Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

Problèmes de Neumann non linéaires sur les variétés riemanniennes

Nonlinear Neumann problems on riemannian manifolds. Let ($\text{M}\text{, g}$) be a C^∞ compact riemannian manifold of dimension n ? 2 whose boundary B is an (n ? 1)-dimensional submanifold and let $\text{M =}\text{M}\text{?B}$ be the interior of $\text{M}$. Study of Neumann problems of the form: $\text{Δφ +?(φ, x) = 0}\text{in}\text{M, (}\text{dφ}\text{dn}\text{) + g(φ, y) = 0}\text{on}\text{B}$, where, for every $\text{(t, x, y) ? R × M × B, ¦?(t, x)¦}\text{and}\text{¦g(t, y)¦}$ are bounded by $\text{C(1 + ¦t¦}{}^{\mathrm{a}}\text{)}\text{or}\text{C}\text{exp}\text{(¦t¦}{}^{\mathrm{a}}\text{)}$. Application to the determination of a conformal metric for which the scalar curvature of M and the mean curvature of B take prescribed values.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

A characterization of first order nonlinear partial differential operators on Sobolev spaces

Nonlinear partial differential operators $\text{G: W}{}^{\mathrm{1,p}}\text{(Ω) → L}{}^{\mathrm{q}}\text{(Ω) (1 ? p, q ∞)}$ having the form G(u) = g(u, D₁u,…, D_Nu), with g?C(R × R_N), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, $\text{W}{}^{\mathrm{1,\infty }}\text{(Ω)}$, and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of $\text{W}{}^{\mathrm{1,p}}\text{(Ω)}\text{and}\text{L}{}^{\mathrm{q}}\text{(Ω)}$.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.     

A remark to the preceding paper by Chernoff

It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C₀^∞(R^m) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C^∞(R^m) and if $\text{T ? ?a ? b ¦ x ¦}{}^2\text{on}\text{C}{}_0{}^{\mathrm{\infty }}\text{(R}{}^{\mathrm{m}}\text{)}$.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

On the spatial theory of von Neumann algebras

If M is a von Neumann algebra in $\text{H}$, each faithful weight ψ on M′ defines an operator-valued weight ψ^?1 of $\text{L(H)}$ on M. For each weight ? on M the positive unbounded operator $\text{d?}\text{dψ}\text{= ? ° ψ}{}^{\mathrm{?1}}$ satisfies all the usual properties of a Radon-Nikodym derivative.     

A multiplication of distributions

If Ω denotes an open subset of $\text{R}$ⁿ (n = 1, 2,…), we define an algebra $\text{g}$ (Ω) which contains the space $\text{D}$′(Ω) of all distributions on Ω and such that $\text{C}{}^{\mathrm{\infty }}\text{(Ω)}$ is a subalgebra of $\text{G}$ (Ω). The elements of $\text{G}$ (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in $\text{G}$(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and $\text{?∈O}{}_{\mathrm{M}}\text{(}\text{R}{}^{\mathrm{2q}}\text{)}$—a classical Schwartz notation—for any G₁,…,G_q∈G(σ), we define naturally an element $\text{?G}{}_1\text{,…,G}{}_{\mathrm{q}}\text{∈G(σ)}$. These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.     

An Lp bound for the Riesz and Bessel potentials of orthonormal functions

Let ψ₁, …,ψ_N be orthonormal functions in $\text{R}$^d and let $\text{u}{}_{\mathrm{i}}\text{= (?Δ)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{ψ}{}_{\mathrm{i}}$, or $\text{u}{}_{\mathrm{i}}\text{= (?Δ + 1)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{ψ}{}_{\mathrm{i}}$, and let $\text{p(x) = ∑¦u}{}_{\mathrm{i}}\text{(x)¦}{}^2$. L^p bounds are proved for p, an example being $\text{∥p∥}{}_{\mathrm{p}}\text{? A}{}_{\mathrm{d}}\text{N}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{for}\text{d ? 3}$, with p = d(d ? 2)^?1. The unusual feature of these bounds is that the orthogonality of the ψ_i, yields a factor $\text{N}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$ instead of N, as would be the case without orthogonality. These bounds prove some conjectures of Battle and Federbush (a Phase Cell Cluster Expansion for Euclidean Field Theories, I, 1982, preprint) and of Conlon (Comm. Math. Phys., in press).     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

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).     

Harmonic analysis on hyperboloids

The regular representation of O(n, N) acting on $\text{L}{}^2\text{(}\text{O(n, N)}\text{O(n, N ? 1)}\text{)}$ is decomposed into a direct integral of irreducible representations. The homogeneous space $\text{O(n, N)}\text{O(n, N ? 1)}$ is realized as the Hyperboloid $\text{H = {(x, t) ? R}{}^{\mathrm{n\; +\; N}}\text{: ¦ t ¦}{}^2\text{? ¦ x ¦}{}^2\text{= 1}}$. The problem is essentially equivalent to finding the spectral resolution of a certain self-adjoint invariant differential operator □_h on H, which is the tangential part of the operator □ = Δ_x ? Δ_t on R^{n + N}. The spectrum of □_h contains a discrete part (except when N = 1) with eigenfunctions generated by restricting to H solutions of □u = 0 which vanish in the region $\text{¦ t ¦ < ¦ x ¦}$, and a continuous part $\text{H}$_?. As a representation of O(n, N), $\text{H}$_? ⊕ $\text{H}$_? is unitarily equivalent to the regular representation on L² of the cone $\text{{(x, t) : ¦ x ¦}{}^2\text{= ¦ t ¦}{}^2\text{}}$, and the intertwining operator is obtained by solving the equation □u = 0 with given boundary values on the cone. Explicit formulas are given for the spectral decomposition. The special case n = N = 2 gives the Plancherel formula for SL(2, R).     

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 $\text{g′,}\text{}\text{?}\text{gf = (}\text{mes}\text{V)}{}^{\mathrm{?1}}\text{∝ φ dV)}$.     

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.