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Register a new userOn the constants in some inequalities for the Sobolev norms and pointwise product
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C. Morosi
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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.
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We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the constants S_{r, n, d}, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r=2 or) n > d/2, r=infinity, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on S_{r,n,d} for n > d/2, 2 < r < infinity; in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.
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.
We show that the first order Sobolev spaces on cuspidal symmetric domains can be characterized via pointwise inequalities. In particular, they coincide with the Hajlasz-Sobolev spaces.
We give fully explicit upper and lower bounds for the constants in two known inequalities related to the quadratic nonlinearity of the incompressible (Euler or) Navier-Stokes equations on the torus T^d. These inequalities are tame generalizations (in the sense of Nash-Moser) of the ones analyzed in the previous works [Morosi and Pizzocchero: CPAA 2012, Appl.Math.Lett. 2013].
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.
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