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Abstract Let d ≥ 2 be an integer and let 2 d / (d − 1) < q ≤ ∞. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality ‖ f σ ˆ ‖ L rad q L ang 2 (R d) ≤ C d , q ‖ f ‖ L 2 (S d − 1 , d σ) , established by L. Vega in 1988. Letting A d ⊂ (2 d / (d − 1) , ∞ ] be the set of exponents for which the constant functions on S d − 1 are the unique extremizers of this inequality, we show that: (i) A d contains the even integers and ∞; (ii) A d is an open set in the extended topology; (iii) A d contains a neighborhood of infinity (q 0 (d) , ∞ ] with q 0 (d) ≤ (1 2 + o (1)) d log ⁡ d. In low dimensions we show that q 0 (2) ≤ 6.76 ; q 0 (3) ≤ 5.45 ; q 0 (4) ≤ 5.53 ; q 0 (5) ≤ 6.07. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions. [ABSTRACT FROM AUTHOR]

Recently, several authors have considered the problem of determining optimal norm inequalities for discrete Hardy-type operators (like Cesàro or Copson). In this work, we obtain sharp bounds for the norms of the difference of the Cesàro operator with either the identity or the shift, when they are restricted to the cone of decreasing sequences in ℓ p (which is closely related to the previously mentioned estimates). Finally, we also address the case of weighted inequalities and find an interesting contrast between the norms of these two difference operators. [ABSTRACT FROM AUTHOR]

The well-known Gevrey classes consist of C ∞ -functions on R d whose derivatives satisfy certain growth conditions. For periodic functions these conditions can be expressed in terms of Fourier coefficients. Motivated by this observation, we introduce Gevrey spaces on the d -dimensional torus and study approximation numbers of their embeddings into L 2. Our special emphasis is on the dependence on the dimension of the underlying domain, which is an important aspect in the numerical treatment of high-dimensional problems. In particular, we determine the exact rate of decay of the approximation numbers, together with optimal asymptotic constants, and establish preasymptotic estimates. Finally we translate our findings into the language of tractability notions from information-based complexity. [ABSTRACT FROM AUTHOR]

Abstract: We obtain sharp estimates of the Hardy–Vitali type total -variation of a function of two variables in terms of its mixed modulus of continuity in . [Copyright &y& Elsevier]

Abstract: In this paper we present two-sided Chernoff-type inequalities for the error of the best approximation of a smooth d-variate function by polynomials of total degree less than k in the -norm. It is supposed that the d-variate weight function w has one-dimensional classical component weights, which satisfy Pearson differential equations. Similarly as in the univariate case, the leading coefficients of the multivariate classical polynomials, orthonormal with respect to the weight function w, play an important role in the presented estimates. [Copyright &y& Elsevier]

Abstract: This paper is devoted to inequalities of Lieb–Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues . We prove that there exists a positive constant , such that, if , then and determine the optimal value of . Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of on all possible potentials V, which is a lower bound for , corresponds to the optimal constant of a subfamily of Gagliardo–Nirenberg inequalities. This explains how (∗) is related to the usual Lieb–Thirring inequality and why all Lieb–Thirring type inequalities can be seen as generalizations of the Gagliardo–Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb–Thirring type where F and G are appropriately related. As a special case corresponding to , (∗∗) is equivalent to an optimal Euclidean logarithmic Sobolev inequality where , is any nonnegative sequence of occupation numbers and is any sequence of orthonormal functions. [Copyright &y& Elsevier]

Abstract: We prove an optimal logarithmic Hardy–Littlewood–Sobolev inequality for systems on compact -dimensional Riemannian manifolds, for any . We show that a special case of the inequality, involving only two functions, implies the general case by using an argument from the theory of linear programing. [Copyright &y& Elsevier]

We prove some sharp Hardy-type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac–Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity. [Copyright &y& Elsevier]

We prove a general optimal Lp-Euclidean logarithmic Sobolev inequality by using Pre´kopa–Leindler inequality and a special Hamilton–Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.). [Copyright &y& Elsevier]

We prove an optimal logarithmic Sobolev inequality in W1,p(Rd). Explicit minimizers are given. This result is connected with best constants of a special class of Gagliardo–Nirenberg-type inequalities. [Copyright &y& Elsevier]

Let Ω be a domain in R d and d Γ the Euclidean distance to the boundary Γ. We investigate whether the weighted Hardy inequality ‖ d Γ δ / 2 − 1 φ ‖ 2 ≤ a δ ‖ d Γ δ / 2 (∇ φ) ‖ 2 is valid, with δ ≥ 0 and a δ > 0 , for all φ ∈ C c 1 (Γ r) and all small r > 0 where Γ r = { x ∈ Ω : d Γ (x) < r }. First we prove that if δ ∈ [ 0 , 2 〉 then the inequality is equivalent to the weighted version of Davies' weak Hardy inequality on Ω with equality of the corresponding optimal constants. Secondly, we establish that if Ω is a uniform domain with a locally uniform Ahlfors regular boundary then the inequality is satisfied for all δ ≥ 0 , and all small r , with the exception of the value δ = 2 − (d − d H) where d H is the Hausdorff dimension of Γ. Moreover, the optimal constant a δ (Γ) satisfies a δ (Γ) ≥ 2 / | (d − d H) + δ − 2 |. Thirdly, if Ω is a C 1 , 1 -domain or a convex domain a δ (Γ) = 2 / | δ − 1 | for all δ ≥ 0 with δ ≠ 1. The same conclusion is correct if Ω is the complement of a convex domain and δ > 1 but if δ ∈ [ 0 , 1 〉 then a δ (Γ) can be strictly larger than 2 / | δ − 1 |. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators. [ABSTRACT FROM AUTHOR]

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case p = 2 , as well as an extension to the coefficient p = 1 that was previously unknown. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings). [ABSTRACT FROM AUTHOR]