Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

Szegö limit theorems in quantum mechanics

We prove a Szegö-type theorem for some Schrödinger operators of the form $\text{H =}\text{?1}\text{2Δ}\text{+ V}$ with V smooth, positive and growing like $\text{V}{}_0\text{¦x¦}{}^{\mathrm{k}}\text{, k > 0}$. Namely, let π_λ be the orthogonal projection of L² onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights $\text{?(x, ξ) = 1, Φ(x, ξ) = (1 + ¦ξ¦}{}^2\text{+ V(x))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and with total symbol a(x, ξ); and let f∈C($\text{R}$). Then

$\text{lim}\text{λ→∞}\text{1}\text{rank}\text{π}{}_{\mathrm{\lambda }}\text{tr}\text{f(π}{}_{\mathrm{\lambda }}\text{Aπ}{}_{\mathrm{\lambda }}\text{)=}\text{lim}\text{λ→∞}\text{1}\text{vol}\text{(H(x, ξ)?λ)}\text{∫}\text{H?λ}\text{f(a(x, ξ))dxdξ}$

(assuming one limit exists).     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Levy's stochastic area formula in higher dimensions

Let (W_t) = (W¹_t,W²_t,…,W^d_t), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from $\text{R}$⁺ into the space of d × d skew-symmetric matrices and x(t) such a function into $\text{R}$^d. A class of stochastic processes (L_t^A,x), a particular example of which is Levy's “stochastic area” $\text{L}{}_{\mathrm{t}}\text{=}\text{1}\text{2}\text{∝}{}_0{}^{\mathrm{?t}}\text{(W}{}^1{}_{\mathrm{s}}\text{,dW}{}^2{}_{\mathrm{s}}\text{? W}{}^2{}_{\mathrm{s}}\text{,dW}{}^1{}_{\mathrm{s}}\text{)}$, is dealt with.The joint characteristic function of W_t and L₁^A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (W_t,L_t^A,x) is given.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Estimates on the first eigenvalue of an integral equation

We show how inequalities of the type $\text{∥F∥}{}_{\mathrm{p}}\text{? C(p, q) a}{}^{\mathrm{<mtext>1\; +\; (</mtext><mtext>1</mtext><mtext>p</mtext><mtext>)?\; (</mtext><mtext>1</mtext><mtext>q</mtext><mtext>)</mtext>}}\text{∥ F ′ ∥}{}_{\mathrm{q\prime }}$ when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝₀^ak(s, z)F(s) ds.     

A comparison of different compactness notions in fuzzy topological spaces

Starting from a defining differential equation $\text{(}\text{?}\text{?t}\text{) W(λ, t, u) = (}\text{λ(u ? t)}\text{p(t)}\text{) W(λ, t, u)}$ of the kernel of an exponential operator $\text{S}{}_{\mathrm{\lambda }}\text{(?, t) = ∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{W(λ, t, u)?(u) du}$ with normalization ∫_?∞^∞W(λ, t, u) du = 1, we determine S_λ for various p(t) including; for example, p(t) a quadratic polynomial, all the known exponential operators are recovered and some new ones are constructed. It is shown that all the exponential operators are approximation operators. Further approximation properties of these operators are discussed. For example, functions satisfying $\text{∥ S}{}_{\mathrm{\lambda }}\text{(?, t) ? ?(t)∥ = O(λ}{}^{\mathrm{?\alpha }}\text{)}$ are characterized. Several results of C. P. May are also improved.     

Asymptotic evaluation of two families of integrals

In this paper the integrals $\text{f}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[?(t + τ)]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t + τ)}{}^{\mathrm{?1}}\text{dt}\text{and}\text{g}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[? ¦ ? τ ¦]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t ? τ)}{}^{\mathrm{?1}}\text{dt}$ are investigated for positive real values of the variable τ. Here, m is a nonnegative integer, v is a complex variable with Re(v) > ?1. Both integrals are related to the complex integral Φ_mγ(z) = ∝₀^∞exp[?(t ? z)]t^?γ(ln t)^m(t ? z)^?1dt with 0 ? Re(γ) < 1, the behavior of which is analyzed in detail. The results are applied to obtain asymptotic representations for f_mn(τ) and g_mn(τ), m and n both nonnegative integers, near τ = 0. The latter integrals play a role in the study of the equations of neutron transport and radiative transfer.     

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

Explicit error terms for asymptotic expansions of Mellin convolutions

Explicit expressions are derived for the error terms associated with the asymptotic expansions of the convolution integral $\text{I(λ) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) h(λt) dt}$, where h(t) and $\text{?(t)}$ are algebraically dominated at both 0⁺ and + ∞. Examples included are Fourier, Bessel, generalized Stieltjes, Hilbert and “potential” transforms.     

A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

Étude d'une équation de Monge-Ampère sur les variétés kählériennes compactes

On a compact Kähler manifold of complex dimension m ? 2, let us consider the change of Kähler metric $\text{g′}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{= g}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{+ ?}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gm</mtext>}}\text{φ}$. Let F?C^∞(V × R) be a function everywhere > 0 and v a real number ≠ 0. When $\text{0 < C}{}^{\mathrm{?1}}\text{? F(x, t) ? C(¦t¦}{}^{\mathrm{a}}\text{+ 1)}$ for all (x, t) ?V × ] ?∞, t₀], where C and t₀ are constants and $\text{1 ? a <}\text{m}\text{(m ? 1)}$, one exhibits a function φ?C^∞ (V) such that $\text{¦g′∥g¦}{}^{\mathrm{?1}}\text{= e}{}^{\mathrm{<mtext>\nu </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>gf</mtext>}}\text{F(x, φ ?}\text{}\text{?}\text{gf) (¦g¦}\text{and}\text{¦g′¦}$ the determinants of the metrics g and $\text{g′,}\text{}\text{?}\text{gf = (}\text{mes}\text{V)}{}^{\mathrm{?1}}\text{∝ φ dV)}$.     

Perturbation theory for eigenvalues and resonances of Schrödinger hamiltonians

Suppose that e^2_?|x|V ∈ ReL^P(R³) for some p > 2 and for g ∈ R, H(g) = ? Δ + gV, H(g) = ?Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is $\text{λ(g) ↗6 0}\text{as}\text{g ↘5 g}{}_0\text{> 0}$. We find a series expansion in powers of $\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, λ(g) = ∑}{}_{\mathrm{n\; =\; 2}}{}^{\mathrm{\infty }}\text{a}{}_{\mathrm{n}}\text{(g ? g}{}_0\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$ whose values for g < g₀ correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.     

Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant

The two lowest eigenvalues E₀(λ), E₁(λ) of a symmetric double well tunnelling problem ?Δ + λ²V as λ → ∞ are considered and they are compared to the two lowest eigenvalues $\text{E}\text{?}{}_0\text{(λ),}\text{E}\text{?}{}_1\text{(λ)}\text{of}\text{?Δ + λ}{}^2\text{(V + W),}\text{where}\text{W}$ is supported away from the well-bottoms of V. We determine the leading exponential splitting of various differences of the four numbers $\text{E}{}_0\text{, E}{}_1\text{,}\text{E}\text{?}{}_0\text{,}\text{E}\text{?}{}_1$. Related problems are discussed.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Coding theory in white Gaussian channel with feedback

The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{F(s, Y}{}_0{}^{\mathrm{s}}\text{, m)ds + W(t)}$. Under the average power constraint, $\text{E[F}{}^2\text{(s, Y}{}_0{}^{\mathrm{s}}\text{, m)] ≤ P}{}_0$, we construct causally the optimal coding, in the sense that the mutual information I_t(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{A(s)[m(s) ?}\text{m}\text{?}\text{(s)] ds + W(t)}$, where $\text{m}\text{?}\text{(s) = E[m(s) ¦ Y(u), 0 ≤ u ≤ s]}$ and A(s) is a positive function such that $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Compositions of singular integral operators

The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L¹(Rⁿ) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on R^{n + 1} homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x₀ = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε₁) Sing(ψ, ε₂)f = ?2π²(∝ φ)(∝ ψ)f + Sing(A, ε₁, + ε₂)f, where

(with notation $\text{¦t¦}{}_0{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{and}\text{¦t¦}{}_1{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{sgn}\text{t}$). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol

$\text{ω}{}_0\text{(x,?)=?πiθ(?)∫θ(x?y)}\text{sgn}\text{y·?dy}\text{if}\text{?}{}_1\text{=1}$

,

$\text{=?2θ(?)∫θ(x?y)}\text{log}\text{|y·?|dy}\text{if}\text{?}{}_1\text{=0}$

where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity.