On Nash and Carlson’s Inequalities for Symmetric <Emphasis Type="Italic">q</Emphasis>-Integral Transforms

The aim of this paper is to prove an \(\mathcal {L}_q^1 \cap \mathcal {L}_q^2\) versions of Nash and Carlson’s inequalities for a class of q-integral operator \(\mathcal {T}_q\) with a bounded kernel. As applications, we give q-analogues of Nash and Carlson’s inequalities for the q-Fourier-cosine, q-Fourier-sine, q-Dunkl and q-Bessel Fourier transforms.     

Wasserstein geometry of porous medium equation

We study the porous medium equation with emphasis on q-Gaussian measures, which are generalizations of Gaussian measures by using power-law distribution. On the space of q-Gaussian measures, the porous medium equation is reduced to an ordinary differential equation for covariance matrix. We introduce a set of inequalities among functionals which gauge the difference between pairs of probability measures and are useful in the analysis of the porous medium equation. We show that any q-Gaussian measure provides a nontrivial pair attaining equality in these inequalities.     

Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q′ = Π _p|q p of the modulus q satisfies log q′ ～ log q. Some applications to zero free regions of Dirichlet L-functions and the Pólya and Vinogradov inequalities are indicated.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings

We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form φ=a+q, where a is an additive mapping and q is a quadratic one.     

On Hanner type inequalities with a weight for Banach spaces

We shall present several Hanner type inequalities with a weight constant and characterize 2-uniformly smooth and 2-uniformly convex Banach spaces with these inequalities. p-Uniformly smooth and q-uniformly convex Banach spaces will be also characterized with another Hanner type inequalities with a weight in the other side term. The best value of the weight in these inequalities will be determined for Lp spaces. Also we shall present a duality theorem between these inequalities in a generalized form.     

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension ${2n-1, n\,\geqslant\,2}$ , and let L ^k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L ^k , for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.     

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight (p,q) inequalities for commutators of potential type integral operators.     

Convexity and concavity constants in Lorentz and Marcinkiewicz spaces

We provide here the formulas for the q-convexity and q-concavity constants for function and sequence Lorentz spaces associated to either decreasing or increasing weights. It yields also the formula for the q-convexity constants in function and sequence Marcinkiewicz spaces. In this paper we extent and enhance the results from [G.J.O. Jameson, The q-concavity constants of Lorentz sequence spaces and related inequalities, Math. Z. 227 (1998) 129-142] and [A. Kamińska, A.M. Parrish, The q-concavity and q-convexity constants in Lorentz spaces, in: Banach Spaces and Their Applications in Analysis, Conference in Honor of Nigel Kalton, May 2006, Walter de Gruyter, Berlin, 2007, pp. 357-373].     

On the inverse problem for three-dimensional potential scattering

We consider a linear perturbation for the wave equation $\text{□u = 0}\text{in}\text{Ω = E}{}^3$ by “repulsive” smooth potentials q(y) that are small at infinity and suitably small (in magnitude). We use a time-dependent approach to prove that the scattering operator S(q) determines uniquenes uniquely the scatterer q (at least in this class). Energy inequalities will play a central role in our discussion.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

On the eigenvalues of (p, 2)-summing operators and constants associated with normed spaces

It is shown that the eigenvalues of (q, 2)-absolutely summing operators are q-th power summable for r > q > 2, but in general not q-th power summable. The method of proof also yields a composition formula for (q, 2)-summing operators which implies that a certain power of these operators is nuclear. Inequalities between (q, r)-summing norms are used to derive estimates for the projection constants and the distance to Hilbert spaces of finite dimensional subspaces of type p and cotype q. One also obtains inequalities between different type and cotype constants of finite dimensional spaces.     

Weak boundedness properties for maximal operators defined on weighted spaces

The purpose of this paper is to obtain characterizations of weak type (1,q) inequalities,q ≥ 1, for maximal operators defined on weighted spaces by means of the corresponding operator acting over Dirac deltas. We present a technical theorem which allows us to obtain characterizations for a pair of weights belonging to the classA ₁ of weights by means of the fractional maximal operator. Analogous results are obtained for the one-sided fractional maximal operator.     

Inequalities for bounds on effective transport coefficients of two-phase media from power expansions at real points

The effective transport coefficient q of two-phase media is uniquely expressed by a special Stieltjes function f. In this paper, we incorporate into bounds on q, the power expansions of f available at a number of real points. To this end we apply the S-continued fraction technique developed in Baker (Essentials of Padé Approximants, Academic Press, New York, 1975) and Tokarzewski (IFTR Rep 4:3–171, 2005). Moreover, we establish the fundamental inequalities for S-bounds on q. As an example of an application the sequences of upper and lower S-bounds on the effective conductivity of a face-centered cubic lattice of spheres are evaluated.     

Gagliardo-Nirenberg inequalities

The paper deals with Gagliardo-Nirenberg inequalities in function spaces of type B _p,q ^s (? ⁿ ) and F _p,q ^s (? ⁿ ).     

On Reverse Hypercontractivity

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Stroock–Varopoulos inequality. A consequence of our analysis is that all simple operators ${L = Id - \mathbb{E}}$ as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all q < p < 1 and every positive valued function f for ${t \geq \log \frac{1-q}{1-p}}$ we have ${\| e^{-tL}f\|_{q} \geq \| f\|_{p}}$ . This should be contrasted with the case of hypercontractive inequalities for simple operators where t is known to depend not only on p and q but also on the underlying space. The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with m-sided dice.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Maximal theorems of Menchoff-Rademacher type in non-commutative Lq-spaces

Estimates for maximal functions provide the fundamental tool for solving problems on pointwise convergence. This applies in particular for the Menchoff-Rademacher theorem on orthogonal series in L₂[0,1] and for results due independently to Bennett and Maurey-Nahoum on unconditionally convergent series in L₁[0,1]. We prove corresponding maximal inequalities in non-commutative L_q-spaces over a semifinite von Neumann algebra. The appropriate formulation for non-commutative maximal functions originates in Pisier's recent work on non-commutative vector valued L_q-spaces.     

Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p→q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces Lp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p→q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p→q-boundedness of the Riesz potential operator.