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1.
Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2) set σΘ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ2, the inequality.
represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the above inequality when Γ is any convex curve and under various geometric assumptions.  相似文献   

2.
A weak type endpoint estimate for the maximal multilinear singular integral operator T*Af(x)=supε>0|(f)(x-y)>ε (Ω(x-y)/(|x-y|(n 1)))(A(x)-A(y)-▽A(y)(x-y))f(y)dy| is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO(Rn). A regularity condition on Ω which implies an LlogL type estimate of T*A is given.  相似文献   

3.
Let Ω be an open bounded domain in ℝN(N ≥ 3) and . We are concerned with two kinds of critical elliptic problems. The first one is
(*)
where 0 ∈ Ω, , 2 < m < 2* and λ > 0. By using the fountain theorem and concentration estimates, if N ≥ 7 and θ > 0, we establish the existence of infinitely many solutions for the following regularization of (*) with small number ϵ > 0
Then if θ > 0 is suitably small, we obtain many solutions for problem (*) by taking the process of approximation. The second problem is
where q ∈ (0, 1), t > 0. By using similar methods as in (*), we prove that if N ≥ 7, and t > 0, there exist infinitely many solutions with positive energy. In particular, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami [1].  相似文献   

4.
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E *(t)=E(t)-2πΔ*(t/2π) with , then we obtain
and
It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of .  相似文献   

5.
We consider two-phase metrics of the form ϕ(x, ξ) ≔ , where α,β are fixed positive constants and B α, B β are disjoint Borel sets whose union is ℝN, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying
. Then we study the closure of the class of two-phase periodic metrics with prescribed volume fraction θ of the phase α. We give upper and lower bounds for the class and localize the problem, generalizing the bounds to the non-periodic setting. Finally, we apply our results to study the closure, in terms of Γ-convergence, of two-phase gradient-constraints in composites of the type f(x, ∇ u) ≤ C(x), with C(x) ∈ {α, β } for almost every x.  相似文献   

6.
The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R n (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).  相似文献   

7.
Let Λα be the analytic Holder class in the unit disk . For f ∈ Λα and , let Mf (I) = max I |f|. Assume that I and J are arcs such that |J| = 2|I| and J and I have the same midpoint. Then
It is proved that this estimate cannot be improved. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 149–154.  相似文献   

8.
For measuresμ on the circleT the quantities , need not be equal; it is shown, however, that they are continuous with respect to each other whenμ varies on bounded subsets ofM(T), the space of measures onT. It is also shown that measuresμ which areɛ-almost idempotent (i.e. ) are the sum of an idempotent measure and of a measureυ satisfying providedɛ is small enough (as a function of ‖μ‖).  相似文献   

9.
Let G be a commutative semigroup and letL be a complete Archimedean Riesz Space. Suppose thatF: G → L satisfies for somee ∈ L + the inequality
Then there exists a unique additive mappingA : G → L such that
As the method of the proof we use the Johnson-Kist Representation Theorem.  相似文献   

10.
Let II be a bounded symmetric domain, ω ⇉ I a bounded subdomain, and let denote the weighted Bergman space of holomorphic square integrable functions on I. Let Tλ, ω be the Berezin-Toeplitz operator on with symbol χΩ and kth eigenvalue λ k (T λ,Ω). We prove that for δ1 sufficiently close to 0 and δ2 sufficiently close to 1 the estimate
holds for all domains ω satisfying the condition |{z ∈ I |d(z, Ω) < ε}| ≤c|Ω|, where d is the invariant distance on I and |ω| is the invariant volume of ω. The proof is based on the fact that the operator norm of the Berezin transform is smaller than 1. Our main technical tool are some of the formulae for the Berezin transform obtained by Unterberger and Upmeier in [11].  相似文献   

11.
Consider the probability spaceW={−1, 1} n with the uniform (=product) measure. Letf: WR be a function. Letff IXI be its unique expression as a multilinear polynomial whereX I iI x i. For 1≤mn let =Σ|I|=m f IXI. LetT ɛ (f)=Σf Iɛ|I| X I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that
This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to the following inequality via duality: For anyq≥2
In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show
where . Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own right. Supported partially by NSF Award Abstract #0071261.  相似文献   

12.
Let A 0, ... , A n−1 be operators on a separable complex Hilbert space , and let α0,..., α n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequality holds for 0 < p ≤ 2. Moreover, we prove that if ω0,..., ω n−1 are the n roots of unity with ω j = e ij/n , 0 ≤ jn − 1, then for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequalities hold for 0 < p ≤ 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso optained.   相似文献   

13.
We obtain a set of necessary and sufficient conditions for to imply for 1 <k ≤ s < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of and .  相似文献   

14.
Let and τ denote the invariant gradient and invariant measure on the unit ball B of ℂn, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C2(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H ϕ(B>) if and only if
Analogous characterizations of Bergman-Orlicz spaces are obtained. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 43–53.  相似文献   

15.
Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {
} or the condition {
} in which the nonnegative function itχ(r) is such that { }. For each α∈(0, 2], we establish an estimate of the generalized Fourier transforms of an arbitrary function { } of the form {
} If, in addition, { }, then, along with this estimate, a similar lower bound is established. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 542–551, April, 1999.  相似文献   

16.
In 1986 S. Axler [3] proved that forfL a 2 the Hankel operator is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for , 1<p<∞. Moreover we prove that is ⋆-compact if and only if as |z|→1.  相似文献   

17.
In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ n (n≥2) with non-smooth boundary ∂Ω and connected complement Ω e =ℝ n . We can prove that if Ω satisfies a certain geometrical condition, then
where ,d n>0 depending only onn, and |·| j (j = n - l, n) is aj- dimensional Lebesgue measure. Research partially supported by the Natural Science Foundation of China and the Grant of Chinese State Education Committee  相似文献   

18.
In this paper, we prove the estimate
, for every δ ∈ (0, ℓN), where C = C(N) is a positive constant depending only on N and . We show that the constant ℓN in this estimate is optimal. We also present a class of maps from into , strictly larger than , on which we can define the notion of degree and for which the previous inequality still holds.  相似文献   

19.
For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A σ ρ of functions g(z) analytic in the unit disk = {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρT[g] = σ, and ρM[g] = ρ, where
__________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 979–995, July, 2007.  相似文献   

20.
The following regularity of weak solutions of a class of elliptic equations of the form are investigated.  相似文献   

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