On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

A q-analogue of the FKG inequality and some applications

Let L be a finite distributive lattice and μ: L → ℝ⁺ a log-supermodular function. For functions k: L → ℝ⁺ let

Orthogonal polynomials associated to a certain fourth order differential equation

We introduce orthogonal polynomials M_j^m,l(x)M_{j}^{\mu,\ell}(x) as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ∈ℂ and ℓ∈ℕ₀.     

Dense analytic subspaces in fractalL 2-spaces

We show that for certain self-similar measures μ with support in the interval 0≤x≤1, the analytic functions {e ^i2πnx :n=0,1,2, …} contain an orthonormal basis inL ² (μ). Moreover, we identify subsetsP ⊂ ℕ₀ = {0,1,2,...} such that the functions {e _n :n ∈ P} form an orthonormal basis forL ² (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝ ^ν , obtained from a given expansivev-by-v matrix and a finite set of translation vectors. We show that the correspondingL ² (μ) has an orthonormal basis of exponentialse ^i2πλ·x , indexed by vectors λ in ℝ ^ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system. Work supported by the National Science Foundation.     

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions     

Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

Denote by γ the Gauss measure on ℝ ⁿ and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} to L ¹γ. This result is in sharp contrast both with the fact that (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H ¹γ to L ¹γ, where H ¹γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case (rI-D)^iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space \mathfrakh¹\mathbbRⁿ{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L ¹ℝ ⁿ . We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L ² μ, and with a kernel satisfying certain analytic assumptions, is bounded from H ¹ μ to L ¹ μ if and only if it is bounded from \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} to L ¹ μ. Here H ¹ μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ ⁿ . We consider integral operators K whose kernels have the form

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let Lf(x)=-\frac1w?_i,j ?_i(a_i,j(·)?_jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying l^-1w(x)|x|² ￡ ?ⁿ_i,j=1a_i,j(x)x_ix_j ￡ lw(x)|x|².{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for V^aL^-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator [b, V^a L^-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMO_w{b\in BMO_\omega} and 0 < α ≤ 1.     

Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L ₂(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L ₂(R) with domain C_c^￥(R){C_c^\infty({\bf R})} and action Hj = -(c j^￠)^￠{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W^1,2_loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{0\}} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if n = n₊ún_-{\nu=\nu_+\vee\nu_-} where n_±(x)=±ò^±1_±x c^-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L₂(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L_￥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L ₂(0,∞) and L ₂(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W^1,￥_loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension.     

Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions

Let S⊂ℝ ^d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW _S , is defined to be the set of all square-integrable functions on ℝ ^d whose Fourier transforms vanish outside S. A sequence (x _j :j∈ℕ) in ℝ ^d is said to be a Riesz-basis sequence for L ₂(S) (equivalently, a complete interpolating sequence for PW _S ) if the sequence (e^-iáx_j,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L ₂(S). Let (x _j :j∈ℕ) be a Riesz-basis sequence for L ₂(S). Given λ>0 and f∈PW _S , there is a unique sequence (a _j ) in ℓ ₂ such that the function

On a Sharper Lower Bound for a Percentile of a Student’s <Emphasis Type="Italic">t</Emphasis> Distribution with an Application

In this communication, we first compare z _α and t _ν,α , the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. We begin with a proof of a well-known result, namely, for every fixed 0 < a < \frac120<\alpha <\frac{1}{2} and the degree of freedom ν, one has t _ν,α  > z _α . Next, Theorem 3.1 provides a new and explicit expression b _ν ( > 1) such that for every fixed 0 < a < \frac120<\alpha < \frac{1}{2} and ν, we can conclude t _ν,α  > b _ν z _α . This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere. A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated well by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by t_n,a/2²z_a/2^-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of t_n,a/2²z_a/2^-2,t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}, namely b_n²,b_{\nu }^{2}, or comes incredibly close to b_n²b_{\nu }^{2} where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under Stein’s fixed-width confidence interval method.     

<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Some Gradient Systems with Measurable Potential

We consider the operator in L ²(B, ν) and in L ¹(B, ν) with Neumann boundary condition, where U is an unbounded function belonging to for some q ∈(1, ∞), B is the possibly unbounded convex open set in where U is finite and ν(dx) = C exp (−2U (x))dx is a probability measure, infinitesimally invariant for N ₀. We prove that the closure of N ₀ is a m-dissipative operator both in L ²(B, ν) and in L ¹(B, ν). Moreover we study the properties of ergodicity and strong mixing of the measure ν in the L ² case.      

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝ ^d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form L_n[f](x): = ?_i=1ⁿ f_i(x)(f(y_i) + á?f(y_i), x-y_i?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y₁,..., y_n\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ ₁, ..., ϕ _n } is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L _n with a smallest constant in an L ^p error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator L _n for approximating f by L _n [f] with respect to the L ^p norm.     

Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets

In this paper we study the boundary limit properties of harmonic functions on ℝ₊×K, the solutions u(t,x) to the Poisson equation

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

L p Norms of Eigenfunctions in the Completely Integrable Case

The eigenfunctions e^iál,x? e^{i\langle\lambda,x\rangle} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions blow up at least at the rate || j_k || L^p 3 C(e)l_k^{[(p-2)/(4p)]-e} \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in L^p unless (M,g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.     

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Let W ì \mathbb C^d{\Omega \subset{\mathbb C}^{d}} be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators Tⁿ_g{T^{\nu}_{g}} with symbol g acting on the standard weighted Bergman space H_n²{H_\nu^2} over Ω with weight ν. Under some conditions on the weights ν and ν ₀ we show that there exists C(ν, ν ₀) > 0, such that the Berezin transform [(g)\tilde]_n0{\tilde{g}_{\nu_{0}}} of g with respect to H²_n0{H^2_{\nu_0}} satisfies: