Gaussian kernels have only Gaussian maximizers

A Gaussian integral kernelG(x, y) onR ⁿ ×R ⁿ is the exponential of a quadratic form inx andy; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound ofG as an operator fromL ^p (R ⁿ ) toL ^p (R ⁿ ) and to prove that theL ^p (R ⁿ ) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1<pq< and also forp>q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young's inequality.Oblatum 19-XII-1989Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A03     

Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.     

Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

For the equationL ₀ x(t)+L ₁x(t)+...+L _n x ⁽ⁿ⁾(t)=O, whereL _k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x ^(1–1) (a)=0, 1=1, ..., p, andx ^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx ^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem

We consider a boundary value problem where f(x) ∈ L_p(R), p ∈ [1,∞] (L_∞(R) ≔ C(R) and 0 ≤ q(x) ∈ L^loc₁( R). Boundary value problem (0.1) is called correctly solvable in the given space L_p(R) if for any f(x) ∈ L_p(R) there is a unique solution y(x) ∞ L_p(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in L_p(R).     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

A White Noise Approach to Stochastic Neumann Boundary-Value Problems

We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)–c(x)U(x)=0, xDR ^d ,(x)U(x)=–W(x), xD, where L is a uniformly elliptic linear partial differential operator and W(x), xR ^d , is d-parameter white noise.     

Bessel Potentials with Logarithmic Components and Sobolev-Type Embeddings

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L ^n/k (log L) (R ⁿ ), are continuously embedded into L (R ⁿ ) if > 1/p, p n/k. In this paper we replace L ^n/k (log L) (R ⁿ ) by the Lebesgue space L ^n/k (R ⁿ ) and increase the smoothness of the functions involved by a "logarithmic" order > 1/p to obtain the continuous embedding into L (R ⁿ ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brézis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.     

On ℓ p -summability of the characteristic values of intergral operators onL(R N )

LetB _p (L ₂) (0<p<) denote the space of all compact linear operatorsK onL(R ^N ) with the following property: the sum ofp-th powers of all positive eigenvalues (repeated according to their multiplicity) of the nonnegative compact operator (K ^* K)^1/2 converges. We prove aB _p -criterion for an integral operatorK onL(R ^N ), for 0<p2. We require that its kernel have sufficient fractional smoothness and decay at infinity. Our proof is based on a suitable factorization of the operatorK which involves fractional powers (H+I)^– 0, 0<1, of the resolvent of the selfadjoint elliptic operatorH=(–) ^m +M _q ^2m onL(R ^N ).     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

An inequality for the volume of inscribed ellipsoids

LetK be a convex body inR ⁿ, and letx* intK be the center of the ellipsoid of the maximal volume inscribed in the body. An arbitrary hyperplane throughx* cutsK into two convex bodiesK ⁺ andK ^–. We show thatw(K ^±)/w(K)0.844..., wherew(·) is the volume of the inscribed ellipsoid.     

The Dyadic Cesàro Operator on R+

The dyadic Cesàro operator C is introduced for functions in the space L ¹ := L ¹(R ₊) by means of the Walsh-Fourier transform defined by

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.     

On the Structure of Spaces of Polyanalytic Functions

Suppose that A_mL_p(D,) is the space of all m-analytic functions on the disk D={z:|z| < 1} which are pth power integrable over area with the weight (1-|z|²), > -1. In the paper, we introduce subspaces A_kL_p ⁰(D,), k=1,2,...,m, of the space A _mL_p(D,) and prove that A _mL_p(D,) is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Some nodes matrices appearing in the numerical analysis for singular integral equations

Let {p _m (w)} be the sequence of Jacobi polynomials corresponding to the weightw(x)=(1–x)(1+x), 0, <1. denote=">x _k (w)=cos _m,k (w),k=1,...,m, the zeros ofp _m (w). If +=0, then the estimates

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M _s is bounded from the weighted Lebesgue spaces L ^p(l ^r, vdx) into L ^q(l ^r, udx) whenever 0 s < n, 1 < p, r < , and 1 q < .     

Optimal quadratures for analytic functions

For integrals _–1 ¹ w(x)f(x)dx with and with analytic integrands, we consider the determination of optimal abscissasx _i ^o and weightsA _i ^o , for a fixedn, which minimize the errorE _n (f)= _–1 ¹ w(x)f(x)dx – _i ⁼¹ⁿ A _i f(x _i ) over an appropriate Hilbert spaceH ²(E ; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA _i *, corresponding to (preassigned) Gaussian abscissasx _i ^G , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA _i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA _i ^G . In each case,A _i ^G =A _i *+O( ^–4n ), . For , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x _i ^G ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x _i ^G ,A _i *;i=1,...,n} to the optimal solution {x _i ^o ,A _i ^o ;i=1,...,n} in terms of _n (), the maximum absolute remainder in the second set ofn normal equations. In each case, _n () is, at least, of the order of ^–4n for large.     

Tangential Convergence of Temperatures and Harmonic Functions in Besov and in Triebel-Lizorkin Spaces

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R₊^N+1 of a distribution in the Besov space B^α_p,q(R^N) or in the Triebel-Lizorkin space F^α_p,q(R^N). In particular, we prove that when Ω= {|_y|^{N/ (N-αp)} < t < 1} the operator M is bounded from F (R^N) into L^p (R^N). The admissible regions for the spaces B (R^N) with p < q are more complicated.     

Some counterexamples for the theory of sobolev spaces on bad domains

It is shown that some well-known properties of the Sobolev spaceL _p ^l () do not admit extension to the spaceL _p ^l () of the functions withl-th order derivatives inL _p (),l>1, without requirements to the domain . Namely, we give examples of such that