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Register a new userCounterexamples for the convexity of certain matricial inequalities
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This paper is withdrawn. We found a mistake in Lemma 4.1
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We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type [ |A f|_{mathcal{X}}^2 leq C |f|_{mathcal{X}} big|A^2 fbig|_{mathcal{X}}, quad f in dombig(A^2big), ] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.
Let $A = (A_1, dots, A_m)$ be an $m$-tuple of elements of a unital $C$*-algebra ${cal A}$ and let $M_q$ denote the set of $q times q$ complex matrices. The joint $q$-matricial range $W^q(A)$ is the set of $(B_1, dots, B_m) in M_q^m$ such that $B_j = Phi(A_j)$ for some unital completely positive linear map $Phi: {cal A} rightarrow M_q$. When ${cal A}= B(H)$, where $B(H)$ is the algebra of bounded linear operators on the Hilbert space $H$, the {bf joint spatial $q$-matricial range} $W^q_s(A)$ of $A$ is the set of $(B_1, dots, B_m) in M_q^m$ for which there is a $q$-dimensional $V$ of $H$ such that $B_j$ is a compression of $A_j$ to $V$ for $j=1,dots, m$. Suppose $K(H)$ is the set of compact operators in $B(H)$. The joint essential spatial $q$-matricial range is defined as $$W_{ess}^q(A) = cap { {bf cl}(W_s^q(A_1+K_1, dots, A_m+K_m)): K_1, dots, K_m in K(H) },$$ where ${bf cl}$ denotes the closure. Let $pi$ be the canonical surjection from $B(H)$ to the Calkin algebra $B(H)/K(H)$. We prove that $W_{ess}^q(A) =W^q(pi(A) $, where $pi(A) = (pi(A_1), dots, pi(A_m))$. Furthermore, for any positive integer $N$, we prove that there are self-adjoint compact operators $K_1, dots, K_m$ such that $${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A) quad hbox{ for all } q in {1, dots, N}.$$ These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the $m=1$ case, and also generalize a result of M{u}ller obtained in case $m ge 1$ and $q=1$. Furthermore, if $W_{ess}^1({bf A}) $ is a simplex in ${mathbb R}^m$, then we prove that there are self-adjoint $K_1, dots, K_m in K(H)$ such that ${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A)$ for all positive integers $q$.
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain sever
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d, formulated in terms of the Laplacian Delta and of the fractional powers D^n := (-Delta)^(n/2) with real n >= 0; we review known facts and present novel results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the L^2 case where, for all sufficiently regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f ||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable values of j,n,theta (with j,n possibly noninteger). In the special cases theta = 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general L^2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and Sobolev sharp constants are confined to quite narrow intervals. Several examples are given.
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