the operators are given by multiplication operators. These will yield bounded operators on L² if the corresponding multiplier function is bounded. To prove boundedness on L^p spaces, Calderón and Zygmund introduced a method of decomposing L¹ functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous operator from L¹ to the space of functions of weak L¹. The Marcinkiewicz interpolation theorem and duality then implies that the singular integral operator is bounded on all L^p for 1 < p < ∞. A simple version of this theory is described below for operators on R. As de Leeuw (1965) showed, results on R can be deduced from corresponding results for T by restricting the multiplier to the integers, or equivalently periodizing the kernel of the operator. Corresponding results for the circle were originally established by Marcinkiewicz in 1939. These results generalize to Rⁿ and Tⁿ. They provide an alternative method for showing that the Riesz transforms, the higher Riesz transforms and in particular the Beurling transform define bounded operators on L^p spaces.^[47]

Calderón-Zygmund decomposition

Let f be a non-negative integrable or continuous function on [a,b]. Let I = (a,b). For any open subinterval J of [a,b], let f_J denote the average of |f| over J. Let α be a positive constant greater than f_I. Divide I into two equal intervals (omitting the midpoint). One of these intervals must satisfy f_J < α since their sum is 2f_I so less than 2α. Otherwise the interval will satisfy α ≤ f_J < 2α. Discard such intervals and repeat the halving process with the remaining interval, discarding intervals using the same criterion. This can be continued indefinitely. The discarded intervals are disjoint and their union is an open set Ω. For points x in the complement, they lie in a nested set of intervals with lengths decreasing to 0 and on each of which the average of f is bounded by α. If f is continuous these averages tend to |f(x)|. If f is only integrable this is only true almost everywhere, for it is true at the Lebesgue points of f by the Lebesgue differentiation theorem. Thus f satisfies |f(x)| ≤ α almost everywhere on Ω^c, the complement of Ω. Let J_n be the set of discarded intervals and define the "good" function g by

\displaystyle{g(x)= \chi_{J_n}(f)\,\,\, (x\in J_n), \,\,\,\,\,g(x) = f(x)\,\,\, (x\in \Omega^c).}

By construction |g(x)| ≤ 2α almost everywhere and

\displaystyle{\|g\|_1 \le \|f\|_1.}

Combining these two inequalities gives

\displaystyle{\|g\|_p^p \le (2\alpha)^{p-1}\|f\|_1.}

Define the "bad" function b by b = f − g. Thus b is 0 off Ω and equal to f minus its average on J_n. So the average of b on J_n is zero and

\displaystyle{\|b\|_1 \le 2\|f\|_1.}

Moreover, since |b| ≥ α on Ω

\displaystyle{m(\Omega)\le \alpha^{-1} \|f\|_1.}

The decomposition

\displaystyle{f(x) =g(x) + b(x)}

is called the Calderón–Zygmund decomposition.^[48]

Multiplier theorem

Let K(x) be a kernel defined on R\{0} such that

W(f)=\lim_{\varepsilon\to 0}\int_{|x|\ge \varepsilon} K(x)f(x)\,dx

exists as a tempered distribution for f a Schwartz function. Suppose that the Fourier transform of T is bounded, so that convolution by W defines a bounded operator T on L²(R). Then if K satisfies Hörmander's condition

A=\sup_{y\ne 0} \int_{|x|\ge 2|y|}|K(x-y) - K(x)|\, dx <\infty,

then T defines a bounded operator on L^p for 1 < p < ∞ and a continuous operator from L¹ into functions of weak type L¹.^[49]

In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if f is smooth of compact support then

m\{x:\, |Tf(x)| \ge 2\lambda\} \le (2A+4\|T\|)\cdot \lambda^{-1} \|f\|_1.

Take a Calderón−Zygmund decomposition of f as above

f(x)=g(x)+b(x)

with intervals J_n and with α = λμ, where μ > 0. Then

m\{x:\, |Tf(x)| \ge 2\lambda\} \le m\{x:\, |Tg(x)| \ge \lambda\} + m\{x:\, |Tb(x)| \ge \lambda\}.

The term for g can be estimated using Chebychev's inequality:

m\{x:\, |Tg(x)| \ge 2\lambda\} \le \lambda^{-2} \|Tg\|_2^2 \le \lambda^{-2}\|T\|^2 \|g\|_2^2 \le 2\lambda^{-1}\mu \|T\|^2 \|f\|_1.

If J* is defined to be the interval with the same centre as J but twice the length, the term for b can be broken up into two parts:

m\{x:\, |Tb(x)| \ge \lambda\}\le m\{x:\, x\notin \cup J_n^*,\,\,\, |Tb(x)| \ge \lambda\} +m(\cup J_n^*).

The second term is easy to estimate:

m(\cup J_n^*)\le \sum m(J_n^*)=2\sum m(J_n) \le 2\lambda^{-1} \mu^{-1} \|f\|_1.

To estimate the first term note that

b=\sum b_n, \qquad b_n =(f - \mathbf{Av}_{J_n}(f))\chi_{J_n}.

Thus by Chebychev's inequality:

m\{x:\, x\notin \cup J_m^*,\,\,\, |Tb(x)| \ge \lambda\}\le \lambda^{-1}\int_{(\cup J_m^*)^c} |Tb(x)|\, dx \le \lambda^{-1} \sum_n\int_{( J_n^*)^c} |Tb_n(x)|\, dx.

By construction the integral of b_n over J_n is zero. Thus, if y_n is the midpoint of J_n, then by Hörmander's condition:

\int_{( J_n^*)^c} |Tb_n(x)|\, dx= \int_{( J_n^*)^c} \left|\int_{J_n} (K(x-y)-K(x-y_n))b_n(y)\, dy\right|\, dx \le \int_{J_n} |b_n(y)| \int_{( J_n^*)^c} |K(x-y)-K(x-y_n)|\, dxdy \le A\|b_n\|_1.

Hence

m \left \{x:\, x\notin \cup J_m^*, |Tb(x)| \ge \lambda \right \} \le \lambda^{-1} A \|b\|_1 \le 2 A\lambda^{-1}\|f\|_1.

Combining the three estimates gives

m\{x:\, |Tf(x)| \ge \lambda\} \le \left (2\mu\|T\|^2 +2\mu^{-1} + 2A \right )\lambda^{-1}\|f\|_1.

The constant is minimized by taking

\mu=\|T\|^{-1}.

The Markinciewicz interpolation argument extends the bounds to any L^p with 1 < p < 2 as follows.^[50] Given a > 0, write

f=f_a + f^a,

where f_a = f if |f| < a and 0 otherwise and f^a = f if |f| ≥ a and 0 otherwise. Then by Chebychev's inequality and the weak type L¹ inequality above

m\{x:\, |Tf(x)| > a\} \le m \left \{x:\, |Tf_a(x)| > \tfrac{a}{2} \right \}+ m \left \{x:\, |Tf^a(x)| > \tfrac{a}{2} \right \}\le 4a^{-2}\|T\|^2 \|f_a\|_2^2 +C a^{-1}\|f^a\|_1.

Hence

\begin{align} \|Tf\|_p^p &= p\int_0^\infty a^{p-1} m\{x:\, |Tf(x)|> a\} \, da \\ &\le p \int_0^\infty a^{p-1} \left ( 4a^{-2}\|T\|^2 \|f_a\|_2^2 +C a^{-1}\|f^a\|_1 \right ) da \\ &=4\|T\|^2 \iint_{|f(x)|<a} |f(x)|^2 a^{p-3}\,dx\, da + 2C\iint_{|f(x)|\ge a} |f(x)| a^{p-2}\,dx\, da \\ &\le \left (4\|T\|^2(2-p)^{-1} + C (p-1)^{-1} \right ) \int |f|^p \\ &=C_p \|f\|_p^p. \end{align}

By duality

\|Tf\|_q \le C_p \|f\|_q.

Continuity of the norms can be shown by a more refined argument^[51] or follows from the Riesz–Thorin interpolation theorem.

Notes

↑ Torchinsky 2004, pp. 65–66
↑ Bell 1992, pp. 14–15
↑ Krantz 1999
↑ Torchinsky 1986
↑ Stein & Rami 2005, pp. 112–114
↑
See:
↑ Garnett 2007, p. 102
↑
See:
↑ Stein & Shakarchi 2005, pp. 213–221
↑ Hörmander 1990
↑ Titchmarsh & 1939 102–105
↑
See:
- Krantz 1999
- Torchinsky 1986
- Duoandikoetxea 2001, pp. 49–51
↑ Stein & Shakarchi 2005, pp. 112–114
↑ Stein & Weiss 1971
↑ Astala, Ivaniecz & Martin 2009, pp. 101–102
↑ Grafakos 2005
↑ Stein & Weiss 1971
↑ Stein & Weiss 1971, p. 51
↑ Grafakos 2008
↑ Stein & Weiss 1971, pp. 222–223
↑ Stein & Weiss 1971
↑ Astala, Iwaniecz & Martin 2009, pp. 93–95
↑ Astala, Iwaniecz & Martin 2009, pp. 97–98
↑ Grafokos 2008, pp. 272–274
↑ Grafakos 2008
↑ Stein & Weiss 1971, pp. 222–223, 236–237
↑ Stein & Weiss 1971
↑ Grafakos 2005, p. 215−216
↑ Grafakos 2005, p. 255−257
↑ Gohberg & Krupnik 1992, pp. 19–20
↑
See:
- Stein & Weiss 1971, pp. 12–13
- Torchinsky 2004
↑ Torchinsky 2005, pp. 41–42
↑ Katznelson 2004, pp. 10–21
↑ Stein & Shakarchi 112-114
↑ Garnett 2007, pp. 102–103
↑ Krantz 1999
↑ Torchinsky 1986
↑ Stein & Shakarchi 2005, pp. 112–114
↑ Arias de Reyna 2002
↑
See also:
↑ Krantz 1999, p. 71
↑ Katznelson 2004, pp. 74–75
↑ Katznelson 2004, p. 76
↑ Katznelson 2004, p. 64
↑ Katznelson 2004, p. 66
↑ Katznelson 2004, pp. 78–79
↑
See:
- Hörmander 1990
- Torchinsky 2005
- Grafakos 2008
- Stein 1970
- Stein & Weiss 1971, pp. 257–267
↑ Torchinsky 2005, pp. 74–76,84–85
↑ Grafakos 2008, pp. 290–293
↑ Hörmander 1990, p. 245
↑ Torchinsky 2005, pp. 87–91

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