The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

Algebras containing a divisible unimodular sequence with dense self adjoint span

We characterize the uniform algebras A on a compact Hausdorff space X which contain a sequence {u_j}_{j = 0}^∞ of unimodular elements with $\text{u}{}_{\mathrm{j}}\text{u}{}_{\mathrm{j\; ?\; 1}}\text{? A}$ and closed span $\text{{u}{}_{\mathrm{j}}\text{u}\text{}}{}_{\mathrm{j\; =\; 0}}{}^{\mathrm{\infty }}\text{= C(X)}$ in terms of the maximal ideal space of A. Roughly, the essential set of A looks like (at most) countably many copies of the boundary of the unit disk, and A looks like the disk algebra on each.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,II

Let L = ∑_{j = 1}^mX_j² be sum of squares of vector fields in $\text{R}$ⁿ satisfying a Hörmander condition of order 2: span{X_j, [X_i, X_j]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X₁,…, X_m are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator $\text{?}{}^2\text{?x}{}^2\text{+ x}{}^2\text{?}{}^2\text{?t}{}^2$ in $\text{R}$². (b) The real part of the Kohn Laplacian on the Heisenberg group $\text{∑}{}_{\mathrm{j\; ?\; 1}}{}^{\mathrm{n}}\text{(}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2\text{+ (}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2$ in $\text{R}$^{2n + 1}. In contrast to non-characteristic points, C^∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.     

Positive definite Hermitian matrices and reproducing kernels

Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{A_j}^m_j=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ $\text{C}$^N, we have the identity

$\text{x}{}^1\text{∑}\text{j=}\text{1}\text{m}\text{A}{}^{-1}{}_{\mathrm{j}}{}^{-1}\text{x = min}\text{∑}\text{j=}\text{1}\text{m}\text{x}{}^1{}_{\mathrm{j}}\text{A}{}_{\mathrm{j}}\text{x}{}_{\mathrm{j}}$

. The minimum is taken here over all the decompositions x =∑^m_j=1x_j. This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces.     

Generalization of a theorem by Hardy,Littlewood, and Pólya

Let τ be a non-expanding map which admits a σ-finite, absolutely continuous invariant measure μ. The measure $\text{μ¦}{}_{\mathrm{E}}$ is approximated as closely as desired by a measure μ_n invariant under the transformation τ_n satisfying $\text{inf}{}_{\mathrm{x}}\text{¦τ′}{}_{\mathrm{n}}\text{(x)¦ > 1}$. The orbit {τ_n^j(x)}_{j = 0}^∞ “exhibits” $\text{μ¦}{}_{\mathrm{E}}$ much more quickly than does the orbit {τ^j(x)}_{j = 0}^∞.     

A complete orthonormal system of divergence

A complete orthonormal system of functions $\text{{Θ}{}_{\mathrm{n}}\text{}}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{,}\text{Θ}{}_{\mathrm{n}}\text{∈L}{}^{\mathrm{\infty }}{}_{\mathrm{[0,1]}}$ defined on the closed interval [0,1] is constructed such that ∑_n=1^∞a_nΘ_n diverges almost everywhere for any {a_n}_n=1^∞?l². For the constructed system the following result is true: Any nontrivial series by the system {Θ_n}_n=1^∞ which converges in measure to zero diverges almost everywhere. To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Extreme value distribution for normalized sums from stationary Gaussian sequences

Let {X_i, i?0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that

for -∞<t<∞ where and $\text{X}{}_{\mathrm{n}}\text{= (2 ln ln n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The result is extended to dependent sequences but assuming that {X_i} is a standard stationary Gaussian sequence with covariance function {r_i}. When {X_i} is moderately dependent (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}_{\mathrm{a}}\text{, 0 < a < 2)}$ we get

where H_a is a constant. In the strongly dependent case (e.g. when $\text{v(∑}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{X}{}_{\mathrm{i}}\text{) ? n}{}^2\text{r(n))}$ we get

for-∞<t<∞.     

An inertial decomposition and related range results

With quasicommutative n-square complex matrices A₁,…,A_s and s-square hermitian G=(g_ij), relationships are given between the image $\text{Σ}{}^{\mathrm{s}}{}_{\mathrm{i,j=1}}\text{g}{}_{\mathrm{ij}}\text{A}{}_{\mathrm{i}}\text{HA}{}^1{}_{\mathrm{j}}$ of a linear transformation on $\text{H}$_n being positive definite and the action of H on generalized inertial decompositions of $\text{C}$ⁿ.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

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.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

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.     

Enumeration of pairs of permutations

Let π = (a₁, a₂, …, a_n), ? = (b₁, b₂, …, b_n) be two permutations of $\text{Z}{}_{\mathrm{n}}\text{= {1, 2, …, n}}$. A rise of π is pair a_i, a_i+1 with a_i < a_i+1; a fall is a pair a_i, a_i+1 with a_i > a_i+1. Thus, for i = 1, 2, …, n ? 1, the two pairs a_i, a_i+1; b_i, b_i+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ω_n denotes the number of pairs with RR forbidden, it is proved that $\text{∑}{}^{\mathrm{\infty }}{}_0\text{ω}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}\text{=}\text{1}\text{?(z)}$, $\text{?(z) = ∑}{}^{\mathrm{\infty }}{}_0\text{(-1)}{}^{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}$. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then $\text{1 + ∑}{}^{\mathrm{\infty }}{}_{\mathrm{n=1}}\text{z}{}^{\mathrm{n}}\text{n!n!}$$\text{∑}{}^{\mathrm{n?1}}{}_{\mathrm{k=0}}\text{ω(n, k)x}{}^{\mathrm{k}}\text{=}\text{(1 ? x)}\text{(?(z(1 ? x)) ? x)}$.     

Solution of a system of functional-differential equations arising from an optimization problem

In connection with an optimization problem, all functions ?: Iⁿ → $\text{R}$ with continuous nonzero partial derivatives and satisfying

$\text{??}\text{?x,}{}_{\mathrm{i}}\text{??}\text{?x}{}_{\mathrm{j}}$

for all x_i, x_j ≠ I, i, j = 1,2,…, n (n > 2) are determined (I is an interval of positive real numbers).     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Uniqueness of solutions of nonlinear Dirichlet problems

In this paper, we consider the uniqueness of radial solutions of the nonlinear Dirichlet problem $\text{Δu + ?(u) = 0}\text{in}\text{Ω}\text{with}\text{u = 0}\text{on}\text{?Ω,}\text{where}\text{Δ = ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}{}^2\text{,?}$ satisfies some appropriate conditions and Ω is a bounded smooth domain in $\text{R}$ⁿ which possesses radial symmetry. Our uniqueness results apply to, for instance, $\text{?(u) = u}{}^{\mathrm{p}}\text{, p > 1}$, or more generally λu + ∑_{i = 1}^ka_iu^pi, λ ? 0, a_i > 0 and p_i > 1 with appropriate upper bounds, and Ω a ball or an annulus.